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About Google Book Search Google's mission is to organize the world's information and to make it universally accessible and useful. Google Book Search helps readers discover the world's books while helping authors and publishers reach new audiences. You can search through the full text of this book on the web at |http: //books .google .com/I b, Google b, Google b, Google b, Google b, Google . L „;..«., Gobble- THE PRINCIPLES OF SCIENCE: A TREATISE ON LOGIC SCIENTIFIC METHOD. W. STANLEY JEYONS, M.A., F.R.S., MACMILLAN AND CO. 1874 / ,iz.db, Google OXFORD: . PtOBABD BALL, AND J. H. DTAC FBINTBK8 TO TBR UNIVEBSITT. by Google ■1!^ CONTENTS. BOOK IV. DJbUCriVE INTESTIGATION. CHAPTER XVm. 0B8BRVATI0K. BBOTIOK 1. Observation 2. Distinction of ObBerration Hjid Experiment . . - > 3. Mental Conditions of Correct Observation . . . ■ 4. Instrumental and Senaual Conditions of Correct Obaervation . 5. External Conditions of Correct Observation , . . • 6. Apparent Sequence of Events .,...■ 7. Negative Arguments founded on the Non-Observaiion ot Phenomena CHAPTER XIX. EXPEBIMKRT. 1. Experiment 2. Exclusion of Indifferent Circumstancea . . . • 3. Simplification of Experiments ...-■• 4. Failure in the Simplification of Experiments 5. Bemoval of Usual Conditions ...■■- 6. Int«rference of Unsuspected Conditions . . ■ • 7. blind or Test Exjterimenta ....'• 8. Negative Kesults of Experiment ....•■ 9. Limits of Expeiiment CHAPTER XX. IlETHOD OF VABIATIOMB. 1. Method of Variationa 2. The Variable and the Variant 3. Measurement of the Variable .,..•' 4. Maintenance of Similar Conditions ..... 6. Collective Experiments .....■■ 6. Periodic Variations .....■•> 7. Combined Periodic Changes 8. Prindple of Forced Vibrations 9. Integrated Variations ...■■■• by Google CHAPTER XXI. THKOET OF APPBOXlHATIOtr, 1. Theory of Approximation .... 2. Substitution of Simple Hypotheses 3. Approximation to Ex&ct Laws . 4. Successive Approximations to Xatoral Conditions 6. DiecoTeiy of Hypothetically Simple Laws . 6. Mathematical Fnnciples of Approximation . 7. Approximate Independence of Small Effects 8. Four MeaniogB of Equality 9. Arithmetic of Approxinuite Quantities . CHAPTER XXII. qUAXTITATIVB INDUCTION. 1. Quantitative Induction ....,,. 106 2. Probable Connexion of Varying Quantities , . . .106 3. Empiiical Mathematical Laws 110 4. Discovery of Rational Formulae . . . , . .118 fi. The Graphical Vethod 116 6. Interpolatbn and Extrapolation 120 7. IHnstiations of Empirical Quantitative Iawb . . 125 8. Kmple Proportional Variation 127 CHAPTER XXia THI VBX OF HTPOTHESIB, 1. The Use of Hypotbems 131 2. Requisites of a good Hypothesis 1 38 3. The First Requiate— Possibility of Deductive Reasoning 140 4. The Second Requisite — Consistency with Established Iawb of Nature 143 6. The Third Requisite — Coofonntty with Facts . . .146 6. Hxperimentum Cruds 148 7. Descriptive HypotbeaiB 153 CHAPTER XXIV. EUFIRICAL KHOWLBDOK, KXFLAMATIOH AKD PBEDIOTIOM. 1. Empirical Knowledge Explanation and Prediction . . 157 2. Empirical Knowledge 158 8. Accidental Discovery ....... 162 4. Empirical Observations subsequently explained . . . 166 6. Overlooked Results of Theory 16S 6. Predicted Discoveries 171 7. Predictions in tbe Science of Light 173 8. Predictions from the Theory of Undalatione . , , 1 76 9. Predictions in other Sciences 178 to. Prediction by Inversion of Cause and Effect . . . 181 11. Facts known only by Theory 186 by Google CHAPTER XXV. ACOOBDAHOZ OF tJUANTITATITB THB0BIE8 AND BXPKHIMXHTB. 1. Aocordance of Quantitative Theories and ExperimentB . 2. Empirical MeasaramentB 9. Qaaatities indicated by Theory, bnt Empirically Ueaflured 4. Explained Eesults of Ueasarement .... 6. Qnantitiee determined by Theory and verified by Heaanrement 6. Quantities determined by Theory and not verified 7. Discordance of Theory and Experiment 8. Accordance of Keasnremento of ABtrouomlcal Distancea 9. Selection of the best Uode of Heasarement . 10. Agreement of Distinct Modes of Measurement 1 1. Residnal Phenomena CHAPTER XXVI. CKABA(TEK OF THE EZPBRtltBNTALUT. 1. Character of the Experimentalist 217 2. Kature of QenioB 219 3. Error of the Baconian Uethod 220 4. Freedom of Theorizing 221 6. The Newtonian Method, the Tme Organum . . . 226 6. Candour and Courage of the Philosophic Hind . . 232 7. The Philosophic Character of Faraday 234 8. Beeervation of Judgment ....... 239 QEKBRALIZATIOS, AKAlOaV, ASB CLASSIFICATION. CHAPTEE XXVII. 1. Qeneralizatiou 242 2. Distinction of Qeneralization and Analogy .... 244 3. Two Meanings of Oeneralisation 246 4. Value of Qeneralization ....... 248 6. Comparative Generality of Phyrical Fropertiea . . . 249 6. Uniform Properties of all Matter 264 7. Variable Properties of Matter 258 8. Extreme Instances of Properties ....•> 269 9. The Detection of Continuity 262 10. The Law of Continuity 268 11. Failure of the Law of Continuity 273 12. Nq^tive Arguments on ^e Principle of Contintiity . . 276 13. Tendency to Hasty Generalization 276 Digit zed by Google CONTENTS. CHAPTER XXVni. ANALOOY. 1. Analogy 283 2. Analogy oB a Qaide in Discovery 286 3. Analogy in ^he Mathematical Say od Probabilitiea,' Cabioet Cyclopaedia, p. iii. Digitized by Google U THE PRINCIPLES OF SCIENCE, and the devices of sorcery and witchcraft often work their own ends. A man dies on the day which he has always regarded as his kst, from his own fears of the day. An incantation effects its purpose, because caie is taken to frighten the intended victim, by letting him know his fate<°. In all such cases the mental condition is the cause of apparent coincidence. In a second class of cases, the event A may make our perception of B follow, which would otherwise happen without being perceived. Thus it was seriously believed as the result of investigation that more comets appeared in hot than cold summers. No account was taken of the fact that hot summers would be comparatively cloudless, and afford better opportunities for the discovery of comets'*. Here the disturbing condition is of a purely external character. Certain ancient philosophers held that the moon's rays were cold-producing, mistaking the cold caused by radiation into space for an effect of the moon, which becomes visible at the time when the absence of clouds permits radiation to proceed. In a third class of cases, our percej^ion of A may make our perception of B foUow. The event B may be con- stantly happening, but our attention may not be drawn to it except by our observing A. This case seems to be illustrated by the fallacy of the moon's influence on clouda The ori^n of this fallacy is somewhat complicated. In the first place, when the sky is densely clouded the moon would not be visible at all ; it would be necessary for us to see the full moon in order that our attention shoidd be strongly drawn to the fact, and this would happen most often on those nights when the sky was />l.^ii^lgsg, Jir, ■^_ Ellis ^, moreover, has ingeniously Lubbock, 'Origin of Civilization,' p. 148, De Morgan's 'Easay,' p. 123. PhaoBophioal Mftgaiine,' 4th Series (1867), vol. xxxiv. p. 64. by Google OJiSERVATIOX. 15 pointed out that there is a general tendency for clouds to disperse at the commencement of night, which is the time when the full moon rises. Thus the change of the sky and the rise of the full moon are likely to attract attention mutually, and the coincidence in time suggests the relation of cause and effect. Mr. Ellis proves from the results of observatiops at the Greenwich Observatory that the moon possesses no appreciable power of the kind supposed, and yet it is remarkable that so acute and sound an observer as the late Sir John Herschel was convinced of the connection. In his ' Results of Obser- vations at the Cape of Good Hope'P, he mentions many evenings when a full moon occurred with a peculiarly clear sky. There is yet a fourth class of cases, in which B is reaUy the antecedent event, hut our perception of A, which is a consequence of B, may he necessary to bring about our perception of B. There can be no doubt, for instance, that upward and downward currents are continually cir- culating in the lowest stratum of the atmosphere during the day-time ; but owing to the transpareucy of the at- mosphere we have no evidence of their existence until we perceive cumulous clouds, which are the consequence of such currents. In like manner an interfiltratiou of bodies of air in the higher parts of the atmosphere is probably in nearly constant progress, but imless threads of cirrous cloud indicate these motions we remain wholly ignorant of their occurrencei. The highest strata of the atmosphere are wholly imperceptible to us, except when rendered P See 'Notes to Meaeuree of Double Stare,' 1204, 1336, 1477, 1686, 1786, 1816, 1835, 1939, 3081, 1186, pp. 165, &c. See also Heracbel'i 'Pomili&r Lectures on Scientific Subjecta, p. 147, and 'Out- lines of Astronomer,' 71)1 ed. p. 285. 1 Jevone, 'On the Cirrons Form of Cloud,' Philosophical Magazine, July, 1857, 4th Series, vol. xiv. p. 33. by Google 16 THE PRINCIPLES OP SCIBNCS. luminous by auroral curroits of electricity, or by the passage of meteoric etones. Tbere are many phenomena in meteorology and other . Bimilar sciences, in which some occurrences depend on others for their visibility. Thus in estimating the com- paratire numbers of meteors seen in different months of the year, it is essentia! to take account of the varying frequency of cloudy weather — or else of the different duration of the daylight which hides all but the most splendid meteors. Observationa of the comparative fre- quency of various kinds of clouds will be complicated by the iact that dense rain clouds necessarily hide those more delicate cirrous clouds which appear in the higher parts of the atmosphere. Most of the visible phenomena of comets probably arise from some substance which, existing previously invisible, becomes condensed or electrified sud- denly into a visible form. Sir John Herschel attempted to explain the production of comet tails in this manner by evaporation and condensation'. Negative Arguments founded on ike Non- observation of Phenomena^ From what has been suggested in preceding sections, it will plainly appear that the nou-obeervation of a pheno- menon is not generally to be taken as proving its non- occurrence. As there are sounds which we cannot bear, rays of light which we cannot feel, indefinite multitudes of worlds which we cannot see, and infinite myriads of minute organisms of which not the most powerful micro- scope can give us a view, we must as a general rule interpret our experience in an affirmative sense only. Accordingly when inferences have been drawn from the non-occurrence of particular facts or objects, more ex- T ' Aatronomy,' 4th ed. p. 358. Digit zed by Google OBSERVATION. 17 tended and. careful examination has often proved their falsity. Not many years since it was qmte a well credited conclusion in geology that no remains of man were foimd in connexion with those of extinct animals, or in any de- pofflt not actually at present in course of formation. Even Babbage accepted this conclusion as strongly confirmatory of the Mosaic accounts'. But when the opinion was yet universally held, flint implements had been found dis- proving any such conclusion, and overwhelming evidence of man's long continued existence has since been found. At the end of the last century when Herschel had searched the heavens with his powerful telescopes, there seemed little probability that planets yet remained unseen vrithin the orbit of Jupiter. But on the first day of this century such an opinion was overturned by the discovery of Ceres, and more than a hundred other small planets have since been added to the lists of the planetary system. The discovery of the Eozoon Canadense in strata of much greater age than any previously known to contain organic remains, has given a severe shock to many groundless opinions concerning the origin of organic forms ; and the oceanic dredging expeditions, under Dr. Carpenter and Professor Wyville Thompson, have further disconcerted geologists by disclosing the continued ex- istence of forms long supposed to be extinct. These and many other cases which might be quoted show the extremely unsafe character of negative inductions. It must not be supposed that negative arguments are of no force and value. The earth's surface, for instance, has been sufficiently searclied to render it highly impro- bable that any terrestrial animals of the size of a camel remain to be discovered. It is believed that no new large animal has been encountered in the last eighteen or twenty ■ BabbRge, ' Ninth Bridgewater Treatise,' p. 67. VOL. II. C Digitized by Google 18 TBE PRINCIPLES OF SCIENCE. centurieB^ and the probability that if existent they would have been seen, iDcreases the probability that they do not exist We may with somewhat lees confidence discredit the existence of any large unrecognised fish, or sea animals, Bucb as the alleged sea-seirpeDt. Bat as we descend to forms of smaller size negative evidence loses weight from the leas probability of our seeing smaller objects. Even the strong induction in favour of the four-fold division of the animal kingdom into Vertebrata, Annulosa, Mollusca, and Ooelenterata, may break down by the discovery of in- termediate or anomalous forms. As civilisation spreads over the surface of the earth, and unexplored tracts are gradually diminished, negative conclusions will in- crease in force ; but we require to learn much yet con- cemiag the depths of the ocean, almost wholly unexamined as they are, and covering three-fourths of the earth's surface. In geology there are a number of assertions to which considerable probability attaches on account of the large extent of the investigations already made, as, for in- stance, that true coal is found only in rocks of a par- ticular geological epoch ; that gold occurs in secondary and tertiary strata only in exceedingly small quantities", probably derived from the disintegration of earHer rocks. In natural history negative conclusions are exceedingly treacherous and unsatisfactoiy. The utmost patience wiU not enable a microscopist or the observer of any living thing to watch the behaviour of the organism under all circumstances continuously for any great length of time. There is always a chance therefore that the critical act or change may take place when the observer's eyes axe with- drawn. This certoinly happens in some cases; for though ' Cuvie/fl 'Eaeajr on the Theory of the Earth,' trauslatioD, p. 6r, &c. " MurcbiBon's ' Silario,' iBt ed. p. 43*- by Google OBSERVATION. 19 the f^-tilization of orchids hy agency of msecto is proved as well as any fact in natural history, Mr. Darwin has never been able by the closest watching to detect an insect in the performance of the operation. Mr. Darwin has him- self, indeed, adopted one conclusion on purely n^;ative evidence, namely that the Orchis pyramidalis and certain other orchidaceous flowers secrete no nectar. But his caution and unweaiying patience in verifying the con- clusion give an impressive lesson to the observer. For twenty-three consecutive days, as he tells us, he examined flowers, in all states of the weather, at all hours, in various localities. As the secretion in other flowers sometimes rapidly takes place and might happen at early dawn, that inconvenient hour of observation was specially adopted. Flowers of diflerent ages were subjected to irritating vapours, to water, and every condition likely to bring on the secretion ; and only after the invariable failure of this exhaustive inquiry was the barrenness of the nectaries assumed to be proved*. In order that a negative argument founded on the non- observation of an object shall have any considerable force, it must be shown to be probable that the object if existent would have been observed, and it is this probability whidi defines the value of the negative conclusion. The fiulure of astronomers to see the planet Tulcan, supposed by some to exist within Mercury's orbit, is no sufficient disproof of its existence. Smilarly it would be very difficult, or even impossible, to disprove tl^e existence of a second satellite of small size revolving round the earth. But if any person make a particular assertion, assigning place and time, then observation will either prove or disprove the alleged' feet. Thus if it is true that when a French observer professed to have seen a planet on the sun's face, an observer in Brazil was carefully scrutinizing the sim and failed to see > Dftnrin'a ' Fertilization of Orchids,' p. 48. C 2 Digitized by Google 20 THE PRINCIPLES OP SCIENCE. it, we have a conclusive negative proof J". On this account, as it has been well said, false facta in science are more mischievous than false theories. A false theory is open to every person's criticism, and is ever liable to be judged by its accordance with facts. But a false or grossly erroneous assertion of a fact often stands in the way of science for a long time, because it may be extremely difficult or even impossible to prove the falsity of what has been once recorded. In other sciences the force of a negative argument will often depend upon the number of possible altemativeswhicb may exist. Thus it was long believed that the character or quality of a musical sound, as distinguished fitim its pitch must depend upon the fonn of the undulation, be- cause no other cause of it had ever been suggested or was apparently possible. The truth of the conclusion was proved by Helmholtz, who applied a microscope to lu- minous points attached to the strings of various instru- ments, and thus actually observed the different modes of undulation*. In mathematics negative inductive arguments have seldom much force, because the possible forms of expres- sion, or the possible combinations of lines and circles in geometry are quite unlimited in number. An enormous number of attempts were made to trisect the angle by the ordinary methods of Euclid's geometry, but their in- variable failure did not establish the impossibility of the task. This was shown in a totfdly different manner, by proving that the problem involves an irreducible cubic equation to which there could be no corresponding plain geometrical solution* This is a case of reductio ad ahsurdum, a form of argument of a totally different r Cbsmbers'B ' AstroDomf ,' i st ed. p. 31. ' 'Th^orie I^fBiologique de la MuBiqne', Paris, 1868, p. 113. 'Algebra,' vol. ii. p. 344. by Google OBSERVATION. 21 character. Similarly no number of failures to obtain a general solution of equations of the fifth degree would eatablieh the impossibility of the task, but in an indirect mode, equivalent to a reductio ad absurdum, the impossi- bility is considered to be proved''. b Peacock, 'Atgabra,' vol. ii. p. 359. Serrct, 'Algebra Sup^rieure/ and ed. p. 389. by Google CHAPTER XIX. EXPERIMENT. Wb now come to conaider the great facilities which we enjoy for esamining the possible combinations of proper- ties and phenomena when objects are within our reach and capable of manipulation. We are said to experiment, when we bring substances together under various con- ditions of temperature, pressure, electric disturbance, molecular attraction, &a, and then record the changes observed. If we denote by A a certain group of antecedent con- ditions, and by X a certain series of subsequent phe- nomena, our object will usually be to ascertain a law of the form A = AX, the meaning of which is that where A is X will happen, and we may sometimes rise to the still simpler and higher law A = X, meaning that where A is, and only where A is, X will happen (see vol, 1. pp. 146, 149.) The great object of the art of experiment is to ascertain exactly those circumBtances or conditions which are re- quisite for the happening of any event X. Now the cir- cumstances which might be enumerated as present in the very simplest experiment are very numerous, in fact almost infinite. Rub two sticks together and consider what would be an exhaustive statement of the conditions. There are the form, hardness, organic structure, and all the chemical qualities of the wood ; the pressure and velo- city of the rubbing ; the temperature, pressure, and all the chemical qualities of the surrounding air ; the proximity Digitized by Google EXPERIMENT. 23 of the earth with Ita attractive and electric properties ; the temperature and other powers of the persons pro- ducing motion ; the radiation from the sun, and to and &om the sky ; the electric excitement possibly existing in any overhanging cloud ; even the positions of the heavenly bodies must be mentioned. Now on d priori grounds it is unsafe to assume that any one of these circumstances is without effect, and it is only on the results of experience that we can finally single out those precise conditions from which the observed heat of friction The great method of experiment consists in removing, one at a time, each of those conditions which may be imagined to have an lEifluence on the result. Our object in the experiment of rubbing sticks is to discover the exact ciroumBtances under which heat appears. Now the presence of air may be requisite ; therefore pr^are a vacuum, and rub the sticks in every respect as before, except that it is done in vacuo. If heat still appoM^ we may say that air is not, in the presence of the other circumstances, a requisite condition. The conduction of heat from neighbouring bodies may be a condition. Prevent this by making all the surrounding bodies ice cold, which is practically what Davy aimed at in rubbing two pieces of ice together. If heat still appears we have eliminated another condition, and so we may go on imtil it becomes apparent that the expenditure of enej^ in the friction of two bodies is the sole condition of the produc- tion of heat. The great diflSculty of experiment arises from the fact that we must not assume an independence to exist among the conditions. Thus previous to experiment we have no right to say that the rubbing of two sticks will produce heat in the same way when air is absent as before. We may have heat produced in one way when air is present. Digitized by Google 24 THE PRINCIPLES OF SCIENCE. and in another when air is absent. The inquiry branches out into two lines, and we ought to try in both cases whether cutting off a supply of heat by conduction pre- vents its evolution in friction. Now the same branching out of the inquiry occurs with regard to every circum- stance which enters into the experiment. Regarding only four circumstances, say A, B, C, D, we ought to test not only the combinations — ABCD, ABCd, ABcD, AJCD, aBCD, but we ought really to go through the whole of the combi- nations given in the fifth column of the Logical Abece- darium. The effect of the absence of each condition should be tried both in the presence and absence of every other condition, and every variety of selection of those conditions. Perfect and exhaustive experimentation would, in short, consist in examining natural phenomena in all their pos- sible combinations and registering all relations between conditions and results which are found capable of exist- ence. Experimentation would thus resemble the exclusion of contradictory combinations carried out in the Indirect Method of Inference (chapter vi, voL i. p. 95), except that the exclusion of any combination is grounded not on prior logical premises, but on d posteriori results of actual trial. The reader will readily perceive, however, that such exhaustive investigation is practically impossible, because the number of requisite experiments would be immensely great. Four circumstances only would require sixteen experiments ; twelve circumstances would require 4096, and the number increases as the powers of two. The t the experimenter has to fall back upon his nd experience in selecting those variations lost likely to yield him significant facts. It )int that logical rules and forms begin to fe,il d. The logical rule is — Try all possible com- by Google EXPERIMENT. 25 binationa; but this being impracticable, the experimentalist necessarily abanJona strict logical method, and trusts to his own insight Analogy, as we shall afterwards see, gives some assistance, and attention will probably be con- centrated on those kinds of conditions which have been found important in like cases. But we are now entirely in the region of probability, and the experimenter, while he is confidently pursuing what he thinks the right clue, may be entirely overlooking the one condition whose im- portance has been hitherto unsuspected. It is an impres- sive lesson, for instance, that Newton pursued all his exquisite researches on the spectrum unsuspicious of the fact that if he reduced the hole in the shutter to a narrow slit, all the mysteries of the bright and dark lines were within his grasp, provided of course that his prisms were sufficiently good to define the rays. In a similar manner we know not what slight alteration in the most familiar experiments may not open the way to realms of new discovery. Many additional practical difficulties encumber the pro- gress of the physicist. It is often impossible to alter one condition without altering others at the same time ; and thus we may not get the pure effect of the condition in question. Some conditions may be absolutely incapable of alteration ; others may be with great difficulty, or only in a certain degree, removable. A very treacherous source of error is the existence of unsuspected conditions, which we of course cannot remove except by accident. These difficulties we will shortly consider in succession. It is often beautiful to observe how the alteration of a single circumstance conclusively explains a phenomenon. An excellent instance is found in Faraday's investigation of the behaviour of Lycopodium spores scattered on a vibrating plate. It was observed that these minute spores collected together at the points of greatest motion, whereas by Google 2G TUB PlilNCIPLES OF SCIENCE. sand aod all heavy particles collect at the nodes, where motion is least. But it happily occurred to Faraday to try the experiment in the exhausted receiver of an air- pump, and it was then foxmd that the light powder behaved exactly like heavy powder. A conclusive proof was thus obtained that the presence of air was the con- dition of importfuice, doubtless because it was thrown into eddies by the motion of the plate, and thus carried the Lycopodium to the points of greatest agitation. Sand was too heavy to he thus carried by the air. Exclusion of Indifferent CircuTtistances. From what has been already said it will be apparent that in the investigation of any new phenomenon the detection and exclusion of indifferent circumstances is a work of great importance, because it allows the concentration of attention upon circiimstaDces which may contain the principal condition. There will always be a multitude of things which we are only too ready to neglect, but many beautiful instances may be given where all the most obvious circumstances have been shown to have no part in the production of a phenomenon. Every person would suppose tliat the peculiar colours of mother-of-pearl were due to the chemical qualities of the substance. Much trouble might have been spent in following out that notion by comparing the chemical qualities of various iridescent substances. But Brewster accidentally took an impression from a piece of mother-of-pearl in a cement of resin and bees -wax, and finding the colours repeated upon the surface of the wax, proceeded to take other impressions in balsam, fusible metal, lead, gum arable, isinglass, &c., and always found tho iridescent colours the same. He thus proved that the chemical nature is wholly a matter by Google EXPERIMENT. . 27 of indiifereDce, and the form of the surface is the condition of such colouTB". Nearly the same may be said of the colours exhibited by thin platee and films. The rings and lines of colour will be of the same character whatever may be the nature of the substance ; nay, a void space, euch as a crack in glass, would produce them even though the air were withdrawn by an air-pump. The conditions are simply the existence of two reflecting surfaces separated by a veiy small space, though it should be added that the refractive index of the intervening substance has some influence on the exact nature of the colour produced at any point. When a ray of light passes close to the edge of an opaque body, a portion of the light appears to be bent towards it, and produces coloured fringes within the shadow of the body. Newton attributed this inflexion of hght to the attraction of the opaque body for the supposed particles of light, although he was aware that the nature of the surrounding medium, whether air or other pellucid substance, exercised no apparent influence on the pheno- mena. Gravesande proved however that the character of the fringes is exactly the same, whether the body be dense or rare, compound or elementary. A wire has exactly the same eflect as a hair of the same thickness. Even the form of the obstructing edge was subsequently shown to be a matter of indifference by Fresnel, and the interference spectrum, or the spectrum seen when light passes through a fine grating is absolutely the same whatever be the form or chemical nature of the bars forming the grating. Thus it appears that the stoppage of a portion of a beam of light is the sole necessary condition for the difiraction or inflexion of light; and the phenomenon is shown to bear no analogy to the reflection and refraction of light, » ' Treatise on Optics,' by Sir D. Brewster, Cabinet Cycloptedia, p. 1 17. Digitized by Google 28 THE PRINCIPLES OF SCIENCE. m which the form and oature of the substance are all important. It ia interesting to observe how carefully Newton, in his researches on the spectrum, observed and proved the indifTerence of many circumstances by actual trial. He says'': 'Now the different magnitude of the hole in the window-shut, and different thickness of the prism where the rays passed through it, and different inclinations of the prism to the horizon, made no sensible changes in the length of the image. Neither did the different matter of the prisms make any : for in a vessel made of polished plates of glass cemented together in the shape of a prism, and filled with water, there is the like success of the ex- periment according to the quantity of the refraction.' But in the latter statement, as I shall afterwards remark (vol. ii. p. 42), Newton assumed an indifference which doea not exist, and fell into an unfortunate mistake. In the science of sound it is shown that the pitch of a sound depends solely upon the number of impulses in a second, and the material exciting those impulses is a matter of perfect indifference. Thus whatever medium, whether air or water, or any gas or liquid, be forced into the Siren, the sound produced is the same ; and the material of which an organ-pipe is constructed does not at all affect the pitch of its sound. In the science of statical electricity it is an important circumstance that the interior of a conducting body is a matter of indifference, resting in a neutral state, while the change is confined to the conducting surface. A hollow sphere takes exactly the same charge as a solid sphere of metal. Some of Faraday's most elegant and successful re- searches were devoted to the exclusion of conditions •• 'Opticks,' 3rd edit. p. 25, Digitized by Google EXPERIMENT. 29 which previous experimenters had thought essential for the production of electrical phenomena. Davy asflerted that no known fluids, except such as contain, water, could be made the medium of connexion between the poles of a battery ; and some chemists believed that water was an essential agent in electro-chemical decomposition. Faraday gives abundant experiments to show that other fluids allow of electrolysis, and attributes the erroneous opinion to the very general use of water as a solvent, and ita presence in most natural bodies". It was, in fact, upon purely negative (vol ii. p. 16) and weak evidence that the opinion bad been founded. Many experimenters attributed peculiar and even myste- rious powers to the poles of a battery, likening them to magnets, which, by their attractive powers, tear apart the elements of a substance. By a most beautiful series of experiments**, Faraday proved conclusively that, on the contrary, the substance of the poles is of no importance, being merely the path through which the electric force reaches the liquid acted upon. Poles of water, charcoal, and many diverse substances, even air itself, produced simi- lar results, or if the chemical nature of the pole entered at all into the question, it was as a disturbing agent. It is a most essential part of the theory of gravitation that the proximity of other attracting particles is wholly without effect upon the attraction existing between any two molecules. Two jwund weights weigh as much to- gether as they do separately. Every pair of molecules in the world have, as it were, a private communication, apart from their relations to all other molecules. Another un- doubted result of experience pointed out by Newton" is that the weight of a body does not in the least depend « 'Experimental Researches in Electricity,* vol. i, pp. 133, 134. d Ibid. vol. i. pp. 137, 162, &c. " ' Prindpia,' bk. iii. Prop. vi. Corollary i. by Google 30 TBE PRINCIPLES OF SCIENCE. upon its form or texture. It may be added that the temperature, electric condition, pressure, state of motion, chemical qualities, and all other circumstances concerning matter, except its mass, are indifferent aa regards its gra- vitating power. As natural science progresses, physicists gain a kind of insight and tact in judging what qualities of a substance are likely to be concerned in any class of phenomena. The physical astronomer treats matter in one point of view, the chemist in another, and the students of physical optics, sound, mechanics, electricity, Ac, make a fair division of the qualities among them. But errors will arise if too much confidence be placed in this independence of various kinds of phenomena, so that it is desirable from time to time, especially when any unexplained discrepancies come into notice, to question the indifference which is assumed to exist, and to test its real existence by appropriate experiments. Simplification of Experiments. One of the most requisite precautions in experimentation is to vary only one circumstance at a time, and to main- tain all other circumstances rigidly unchanged. There are two distinct reasons for this rule, the first and most ob- vious being that if we vary two conditions at a time, and find some effect, we cannot tell whether the effect is due to one or the other, or to both jointly. A second reason is that if no effect ensues we cannot safely conclude that either of them is indifferent ; for the one may have neu- tralized the effect of the other. In our logical formuke, A (B + 6) is identical with A (see vol. i p. r 1 2), and B may be indifferently present or absent ; but A (BC 1 6c) is not identical with A, and none of our logical processes enabled us to make the reduction. by Google EXPERIMENT. 31 If we want to prove that oxygen is neceesaiy to life, we must not put a rabbit into a veseel frora which the oxygen has been exhausted by a bunmig candle. We should then have not only an absence of oxygen, but an addition of carbonic acid, which may have been the destruotive agent. For a similar reason Lavoisier avoided the use of atmo- spheric air in experiments on combustion, because air was not a simple substance, and the presence of nitrogen might impede or even alter the effect of oxygen. As Lavoisier expressly remarks^, 'In performing experiments, it is a necessary principle, which ought never to be deviated from, that they be simplified as much as possible, and that every circumstance capable of rendering their resulte com- plicated be carefully removed.' It has also been well said by CuvierS that the method of physical inquiry consists in isolating bodies, reducing them to their utmoat simplicity, and in bringing each of their properties separatdy into action, either mentally or by experiment. The electro-magnet has been of the utmost service in the investigation of the magnetic properties of matter, by allowing of the production or removal of a moat powerful magnetic force without disturbing any of the other ar- rangements of the experiment. Many of Faraday's most valuable experiments would have been irustrated had it been necessary to introduce a heavy permanent magnet, which could not be suddenly moved without shaking the whole apparatus, disturbing the air, producing currents by differences of temperature, &c. The electro-magnet is perfectly under control, and its influence can be brought into action, reversed, or stopped by merely touching a button. Thus FaiBday was enabled to prove the rotation of the plane of circular polarized light by the feet that a certain light ceased to be visible when the electric current f LavoiBier's 'Chemistry,' translated hy Eerr, p. 103. s Cuvier'B 'Animal Kingdom,' introdaction, pp. i, 2. by Google 32 THB PRINCIPLES OF SCIENCE. of the magnet was cut off, and vice versd the light ap- peared when the current was re-made. ' These pheno- menEi,' he says, ' coidd be reversed at pleasure, and at any instant of time, and upon any occasion, showing a perfect dependence of cause and effect ••.' Another elegant experiment by Faraday Illustrates the maintainance of similar conditions He proved that liquids may conduct electricity when solids will not, by putting the poles of a battery in melted nitre, when a strong current was shown to exist by the galvanometer. But as soon as the nitre was allowed to solidify, the current ceased. Everything else remaining the same, the current existed when the nitre was liquid, and not when the nitre was solid'. It was Newton's omission to obtain the solar spectrum under the simplest conditions which prevented him from discovering the dark lines. Using a broad beam of light which had passed through a round hole or a triangular slit, be obtained a brilliant spectrum, but one in which many different coloured rays overlapped each other. In the i-ecent history of the science of the spectrum, one main difficulty has consisted in the mixture of the lines of several different substances, which are usually to be found in the light of any flame or spark. It is seldom poesible to obtain the light of any element in a perfectly simple manner. Angstrom greatly advanced this branch of science by examining the light of the electric spark when formed between poles of various metals, and in the presence of various gases. By varying the pole alone, or the gaseous medium alone, he was able to discriminate correctly be- tween the lines due to the metal and those due to the surrounding gas ''. >> ' Eiperimental Researches in Electricity,' vol. iii. p. 4. ' ' Life of Faraday,' vol. ii. p. 34. ^ ' Philosophical Magnzioe,' 4th ScricB, vol. ix. p. 327- by Google EXPERIMENT. 33 Failure in the SimpliJiccUion of Experiments. Id some cases it seems to be impossible to cany out the rule of vaiying one circumstance at a time. When we attempt to obtain two instances or two forms of experi- ment in which a single circumstance shall be present or absent, it may be found that this single circumstance entails one or more others. Benjamin Franklin's experi- ment concerning the comparative absorbing powers of different colours is well known. 'I took,' he says, 'a number of little square pieces of broadcloth from a tailor's pattern card, of various colours. They were black, deep blue, lighter blue, green, purple, red, yellow, white, and other colours and shades of colour. I laid them all out upon the snow on a bright sunshiny morning. In a few hours, the black being most warmed by the sun, was sunk so low as to be below the stroke of the sun's rays ; the dark blue was almost as low ; the lighter blue not quite so much as the dark ; the other colours less as they were lighter. The white remained on the surface of the snow, not having entered it at all.' This is a very -elegant and apparently simple experiment ; but when Leslie had completed his series of researches upon the nature of heat, he carae to the conclusion that the colour of a surface has very little effect upon the radiating power, tiie mechanical nature of the surface appearing to he more influential. He remarks* that ' the question is incapable of being posi- tively resolved, since no substance can be made to assume different colours without at the same time chan^g its internal structure.' More recent investigation has shown that the subject is one of considerable complication, be- cause the absorptive power of a surface may be different according to the character of the rays winch fall upon it ; ' ' Inquiry into the Nature of Heat,' p. 55. VOL. IL D Digitized by Google 34 TEE PRINCIPLES OF SCIENCE. but there can be no doubt as to the acuteness with whiuh Leslie points out the diflBculty. In Well's inveetigations concerning the nature of dew, we have, again, very complicated conditions. If we expose plates of various material, such as rough iron, glass, polished roetal, to the midnight sky, they will be dewed in various degrees ; but since these plates difier both in the nature of the surface and the conducting power of the material, it would not be plain whether one or both circumstances were of importance. We avoid this diflSculty by exposing the same material polished or varnished, so as to present dif- ferent conditions of surface™ ; and ^ain by exposing different substances with the same kind of surface. When we are quite unable to isolate circumstances we must resort to the procedure deBcribed by Mr. J. S. Mill under the name of the Joint Method of Agreement and Difference. We must collect as many instances as possible in which a given circumstance produces a g^ven result, and as many as possible in which the absence of the circumstance is followed by the absence of the result. To' adduce his example, we cannot experiment upon the cause of double refraction in Iceland spar, because we cannot idter its crystalline condition without altering it altogether, nor can we find substances exactly like calc spar in every circum- stance except one. We can only resort therefore to the method of comparing together all known substances which have the property of doubly-refracting light, and we find that they agree in being crystalline". This in- deed is nothing but an ordinary process of perfect or probable induction, already partially described, and to be further discussed under the subject of Classification. It may be added, however, that the subject does admit of ■" Herscbel, ' Frelinuoaiy Discourse on the Btudj of Natural Philo- Bophy,' p. i6i. ■> ' Syatcm of Logic/ bk. III. cliop. viii. § 4. 5tli. eil. vol. i. p. 433. Digit zed by Google EXFERIMENT. 35 perfect experimental treatment, since glass, when strongly compressed, and so long only as it is compressed in one direction, becomes capable of doubly -refracting light, and as there is probably no alteration in the glass bat change of elasticity, we leam that the power of double refraction is very probably due to a difference of elasticity in different directions. Removal of Usual Conditions. One of the great objects of experiment is to enable us to judge of the behaviour of substances under conditions widely different from those which prevail upon the surface of the earth. We live in an atmosphere which does not vary beyond certain narrow limits in temperature or pressure. Many of the powers of nature, such as gravity, which constantly act upon us, are of almost fixed amount. Now it will atVerwards be shown that we cannot apply a quantitative law to circumstances much different from those in which it was observed, without considerable risk of error. In the other planets, the sun, the stars, or remote parts of the Universe, the conditions of existence must often be widely different from what we commonly experience here. Hence our knowledge of nature must remain very re- stricted and hypothetical, unless we can subject substances to very unusual conditions by suitable experiment. The electric arc is an invaluable means of exposing metals or other conducting substances to the highest known temperature. By its aid we leam not only that all the met^Js can be vaporized, but that they all give off distinctive rays of light. At the other extremity of the scale, the intensely powerful freezing mixture devised by Faraday, consisting of solid carbonic acid and ether mixed in vacuo, enables us to observe the nature of sub- stances at temperatures immensely below any we meet with naturally on the earth's surface. D 2 Digitized by Google 36 THE PRINCIPLES OF SCIENCE. We can hardly realize now the importance of the in- vention of the air-pump, previous to which it was exceed- ingly diflScult to make any experiment except under the ordinary pressure of the atmosphere. The Torricellian' vacuum had been employed by the philosophers of the Accademia del Cimento to show tiie behaviour of water, smoke, sound, magnets, electric substances, &c., in vacuo, but their experiments were ofl«n unsuccessful from the difficulty of excluding air". Among the most constant circumstsuices under which we live is the force of gravity, which does not vary, except by a slight fraction of its amount, in any part of the earth's crust or atmosphere to which we can attain. Now this force is often sufficient to overbear and disguise various actions ; for instance, the mutual gravi- tation of small bodies. It was an interesting experi- ment of Plateau to withdraw substances from the action of gravity by suspending them in liquids of exactly the same specific gravity. Thus a quantity of oil poured into the middle of a suitable mixture of alcohol and water, assumes a spherical shape which, on being made to rotate, becomes spheroidal, and then successively sepa- rates into a ring and a group of spherules. Thus we have at least an illustration of the mode in which the planetary system may have been produced?, though it is to be remembered that the extreme difference of scale prevents our arguing with confidence from the experiment to the conditions of the nebular theoiy. It is possible that the so-called elements are elementary only to us, because we are restricted to temperatures at which they are fixed. Lavoisier carefully defined an element as a substance which cannot be decomposed hy ° ' Essayes of Natural Experiments made in the Accademia del Cimento.' Englished by Richard Waller, 1684, p. 40, &c. P Plateau, Taylor's ' Scientific lUemoirB,' vol. iv. pp. 16-43. by Google EXPERIMENT. 37 any known means ; but it seems almost certain that some series of elements, for instance Iodine, Bromine, and Chlo- rine, WB really compounds of a simpler substance. We must doubtless look to the production of intensely high temperatures, as yet quite beyond our means, for the de- composition of these so-called elementa But it may very possibly be found that, in this age and part of the uni- verse, the dissipation of energy has so fer proceeded that there are no sources of heat left to us sufiSciently intense to e&ct the decompo^tion of the supposed elements. Interference of Unsuspected Conditions. It may oilen happen that we are not aware of all the conditions imder which our researches are made. Some Bubstance may be present or some power may be in action, which escapes the most vigilant examination. Not being aware of its existence, we are of course unable to take proper measures to exclude it, and thus determine the share which it may have in the results of our experiments. There can be little doubt that the alchemists were often misled and encoiu^ged in their vain attempts by the un- suspected presence of traces of gold and silver in the substances they proposed to transmute. Lead, as drawn ft'om the smelting furnace, almost always contains some silver, and gold is associated with many other metals. Thus small quantities of noble metal would often appear as the result of experiment and raise delusive hopes. In more than one case the unsuspected presence of common salt in the air has caused great trouble. In the early experiments on electrolysis it was found that, when water was decomposed, an acid and an alkali were produced at the poles, together with oxygen and hy- drogen. . In the absence of any other explanation for this singular residt, some chemists rushed to tiie conclusion DigitizedbyGOOgle 38 THE PRINCIPLES OF SCIENCE. that electricity must have the power of generating acids and alkalis, and one chemist thought he had discovered a new substance called electric add. But Davy proceeded to a systematic investigation of the circumstances, by varying the conditions. Changing the glass vessel for one of agate or gold, he found that far less alkali was produced ; excluding impurities by the use of very care- fully distilled water, he found that the quantities of acid and alkali were still further diminished ; and having thus obtained a clue to the cause he completed the exclusion of impurities by avoiding contact with his fingers, and by placing the apparatus under an exhausted receiver, no acid or alkali being then detected. It would be difficult to meet with a more elegant or successful case of the detection of a condition previously unsuspected i- It is highly remarkable that the presence of common salt in the ab-, proved to exist by Davy, nevertheless continued a stumbling-block in the science of spectrum analysis, and probably prevented men, such as Brewster, Herschel, and Talbo^ from anticipating by thirty years the discoveries of Bunsen and KirchhofF. As I have else- where pointed ouf, the utility of the spectrum was known in the middle of the last centuty to Thomas Melvill, a talented Scotch physicist, who died at the early age of 27 years^. But Melvill was struck in his examina- tion of various coloured flames by the extraordinary pre- dominance of homogeneous yellow light, which was due to some circumstance escaping his attention. Wollaston and t cxlix. p. 880, Ac, where they describe a coDBtant flame of carbon monoxide gas. by Google METHOD OF VARIATIONS. Fizeau's method of measuring the velocity of light enabled him to appreciate the time occupied by light in travelling through a distance of eight or nine thousand metres. But the revolving mirror of Wteatstone sub- sequently enabled Foucault and Fizeau to measure the velocity in a space of four metres. In this latter method there was the obvious advantage that various media could be substituted for air, and the temperature, density, and other conditiofis of the experiment accurately governed or defined. Measurement of the Variable. There is little use in obtaining exact measurements of an effect unless we can also exactly measure the conditions with which the effect is to be connected. It is absurd to measure the electrical resistance of a piece of metal, its elasticity, tenacity, density, or other physical qualities, if these vary in degree, not only with the minute and almost inappreciable impurities of the metal, but also with its physical condition. If the same bar changes its properties by being heated and cooled, and we cannot exactly define the state in which it is at any moment, our care in measuring will be wasted, because it can lead to no law. It is of little use to determine very exactly the electric conductibility of carbon, which as graphite or gas carbon conducts like a metal, as diamond is almost a non-con- dnctor, and in several other forms possesses variable and intermediate powers of conduction. It will be of use only for immediate practical applications. Before measuring these we ought to have something to measure of which the conditions are capable of exact definition, and to which at a fiiture time we or others can recur. Similarly the accuracy of our measurement need not much surpass the accuracy with which we can define the conditions of the object treated. by Google 84 THE PRINCIPLES OF SCIENCE. The speed of electricity in passing through a conductor mainly depends upon the inductive capacity of the sur- rounding substances, and, except for technical or special purposes, there is little use in measuring velocities which in some cases are one hundred times as great as in other cases. But the maximum speed of electric conduction is probably a constant quantity of great scientific importance, and according to Prof. Clerk Maxwell's determination in 1868 is 174,800 miles per second, or little less than that of light. The true boiling point of water is a point on which all practical thermometry depends, and it is highly important to determine that point in relation to the ab- solute thermometric scale. But when water free from air and impurity is heated there seems to be no definite limit to the temperature it may reach, a temperature of 356° Fahr. having been actually observed. Such temperatures, therefore, do not require very accurate measurement. All meteorological measurements depending on the accidental condition of the sky are of infinitely leas importance than physical measurements in which such accidental conditions do not intervene. Many profound investigations depend upon our knowledge of the radiant energy continually poured upon the earth by the sun ; but this must be measured when the sky is perfectly clear, and the absorp- tion of the atmosphere at its minimum. The slightest interference of cloud destroys the value of such a measure- ment, except for meteorolc^cal purposes, which are of vastly less generality and importance. It is seldom use- ful, again, to measure such a quantity as the height of a snow-covered mountain within a foot, when the thick- ness of the snow alone may cause it to vary 25 feet or more, when in short the height itself is indefinite to that extent^. c Humboldt's ' Cosmos' (Bohn), vol. i. p. 7. by Google METHOD OF VARIATIONS. 55 Maintenance of Similar Conditions. Our ultimate object in induction mu^t be to obtain the complete relation between the conditions and the effect, but this relation ^11 generally be so complex that we can only attack it in detail. We must, as far as possible, con- fine the variation to one condition at a time, and establish a fieparate relation between each condition fmd the eflect. This will be at any rate the first step in approximating to the complete law, and it will be a subsequent question how far the simoltaneous variation of several conditions modifies their separate actions. In many of the most im- portant experiments, indeed, it is only one condition which we wish to study, and the others are merely interfering forces which we would gladly avoid if possible. One of the conditions of the motion of a pendulum is the reast- anee of the air, or other medium in which it swings ; but when Newton was desirous of proving the equal gravita- tion of all substances, he had no interest in so entirely difierent a force as the efiect of the air. His object was then to observe a single force only, and so it is in a great many other experiments. Accordingly one of the most important methods of investigation consists in maintaining all the conditions of like magnitude except that which is to be studied. As that admirable experimental philosopher, Gilbert, expressed it^ 'There is always need of similar preparation, of similar figure, and of equal magnitude, for in dissimilar and unequal circumstances the experiment is doubtful.' In Newton's decirive experiment similar conditions were provided for, with the usual simplicity which characterizes the highest art. The pendulums of which the oscillations were compared consisted of exactly equal boxes of wood, hanging hy equal threads, and filled with different sub- ' Gilbert, 'De Uagneto,' p. 109 by Google 56 THE PRINCIPLES OF SCIENCE. stances, so that the total weights should be exactly equal and the centres of oscillation at the same distance from the points of suspension. Hence the resistance of the air became approximat-ely a matter of indifference ; for the outward size and shape of the pendulums being exactly the same, the absolute force of resistance would be the same, so long as the pendulums vibrated with equal velocity ; and the weights being equal the force would diminish the velocity in like degree. Hence if any in- equality were observed in the vibrations of the two pen- dulums, it must arise from the only circumstance which was different, namely the chemical character of the matter within the boxes. No inequality being observed, the chemical nature of substances can have no appredable influence upon the force of gravitation?. A beautiful experiment was devised by Dr. Joule for the purpose of showing that the gain or loss of heat by a gas is connected, not with the mere change of ita volume and density, but with the energy received or given out by the gas. Two strong vessels, connected by a tube and stop- cock, were surrounded entirely with water after the air had been exhausted from one vessel and condensed in the other to the extent of twenty atmospheres. The whole apparatus having been brought to a uniform temperature by agitating the water, and the temperature having been exactly observed, the stop-cock was opened, so that the air at once expanded and Med the two vessels uniformly. The temperature of the water being again noted was found to be almost entirely unchanged. The experiment was then repeated in an exactly similar manner, except that the strong vessels were placed in separate portions of water. It was then discovered that cold was produced in the vessel from which the air rushed, and an almost exactly equal quantity of heat appeared in that to which t ' Principia,' bk. III. Prop. vi. by Google METHOD OF VARIATION'S. 67 it was conducted. Thus Dr. Joule clearly proved that rarefaction produces as much heat as cold, and that only when there is a disappearance of mechanical energy will there be production of heat''. What we have to notice, however, is not so much the result of the experiment, as the admirably simple manner in which a single change in the apparatus, the separation of the portions of water siuToimding the strong air vepsels, is made to give indi- cations of the utmost significance. Collective Experiments, There is an interesting class of experiments which enable us to observe an indefinite number of quantitative rraults in one act. Generally speaking, each experiment yields us but one number, and before we can approach the real processes of reasoning we must laboriously repeat measurement after measurement, until we can lay out a pretty complete curve of the variation of one quantity as depending on another. Now we can sometimes abbreviate this labour, by making one quantity vary in different parts of the same apparatus through every required amount. Thus in observing the height to which water rises by the capillary attraction of a glass vessel, we may take a series of glass tubes of different bore, and measure the height through which it rises in each. But if we take two glass plates, and place them vertically in water, so as to be in contact at one vertical side, and sUghtly separated at the other side, the interval between the plates varies through every intermediate width, and the water rises to a corresponding height, producing at its upper surface a hyperbolic curve. The absorption of light in passing through a coloured liquid may be beautifully shown by enclosing the liquid 1" ' PhiloBOpbical Magazine,' 3rd Series, vol. xxvi. p. 375. Digitized by Google 58 THE PRINCIPLES OF SCIENCE. in a wedge-shaped glass, eo that we have at a single glance an infinite variety of thicknesses in view. As Newton himself remarked, a red liquid viewed in this manner is found to have a pale yellow colour at the thinnest part, and it passes through orange into red. which gradually hecomes of a deeper and darker tint'. The effect may be noticed even in a common conical wine- glass. The prismatic analysis of light from such a wedge- shaped vessel discloses the reason, by exhibiting the pro- gressive absorption of different rays of the spectrum as investigated by Dr. J. H. Gladstone''. A moving body may sometimes be made to mark out its own course, like a shooting star which leaves a tail behind it. Thus an inclined jet of water exhibits in the clearest manner the parabolic path of a projectile. In Wheatstone's Kaleidophone the curves produced by the . combination of vibrations of different ratios are shown by placing bright reflective buttons on the tops of wires of various forms. The motions are performed so quickly that the eye receives the impression of the path as a com- plete whole, just as a burning stick whirled round pro- duces a continuous circle. The laws of electric induction are beautifully shown when iron filings are brought under the influence of a magnet, and fall into curves correspond- ing to what Faraday called the Lines of Magnetic Force. When Faraday tried to define what he meant by his lines of force, he was obliged to refer to the filinga ' By magnetic ciures,' he says', ' I mean lines of magnetic forces which would be depicted by iron filings.' Robison had previously produced similar curves by the action of irictional electricity™, and from a mathematical investiga- ' *0pticb8,' 3rd edit. p. 159. •t Watte, ' Dictionary of Chemistry,' vol. iii. p. 637. ' ' Faraday's Life,' by Bence Jones, vol. ii. p. 5. "> Watte ' DictioDary of Chemistty,' vol. ii. pp. 402, 403. by Google ' " METHOD OF VARIATIONS. 59 tion of the forma of such curves we may infer that mag- netic and electric attractions obey the general law of emanation, that of the inverse square of the distjince. !□ the electric brush we have another similar exhibition of the laws of electric attraction. There are several branches of science in which col- lective experiments have been used with great ad- vantage. Lichtenberg's electric figures, produced 1^ scattering electrified powder on an electrified resin cake, so as to show the condition of the latter, suggested to Chladni the notion of discovering the state of vibration of plates by strewing sand upon them. The sand collects at the points where the motion is least, and we gain at a glance a comprehension of the general form of undulation of the whole plate. To this method of e;xperiment we owe the beautiful observations of Savart. The exquisite coloured figures exhibited by plates of crystal, when ex- amined by polarized light, afibrd a more complicated example of the same kind of investigation. They led Brewster and Fresnel to a successful explanation of the properties of the optic axes of crystala The unequal conduction of heat in crystalline substances has also been shown in a similar manner, by spreading a thin layer of wax over the plate of crystal, and applying heat to a single point. The wax then melts in a circular or elliptic area according as the rate of conduction is uniform or not. Nor should we forget that Newton's rings were an early and most important instance of investigations of the same kind, showing the efiects of interference of light undula- tions of all magnitudes at a single view. Sir John Herschel gave to all snob opportunities of observing directiy the results of a general law, the name of Col- lective Instances^, and I propose to adopt the name Collective Experiments, " ' Preliminaiy DiBcourse,' &c., p. 185. by Google 60 THE PRINCIPLES OF SCIENCE. Such experiments will iu many subjects only give the first hint of the nature of the law in question, but will not admit of any exact measurements. The parabolic form of a jet of water may well have su^ested to Galileo his views concerning the path of a projectile ; but it would not serve now for the exact investigation of the laws of gravity. It is not likely too that capUlary attraction could be exactly measured by the use of inclined plates of glass, and the tubes would probably be better for precise investigation. As a general rule, these collective experiments would be most useftd for popular instruction and illustration of the laws of science. But when the ciu^es and figures produced are of a precise and per- manent character, as in the coloured figures produced by crystalline plates, they may admit of exact measurement, and may often be the only mode of approaching the ques- tion. Newton's rings, diffraction fringes, and other effects of the interference of light, allow of very accurate measurements. Under the class of collective experimenta we may per- haps place those in which we render visible the motions of a mass of gas or liquid by diffusing some opaque substance in it. The bebavioxu: of a body of air may often be studied in a beautiful way by the use of smoke, as in the production of smoke rings and jets. In the case of liquids lycopodium powder is sometimes employed. To detect the mixture of currents or strata of Uquid, I em- ployed exceedingly dilute solutions of common salt and silver nitrate, which produce a very visible cloud wherever they come into contact". Atmospheric clouds often reveal to us the movements of great volumes of air which would otherwise be quite unapparent. " ' PbUosopIiical Magazine,' July, 1857, 4th Series, vol xiv. p. 24. by Google METHOD OF VARIATIONS. Periodic Variations. A very large and important class of investigations are con- cerned with Periodic Variations. We may define a periodic phenomenon as one which, with the constant and uniform change of the variable, returns time after time to the same value. If we strike a pendulum it presently returns to the point from which we disturbed it, and with the uniform progress of time goes on making excursions and returning, until stopped by the dissipation of its energy. If one body in space approaches by gravity towards another, they will revolve round each other in an elhptic orbit, and return for an indefinite number of times to the same relative positions. On the other hand a single body projected into empty space, away from the action of any extraneous force, would go on moving for ever in a straight tine, according to the first law of motion. In the latter case the variation is called secular, because it pro- ceeds during ages in a similar manner, and suffers no iTfploioi or going round. It may be doubted whether there really is any motion in the universe which is not periodical. Mr. Herbert Spencer long since adopted the doctrine that all motion is ultimately rhythmical p, and abxmdance of evidence may be adduced in fevour of his view. The so-called secular acceleration of the moon's motion is certainly periodic, and as, so far as we can tell, no body is beyond the attractive power of other bodies, rectilinear motion becomes purely hypothetical, or at least infinitely improbable. All the motions of all the stars must tend to become periodic. Though certain disturb- ances in the planetary sj-stem seem to he uniformly pro- gressive, Laplace is considered to have proved that they really have their limits, so that after an almost infinitely great time, all the planetary bodies might return to the p 'Firet Principles,' 3rd edit, chap. %. p. 253. Digitized by Google 62 TUB PRINCIPLES OF SCIENCE. same exact places, and the stability of the system be eeta- blished. But any such theory of periodic stability is really hypo- thetical, and does not take into account a multitude of phenomena resulting in the dissipation of energy, which may be a really secular process incapable of restoration. For our present purposes we really need not attempt to form any opinion on such lofty questions. Any change which does not present the appearance of a periodic character will be empirically regarded as a secular change for the present, so that there will be an abundant supply of non-periodic variations. The variations which we produce experimentally will often be non-periodic. When we communicate beat to a gas it increases in bulk or pressure, and as far as we can go the higher the temperature the higher the pressure. Our experiments are of course restricted in temperature both above and below, but there is every reason to believe that the bulk being the same, the pressure would never return to the same point at any two different tempera- turea We may of course repeatedly raise and lower the temperature at regular or irregular intervals entirely at our will, and the pressure of the gas will vary in like manner and exactly at the same intervals, but such an arbitrary series of changes would not constitute Periodic Variation. It would constitute a succession of distinct experiments, which would place beyond reasonable doubt the connexion of cause and effect. Whenever a phenomenon recurs at equal or nearly equal intervals, there is, according to the theory of pro- bability, considerable evidence of connexion, because if the recurrences were entirely casual it is exceedingly unlikely that they would happen at equal intervals. Thus the mere fact that a brilliant comet had appeared in the years 1301, 1378, 1456, 1531, 1607, and 1682, gave con- by Google METHOD. OF VARUTTONS. 63 Biderable presumption in favour of the identity of the body apart from the similarity of the orbit. There is nothing which so strongly fascinates the attention of men as the recurrence time after time of some unusual event. Things and appearances which remain ever the same, like mountains and valleys, fail to excite the curiosity of a primitive people. It has been remarked by Laplace that even in his day the rising of Venus in its brightest phase never faded, to excite surprise and interest. So there is little doubt that the first germ of physical science arose in the attention given by Eastern people to the changes of the moon and the motions of the planets. One of the earliest astronomical discoveries must have coDMsted in proving the identity of the morning and evening stars, on the ground of their similarity of aspect and invariable alternation^. Periodical changes of a somewhat complicated kind must have been understood by the ChaldEeans, because they were aware of the cycle of 6585 days or 19 years which brings round the new and full moon upon the same days, hours, and even minutes of the year. The earliest efforts of scientific prophecy were founded upon this knowledge, and if at present we cannot help wondering at the precise antici- pations of the nautical almanack, we may readily imagine the wonder excited by such successful predictions in early times. Combined Periodic Changes. We shall seldom or never find a body subject to a single periodic variation, and free from any other disturbances. As a generd! rule we may expect the periodic variation itself to undergo variation, which may possibly be secular or incapable of repetition, but is more likely to prove 1 Laplace, ' System of the World,' vol. i. pp. 50, 54, &c. Digitized by Google 64 THE PRINCIPLES OF SCIENCE. periodic likewise ; nor is there any limit to the complica- tion of periods beyond periods, or periods within periods, which may ultimately be dieclosed. Itt studying, then, a phenomenon of rhythmical character we have a succession of questions to ask. Is the periodic variation uniform ? If not, is the change uniform \ If not, is the change itself periodic \ Is that new period uniform, or subject to any other change, or not ? and so on ad infinitum. In some cases there may be many distinct causes of periodic variations, and according to the principle of the superposition of small efiecta, to be afterwards more fully considered, these periodic effects will be simply added together, or at least approximately so, and the joint result may present a very complicated subject of inveRtigation. Thus the tides of the ocean consist of a series of super- imposed undulations, of which the number and character have by no means been determined as yet. Not only are there the ordinary and very obvious semi-diurnal tides caused by sun and moon, but a series of minor tides, such as the lunar diurnal, the solar diurnal, the lunar monthly, the lunar fortnightly, the solar annual and solar semi-annual are gradually being disentangled by the labours of Sir W. Thomson and others''. Variable stars present very interesting periodic pheno- mena ; while some stars, S Cephei for instance are subject to very regular and equal variations, others, like Mira Ceti, are less constant in the degrees of brilliancy which they attain or the rapidity of the changes, pos- sibly on account of some much longer periodic variation". The star ^ Lyrse presents a double maximum and minimum in each of its periods of nearly i^ days, and since the discovery of this variation the period in a periotl has probably been on the increase. 'At first the varia- ' 'BriliRli Association Report,' 1870, p. 120. ' Herscliel's 'Outlines of Astroiiomy,' 4th edit. pp. 555-557. Digit zed by Google METHOD OF VARIATIONS. bility was more rapid, then it became gradually slower ; and this decrease in the length of time reached its limit between the years 1840 and 1844. During that time ita period was nearly invariable ; at present it is again decidedly on the decrease*." It is evident that the tracing out of such complicated variations presents an almost unlimited field for interesting investigation. The number of such variable stars already known is consider- able, and there is no reason to suppose that any appreciable fraction of the whole number has yet been detected. Principle of Forced Vibrations. All investigations of the connection of periodic causes and effects rest upon a most important and general prin- ciple, which has been demonstrated by Sir John Herscbel for some spedal cases, and clearly explmned by him in several of his works". The principle may be formally stated in the following manner : ' If one part of any system connected together either by material ties, or by the mutual attractions of its members, be continually mfuntained by any cause, whether inherent in the consti- tution of the system or external to it, in a state of regular periodic motion, that motion will be prop^ated through- out the whole systems, and will give rise, in every member of it, and in every part of each member, to periodic move- ments executed in equal period, with that to which they owe their origin, though not necessarily synchronous with them in their maxima and minima.' The meaning of the proposition is that the effect of a periodic cause will be periodic, and will recur at intervals equal to those of the t Humboldt's 'Coamoe' (BoIid), vol. iii. p. 239. " ' EacyclopRdia Metropolitaua,' art. Simtid, § 333; 'Outlines of AgtroDomy,' 4th edit. § 650, pj>. 410, 487-88 ; ' Meteorology,' fieprint, p- m- VOL, II. F by Google 66 THE PBINGIPLES OF SCIENCE. cause. Accordingly whenever we find any two phenomena which do proceed, time after time, through changes of exactly the same period, there is much probability that they are connected. It was in this manner, douhtleas, that Piiny correctly conjectured that the cause of the tides lay in the sun and moon, the intervals between suc- cessive high tides being equal to the intervals between the moon's passage across the meridian. Kepler and Descartes too admitted the connection previous to Newton's demonstration of its precise nature. When Bradley discovered the apparent motion of the stars arising from the aberration of light, he was soon able to attribute it to the earth's annual motion, because it went through aU its phases in exactly a year. The most extensive and beautiful instance of induction concerning periodic changes which can be cited, is that of the discovery of an eleven-year period in various meteoro- logical and astronomical phenomena. It would be difficult to mention any two things apparently more disconnected than the spots upon the sun and auroras. As long ago as 1826, Schwabe, of Dessau, commenced a regular series of observations of the spots upon the sun, which has been continued to the present time, and he was able to show that at intervals of about eleven years the spots increased much in size and number. Hardly was this discovery made known, than Dr. Lamont pointed out a nearly equal period of variation in the magnetic needle as regards declination. The occasional magnetic storms or sudden irregular disturbances of the needle were next shown to take place most frequently at the times when sun spots were prevalent, and as auroras are generally coincident with magnetic storms, these strange phenomena were brought into the cycle^. It has since been shown by » 'Nature,' vol 1. p. 184; Quetelet, 'Sur k Pbysique du Globe,' pp. 14S, 363-64, &c. by Google METHOD OF VARIATIONS, Professor Piazzi Smyth and Mr. E. J. Stone, that the temperature of the earth's surface as indicated by sunken thermometers gives some evidence of a like period. The existence of a periodic cause having once been established, it is quite to be expected, according to the principle of forced vibrations, that its influence wiU be more or less considerable in all meteorological phenomena^ Perhaps the most mysterious part of these investiga- tions is that which refers the phenomena to the planetary configurations as an ulterior cause. Professor Balfour Stewart, with Messrs. Warren de la Eue and Loewy, by laborious researches discovered a periodic change of , 584 days in the sun spots, coincident with changes in tha relative positions of the Earth, Jupiter, and Venus. It has since been rendered probable by the researches of Dr. Kirkwood and others, that Schwabe's eleven-year period is due to the action of Mercury. Several other periods of more or less importance have been supposed to exist, but the subject is yet open to much more inquiry. Integrate Variations. In GoAsidering the infinite variety of modes in which one effect may depend upon another, we must set apart in a distinct class those which arise from the accumulated effects of a constantly acting cause. When water runs out of a cistern, the velocity of motion depends, according to TorriceUi's theorem, on the height of the surface of the water above the vent ; but the amount of water which leaves the cistern in a given time depends upon the aggregate result of that velocity, and is only to be ascertained by the mathematical process of integration. When one gravitating body falls towards another, the force of gravity varies according to the inverse square of the distance ; to obtain the velocity produced we by Google C8 THE PRINCIPLES OF SCIENCE. must integrate or sum the effects of that law ; and to obtain the space passed over by the body in any given time, we must again integrate with regard to the variable velocity. In periodic variatioos the same distinction must be drawn. The heating power of the sun's rays at any place on the earth varies every day with the height attained, and is greatest about noon ; but it does not follow that the temperature of the air is greatest at the same time. This temperature is an integrated effect of the sun's heat- ing power, and as long as the sun is able to give more heat to the air than the air loses in any other way, the temperature continues to rise, so that the maximum is deferred until about 3 p.m. Similarly the hottest day of the year falls, on an average, about one month later than the summer solstice, and all the seasons 1^ about a month behind the motions of the sun. In the case of the tides, too, the effect of the sun's or moon's attractive power is never greatest when the power is greatest ; the effect always lags more or less behind the cause. Yet the in- tervals between the successive tides are exactly equal, in the absence of disturbfince, to the intervals between the passage of the sun or moon across the meridian. Thus the principle of forced vibrations holds true of all such cases. In periodic phenomena, however, very curious results will sometimes follow from the integration of effects. If we strike a pendulum, and then repeat the stroke time after time when it is in the same part of the vibration, every stroke concurs with every other one in adding to the momentum, and we can thus increase the extent and violence of the vibrations to any degree. We can stop the pendulum again by strokes apphed when it is moving in the opposite direction, and the successive effects being added together will soon bring it to rest. Now if we Digitized by Google METHOD OF VARIATIONS. 69 alter the intervals of the strokes so that each two suo- ceasive strokes act in opposite manners they will exactly neutralize each other, and the energy expended will be turned into heat or sound at the point of percussion. Exactly similar effects occur in all cases of rhythmical motion. If the musical note C is sounded in a room con- taining a piano, the string corresponding to it will be thrown into vibration, because every successive stroke of the air-waves upon the string finds it in like position as regards the vibration, and thus adds to its energy of motion. But the other strings being incapable of vibrating with the same rapidity are struck at various periods of their vibrations, and one stroke will sooner or later be opposed by one contrary in effect. All phenomena of resonance arise from this coincidence in tirae of undu- lation. The air in a pipe closed at one end, and about 12 inches in length, is capable of vibrating 513 times in a second. If, then, the note C is sounded in front of the open end of the pipe, every successive vibration of the air is treasured up as it were in the motion of the air. In a pipe of different length the pulses of air would strike each other, and the mechanical energy would be transmuted into heat and become no longer perceptible as sound. These accumulated vibrations may sometimes become so intense as to lead to unexpected results. A glass vessel if touched with a violin bow at a suitable point may be fractured with the excess of vibration. In the same way a suspension bridge may readily be broken down if a com- pany of soldiers walk across it in steps the intervals of which happen to agree with the intervals of vibration of the bridge itself. But if they break the step or march with very different time, they may have no perceptible effect upon the bridge. In fact if the impulses com- municated to any vibrating body are exactly synchronous Digitized by Google 70 THE PRINCIPLES OF SCIENCE. with ite vibrations, the energy of those vibrations will be unlimited, and may fracture any body. Let us now consider what will happen if the strokes be not exactly at the same intervals as the vibrations of the body, but, eay, a very little slower. Then a succession of strokes will meet the body in nearly but not quite the same position, and their effects wiU be accumulated. Afterwards the strokes will be^n to fall when the body is in the opposite phase. Thus imagine that one pen- dulum moving exactly from one extreme point to another in a second, should be struck by another pendulum which makes 6i beats in a minute; then, if the pendulums commence together, they will at the end of 30J beats be moving in opposite directions. Hence whatever energy was communicated in the first half minute will be neutra- lized by the opposite effect of that given in the second half The effect of the strokes of the second pendulum will therefore be alternately to increase and decrease the vibrations of the first, so that a new kind of vibration will be produced running through all its phases in 61 seconds. An effect of this kind was actually observed by EUicott, a member of the Royal Society, in the case of two clocksy. He found that through the wood-work by which the clocks were connected a slight impulse was transmitted, and each pendulum alternately lost and gained momentum. Each clock, in fact, tended to stop the other at regular in- tervals, and in the intermediate times to be stopped by the other. Many of the most important disturbances in the planetary system depend upon the same principle ; for if one planet happens always to pull another in the same direction in similar parts of their orbits, the effects, how- ever slight, will be accumulated, and a disturbance of large ultimate amount and of long period will be produced The 7 ' Philosophical Transactions' (>;39), vol. xU. p. 136. Digit zed by Google METHOD OF VARIATIONS. 71 long inequality in the motions of Jupiter and Saturn is thus due to the fact that five times the mean motion of Saturn is very nearly equal to twice the mean motion of Jupiter, causing a coincidence in their relative positions and disturbing powers'^. » Gront'e ' History of Physical Astronomy,' p. 59. by Google CHAPTER XXI. THEORY OF APPROXIMATION. In order that we may gain a true understanding of the kind, degree, and value of the knowledge which we ac- quire by experimental investigation, it is requisite that we should be fiilly consciouB of its approximate character. We must learn to distinguish between what we can know and cannot know — between the questions which admit of solution, and those which only seem to be solved. Many persons may be misled by the expression exact science, and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws, exact to the last degree. There is even a prevailing impression that when once mathematical forraulsB have been successfully applied to a branch of science, this por- tion of knowledge assumes a new nature, and admits of reasoning of a higher character than those sciences which are still unmathematical. The very satisfactory d^ee of accuracy attained in the science of astronomy gives a certain plausibility to erro- neous notions of this kind. Some persons no doubt con- sider it to be proved that planets move in ellipses, in such a manner that ail Kepler's laws hold exactly true ; hut there is a double error in any such notions. In the first place, Kepler's laws are not proved, if by proof we mean certain demonstration of their exact truth. In the next place, even assuming Kepler's laws to he exactly true in a by Google THEORY OF APPROXIMATION. 73 theoretical point of view, the planets never move according to those laws. Even if we could observe the motions of a planet, of a perfect globular form, free from all perturbing or retarding forces, we could never perfectly prove that it moved in an ellipse. To prove the elliptical form we should have to measure inBnitely small angles, and in- finitely small fractions of a second ; we should have to perform impossibilities. All we can do is to show that the motion of an unperturbed planet approaches very nearly to the form of an ellipse, and the more nearly the more accurately our observations are made. But if we go on to assert that the path is an ellipse we pass beyond our data, and make an assumption which may be more or less probable, but cannot be proved, in the strict sense of that term. But, secondly, as a matter of fact no planet does move in a perfect ellipse, or manifest the truth of Kepler's laws exactly. The very law of gravity prevents its own results from being clearly exhibited, because the mutual pertur- bations of the planets distort the elliptical paths. Those laws again hold exactly true only of infinitely small planetaiy bodies, and when two great globes, like the sun and Jupiter, attract each other, the law must be modified. The periodic time is then shortened in the ratio of the square root of the number expressing the sun's mass, to that of the sum of the numbers expressing the masses of the Bun and planet, as was shown by- Newton*. Even at the present day discrepancies exist between the observed dimensions of the planet's orbits and their theoretical magnitudes, afrer making allowance for all disturbing causes'*. Nothing, in fact, is more certain in scientific method than that approximate coincidence can aloue be expected. In the measurement of continuous quantity » 'Principia,' bk. III. Prop. 15. ^ See Lockyer's 'Lessons in Elementary Astronomy,' p. 301. by Google 74 THE PRINCU'LES OF SCIENCE. perfect correspondence must be purely accidental, and should give rise to suspicion rather than to satisfaction. Oiie remarkable result of the approximate character of our observations is that we never could prove the esistence of perfectly circular or parabolic movement, even if it existed. The circle is a singular case of the ellipse, for which the eccentricity is zero ; it is infinitely improbable than any planet, even if undisturbed by other bodies, should have a circle for its orbit ; but if the orbit were a circle we could never prove the entire absence of ec- centricity. All that we could do would be to declare the divergence from the circular form to be inappreciable. Delambre was unable to detect the slightest ellipticity in the orbit of Jupiter's first satellite, but he could only infer that the orbit was nearly circular. The parabola is the singular limit between the elhp&e and the hyperbola. As there are elliptic and hyperbolic comets, so we might conceive the existence of a parabolic comet. Indeed if an undisturbed comet fell towards the sun from an infinite distance it would move in a parabola ; but we could never prove that it so moved. Substitution of Simple Hypotheses. ' In truth men never can solve problems fulfilling the complex circumstances of nature. AU laws and explana- tions are in a certain sense hypothetical, and apply exactly to nothing which we can know to exist. In place of the actual objects which we see and feel, the mathematitaan invariably substitutes imaginary objects, only partially resembling those represented, but so devised that the discrepancies may not be of an amount to alter seriously the character of the solution. When we probe the matter to the bottom physical astronomy is as hypothetical as Euclid's elements. There may exist in nature perfect by Google THEORY OF APPROXIMATION. 75 straight lines, triangles, circles, and other regular geo- metrical figures ; to our scieuce it is a matter of indif- ference whether they do or do not exist, because in any case they must be beyond our powers of appreciation. If ■we submitted a perfect circle to the most rigorous scrutiny and measurement, it is impossible that we should discover whether it were perfect or not. Nevertheless in geometry we argue concemipg perfect rectilineal figures and curves, and the conclusions apply to existing objects so &r as we can assure ourselves that they Si^cee with the hypothetical conditions of our reasoning. Now this is in reality all that we can do in the most perfect of the sciences of nature. Doubtless in astronomy we meet with the nearest ap- proximation to actual conditiona The law of gravity is not a complex one in itself, and we believe it with much probability to be exactly true ; but we cannot calculate out in any one case its accurate results. The law asserts that every particle of matter in the imiverse attracts every other particle, with a force depending on the masses of the particles and their distance. We cannot then know the force acting on any one particle unless we know the masses and distances and positions of all the other particles in the imiverse. The physical astronomer has from the first made a sweeping assumption, namely, that all the other millions of existing systems exert no perturbing effects in our planetary system, that is to say, no effects in the least appreciable. Thus the problem becomes at once hypo- thetical, because there is little doubt that gravitation be- tween our sun and planets and other systems must exist in some degree. But even when they consider the re- lations of our planetary bodies ivler se, all their processes are grossly approximative. In the first place they assume that each of the planets is a perfect ellipsoid, with a smooth surface and a homogeneous interior. That this assumption is untrue every moimtaiu and valley, every Digitized by Google 76 * TUB PRINCIPLES OF SCIENCE. sea, every miDe affords conclusive evidence. If the astro- nomer is to make his calculations perfelct, be must not only take account of the Himalayas and the Andes, the Atlantic and Pacific, but the attraction of every hill, nay, every ant-hill, must be separately calculated, nor must the attractive power of any grain of aand he neglected. So for are they from having yet considered any local inequality of the surface, that they have not yet decided upon the general form of the earth ; it is yet a matter of specula- tion whether or not the earth is an ellipsoid with three unequal axe8*=. If, as is probable, the globe is proved to be irregularly compressed in some directions, the calcular tions of astronomers will liave to be repeated and refined, in order that they may approximate to the attractive power of such a body. If we cannot accurately learn the form of our own earth, how can we expect to ascertain that of the moon, the sun, and other planets, in some of which are probably irregularities of greater proportional amount. The science of physical astronomy is yet in a further way merely approximative and hypothetical. Given perfectly homogeneous ellipsoids acting upon each other according to the law of gravity, the best mathematicians have never and perhaps never will determine exactly the resulting movements. Even when three bodies simul- taneously attract each other the complication of eifects is so great that only approximate calculations can be made. Astronomers have not even attempted the general problem of the simultaneous attractions of four, five, six, or more bodies, resolving the general problem into so many dif- ferent problems of three bodies. The principle upon which the calculations of physical astronomy proceed, is to neglect every effect which could not lead to any quantity appreciable in observation, and the quantities rejected '^ Thomeon nnd Tait, * Treatise on Nutur&l Fliiloeuphy,' vol. i. p. 646. Digitized by Google TEE0R7 OF APPROXIMATION. 77 are indefinitely more numerous and complex than the few larger terms which are retained. All then Is merely approximate. Concerning other branches of physical science the same general statements are even more evidently true. We speak and calculate about inflexible bars, inextensible lines, heavy points, homogeneous substances, uniform spheres, perfect fluids and gases, and we deduce an infinite number of beautiful theorems ; but all is hypothetical. There is no such thing as an inflexible bar, an inextensible line, nor any one of the other perfect objects of mechanical science; they are to be classed with those other almost mythical existences, the straight line, triangle, circle, rectangle, &c., about which Euclid so freely discoursed. Take the simplest operation considered in statics — the use of a crowbar in raising a heavy stone, and we shall find, as Thomson and Tait have pointed out, that we neglect far more than we observe**. If we suppose the bar to be quite rigid, the fulcrum and stone perfectly hard, and the points of contact real points, we might give the true re- lation of the forces. But in reality the bar must bend, and the extension and compression of different parts in- volve us in difficulties. Even if the bar be homogeneous in all its parts, there is no mathematical theory capable of determining with accuracy all that goes on ; if, as is in- finitely more probable, the bar is not homogeneous, the complete solution will be indefinitely more complicated, but hardly more hopeless. No sooner had we determined the change of form according to simple mechanical prin- ciples, than we should discover the interference of thermo- dynamic principles. Compression produces heat and extension cold, and thus the conditions of the problem are modified throughout. In attempting a fourth ap- proximation we should have to allow for the conduction ^ 'Treatise on Natural Philoaopby,' vol. i. pp. 337, &c. Digitized by Google 78 THE PRINCIPLES OF SCIENCE. of heat from one part of the bar to another. All these eifecte are utterly inappreciable in a practical point of view, if the bar be a good stout one ; but in a theoretical point of view they entirely prevent our saying that we have solved a natural problem. The faculties of the human mind, even when aided by the wonderful powers of abbreviation conferred by analytical methods, are utterly unable to cope with the complications of any one real pro- blem. And had we exhausted all the known phenomena of a mechanical problem, how can we tell that hidden phenomena, as yet undetected, do not intervene in the commonest actions. It is plain that no phenomenon comes within the sphere of our senses luiless it possesses a certain momentum or magnitude capable of irritating the appropriate nerves. There may then, and, in tact, must be indefinite worlds of phenomena too slight to rise within the scope of our consciousness. All the instruments with which we perform our measure- ments are fallible and faulty. We assume that a plumb- line gives a perfectly vertical line ; bat this is never true in an absolute sense, owing to the attraction of mountains and other inequalities in the surface of the earth. In an accurate trigonometrical survey, the divergencies of the plumb-line must be approximately determined and allowed for*". We assume a surface of mercury to be perfectly plane, but even in the breadth of 5 inches there is a cal- culable divergence from a true plane of about one ten- millionth part of an inch ; and this surface further diverges from true horizontality as the plumb-line does from true verticality. That most perfect instrument, the pendulum, is not even theoretically perfect, except for infinitely small arcs, and the delicate experiments performed with the torsion balance proceed on the assumption that the force of torsion of a wire is proportional to the angle of • Pratt, ' Philosophical Transactions,' vol. cxlvi. p. 31. Digit zed by Google THEORY OF APPROXIMA T/OA: 79 torsion, which is again only true for infinitely small angles'. We need to take great care that in simplifying a problem we do not overlook some circumstance which from peculiar mathematical conditions is of importance. Thus in experiments upon the density of the earth we may treat irregularities of its contour as producing in- considerable effects. But a like assumption niust not be made concerning irregularities in the strata of the earth at a short distance below the point of experiment e. Such is the purely approximate character of aJI our operations that it is not uncommon to find the theo- retically worse method giving truer results than the theo- retically perfect method. The common pendulum which is not isochronous is better for practical purposes than the cycloidal pendulum which is isochronous in theory, but subject to mechanical di£5culties. The spherical form is not the correct form for a speculum or lense, but it differs so slightly from the true form, and is so much more easily produced mechanically, that it is generally best to rest content with the spherical surface. Even in a six-feet mirror the difference between the parabola and the sphere is only about — ■ of an inch, a thickness whicb would be taken off in a few rubs of the polisher. Watts' ingenious parallel motion was intended to produce recti- linear movement of the piston rod. In reality the motion was always curvilinear, but a certain part of the curve approximated sufficiently for his purposes to a straight line. Approximation to Exact Laws. Though we can never prove any numerical law with perfect accuracy, it would be a great mistake to suppose ' Baily, ' Hemoira of the Koyal Astronomical Societ}',' vol. xiv, p. 99. e Airy, PhiloBophJcnl Trnnsacttotie,' vol. cxlvi, p. 334. Digitized by Google 80 TBB PRmClPLES OF SCIENCE. that there is any iDexactaess in the laws of nature. We may even discover a law which we believe to represent the action of forces with perfect exactness. The mind may seem to pass in advance of its data, and choose out certain numerical results as absolutely true. We can never really pass beyond our data, and so far as assump- tion enters in, so far want of certainty will attach to our conclusions ; nevertheless we may in many cases rightly prefer a probable assumption of a predse law to numerical results, which are at the best only approximative. We must accordingly draw a strong distinction between the laws of nature which we believe to be accurately stated in our formulas, and those to which our statements only make an approximation, so that at a future time the law will be differently stated. The law of gravitation is expressed in the form F=-jj7, meaning that gravity is proportional directly to tiie product of the gravitating masses, and indirectly to the square of their distance. The latent heat of steam, again, is expressed by the equation log Y = a + ha' + c^, in which are five quantities a, h, c, a, |8, to be deter- ■ mined by experiment. Now there is every reason to believe that in the progress of science the law of gravity will remain entirely unaltered, and the only effect of further inquiry will be to render it a more and more probable expression of the absolute truth. The law of the latent heat of steam, on the other hand, will be modified by every new series of experiments, and it may not improbably be shown that the assumed law can never be made to agree with the results of experiment. Philosophers have by no means always supposed that the law of gravity was exactly true. Newton, though he had the highest confidence in its truth, admitted that there were motions in the planetary system which he by Google THEORY OF APPROXIMATION. 81 could not reconcile with the law, Euler and Clairaut who were, with D'Alembert, the first to apply the full powers of mathematical analysis to the theory of gravita- tion as explaining the perturbations of the planets, did not treat the law as sufficiently established to attribute all discrepancies to the errors of calculation and obser- vation. In short, they did not feel certain that the force of gravity exactly obeyed the well known rule. The law might have involved other powers of the distance. It might have been expressed, for example, in the form and the coefficients a and e might have been so small that those terms woidd only become apparent in very accurate comparisons with fact. Attempts have been made from time to time to account for difficulties, by attributing value to such neglected terms. Gauss at one time thought that the even more fundamental prin- ciple of gravity, that the force is dependent only on mass and distance, might not be exactly true, and he undertook accurate pendulum experiments to test this opinion. Only as these repeated doubta have been time after time resolved in favour of the law of Newton, has it been assumed as precisely correct. But this belief does not rrat on experiment or observation only. The calculations of physical astronomy, however accurate, could never show that the other terms of the above general expression were absolutely devoid of value. It could only be shown that they had such slight value as never to become apparent. There are, however, other theoretical reasons why the law is probably complete and true as commonly stated. "Whatever influence or power spreads from a point, and expands uniformly through space, will doubtless vary in- versely in intensity as the square of the distance, simply because the area over which it ia spread increases as the VOL, n. o Digitized by Google 82 THE PRINCIPLKS OF SCIENCE. square of the radius. This part of the law of gravity may be considered aa due to the properties of space, and there is a perfect analogy in this respect between gravity and all other emanating forces or substances, as was pointed out in a most comprehensive and clear manner by Keill *•. Thus the undulations of light, beat, sound, and the attrac- tions of electricity or magnetism obey the very same law so far as we can ascertain. If the molecules of a gas or the particles of matter constituting odour w^ere to start from a point and move from it in straight lines uniformly, their distances would increase and their density decreaae according to the same principles. The other known laws of nature stand in a precisely similar position. Dalton's laws of definite combining proportions never have been, and never can be exactly proved ; but chemists having shown, to a considerable degree of approximation, that all the more common elements combine together as if each element had atoms of an invariable mass, assume that this is ex- actly true. They go even further. Prout pointed out in 1815 that the equivalent weights of the elements appeared to be simple commensurable numbers j and Dumas, Pelouze, Marignac, Erdmann, Stas, and others have gradually rendered it likely that the atomic weights of hydrogen, carbon, oxygen, nitrogen, chlorine, and silver, are in the ratios of the numbers 1, 12, 16, 14, 35"5, and 108. Chemists then step beyond their data; they throw aside their actual experimental numbers, and assume that the true ratios are not those exactly indicated by any weighings, but the simple ratios of these numbers. They boldly assume that the discrepancies are due to experimental errors, and they are justified by the fact that the more elaboiute and skilful the researches on the subject, the more nearly their assumption is verified. ^ 'An Introduction to Natural Philosophy,' 3rd, edit., 1733, p. 5. by Google THEORY OF APPROXIMATION. 83 Potaseium is the ooly element whose atomic weight has been determined with great care, but which has not shown an approach to a simple ratio with the other ele- ments. This exception may be due to some unsuspected cause of error ', A similar assumption is also made in the law of definite combining volumes of gases, and Sir B. C. Brodie has clearly pointed out the line of argument by which the chemist, observing that the discrepancies be- tween the law and fact are within the limits of experi- mental error, assumes that they are due to error"'. Faraday, in one of his researches, expressly makes an assumption of the same kind. Having shown, with some degree of experimental precision, that there exists a simple proportion between quantities of electrical energy and the quantities of chemical substances which it can decompose, 80 that for every atom dissolved in the battery cell an atom ought theoretically, that is without regard to dissi- pation of some of the energy, to be decomposed in the electrolytic cell, he does not stop at his numerical results. ' I have not hesitated,' he says, ' to apply the more strict results of chemical analysis to correct the numbers ob- tained as electrolytic results. This, it is evident, may be done in a great number of cases i, without using too much liberty towards the due severity of scientific research.' The law of the conservation of enei^ itself, one of the widest of all physical generalizations, must rest upon the same footing. The most that we can do by experiment is to show that the energy entering into any experimental combination is almost exactly equal to what comes out of it, and more nearly so the more accurately we perform all the measurements. Absolute equality is always a matter of assumption. We cannot even prove the indestructibility ' Watts, 'Dictionary of Chemiatty,' vol, L p. 455. ^ ' Philosophical TransactioiiB,' (i86ti) vol. clvi. p. 809. ' ' Experimental Bcsearches in Electricity,* vol, 1. p. 246. a 2 by Google 84 THE PRINCIPLES OF SCIENCE. of matter ; for were an exceedingly minute fraction of existing matter to vanish in any experiment, Bay one part in ten millions, we could never detect the loes. Successive Approximations to Natural Conditions. When we examine the history of scientific problems, we find that one man or one generation is usually able to make but a single step at a time. A problem is always solved for the first time by making some bold hypothetical simplification, upon which the next investigator makes hjrpothetical modifications approaching more nearly to the truth. Errors are successively pointed out in previous solutions, until at last there might seem little more to be desired. Careful examination, however, will show that an indefinite series of minor inaccuracies remain to be cor- rected and explained, were our powers of reasoning suffi- ciently great, and the purpose adequate in importance. Newton's successful solution of the problem of the planetary movements entirely depended at first upon a great but hypothetical simplification. The law of gravity only applies directly to two infinitely small particles, so that when we deal with vast globes like the earth, Jupiter, or the sun, we have an immense aggregate of separate attractions to deal with, and the law of the aggregate need not coincide with the law of the elementary particles. But Newton, by a great effort of mathematical reasoning, was able to show that two homogeneous spheres of matter act as if the whole of their masses were concen- trated at the centres; in short, that such spheres are a^regates which manifest the simple law of gravity or are centrobaric bodies (vol. i. p. 423). He was then able with comparative ease to calculate the motions of the planets on the hjrpothesis of their being spheres, and to show that the results roughly agreed with observation. Digitized by Google THEORY OF APPROXIMATION. 85 Newton, indeed, wa8 one of the few men who could make two great steps at once. He did not rest contented with the spherical hypothesis ; having reason to believe that the earth was really a spheroid with a protuberance around the equator, he proceeded to a second approxima- tion, and proved that the attraction of the protuberant matter upon the moon accounted for the precession of the equinoxes, and led to various complicated effects. But, as I have already mentioned (vol. ii. p. 76), even the spheroidal hypothesis is for from the truth. It takes no account of the irregularities of surface, the great protu- berance of land, for instance, in Central Asia and South America, and the deficiency in the bed of the Atlantic. To determine the law according to which a projectile, such as a cannon ball, moves through the resisting atmo- sphere is a problem very imperfectly solved at the present day, but in which many successive advances have been made. So little was known concerning the subject three or four centuries ago that a cannon ball was supposed to move at first in a straight line, and only after a time to be deflected into a curve. Tartaglia ventured to maintain that the path was curved throughout, as by the principle of continuity it should be ; but the ingenuity of Galileo was required to prove this opinion, and to show that the curve was approximately a parabola. It is only, however, under several forced hypotheses that we can assert the path of a projectile to be truly a parabola : the path must be through a perfect vacuum, where there is no resisting medium of any kind ; the force of gravity must be equal and act in parallel lines; and the moving body must be either a mere point, or a perfect centrobaric body, that is a body possessing a definite centre of gravity. None of these conditions can be really fulfilled in practice. The next great step in the problem was made by Newton and . Huyghens, the latter of whom asserted that the atmo- Digitized by Google 86 THE PRINCIPLES OF SCIENCE. sphere would offer a reaietance proportional to the velocity of the moving body, and concluded that the path would have in consequence a logarithmic character. Newton investigatod in a general manner the subject of resisting media, and came to the coDclusion that the resistance was more nearly proportional to the square of the velocity. The subject then fell into the hands of Daniel Bernouilli, who pointed out the enormous resistance of the air in cases of rapid movement, and calculated that a cannon ball, if fired vertically in a vacuum, would rise eight times aa high as in the atmosphere. In more recent times an immense amount both of theoretical and experimental in- vestigation has been spent upon the subject, smce it is one of great importance in the art of war. Successive approximations to the true law have been made, but nothing like a complete and final solution has been achieved or even hoped for"". It is quite to be expected that the earliest experi- menters in any branch of science will overlook corrections which afterwards become most apparent. The AraHan astronomers determined the meridian by taking the middle point between the places of the sun when at equal altitudes on the same day. They overlooked the fact that the sun has its own motion among the stars in the time intervening between the observations. Newton thought that the mutual disturbances of the planets might be disregarded, excepting perhaps the effect of the mutual attraction of the greater planets, Jupiter and Saturn, near their conjunction ". The expansion of quicksilver was long used as the measure of temperature, in ignorance or dis- regard of the fact that the rate of expansion increases with the temperature. Rumford, in the first experiment leading to a determination of the mechanical equivalent of ■" Hntton'e ' Mathematical Dictionary,' vol. ii. pp. 387-292. 1 ' Principia,' hk. iii. Prop. 13. Digitized by Google THEORY OF APPROXIMATION. 87 heat, disregarded the heat absorbed by the box containing the water heated and by other parts of the apparatus, otherwise he would in Dr. Joule's opinion, have come nearly to the correct result. It 18 surprising to learn the number of causes of error which enter into even the simplest experiment, when we strive to attain the most rigid accuracy. Thus we cannot perform the simple experiment of compressing a portion of gas in a bent tube by a column of mercuiy, in order to test the truth of Boyle's Law, without paying regard to, — (i) the variations of atmospheric pressure, which are com- municated to the gas through the mercury; {2) the compressibility of mercury, which causes the column of mercury to vary in density ; (3) the temperature of the mercury thronghout the column ; (4) the temperature of the gas which is with difficulty maintained invariable ; (5) the expansion" of the glass tube containing the gas. Although Regnault took all these circumstances into ac- count in his accurate examination of the law °, there is no reason for supposing that he exhausted the sonrees of inaccuracy. All the eai'lier investigations concerning the nature of waves in elastic media proceeded upon the assiunption that waves of different length would travel with equal speed. Newton's theoiy of sound had led him to this conclusion, and experiment, or indeed the commonest observations (see vol. i. p. 344) had sufficiently verified the inference. When the undulatory theory came to be applied at the commencement of this century to explain the phenomena of light, a great difficulty was encoimtered. The angle at which a ray of light is refracted in entering a denser medium depends, according to that theory, on the velocity with which the wave travels, so that if all waves of light were to travel with equal velocity in the same ° JamiD, ' Couru de Physique,' vol. i. pp. 282-3. by Google 88 TSE PRINCIPLES OF SCIENCE, medium, the disperaion of mixed light by the prism and the production of the spectrum could not take place. Some of the most striking phenomena were thus in direct conflict with the theory. The great French mathema- tician, Cauchy, first pointed out the true explanation, namely that all previous investigators had raade an arbitrary assumption for the sake of simplifying the calculations. They had assumed that the particles of the vibrating medium are bo close together that the intervals are qnite inconsiderable compared with the length of the wave, or in other terms infinitely small. This hypothesis happened to be approximately true in the CEise of air, so that no error was discovered in experiments on sound. Had it not been so, the earlier analysts would probably have failed to give any solution, and the pro- gress of the subject might have been retarded. Cauchy was able to make a new approximatidn to truth under the more difBcult supposition, that the particles of the Tibrating medium are situated at considerable distances, and act and react upon the neighbouring particles by attractive and repulsive forces. To calculate the rate of propagation of a disturbance in such a medium is a work of excessive difficulty. The complete solution of the problem appears indeed to be beyond human power, so that we must be content, as in the case of the planetary motions, to look forward to successive approximations. All that Cauchy could do was to show that certain mathe- matical terms or quantities, neglected in previous theories, became of considerable amount under the new conditions of the problem, so that there will exist a relation between the length of the wave, and the velocity at which it travels. To remove, then, the difficulties in the way of the undulatoiy theory of light, a new approach to pro- bable conditions was needed P. p Lloyd's 'Lectures on the Wave Theory,' pp. ai, 23. by Google THEORY OF APPROXIMATION. 89 In a Bimilar manner Fourier's theory of the conduction and radiation of heat was baaed upon the hypothesis that the quantity of heat passing along any line is simply pro- portional to the rate of change of temperature. But it has since been shown by Forbes that the conductivity of a body diminishes as its temperature increases. All the details of Fourier's solution therefore require modification, and the results are in the meantime to be regarded as only approximately true i. We ought to distinguish between those problems which are phyacally and those which are merely mathematically incomplete. In the latter case the physical law is cor- rectly seized, but the mathematician neglects, or is more often unable to follow out the law in all its results. The law of gravitation and the principles of harmonic or undula- tory movement, even supposing the data to be correct, can never be followed into all their ultimate results. Dr. Young explained the production of Newton's rings by supposing tliat the rays reflected from the upper and lower surfaces of a thin film of a certain thickness were in opposite phases, and thus neutralized each other. It was • pointed out, however, that as the light reflected from the nearer surface must be undoubtedly a little brighter than that from the further surface, the two rays ought not to neutralize each other so completely as they are observed to do. It was finally shown by Poisson that the dis- crepancy arose only from incomplete solution of the problem ; for the light which has once got into the film must be to a certain extent reflected backwards and forwards ad iitfinitum ; and if we follow out this course of the light by a perfect mathematical analysis, absolute dark- ness may be shown to result from the interference of the rays ^ In such a case as this we nsed no physical laws 1 Tail's ' Thermodynamics,' p. lo. I Lloyd's ' Lectures on tlie Wave Theory,' pp. 82, 83. Digitized by Google 90 THE PRINCIPLES OF SCIENCE, but those of reflection and refraction, and the only diffi- culty consisted in developing their full consequences. There is one inatructive result of the theory of error which should always be borne in mind, namely that when a large variable error is combined with a small variable error, the uncertainty of the final result, as measured by its probable error, is scarcely at all affected by the small variable error'. Accordingly our efforts at accuracy must be devoted to the sources of error in the order of their magnitude. There is no use in making instruments to measure the heat of the sun with the last degree of accuracy, when the varying transparency of the atmo- sphere produces uncertainties of far greater amount. It is needless to observe a comet or other heavenly body with the very finest instruments if it appears low down on the horizon, where the atmospheric refraction is not accurately determinate. In short, minuter variable sources of error may be entirely neglected, so long as those of a consider- ably greater amount remain beyond our powers of correc- tion. Discovery of Hypotheticaily Simple Laws. In some branches of science we meet with natural laws of a simple character which are in a certain point of view exactly true and yet can never be manifested as exactly true in natural phenomena. Such, for instance, are the laws concerning what is called a perfect gas. The gaseous state of matter is that in which the general properties of matter are exhibited in the simplest and most general manner. There is much advantage accordingly in ap- proaching the question of molecular mechanics from this side. But when we ask the question — What is a gas ? the answer must be a hypothetical one. Finding that ' Airy, ' Fhiloeophical Transact ions,' (1856) vol. czlvi. p. 324. Digitized by Google TBEOBT OF APPROXIMATION. 91 gases nearly obey the law of Boyle and Marriotte ; that they nearly expand by heat at the uniform rate of one part in 272-9 of their volume at 0° for each degree centi- gi-ade ; and that they more nearly ful61 these conditions the more distant the point of temperature at which we examine them from the liquefying point, we pass by the principle of continuity to the conception of a perfect gae. Such a gas would probably consist of atoms of matter at 80 great a distance from each other as to exert no attrac- tive forces upon each other ; but for this condition to be exactly fulfilled the distances must be infinite, so that an absolutely perfect gas cannot exist. But the perfect gas is not merely a limit to which we may approach, it is a limit passed by at least one real gas. It has been shown by Despretz, Pouillet, Dulong, Ar^o, and finally Regnault, that all gases diverge from the Boylean law, and in nearly all cases the density of the gas increases in a somewhat greater ratio than the pressure, indicating a tendency on the part of the molecules to approximate of their own accord, and condense into liquid. In the more condensible gases such as sulphurous acid, ammonia, and cyanogen, this tendency is strongly apparent near the liquefying point. Hydrogen on the contrary diverges from the law of a perfect gas in the opposite direction, that is, the density increases less than in the ratio of the pressure *. This is a singular exception, the bearing of which I am unable to All gases diverge again from the law of uniform ex- pansion by heat, but the divergence is less as the gas in question is less condensible, or examined at a temperature more removed from its liquefying point. Thus the perfect gas in this respect must have an infinitely high tempera- ture. According to Dalton's law each gas in a mixture re- tains its own properties wholly unaffected by the presence ' Jamin, ' Cours de Physique,' vol. i. pp. 383-a88. Digitized by Google THE pRiarciPLEs 6f science. of any other gas". This law is probably true only by approximation, but it is obvious that it would be true of the perfect gas with infinitely distant partidea *. Mathematical Principles of Approximation. The whole subject of the approximate character of physical science will be rendered more plain if we con- sider it from a general mathematical point of view. Throughout quantitative investigations we deal with the relation of one quantity to certain other quantities, of which it is a function ; but the subject is quite sufficiently complicated if we view one quantity as a function of one other. Now, as a general rule, a function can be developed or expreesed as the sum of certain other quanti- ties, the values of which depend upon the successive powers of the variable quiintity. Thus, if y be the one quantity which is regarded as a function of x, then we may say that y = A + Ba;+Ca!" + Da:' + EiB* + .... In this equation. A, B, C, D, &c., are fixed quantities, of different values in diiferent cases. The terms may be infinite in number or after a time "may cease to have any value. Any of the co-efficients A, B, C, Ac, may be zero or negative ; but whatever they may be they are fixed. The quantity x on the other hand may be made what we like, being variable at our will. Suppose, in the first place, that x and y are both measurable lengths. Let us assume that ,„ „„„ part of an inch is the least that we can take note of. Then when x is one hundredth of an inch, we have x* = rrWs' ^^^ if C be less than xmity, the term Car* will be inappreciable, being less than we " Joule and Thomaon, ' Phtlosopbical TraDsactionB,' i8S4> ^^^l- c^>t. P- 337- ' The properties of a perfect gas have been described by Ranking 'Transactions of the Royal Society of Edinbui^h,' vol. zxv. p. 561. by Google THEORY OF APPSOXIMATION. 93 can measure. Unlees any of the quantities D,E, kc, should happen to be very great, it is evident that all the suc- ceeding terms will also be inappreciable, because the powers of x become rapidly smaller in geometrical ratio. Thus when x is made small enough the quantity y seems to obey the equation y = A + Ba;. If X should be made still less, if it should become so small, for instance, as i T ooo.ooo ^^ ^" inch, and B should not be very great, then y would appear to be the fixed quantity A, and would not seem to vary with x at all. On the other hand, were x to grow greater, say equal to -,*5 inch, and C not be very small, the term C 3? would become appreciable, and the law would now be more We can invert the mode of viewing this question, and suppose that while the quantity y undergoes variations depending on many powers of x, that our power of de- tecting the changes of value is more or less acute. While our powers of olraervation remain very rude and imperfect we may even be unable to detect any change in the quantity at all, that is to say Ba; may always be smaller than to come within our notice, just as in former days the fixed stars were so called because they remained at apparently fixed distances from each other. With the use of telescopes and micrometers we become able to de- tect the existence of some motion, so that the distance of one star from another may be expressed by A + B x, the term including 3? being still inappreciable. Under these circumstances the star will seem to move uniformly, or in simple proportion to the time, x. With much improved means of measurement it will probably be found that this uniformity of motion is only apparent, and that there exists some acceleration or retardation due to the next term. More and more careful investigation will show Digitized by Google 94 THE PRINCIPLES OF SCIENCE. the law to be more and more complicated than was pre- viously supposed. There is yet aoother way of explaining the apparent results of a complicated law. If we take any curve and regard only a portion of it free from any kind of discon- tinuity, we may represent the character of such portion by an equation of the form y = X + Ba: + Ca;* + Tfa? + Restrict the attention to a very small portion of the curve, and the eye will be unable to distinguish its difference from a straight line, which amounts to saying that in the portion examined the term Ca;* has no value appreciable by the eye. Take a larger portion of the curve and it will be apparent that it possesses curvature, but it will be possible to draw a parabola or ellipse so that the curve shall be apparently coincident with a portion of' that parabola or ellipse. In the same way if we take lai^r and larger arcs of the curve it will assume the character successively of a curve of the third and fourth degrees ; that is to say, it coiresponds to equations involving the third and fourth powers of the variable quantity. We have arrived then at the conclusion that every phe- nomenon, when its amount can only be rudely measured, will either be of fixed amount, or will seem to vary uni- formly like the distance between two inclined straight lines. More exact measurement may show the error of this first assumption, and the variation will then appear to he like that of the distance between a straight line and a pai"abola or ellipse. We may afterwards find that a curve of the third or higher degrees is really reqiiired to represent the variation. I propose to call the variation of a quantity linear, elliptic, cuUc, quartic, quintic, &c., according as it is discovered to involve the first, second, third, fourth, fifth or liigher powers of the variable. It is a general rule in quantitative investigation that we com- by Google THEORY OF APPROXIMATION. 95 Dieoce by discovering linear, and afterwards proceed to elliptic or more complicated laws of variation. The ap- proximate curves whicb we employ are all, according to De Morgan's use of the name, parabolas of some order or otber ; and since the common parabola of the second order is approximately the same as a very elongated ellipse, and is in fact an infinitely elongated ellipse, it is convenient and proper to call variation of the second order elliptic. It might also be called quadric variation. As regards many important phenomena we are yet only in the first stage of approximation. We know that the sun and many so-called fixed stars, especially 6 1 Cygni, have a proper motion through space, and the direction of this motion at the present time is known with some degree of accuracy. But it is hardly consistent with the theory of gravity that the path of any body should really be a straight line. Hence, we must regard a rectilinear path as only an approximate and provisional description of the motion, and look forward to the time when its curva- ture will be ultimately detected and measured, though centuries perhaps must first elapse. On the surface of the earth we are accustomed to assume that the force of gravity is uniform at all ordinary heights above or below the surface, because the variation is of so slight an amount that we are scarcely able to detect it. But supposing we could measure the variation, we should find it simply proportional to the height. Taking the earth's radius to be unity, let h be the height at which we measure the force of gravity. Then by the well-known law of the inverse square, that force will be proportional to j^~kf or to y (i - 2 A + 3 /i* - 4 A' + ). But at all heights to which we can attain h will be so small a fraction of the earth's radius that 3 A' will be in- by Google 96 THE PRtNClPLSS OF SCIENCB. appreciable, and the force of gravity will seem to follow .the law of linear variation, being proportional to i — 2 A. When the circumstances of an experiment are much altered, different powers of the variable may become pro- minent. The resistance of a liquid to a body moving through it may be approximately expressed as the sum of two terms respectively involving the first and second powers of the velocity. At very low velocities the first power is of most importance, and the resistance, as Pro- fessor Stokes has shown, is nearly in simple proportion to the velocity. When the motion is rapid the resistance increases in a still greater degree, and is more nearly pro- portional to the square of the velocity. Approximate Independence of Small Effects. One result of the general theory of approximation possesses such great importance in physical science, and is so often applied, that we may consider it separately. The investigation of causes and effects is immensely simplified when we may consider each cause as producing its own effect invariably, whether other causes are acting or not. Thus, if the body P produces the effect x, and Q produces y, the question is whether P and Q acting to- gether will produce simply the sum of the separate effects, x-i-y. It is under this supposition that we treated the methods of eliminating error (Chap. XV,), and errors of a less amount would still remain if the supposition was a forced and unnatural one. There are probably some parts of science in which the supposition of independence of effects holds rigidly true. The mutual gravity of two bodies, for instance, is entirely unaffected by the presence or absence of other gravitating bodies. People do not usually consider that this important prbciple is involved in such a simple thing as putting two pound weights in by Google THEORY OF APPROXIMATION. 97 the scale of a balance. How do we know that two pound weights together will weigh twice as much as one ? Do we know it to be exactly bo % Like other results founded on induction we cannot prove it certainly and absolutely, but all the calculations of physical astronomy proceed upon the aseumption, bo that we may consider it proved to a very high degree of approximation. We may, in fact, aasume with much probability that bodies gravitate in entire independence of each other. Had not this been true the calculations of physical astronomy would have been almost infinitely more complex than they actually are, and the progress of knowledge would have been vastly Blower, The science of the spectrum again ia much simplified by the fact that elements do not apparently interfere with each other in the production of light. The spectrum of sodium chloride is the spectrum of sodium superposed upon that of chlorine. Were it otherwise, we should have as many distinct spectra as there are distinct com- pounds in chemistry, and the subject would be almost hopelessly complex. The spectrum of a substance would then no more enable us to tell its componentB than the appearance of a new mineral indicates its composition. But it would probably be too early to assert the entire absence of any joint spectra. There is so much yet unexplained in the subject that some efiects due to the mutual action of elements may possibly be discovered, and the independence will then be only approximate. It is a general principle of scientific method that if efiects be of small amount, comparatively to our means of observation, all joint efiects will be of a higher order of smallness, and may therefore be rejected in a first ap- proximation. This principle was distinctly employed by Daniel Bemomlli in the theory of sound, under the title of 'The Principle of the Coexistence of Small Vibrations.* VOL, II. H Digitized by Google »8 THE PRINCIPLES OF SCIENCE. He showed that if a atring is affected by two kinds of vibrations, we may consider each to be going on as if the other did not exist We cannot perceive that the sonnd- ing of one musical instrument prevents or even modifies the sound of another, so that all sounds would seem to travel through the air, and act upon the ear in independ- ence of each other. An exactly similar assumption is made in the theoiy of tides, which are really great waves. One wave is produced by the attraction of the moon, and another by the attraction of the sun, and the question arises, whether when these waves coincide, as at the time of spring tides, the joint wave will be simply the sum of the separate waves. On the principle of Bemouilli this will be BO, because the tides on the ocean are almost indefinitely small compared with the depth of the ocean. The principle of Bemouilli, however, is only approxi- mately true. A wave never is exactly the same when another wave is interfering with it, but the less the dis- placement of particles due to each wave, the less in a still higher degree is the effect of one wave upon the other. In recent years Helmholtz was led to suspect that some of the phenomena of sound might after all be due to resultant effects overlooked by the assumption of previous physicists. He investigated the secondary waves which would arise from the interference of considerable disturb- ances, and was able to show that certain summation or resultant tones ought to be heard, and experiments subse- quently devised for the purpose showed that they might be heard. Throughout the mechanical sciences the Principle of the Superposition of Small Motions is of fundamental im- portance y, and it may be thus explained. Suppose that two forces, acting from the points B and 0, are simidtaneously moving a body A, Let the force acting J See Thomaon and Toil's ' Natural Philosophy,' vol. i p. 60. Digitized by Google THEORY OF APPROXIMATION. 99 from B be such that in one second it would move A to p, and similarly let the second force, acting alone, move A to r. The question arises, a p ^ then, whether their joint action will urge A to j along the diagonal of the parallelogram. May we say that A will move the distance Aj> in the direction AB, and Ar in the direction AC, or, what ia the same thing, along the parallel line pq ? In all strictness we cannot say so ; for when A has moved towards p, the force from C will no longer act along the line AC, and similarly the motion of A towards r will modify the action of the force fitim B. This interference of one force with the line of action of the other will evidently be greater the lai^er is the extent of motion considered ; on the other hand, as we reduce the paral- lelogram kpqr, compared with the distances AB and AC, the less will be the interference of the forces. Accord- ingly mathematicians avoid all error by considering the motions as infinitely small, so that the inteiference be- comes of a still higher order of infinite smallness, and may be entirely neglected. By the resources of the Differ- ential Calculus it is possible to calculate the motion of the particle A, as if it went through an infinite number of infinitely small diagonals of parallelograms. The great discoveries of Newton really arose from applying this method of calculation to the movements of the moon round the earth, which, while constantly tending to move onward in a straight line, is also deflected towards the earth by gravity, and moves through an elliptic curve, composed as it were of the infinitely small diagonals of infinitely small parallelograms. The mathematician, in his investigation of a curve, always in fact treats it as made up of a great number of short straight lines, and it H 2 Digitized by Google 100 THE PRINCIPLES OF SCIENCE. may even be doubtful whether he could treat it in any other manner. Nevertheless there is no error in the final results, because having obtained the formulse flowing from this supposition, each straight line is then regarded as be- coming infinitely small, and the polygonal line becomes undistinguishable from a perfect curve ^. In abstract mathematical theorems the approximation to absolute truth is perfect, because we can treat of in- finitesimals. In physical science, on the contrary, we treat of the least quantities which are perceptible. Neverthe- less, while carefully distinguishing between these two dif- ferent cases, we may fearlessly apply to both the principle of the superposition of small motions or effects. In physical science we have only to take care that the eflects really are so small that any joint efiect will be unquestion- ably imperceptible. Suppose, for instance, that there is some cause which alters the dimensions of a body in the ratio of I to i + a, and another cause which produces an alteration in the ratio of i to i + ;S, If they both act at once the change will be in the ratio of i to (i + a) (i + ^), or as r to i +n+^ + a^. But if a and j8 be both very small fractions of the total dimensions, a^ will be yet far smaller and may be disregarded ; the ratio of change is then approximately that of i to i+a+j8, or the joint effect ia the sum of the separate effects. Thus if a body were subjected to three strains at right angles to each other, the total change in the volume of the body would be approximately equal to the sum of the changes pro- duced by the separate strains, provided that these are of very small amount. In like manner not only is the ex- pansion of every solid and hquid substance by heat approximately proportional to the change of temperature, when this change is very small in amount, but the cubic z Challis, ' Notes on tbe Principles of Pure and Applied Calculation,' 1869, p. 83. by Google THEORY OF APPROXIMATION. 101 expansion may also be considered as being three times as great as the linear expansion. For if the increase of tem- perature expands a bar of metal in the ratio of i to j + a, and the expansion be equal in all directions, then a cube of the same metal would expand as i to (i +af, or an I to 1 + jd + 3a* + a". When a is a very small quantity the third term 30' will be imperceptible, and still more so the fourth term a*. The coeflScients of expansion of solids are in fact so small, and so imperfectly determined, that physicists seldom take into account their second and higher powers. It is an universal and important result of these prin- ciples that all very small errors may be assumed to vary in simple proportion to their causes ; a new reason why, in eliminating errors, we should first of all make them as small as possible. Let us suppose, with De Morgan, that there is a right-angled triangle of which the two sides containing the right aogle are really of the lengths 3 and 4, so that the hypothenuse is ^y + 4' or 5. Now if in two measurements of the first side we commit slight errors, making it successively 4'ooi and 4*002, then calcu- lation will g^ve the lengths of the hypothenuse as almost exactly 5-0008 and 5-00016, so that the error in the hypothenuse will seem to vary in simple proportion to that of the side, although it does not really do so with perfect exactness*. The logarithm of a number does not vary in proportion to that number — nevertheless we should find the difference between the logarithms of the numbers 1 00000 and looooi to be almost exactly equal to that between the numbers looooi and 100002. It is thus a general rule that very small differences between suc- cessive values of a function are approximately proportional to the small differences of the variable quantity. ■ De Morgan's ' Differential Calculus.' ■ DigitizedbyGOOgle 102 THE PRINCIPLES OF SCIENCE. Four Meanings of Equality. Although it might seem that there are few terms more free 'from ambiguity than the term equal, yet scientific men do as a matter of fact employ it with four meanings, which it is very desirable to distinguish carefully. These meanings I may briefly describe as (r) Absolute Equality. (2) Sub-equality. (3) Apparent Equality. (4) Probable Equality. By absolute equality we Kgnify that which is complete and perfect to the last degree ; but it is obvious that we can only know such equality in a theoretical or hypothe- tical manner. The areas of two triangles standing upon the same base and between the same parallels are abso- lutely equal. Hippocrates beautifully proved that the area of a lunula or figure contained between two seg- ments of circles was absolutely equal to that of a certain right-angled triangle. As a general rule all geometrical and other elementary mathematical tJieorems involve ab- solute equality. De Morgan proposed to describe as sub-equal those quantities which are equal within an infinitely small quantity, so that x is sub-equal to a; + dx. The whole of the differential calculus may, as I apprehend it, be said to arise out of the neglect of infinitely small quantities ; with this subject however we are not in this place much concerned. In mathematical science many other subtle distinctions may have to be drawn between kinds of equality, as De Morgan has shown in a remarkable memoir ' On Infinity ; and on the Sign of Equality ' \ Apparent equality is that with which physical science deals. Those magnitudes are practically equal which •■ 'Cambridge Philosophical TranBactions,' [1865] vol. xi, Part I. DigitizedbyGOOgle TUEORT OF APPROXIMATION. 103 differ only by an imperceptible quantity. To tbe car- penter anything less than the hundredth part of an inch is non-existent ; there are few arte or artists to which the hundred-thousandth of an inch is of any account. Since all coincidence between physical magnitudes is judged by one or other sense, we must be restricted to a knowledge of apparent equality. In reality even apparent equality is rarely to be ex- pected. More commonly experiments will give only probable equtdity, that is results will come bo near to each other that the difference may be ascribed to un- important disturbing causes. Thus physicists often assume quantities to be equal provided that they fall within the limits of probable error of the processes employed. We cannot expect observations to agree with theory more closely than they agree with each other, as Newton re- marked of his investigations concerning nancy's Comet. Arithmetic of Approximate Quantities. ConMdering that almost all the quantities which we treat in physical and social science are approximate only, it seems desirable that some attention should be paid in the teaching of arithmetic to the correct interpretation and treatment of approximate numerical statements. We ought carefully to distinguish between 2'5 when it means exactly two and a half, and when it means, as it usually does, anything between 2*45 and 2-55 It would be better in the latter case to write the number as 2-5 ... . and we might then distinguish 2-50 .... as meaning anything between 2-495 ■ ■ ■ ■ and 2*505. When approximate numbers are added, subtracted, multiplied, or divided, it becomes a matter of some complexity to determine the degree of accuracy of the result. There are few persons, for instance, who could assert straightway that Digitizedby Google 104 THE PRINCIPLES OF SCIENCE, the sum of the approximate numbers 34'70, 52*693, 80T, is i67'5 within less than 'oy. So far as I know Mr. Sandeman is the only mathematician who has traced out the rules of approximate arithmetic, and his directions are worthy of careful attention''.. Although the accuracy of measurement has so much advanced Eonce the time of Leslie, it is not superfluous to repeat his protest against the unfairness of affecting by a display of decimal frac- tions a greater degree of accuracy than the nature of the case requires and admits '^. I have known a scientific man to register the barometer to a second of time when the nearest quarter of an hour would have been amply sufficient. Chemists often publish results of analysis to the ten-thousandth or even the miUionth part of the whole, when in all probability the processes employed can- not be depended on beyond the hundredth part. It is seldom desirable to give more than one place of figures of imcertain amount ; but it must he allowed that a nice per- ception of the degree of accuracy possible and desirable is requisite to save misapprehension and needless computa- tion on the one hand, and to secure all attainable exact- ness on the other hand. ' Sandeman, ' Pellcotetica,' p. 214. d Leslie, ' Inquiry into the Nature of Heat,' p. 505. by Google CHAPTER XXII. QUANTITATIVE INDUCTION. Let it be observed that we have not yet formally con- sidered any processes of reasoning which have for their object to disclose general laws of nature expressed in quantitative formulae or equations. We have been in- quiring into the modes by which a phenomenon may be measured, and, if it be a composite phenomenon, may be resolved, by the aid of several measurements, into its component parts. We have rlso considered the precau- tions to be taken in the performance of observations and experiments in order that we may know what phenomena we really do measure and record. In treating of the approximate character of all .observations, we have par- tially entered upon the subject of Quantitative Induction proper, but we must remember that no number of &cts and observations can by themselves constitute science or general knowledge. Numerical facts, like other facta, are but the raw materials of knowledge, upon which our reasoning faculties must be exerted in order to draw forth the secret principles of nature. It is by an inverse process of reasoning that we can alone discover the mathe- matical laws to which varying quantities conform. By well- conducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. In the usual pro- gress of a science three questions will have to be answered as regards every important quantitative phenomenon : — by Google 106 TBE PRUfCIPLES OF SCIENCE. (i) Is there any coDstant relation between the variable and variant 1 {2) What is the empirical formula expressing this re- lation t {3) What is the rational formula expressing the law of nature involved ? Prohable Connexion of Varying Quantities. We find it stated in Mr. Mill's System of Log^c * that ' Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.' This assertion may be considered true when it is interpreted with sufficient caution ; but it might otherwise lead us Into great errors. There is nothing whatever in the nature of things to prevent the existence of two variations which should apparently follow the same law, and yet have no connexion with each other. One binary star might be going through a revolution which, so far as we could tell, was of apparently equal period with that of another binary star, and according to the above rule the motion of one would be the cause of the motion of the other, which would not be really the case. Two astronomical clocks might conceivably be made so nearly perfect that, for several years, 00 difference could be detected, and we might then infer that the motion of one clock was the cause or effect of the motion of the other. This matter really requires the most careful discrimination. We must always bear in mind that the continuous quantities of space, time, force, &c., which we measure, are made up of an in6nite number of infinitely small units. We may then meet with two variable phenomena which follow • Book iii. chap, viii, § 6, Digitized by Google QUANTITATIVE INDUCTION, laws BO nearly the same, that in no part of the variations open to our observation can any discrepancy be discovered. I grant that if two clocks could be shown to have kept exactly the same time during one year, or any finite interval of time, the probability would become infinitely high that there was a connexion between their motions. But it is apparent that we can never absolutely prove such coincidences to exist. Allow that we may observe a difference of one tenth or one hundredth of a second in their time, yet it is just possible that they were independ- ently regulated so as to go together within less than that quantity of time. In short it would require either an in- finitely long time of observation, or infinitely acute powers of measuring a discrepancy to decide positively whether two clocks were or were not in relation with each other. A similar question actually occurs in the'case of the moon's motion. We have absolutelyno record that any other portion of the moon was ever visible to men than such as we now see. This fact sufficiently proves that within the historical period the rotation of the moon on its own axis has coincided with its revolutions round the earth. Does this coincidence prove a relation of cause and effect to exist between these motions 1 The answer must be in the negative, because there might have been so slight a discrepancy between the motions that there has not yet been time to produce any appreciable effect. There may nevertheless be a high probability of con- nexion. The whole question of the relation of quantities thus resolves itself into one of probability. When we can only rudely measure a quantitative result, we can assign but shght importance to any correspondence. Because the brightness of two stars seems to vary in the same manner there is no appreciable probability that they have any relation with each other. Could it be shown that by Google 108 THE PRINCIPLES OF SOIElfGE. their periods of variation were the same even to infinitely small quantities it would be certain, that is infinitely pro- bable, that they were connected, however unlikely this might be on other grounds. - The general mode of esti- mating such probabilities is identical with that applied to other inductive problems. Thus, if the two periods of variation were assigned by pure chance and entirely inde- pendently of each other, the probability would be about one in ten million that they would agree to the one ten- millionth part ; but if the periods be observed to agree to less than that part then there is a probability of at least ten million to one in favour of the opposite hypothesis of connexion. That any two periods of variation should by chance become absolutely equal is infinitely improbable ; hence if, in the case of the moon or any other change, we could prove 'absolute coincidence, we should have certainty of connexion''. With approximate measurements, which alone are within our power, we must hope for approximate certainty at the most. The general principles of inference and probability, ac- cording to which we treat causes and efiects varying in amount, are exactly the same as those by which we treated simple experiments. Continuous quantity, how- ever, affords us an infinitely more extensive sphere of observation, because every different amount of cause, however httle different, ought to be followed by a dif- ferent amount of effect. If we can measure temperature to the one hundredth part of a degree centigrade, then even between o° and ico" we have 10,000 possible dis- tinct trials. If the precision of our measurements is increased, so that the one thousandth part of a degree can be appreciated, our trials may be increased tenfold. The probability of connexion will be proportional to the accuracy of our measurements. ^ laplace, ' System of the World,' transl. by Harte, vol. ii. p. 366. Digitized by Google QUANTITATIVE INDUCTION. When we have the power of varying the quantity of a cause efttirely at our will it is easy to discover whether a certain effect is due to that cause or not. We can then make as many regular or irregular changes as we like, and it is quite incredible that the supposed effect should by chance go through exactly the corresponding series of changes unless by dependence. Thus, if we have a bell ringing in vacuo, the sound increases as we let in the air, and it decreases again as we exhaust the air. Tyndall's singing flames evidently obeyed the directions of his own voice; and Faraday when he discovered the relation of magnetism and light found that, by making or breaking or reversing the current of the electro-magnet, he had complete command over a ray of light, proving beyond all reasonable doubt the dependence of cause and effect In such cases it is the perfect coincidence in time between the change in the effect and that in the cause which raises a high improbability of casual coincidence. It is by a very simple case of variation that we infer the existence of a material connexion between two bodies moving with exactly equal velocity, such as the locomotive engine and the train which follows it. Elaborate observa- tions were requisite before astronomers could all be con- vinced that the red hydrogen flames seen during solar eclipses belonged to the sun, and not to the moon's atmo- sphere as Flamsteed assumed. As early as 1706, Captain Stannyan noticed a blood red streak in an eclipse which he witnessed at Berne, and he asserted that it belonged to the sun ; but his opinion was not finally established until photographs of the eclipse in i860, taken by Mr. De la Eue, showed that the moon's dark body gradually covered the red prominences on one side, and uncovered those on the other, in short, that these prominences moved precisely as the sun moved and not as the moon moved. by Google no THE PRINCIPLES OF SCIENCE. Even when we have no means of accuratelj measuring the variable quantities we may yet be convinced of their connexion, if one always varies perceptibly at the same time as the other. Fatigue increases with exertion ; hunger with abstinence from food ; desire and degree of utiUty decrease with the quantity of commodity con- sumed. We know that the sun's heating power depends upon his height in the sky ; that the temperature of the air falls in ascending a mountain ; that the earth's crust is found to be perceptibly warmer as we sink mines into it; we infer the direction in which a sound comes from the change of loudness as we approach or recede. The facility with which we can time after time observe the increase or decrease of one quantity with another suf- ficiently shows the connexion, although we may be un- able to assign any precise law of relation. The probabiHty in such cases depends upon frequent coincidence in time. Empirical McUkematical Laws. It is important to acquire a clear comprehension of the part which is played in scientific investigation by em- pirical formulffi and laws. If we have a table containing certain values of a variable and the corresponding values of the variant, there are certain mathematical processes by which we can infallibly discover a mathematical formula yielding numbers in more or less exact agreement with the table. We may generally assume that the quantities will approximately conform to a law of the form y = A + Bx-fCx*, in which X is the variable and y the variant. We can then select from the table three values of y, and the cor- responding values of z ; inserting them in the equation, we obtain three equations by the solution of which we gain the values of A, B, and C. It will be found as a Digitized by Google QUANTITATIVE INDUCTION. general rule that the formula thus obtained yields the other numbers of the table to a considerable degree of approximation. In many cases even the second power of the variable will be unnecessary ; thus Regnault found that the results of his elaborate inquiry into the latent heat of steam at different pressures were represented with suflBcient ac- curacy by the empirical formula A = 6065 H- 0-305 t, in which X is the total heat of the steam, and t the tem- perature*'. In o^er cases it maybe requisite to include the third power of the variable. Thus physicists assume the law of the dilatation of liquids to be of the form it= at + 6(» + cf, and they calculate from results of observation the values of the three constants a, h, c, which are usually small quantities not exceeding one hundredth part of a unit, but requiring to be determined with great accuracy •*. Theoretically speaking, this process of empirical repre- sentation might be applied with any degree of accuracy ; we might include stiU higher powers in the formula, and with sufl&cient labour obtain the values of the constants, by using an equal number of experimental results. In a similar manner all periodic variations may be repre- sented with any required degree of accuracy by formulee involving the sines and cosines of angles and their mul- tiples. The form of any tidal or other wave may thus be expressed, as Sir G. B. Airy has explained^. Almost all the phenomena registered by meteorolo^ts are periodic in character, and when freed from disturbing causes may be embodied in empirical formulfie. Bessel has given a E ' Chemical Reports and Memoirs,' Cavendish Society, p. 294. ^ Jamin, ' Cours de Physique,' vol. ii. p. 38. * ' On Tides and Waves,' Encyclopeedia Uetropolitona, p. 366*. by Google 112 THE PRIlfCIPLES OF SCIENCE. rule by which from any regular series of observations we may, on the principle of the method of leaat squares, calculate out with a moderate amount of laboiu: a formula expressing the variation of the quantity observed, in the most probable manner. In meteorology three or four terms are usually suflScient for representing any periodic phenomenon, but the calculation might be carried to any higher degree of accuracy. As the details of the process have been described by Sir John Herscbel in his admirable treatise on Meteor* logy f, I need not further enter into them. The reader might be tempted to think that in these processes of calculation we have an infallible method of discovering inductive laws, and that my previous state- ments (Chap. VII.) as to the purely tentative and inverse character of the inductive process are negatived. Were there indeed any general method of inferring laws from facts it would overturn my statement, but it must be carefidly observed that these empirical formulae do not coincide with natural laws. They are only approximations to the results of natural laws founded upon the general principles of approximation. It has already been pointed out that however complicated be the nature of a curve we may examine so small a portion of it, or we may ex- amine it with such rude means of measurement, that its divergence from an elliptic curve will not be apparent. As a still ruder approximation a portion of a straight line will always serve our purpose ; but if we need higher pre- cision a curve of the third or fourth degree will almost certainly be sufficient. Now empirical formulae really re- present these approximate curves, but they give us no information as to the precise nature of the curve itself to which we are approximating. In another mode of ex- pression we may say that we do not learn what function ^ ' Encyclopwdift Britannica,' art. Meteorology. Reprint §§ 151-156. by Google QUANTITATIVE INDUOTION. the variaDt is of the variable, but we obtain another fuDc- tion which, within the bounds of our observation, gives nearly the same series of values. Discovery of Rational FormulcB. Let UB now proceed to consider Hie modes in which £rom numerical results we can establish the actual relation between the quantity of the cause and that of the effect. What we want is a rationed formula or function, which may exhibit the reason or exact character and origin of the law in question. There is no word more frequently used by mathematicians than the word /ufu^ion, and yet it is di£Scult to define ita meaning with perfect acciuac^. Originally it meant performance or execution, being equi- valent to the Greek Xetrovpyta or rAeff^a. Mathematicians at first used it to mean any power of a quantity, hut afterwards generalized it so as to include ' any quantity formed in any .manner whatsoever from another quantity?.' Any quantity, then, which depends upon and varies with another quantity may be called a function of it, and eith^ may be considered a function of the other. Given the quantities, we want the function of which they are the values. It may first of all be pointed out that simple iaspection of the numbers cannot as a general rule disclose the function. In an earlier part of this work (vol. i. p. 142) I put before the reader certain numbers, and requested him to point out the law which they obey, and the same question will have to be asked in eveiy case of quantitative induction. There are perhaps three methods, more or less distinct, by which we may hope to obtain an answer : (i) By purely haphazard trial (2) By noting the general character of the variation of B Lagraoge, ' Lemons sur le Calcnl dea Fonctione,' 1806, p. 4. VOL. II. I by Google 114 THE PRINCIPLES OF SCIENCE. the quantities, and trying by preference functiona which give a similar form of variation. (3) By deducing from previous knowledge the form of the function which is most likely to suit. Having certain numerical results we are always at perfect liberty to invent any kind of mathematical formula we like, and then try whether, by the suitable selection of values for the unknown constant quantities we can make it give the required results. If ever we fell upon a formula which does so, to a fair degree of approximation, there is a presumption in favour of ita being the true function, although there is no certainty whatever in the matter. In this way I happened to discover a simple mathematical law which closely agreed with the results ■of certain experiments on muscular exertion. This law was afterwards shown by Professor Haughton to be the true rational law according to his theory of muscular action**. But the chance of succeeding in this manner is usually very small. The number of possible functions is certainly infinite, and even the number of comparatively simple functions is so very large that the probability of falling upon the correct one by mere chance is very slight. Let the reader observe that even when we can thus obtain the law it is by a deductive process, not by showing that the numbers give the law, but that the law gives the numblers. In the second place, we may, by a survey of the numbers, gain a general notion of the kind of law they are likely to obey, and we may be much assisted in this process by drawing them out in the form of a curve, as will be presently considered. We can in this way ascer- tain with some probability whether the curve is likely to *^ Haughton, 'Principles of Animal Mechanica,' 1873, pp. 444-450, Natwre, 30th of Jnne, 1870, vol. ii. p. 138. Digitized by Google QUANTITATIVE INDUCTION. be a closed one, or whether it has infinite branches ; whether such branches are asymptotic, that is, approach indefinitely towards straight lines ; whether it is logar rithmic in character, or trigonometric. This indeed we can only do if we remember the results of previous in- vestigations. The process is still inversely deductive, and consists in noting what laws gave particular curves, and then inferring inversely that such curves belong to such laws. If we can in this way discover the class of funo- tions to which the required law belongs, our chances of complete success are much increased, because our hap- hazard trials are now reduced within a narrower sphere. But, unless we have almost the whole curve before us, Uie identification of its character must be a matter of great uncertainty ; and if, as in most physical investigations, we have a mere fragment of the curve, the assistance given would be quite illusory. Curves of almost any character can be made to approximate to each other for a limited extent, so that it is only by a kiiid of divination that we can fall upon the actual function, imlees we have theoretical knowledge of the kind of function applicable to the case. When we have once obtained what we believe to be the correct form of fimction, the remainder of the work is mere mathematical computation to be performed infallibly according to fixed rules', which include those employed in the determination of empirical formulsB (voL ii. p. no). The function wilJ involve two or three or more unknown constants, the vdues of which we need to determine by our experimental resulta Selecting some of our results widely apart and nearly equidistant, we must form by means of them as many equations as there are constant quantities to be determined. The solution of these equa- tions win then give us the constants required, and having ' See Jamin, ' Coura de FbyBique/ Tol. ii. p. 50. Digit zed by Google 116 THE PRINCIPLES OF SCIENCE. now the actual function we can try whether it gives with sufficient accuracy the remainder of our experimental resultfl. If not, we must either make a new selection of results to give a new set of equations, and thus obtain a new set of values for the constants, or we must acknow- ledge that our form of function has been wrongly chosen. If it appears that the fonn of function has been correctly ascertained, we may regard the constants as only approxi- mately accurate and may proceed by the Method of Least Squares (vol. i p. 458) to determine the most probable values as given by the whole of the experimental results. In most cases we shall find ourselves obliged to fall back upon the third mode, that is, anticipation of the form of the law to be expected on the ground of previous knowledge. Theory and analogical reasoning must be our guides. The general nature of the phenomenon will often indicate the kind of law to be looked for. If one form of energy or one kind of substance is being converted into another, we may expect the law of direct simple pro- portion. In one distinct class of cases the effect already produced influences the amount of tiie ensuing effect, as for instance in the cooling of a heated body, when the law will be of an exponential form. When the direction in which a force acta influences its action, trigonometrical functions must of course enter. Any force or influence which spreads freely through tridimensional space will be subject to the law of the inverse square of the distance. From such considerations we may sometimes arrive deduc- tively and analogically at the general nature of the mathe- matical law required. The Graphical Method. In endeavouring to discover the mathematical law obeyed by experimental results it is often necessary. by Google QUANTITATIVE INDUCTION. and almost always desirable, to call in the aid of space- representations. Every equation involving two variable quantities corresponds to some kind of plane curve, and every plane curve may be represented symbolically in an equatirai of a more or leas complex character, containing two unknown quantities. Now in an experimental re- search we obtain a number of values of the variant cor- responding to an equal number of values of the variable ; but all the numbers are affected by more or less error, and the values of the variable will often be irr^ularly disposed. Even if the numbers were absolutely correct and disposed at regular intervals, there is, as we have seen, no direct mode of discovering the law, but the dif- ficulty of discovery is much increased by the uncertainty and insularity of the results. Under such circumstances, the best mode of proceeding is to procure or prepare a paper divided into small equal rectangular spaces, a convenient size for the spaces being one-tenth of mi inch square. The values of the variables being marked off along the scale formed by the lowest horizontal line, a point is marked for each corresponding value of the variant perpendicularly above that of the variable, and at such a height as corresponds to the amoimt of the variant. The exact scale of the drawing is not of much im- portance, but it may require to be adjusted according to circumstances, and different values must often be attri- buted to the upright and horizontal divisions, so as to make the variations conspicuous, but not excessive. If now a curved line be drawn through all the extremities of the ordinates, it will probably exhibit many irregular inflections, owii^ to the errors which affect all the numbers. But, when the results are numerous, it soon becomes ap- parent which results are more divergent than others, and guided by a so-called seme of continuity, it becomes pos- by Google TUB PRINCIPLES OF SCIENCE. fflble to trace a line among the points which will approxi- mate to the true law more nearly than the points them- Belvea The accompanjing figure sufficiently ezpUuns itself. Perkins employed this graphical method with much care in exhibiting the results of his experiments on the compreewon of water^. The numerical results were marked upon a sheet of paper very exactly ruled at intervals of one-tenth of an inch, and the original marks were left in order that the reader might judge of the correctness of the curve drawn, or choose another for himself. Regnault carried the method to perfection by laying off the points with a small screw dividing engine' ; and he then formed a table of results by drawing a con- tinuous curve, and measuring its height for equidistant values of the variable. Not only does a curve drawn in this manner enable us to assign by measurement numerical results more free from accidental errors than any of the numbers obtained directly irom experiment, but the form of the curve sometimes indicates the class of functions to which our results belong k ' Philoeopbicat Traniactions,' i8i(>, p. 544. 1 Jamin, 'Cours de Physique,' vol. ii p, 34, kc. by Google QUANTITATIVE INDUCTION. Engraved sheets of paper ready prepared for the draw- ing of curves may be obtained from Mr. Stanford, at 6 and 7 Charing Cross, or from Messrs. W. and A. K. Johnston, of London and Edinburgh. When -we do not require great accuracy, paper ruled by the common machine-ruler into equal squares of about one-fifth or one- sixth of an inch square will serve well enough. I have found Vere Foster's Exercise Book, No. i?"", which is ruled in this way, very useful for statistical or other numerical purposes. I have also met with engineers' and surveyors' memorandum books ruled with one-twelfth inch squares. When a number of complicated curves have to be drawn, I have found it best to rule a good sheet of drawing paper with lines carefully adjusted at the most convenient distances, and then to prick the points of the curve through it upon another sheet fixed underneath. In this way we obtain an accurate curve upon a blank sheet, and need only introduce such division lines as are requisite to the xuiderstanding of the curve. In some cases our numerical results will correspond, not to the height of single ordinates, but to the area of the curve between two ordinates, or the average height of ordinates between certain limits. If we measm-e, for instance, the quantities of heat absorbed by water when raised in temperatiare from 0° to 5°, from 5° to 10°, and so on, these quantities will really be represented by areas of the curve denoting the specific heat of water ; and, since tJie specific heat varies continuously between every two points of temperature, we shall not get the correct curve by simply laying off the quantities of heat at the mean temperatures, namely 2 J", 'j\°, and so on. Mr. J. W. Strutt has shown that if we have drawn such an incorrect curve, we can with little trouble correct it by a simple " Published by WhitUker & Co., London. Digitized by Google 120 THE PRINCIPLES OF SCIENCE. geometrical process, and obtain to a very close approxi- mation the true ordinates instead of those denoting Interpolation and EoUrapolation. When we. have by experiment obtained two or more numerical results, and endeavour, without further resort to experiment, to infer and calculate intermediate results, we are said to interpolate. If we wish to assign by reasoning results lying beyond the limits of experiment, we may be s^d, using an expression of Sir George Airy, to extrapolate. These two operations are to a certain extent the same in principle, but differ in practicability. It is a matter of great scientific importance to appre- hend precisely how far we can interpolate or extend experimental results by extrapolation, and on what In the first place, if the interpolation is to be more than empirical and speculative, we must have not only the experimental results, but the laws which they obey — we must in fact go through the complete process of scien- tific investigation. Having discovered the laws of nature applying to the case, and verified them by showing that they agree with the experiments in question, we are then in a fiiir position to anticipate the results of any similar experiments. Our knowledge even now is not certain, because we cannot completely prove the truth of any assumed law, and we cannot possibly exhaust all the cir- ctmistances which may more or less afiect the result. Even at the best then our interpolations wiU partake of the want of certainty and precision attaching to all our knowledge of nature. Yet having the supposed laws, our » J. W. StniU, 'Od a correction sometimee required in curves pro- feesing to represent the connexion between two physical magnitudes.' 'Philosophical Magazine,' 4tb Series, vol. zlii. p. 441. by Google QUANTITATIVE INDUCTION. results will be as sure and accurate as any we can attain to. But sQch a complete procedure is more thau we generally mean by interpolation, wbich generally denotes the em- ployment of Bome general metbod of eetimating in a merely approximate and probable manner the results whidi might have been expected independently of any complete theoretical investigation. Regarded in this light, interpolation is in reality an indeterminate problem. From given values of a function it is impossible to determine that function ; for we can always invent an infinite number of functions which would give those values if we are not restricted by any other conditions, just as through a given series of points we can always draw an infinite number of curves, if we may di- verge between or beyond the points into bends and cusps as we think fit". In any process of interpolation we must in fact be guided more or less by d priori condderations; we must know, for instance, whether or not periodical fluctuations are to be expected, and we must be guided accordingly in the choice of mathematical formulae. Sup- posing, for the present, that the phenomenon is non- periodic, we next pioceed to assume that the function can be expressed in a limited series of the powers of the variable. The number of powers which can be included depends upon the number of experimental results avail- able, and must be at least one less than this number. By processes of calculation, which have been idready alluded to in the section on empirical formulae, we can then calculate the coefficients of the powers, and obtain an empirical ■formiJa which will give the required intermediate results. In reality, then, we return to the methods treated undt^r the head of approximation and empirical formulee ; and interpolation, as commonly understood, consists in assum- Henchel, 'Appendix to Translatjon of Lacroix' Differential Calculus,' P- 55«- by Google THS PBINOIPLBS OF SCIENCE. ing that a curve of Bimple character is to pase through certain determined points. If we have, for instance, two experimental results, and only two, we must assume that the curve is a straight line ; for the parabolas which can be passed through two points are infinitely various in magnitude, and quite indeterminate. One straight line alone can pass through two points, and it will havs an equation of the form y=wui; + n, the constant quantities of which can be readily determined from two results. Thus, if the two values for x, 7 and 1 1, give the values for y, 35 and 53, the solution of two simple equations gives i/=4'5xa;+3'5 as the equation, and for any other value of X, for instance 10, we get a value of y, 48'5. When we take an exactly intermediate value of x, namely 9, this process yields a simple mean result, namely 44. Three experimental results being given, we may assume that they fall upon a portion of a parabola, and simple algebraic calculation readily gives the poBition of any intermediate point upon the parabola. Concerning the process of interpolation as practised in the science of meteorology the reader will find some directions iu the French edition of Eiemtz' Meteorol(^P. When we have, either directly by experiment or by the use of a curve, a series of values of the variant for exactly equidistant values of the variable, it is often very instructive to take the dLfTerences between each value of the variant and the next, and then the differences between those differences, and so on. If any series of dtfTerences approach^ closely to zero it is an indication that the numbers may be correctly represented by a finite em- pirical formula ; if the nth differences are zero, then the formula will contain only the first n-i powers of the vBriable, Indeed we may sometimes obtain by the Cal- F * Coon oomplet Jniniu, ' Conn de Phjuqae,' vol. ii. p. 138. by Google QUANTITATirS INDUCTION. laboured on the subject without exhausting it, and Brink- ley and Ivory have since treated it. A closely connected problem, that r^arding the relation between the pressure and elevation in difTerent strata of Uie atmosphere, has received ^e attention of a long succession of physicists and was most carefully investigated by Laplace. Tet no invariable and general law has been detected. The same may be said concerning the law of human mortality ; abundant statistics on this subject are available, and many hypotheses more or less satisfactory have been put for- ward as to the general form of the curve of mortality, but it seems to be impossible to discover more than an approximate law. It may perhaps be urged that in such subjects no single invariable law can be expected. The atmosphere may be divided into several variable strata which by their xmcon- nected changes frustrate the exact calculations of astro- nomers. Human life may be subject at different ages to a succession of diffeirent influences incapable of reduction under any one law. The results observed may in fact be ag^egates of an immense number of separate results each governed by their own separate laws, so that the subjects may be complicated beyond the possibility of complete resolution by empirical methoda This is certainly true of the mathematical functions which must some time or other be introduced into the science of political economy. Simple Proportional Variation. When we first treat nxunerical results in any novel kind of investigation, our impression will probably be that one quantity varies in simple proportion to another, so as to obey the law y = 77ix+n. We must learn to distinguish ■carefidly between the cases where this proportionality is jeally, and where it is only apparently true. When con- by Google 128 TBE PRINCIPLES OF SCIENCE. sideling the principles of approximation we found that a smfdl portion of any curve will appear to be a straight line. Whenever our modes of measurement are compara- tively rude, we must expect to be unable to detect the curvature. Thus Eepler made meritorious attempts to discover the law of refraction, and he slightly approxi- mated to it when he observed that the angles of incidence and re&action if small bear a constant ratio to each other. Angles when small are very nearly as their sines, so that he reached an approximate result of the true law. Cardan assxmaed, probably as a mere guess, that the force required to sustain a body on an inclined plane was simply propor- tional to the angle of elevation of the plane. This is approximately the case when the angle is very small, and it becomes true again when the angle is a right angle ; but in reality the law is much more (implicated, the power required being proportional to the sine of the angle. The early thermometer-makers were quite unaware whether the expansion of mercury waa exactly propor- tional or not to the heat communicated to It, and it is only in the present century that we have learnt it to be not so. We now know that even gases obey the law of uniform expansion by heat only in an approximate man- ner. Until some reason to the contrary is shown, we should do wdl to look upon every law of simple propor- tion as only provisionally true. Nevertheless, there are many of the most important laws of nature whidi are in the form of simple propor- tions. Wherever a imiform cause acts in independence of its previous effects, we may expect this relation. Thus, an accelerating force acts equally upon a moving and a motionless body. Hence the velocity produced is always in simple proportion to the force, and also to the duration of its uniform action. As gravitating bodies never in- terfere with each other's gravity, this force is in direct by Google QUANTITATIVE INDUCTION. simple proportion to the mass of each of the attracting bodies, the mass being measured by, or proportional to inertia. Similarly, in all cases of * direct - unimpeded action,' as Sir J. Herschel has remarked ^, we may expect simple proportion to manifest itself. In such cases the equation expressing the relation may have the still simpler form y=ni^. A Mmilar simple relation holds true wherever there is a conversion of one Bubstance or form of enei^ into another. The quantity of chloride of silver is propor- tional to the quantity either of chlorine or silver. The amount of heat produced in friction is exactly propor- tional to the mechanical energy absorbed. It was ex- perimentally proved by Faraday that ' the chemical power of the current of electricity is in direct proportion to the quantity of electricity which passes.' When an electric current is produced, the quantity of electric energy is simply proportional to the weight of metal dissolved. If electricity is turned into heat, there is again simple proportion. Wherever, in fact, one thing is but another thing with a new aspect, we may expect to find the law of simple proportion. It is only among the most elementary causes and effects that this simple re- lation will hold true. Simple conditions do not, generally speaking, produce dmple results. The planets move in approximate circles round the sun, but the apparent motions, as seen from the earth, are so various, that men have not believed in such a simple view of the matter for more than about two centuries and a half. All those motions, again, are summed up in the law of gravity, of no great complexity, yet men never have, and never can be, able to exhaust the complications of action and reaction, even among a small number of planets. We should be on our guard against a tendency to assume that > ' Freliminary Discourse,' &c. p. i g3. by Google 130 TUB PRINCIPLES OF SCIENCE. the connexion of cause and effect is one of direct pro- i:iortion. Bacon reminds us of tlie woman in ^sop'B fable, who expected that her ben, with a double measure of barley, would lay two eggs a day instead of one, whereas it thereby grew fat, and ceased to lay any eggs at alL Digit zed by Google CHAPTER XXIII. THE USE OF HYPOTHESIS. Ip the views of induction upheld in this work be correct, all inductive investigation consists in a marriage of hypothesis and experiment. When facta are already in oiir possession, we frame an hypothesis to explain their mutual relatione, and by the success or non-success of this explanation is the value of the hypothesis to be entirely judged. In the framing and deductive treatment of such hypotheses, we must avail ourselves of the whole body of scientific truth already accumulated, and when once we have obtained a probable hypothesis, we must not rest until we have verified it by comparison with new facts. By deductive reasoning and calculation, we must endeavour to anticipate such new phenomena, especially those of a singular and exceptional nature, as would necessarily happen if the hypothesis be true. Out of the infinite number of observations and experiments which are possible at everj' moment, theory must lead us to select those few critical ones which are suitable for confirming or negativing our anticipations. This work of inductive investigation cannot be guided by any system of precise and infallible rules, like those of deductive reasoning. There is, in fact, nothing to which we can apply rules of method, because the laws of natxure to be treated must be in our possession before we can treat them. If, indeed, there were any single rule of by Google 132 THE PRINCIPLES OF SCIENCE. inductive method, it would direct us to make an ex- haustive arrangement of facts in all possible orders. Given a certain number of specimens in a museum, we might arrive at the best possible classification by going syatematically through all possible classifications, and, were we endovfed with infinite time and patience, this would be an effective method. It doubtless is the method by which the first few simple steps are taken in every incipient branch of science. Before the dignified name of science is applicable, some coincidences will chance to force themselves upon the attention. Before there was a science of meteorology, or any comprehension of the true conditions of the atmosphere, all observant persons learned to associate a peculiar clearness of the atmosphere with coming rain, and a colourless sunset with fine weather. Knowledge of this kind is called empirical, as seeming to come directly from experience ; and there is doubtless a considerable portion of our knowledge which must always bear this character. We may be obliged to trust to the casual detection of coincidences in those branches of knowledge where we are deprived of the aid of any guiding notions ; but a very little reflection will show the utter insufficiency of haphazard experiment, when applied to investigations of a complicated nature. At the best, it will be the simple identity, or partial identity, of classes, as illus- trated in pp. 146-154 of the first volume, which can be thus detected. It was pointed out that, even when a law of nature involves only two circumstances, and there are one hundred distinct circumstances which may possibly be connected, there will be no less than 4950 pairs of circumstances between which a coincidence may exist. When a law involves three or more circum- stances, the possible number of coincidences becomes vastly greater still. When considering, again, the subject Digitized by Google THE USE OP HYPOTUESIS. 133 of combinations and permutations, it became apparent that we could never cope with the possible variety of nature. Ad exhaustive examination of the metallic alloys, or chemical compounds which can be formed, was foQod to be out of the question (vol. i. p. 218). It is on such considerations that we can explain the very small addi- tions made to our knowledge by the alchemists. Many of them were men of the greatest acuteness, and their indefatigable labours were pursued through many cen- turies. A few of the more common compoimds and phenomena were discovered by them, but a true insight into the principles of nature, now enables chemists to discover far more useful facte in a single year than were yielded by the alchemists during many centuries. There can be no doubt that Newton was really an alchemist, and often spent his days and nights in laborious experiments. But in trying to discover the secret by which gross metals might be rendered noble, his lofty powers of deductive investigation were wholly useless. Deprived of all guiding clues, his experiments must have been, like those of all the alchemists, purely tentative and hap- hazard. While his hypothetical and deductive investiga- tions have given us the true system of nature, and opened the way in almost every one of the great branches of natural philosophy, the whole results of his tentative experiments fUB comprehended in a few happy guessoe, given in his celebrated ' Queries.' Even when we are engaged in apparently passive observation of a phenomenon, which we cannot modify experimentally, it is advantageous that our attention should be guided by some theoretical anticipations. A phenomenon which seems simple is, in all probability, really complex, and unless the mind is actively engaged in looking for particidar details, it is quite likely that the most critical circumstances will be passed over. Bessel Digit zed by Google 134 THE PRWCIPLES OF SCIENCB. regretted that no distinct theory of the constitution of cometa had guided his observatioiis of Halley's comet*; in attempting to verify or refute any good hypothesis, not only would there have been a chance of establishing a true theory, but if confuted, the very confutation would pro- bably have involved a large store of useful observations. It woxJd be an interesting work, but one which I can- not undertake, to trace out the gradual reaction which has taken place in recent times against the purely empirical, or Baconian, theory of induction. Francis Bacon, seeing the futility of the scholastic logic, which had long been predominant, asserted that the accumulation of facts and the careful and orderly abstraction of axioms, or general laws from them, constituted the true method of induction. This method, as far as we can gather its exact natxire from Bacon's writings, would correspond to the process of exhaustive examination and classification to which I have just alluded. The value of this method might be estimated historically l^ the fact that it has not been followed by any of the great masters of science. Whether we look to Galileo, who preceded Bacon, to Gilbert, his contemporary, or to Newton and Descartes, his successors, we find that discovery was achieved by the exactly opposite method to that advocated by Bacoa Throxigh- out Newton's works, as I shall more fully show in suc- ceeding pages, we find deductive reasoning wholly pre- dominant, and experiments are employed, as they should be, to confirm or refiite hypothetiad anticipations of nature. In my 'Elementary Lessons in Logic' (p. 258), I stated my belief that there was no kind of reference to Bacon in Newton's works. I have since found that Newton does once or twice employ the • Tyndsll, ' On Cometary Theory,' Philosophicfll Magazine, April, 1869. ^ti Series, vol, xxxvii p. 243. by Google THE USE OF HYPOTHESIS. 135 expression eocpei-imentum crucis in his ' Opticks,' but this is the only expression, so far as I am aware, which could indicate on the pact of Newton direct or indirect acquaintance with Bacon's writings *•. Other great physicists of the same age were equally prone to the use of hypotheses rather than the bUnd accumulation of facts in the Baconian manner. Hooke emphatically asserts in his posthumous work on Philo- sophical Method, that the first requisite of the Natural Philosopher is readiness at guessing the solution of many phenomena and making queries. ' He ought to be very well skilled in those several kinds of philosophy already known, to understand their several hypotheses, sup- positions, collections, observations, &c., their various ways of ratiocinations and proceedings, the several failings and defects, both in their way of raising, and in their way of managing their several theories : for by this means the mind will be somewhat more ready at guessing at the solution of many phenomena almost at first sight, and thereby be much more prompt at making queries, and at tracing the subtlety of Nature, and in discovering and searching into the true reason of things.' We find Horrocks, again, than whom no one was more filled with the scientific spirit, telling us how he tried theoiy after theory in order to discover one which was in accordance with the motions of Mars''. It might readily be shown again that Huyghens, who possessed one of the most perfect philosophical intellects, followed the deductive process combined with continual appeal to experiment, with a skill closely analogous to that of Newton. As to Descartes and Leibnitz, their investigations were too much opposed to the Baconian rules, since they too often l" See ' Fhilosopliical Transactions,' abridged by Lowthorp. ♦th edit, vol. i. p. 130. c Horrocks, 'Opera Posthuma' (1673), p. a^6. by Google THE PRINCIPLES OF SCIENCE. adopted hypothetical reasoning to the exclusion of ex- perimental verification. Throughout the eighteenth cen- tury science was supposed to be advancing by the pur- suance of the Baconian method, but in reality hypothetical investigation was the main instrument of prc^ess. It is only in the present century that physicists began to recog- nise this truth. So much opprobrium had been attached by Bacon to the use of hypotheses, that we find Young speaking of them in an apologetic tone. ' The practice of advancing general principles and applying them to par- ticular instances is so far from being £ital to truth in all sciences, that when those principles are advanced on suf- ficient grounds, it constitutes the essence of true phi- losophy'''; and he quotes cases in which Sir Humphry Davy trusted to his theories rather than his experimenta The late Sir John Herschel, who was both a practical physicist and an abstract logician, always entertained the deepest respect for Bacon, and made the ' Novum Organum ' as far as possible the basis of hb admirable ' Discoiu*Be on the Study of Natural Philosophy.' Yet we find him in Chapter Til fully recognising the part which the forma- tion and verification of theories forms in the higher and more general investigations of physical science. The late Mr. J, & Mill carried on the reaction by recognising as a distinct method Ihe Deductive Method in which Ratio- cination, that is, deductive reasoning, is employed for the discovery of new opportunities of testing and verifying a hypothesis. His main error consisted in the fact that throughout the other parts of his system he inveighed against the value of the deductive process, and even asserted from time to time that eveiy process of reasoning is inductive. In fact Mill fell into much confusion in the use of the words induction and deduction, because he ' Young's WorkF, vol. I p. 593. Digitized by Google TUB USE OF HYPOTUESIS. 137 failed to observe that the inverse use of deduction con- stitutes induction. Even Francis Bacon was not wholly unaware of the value of hypothetical anticipation. In one or two places he incidentally acknowledges it, as when he remarks that the subtlety of nature surpasses that of reason, adding that ' axioms abstracted &om particular facts in a careful and orderly manner, readily suggest and mark out new particulars.' The true course of inductive procedure is that which has yielded all the more lofty and successfitl results of science. It consists in Anticipating Nature, in the sense , of forming hypotheses as to the laws which are probably in operation ; and then observing whether the combi- nations of phenomena are such as would follow from the laws supposed. The investigator begins with facts and ends with them. He uses such facts as are in the first place known to him in suggesting probable hypotheses ; deducing other facts which would happen if a particular hypothesis is true, he proceeds to teat the truth of his notion by fresh observations or experiments. If any result prove different from what he expects, it leads him either to abandon or to modify his hypothesis ; but every new fact may give some new suggestion as to the laws in action. Even if the result in any case agrees with his anticipations, he does not regard it as finally confirmatory of his theory, but proceeds to test the truth of the theory by new deductions and new trials. The investigator in such a process is assisted by the whole body of science previously accumulated. He may employ analogy, as I shall point out, to guide him in the choice of hypotheses. The manifold connexions between one science and another may give him strong clues to the kind of laws to be expected, and he thus always selects out of the infinite number of possible hypotheses those Digitized by Google 138 THE PRINCIPLES OF SCIENCE. which are, as far as can he foreseen at the moment, most prohahle. Each experiment, therefore, which he performs is that most likely to throw light upon his subject, and even if it frustrate his first views, it probably tends to put him in possession of (Jie correct clue. Requisites of a Good Hypothesis. There will be no difficulty in pointing out to what conditions, or rather to what condition an hypothesis must conform in order to be accepted as valid and probable. That condition, as I conceive, is the single one of enabling us to infer the existence of phenomena which occur in our experience. Agreement with fact is the one sole and sujftcient test of a true hypothesis. Hohbes, indeed, has named two conditions which he considers requisite in an hypothesis, namely, (i) That it should he conceivable and not absurd ; (2) That it should allow of phenomena being necessarily inferred. Boyle, in noticing Hobbes' views, proposed to add a third condition, to the effect that the hypothesis should not be inconsistent with any other truth or phenomenon of nature*. Of these three conditions, I am inclined to think that the first cannot be accepted, unless by inconceivable and absurd we mean self-contradictory or inconsistent with the laws of thought and nature. I shall have to point out that some of the most sure and satisfactory theories involve suppositions which are wholly inconceivable in a certain sense of the word, because the mind cannot sufficiently extend its ideas to frame a notion of the actions supposed to exist. That the force of gravity should act instan- taneously between the most distant parts of the planetary system, or that a ray of violet light should consist of e Boyle's * Physical Exaraen,' p. 84, Digitized by Google TUB USE OF UYPOTUESJS. 139 about 700 billioQS of vibrations in each second, are state- ments of an inconceivable and absurd character in one sense ; but they are so far from being opposed to fact that we cannot on any other suppositions account for the phe- nomena observed. But if an hypothesis involve self-con- tradiction, or is inconsistent with known laws of nature, it is so far self-condemned. We cannot even apply processes of deductive reasoning to a self-contradictoiy notion ; and being entirely opposed to the most general and certain laws known to us, the primary laws of thought, it thereby conspicuously fails to agree with facta. Since nature, again, Is never self-contradictory, we cannot at the same time accept two theories which lead to contradictory results. If the one agrees with nature, the other cannot. Hence if there be a law which we believe with high pro- bability to be verified in observation, we must not frame an hypothesis in conflict with it, otherwise the hypothesis will necessarily be in disagreement with observation. Since no law or hypothesis is proved, indeed, with ab- solute certainty, there is always a chance, however slight, that the new hypothesis may displace the old one ; but the greater the probability which we assign to that old hypothesis, the greater must be the evidence required in favour of the new and conOicting one. A decisive ex- perimeMuni crucia to negative the one, and establish the other, will probably be requisite to allay the strife. I am inclined to assert, then, that there is but one test of a good hypothesis, namely, its conformity with observed facta; but this condition may be said to involve, at the same time, three minor conditions, nearly equivalent to those suggested by Hobbes and Boyle, namely : — ( 1 ) That it allow of the application of deductive reason- ing and the inference of consequences. (2) That it do not conflict with any laws of nature, or of mind, which we hold as true. Digitized by Google THE PRINCIPLES OF SCIENCE. (3) That the consequences inferred do agree with facte of obeeiration. The First Requisite — Possibility of Deductive Reasoning. Ab the truth of an hypothesis is to be proved by ita con- formity with fact, the first condition is that we be able to apply methods of deductive reasoning, and learn what should happen according to such an hypothesis. Even if we could imagine an object acting according to laws wholly unknown in other parts of nature, it would be useless to do so, because we could never decide whether it existed or not. We can only infer what would happen xinder supposed conditions by applying what knowledge we possess of nature to those conditions. Hence, as Bc^ covich truly said, we are to understand by hypotheses 'not fictions altogether arbitrary, but suppositions con- formable to experience or analogy.' It follows that every hypothesis worthy of consideration must suggest some likeness, analogy, or common law, acting in two or more things. If, in order to explain certain facte, a, a', a", &c., we invent a cause A, then we must in some degree appeal to experience as to the mode in which A will act. As the objects and laws of nature are certainly not known to the mind intuitively, we must point out some other cause B, which supplies the requisite notions, and all we do is to invent a fourth term to aji analogy. As B is to its efiects h, y, y, &c., so is A to its efiects a, a', a", &c When, for instance, we attempt to explain the passage of light and heat radiations through space unoccupied by matter, we imagine the existence of the so-called ether. But if this ether were wholly difierent firom anything else known to us, we should in vain try to reason about it. We must at least apply to it the laws of motion, that is, we must by Google THE USE OF HYPOTHESIS. HI 80 far likea it to matter. And as when applying those laws to the elastic medium air, we are able to infer the phenomena of sound, bo by aiding in a similar manner ■ concerning ether we are able to infer the existence of light phenomena corresponding to what do occur. All that we do is to take a material elastic substance, increase its elasticity in an almost indefinite degree, and denude it of gravity and some others of the ordinary properties of matter, but we must retain sufficient likeness to matter to allow of deductive calculations. The force of gravity is in some respects an almost in- comprehensible existence, but in other respects entirely conformable to experience. We can distinctly observe that the force is proportional to mass, and that it acts in entire independence of the other matter which may be present or intervening. The law of the decrease of in- tensity as the square of the distance increases, may be observed to hold true of light, sound, and any other influences emanating from a point, and spreading uni- formly through space. The law is doubtless connected at this point with the primary properties of space itself, and is so far conformable to our necessary ideas. It may well be said, however, that no hypothesis can be so much aa framed in the mind unless it be more or less conformable to experience. As the material of our ideas is undoubtedly derived from sensation, so we cannot figure to ourselves any existence or agent, but as endowed with some of the properties of matter. All that the mind can do in the creation of new existences is to alter com- binations, or by analogy to alter the intensity of sensuous properties. The phenomenon of motion is familiar to sight and touch, and different degrees of rapidity are also familiar : we can pass beyond the limits of sense, and suppose the existence of rapid motion, such as our senses could not measure or observe. We know what is elasticity. Digitized by Google 142 THE PRINCIPLES OF SCIENCE. and we can therefore in a certain sense figure to ourselves elasticity a thousand or a million times greater than any which is aeneuously known to us. The waves of the ocean are many times higher than our own bodies ; other waves, we may observe, are many times less ; continue the pro- portion, and we may ultimately arrive at waves as small as those of light. Thus it is that from a sensuous basis the powers of mind enable us to reason concerning agents and phenomena different in an xmlimited degree. If no hypothesis then can be absolutely opposed to sense, accordance with experience must always be a question of degree. In order that an hypothesis may allow of satisfactory comparison with experience, it must possess a certain definiteness, and, generally speaking, a certain mathe- matical exactness allowing of the precise calculation of results. We must be able to ascertain whether it does or does not agree with facts. The theory of vortices, on the contrary, did not present any mode of calculating the exact relations between the distances and periods of the planets and satellites ; it could not, therefore, -undergo that rigorous testing to whicli Newton scrupulously submitted his theory of gravity before its promulgation. V^ueness and incapa- biUty of precise proof or disproof often enables a false theory to live ; but with those who love truth, such v^ueness should excite the highest suspicion. The up- holders of the ancient doctrine of Nature's abhorrence of a vacuum, had been unable to anticipate the important fact that water would not rise more than 33 feet in a common suction pump. Nor when the fact was pointed out could they explain it, except by introducing a special alteration of the theory to the effect that Nature's ab- horrence of a vacuum was limited to 33 feet. Digit zed by Google TUK USE OF HYPOTHESIS. H3 The Second Requisite — Consistency mth establislied Laws of Nature. In the second place an hypothesis must not be contra- dictory to what we believe to be true concerning Nature. It must not involve self-inconsistency which is opposed to the highest and simplest lawsj namely, those of liOgic. Neither ought it to be irreconcileable with the simple laws of motion, of gravity, of the conservation of energy, or any parts of physical science which we consider to be established beyond reasonable doubt. Not that we are absolutely forbidden to adopt such an hypothesis, but if we do so we must be prepared to disprove some of the best demonstrated truths in the possession of mankind. The fact that conflict exists means that the conse- quences of the theory are not verified if previous dis- coveries are correct, and we must therefore show that previous discoveries are incorrect before we can verify our theory. An hypothesis will be exceedingly improbable, not to say invalid, if it supposes a substance or agent to act in a manner unknown in other cases; for it then fails to be verified in our knowledge of that substance or agent. Several physicists, especially Euler and Grove, have sup- posed that we might dispense with any ethereal basis of light, and infer from the interstellar passage of rays that there was some kind of rare gas occupying space. But if so, that gas must be excessively rare, as we may infer from the apparent absence of an atmosphere around the moon, and from many other facts and laws known to us Concerning gases and the atmosphere ; and yet at the same time it must possess an elastic force at least a billion times as great as atmospheric air at the earth's surface, in order to account for the extreme rapidity of the light Digitized by Google 144 THE PRINCIPLES OF SCIENCE. rays. Such an hypothesis then is inconsistent with the main body of our knowledge concerning gases. Provided that there be no clear and absolute conflict with known laws of nature, there is nothing so im- probable or apparently inconceivable that it may not be rendered highly probable, or even approximately certain, by a sufficient number of concordances. In fact the two best founded and most conspicuously successful theories in the whole range of physical science involve the most absurd suppositions. Gravity is a force which appears to act between bodies through vacuous space ; it is in positive contradiction to the old dictum that nothing could act but through some intervening medium or sub- stance. It is even more puzzling that the force acts in perfect indiflerence to all intervening obstacles. Light in spite of its extreme velocity, shows much respect to matter, for it is almost instantaneously stopped by opaque substances, and to a considerable extent absorbed and de- fleeted by transparent ones. But to gravity all media are, as it were, absolutely transparent, nay non-existent ; and two particles at opposite points of tbe earth affect each other exactiy as if the globe were not between. To complete the apparent impossibility, the acUon is, so far as we can ob- serve, absolutely instantaneous, so that every particle of the universe is at every moment in separate cognizance, as it were, of the relative position of every other particle throughout the universe at that same moment of absolute time. Compared with such incomprehensible conditions, the theory of vortices deals with common-place realities. Newton's celebrated saying, hypotheses non jingo, bears the appearance of pure irony; and it was not without apparent grounds that Leibnitz and the greatest con- tinental philosophers charged Newton with re-introducing occult powera and qualitiea The undulatory theory of light presents almost equal Digitized by Google THE USE OF HYPOTHESIS. 146 difficulties of conception. We are asked by physical philosophers to give up all our ordinary prepoesessions, and believe that the interstellar space which seemed so empty is not empty at all, but filled with soTtiething immensely more solid and elastic than steeL As Dr. Yoimg himself remarked f, 'the luminiferous ether, per- vading all space, and penetrating almost all substances, is not only highly elastic, but absolutely solid ! I ! ' Sir John Herschel has calculated the amount of force which may be supposed, according to the undulatory theory of light, to be exerted at each point in space, and finds it to be 1,148,000,000,000 times the elastic force of ordinary air at the earth's surface, so that the pressure of the ether upon a square inch of surface must be about 1 7,000,000,000,000, or seventeen bilHons of pounds^. Yet we live and move without appreciable resistance tbroxigh this medium, in- definitely harder and more elastic than adamant. All our ordinary notions must be laid aside in contemplating such an hypothesis ; yet they are no more than the observed phenomena of light and heat force us to accept. We cannot deny even the strange suggestion of Dr. Young, that there may be independent worlds, some possibly existing in different parts of space, but others perhaps pervading each other unseen and iinfcnown in the same space"". For if we are bound to admit the conception of this adamantine firmament, it is equally easy to admit a plurality of such. We see, then, that mere difficulties of conception must not in the least discredit a theory which otherwise agrees with facts, and we must only reject hypotheses which are inconceivable in the sense of break- ing distinctly the primary laws of thought and nature. ' TouDg'a 'WorkB,' vol i. p. 415. ■ 'Fsmilifu- Lectures on Scientific Subjects,' p. 281. l> Toung'B 'Works,' vol. i, p. 477. by Google 146 THE PRINCIPLES OF SCIENCE. The Third Requisite — Covformity with Facts. Before we accept a new hypothesis, it must furnish us with distinct credentials, consisting in the deductive anti- cipation of a series of fiicts, which are not akeady con- nected and accounted for by any equally probahle hypo- thecs. We cannot lay down any precise rule as to the number of accordances which can establish the truth of an hypothesis, because the accordances will vary much in value. While, on the one hand, no finite number of accordances will give entire certainty, the probability of the hypothesis will increase very rapidly with the number of accordances. Seldom, indeed, shall we have a theory free from difficulties and apparent inconsistency with facts. Though one real and undoubted inconsistency would be sufficient to overturn the most plausible theory, yet there is usually some probability that the &ct may be misin- terpreted, or that some supposed law of nature, on which we are relying, may not be true. Almost every problem in science thus takes the form of a balance of probabilities. It is only when difficulty after difficulty has been success- fully explained away, and decisive experimenta cruets have, time after time, resulted in favour of our theory, that we can venture to assert the falsity of all objections. The sole real test of an hypothesis is its accordance with fact. Descartes' celebrated system of vortices is exploded and rejected, not because it was intrinsically absurd and mconceivahle, bat because it could not give results in accordance with the actual motions of the heavenly bodies. The difficulties of conception involved in the apparatus of vortices, are mere child's play compared with those of gravitation and the undulatory theory already described. The vortices are on the whole plausible suppositions ; for the planets and satellites bear at first sight much re- semblance to objects carried round in whirlpools, an Digitized by Google TUB USE OF HYPOTHESIS. \il analogy which doubtleea suggested the theory. The failure was in the first and third requisites ; for, as already remarked, the theory did not allow of any precise cal- culation of planetary motions, and was so far incapable of rigoi-QxiB verification. But so far as we can institute a comparison, facts are entirely against the vortices. Newton carefully pointed out that the Cartesian theory was incon- sistent with the laws of Kepler, and would represent the planets aa moving more rapidly at their aphelia than at their perihelia'. Newton did not ridicule the theory as absurd, but showed^ that it was ' pressed with many diflSculties.' The rotatory motions of the sun and planets on their own axes are in striking conflict with the revo- lutions of the satellites carried round them ; and comets, the most flimsy of bodies, calmly pursue their courses in elliptic paths, altogether irrespective of the vortices which they intersect. We may now also point to the inter- lacing orbits of the minor planets as a new aiid insuper- able difBculty in the way of the Cartesian ideas. Newton, though he established the best of theories, was also capable of proposing one of the worst ; and if we want an instance of a theory decisively contradicted by facts, we have only to turn to his views concerning the origin of natural colours. Having analysed, witli incom- parable skill, the origin of the colours of thin plates, he su^ests that the colours of all bodies and substances are determined in like manner by the size of their ultimate particles. A thin plate of a definite thickness will reflect a definite colour ; hence, if broken up into Iragments it will form a powder of the same colour. But, if this be a sufficient explanation of coloured substances, then every coloured fluid ought to reflect the complementary colour of that which it transmits. Colourless transparency arises, i ' Principta,' bk. II. Sect. ix. Prop. 53. k Ibid. bk. III. Prop. 43. Qenenl Scholium. L 2 Digitized by Google 148 THE PRINCIPLES OF SCIENCE. according to Newton, from all the particles being too minute to reflect light; but if so, every transparent sub- stance should appear perfectly black by reflected light, and, vice versd, every black substance should be trans- parent. Newton himself so acutely felt this last difliculty as to suggest that true blackness is due to some internal refraction of the rays to and fro, and an ultimate stifling of them, which he did not attempt further to explain. Unless some other process came into operation, neither refraction nor reflection, however often repeated, would destroy the energy of light. The theory gives no account, therefore, as Brewster shows, of 24 parts out of 25 of the light which faUs upon a black coal, and the — th part which is reflected from the lustrous surface is equally in- conmstent with the theory, because fine coal-dust is almost entirely devoid of reflective power'. It is now generally believed that the colours of natural bodies are due to the unequal absorption of rays of light of different refrangi- bility. Experimentum Crucis. As we deduce more and more conclusions from a theory, and find them verified by trial, the probability of the theory increases in a most rapid manner ; but we never escape the risk of error altogether. Absolute certainty is beyond the power of inductive investigation, and the most plausible suppositions may ultimately be proved false. Such is the groundwork of similarity in nature, that two very different conditions may often give closely similar results. We sometimes find ourselves therefore in possession of two or more hypotheses which both agree ' Brewster's ' life of Newton,' ist edit. chap. vii. Digitized by Google THE USE OP HYPOTHESIS. 149 with so many experimental facts as to have great appear- ance of truth. Under such circumstances we have need of some new experiment, which shall give results agreeing with one hypothesis but not with the other. Any such experiment which decides between two rival theories may be called an Experimentum Crucis, an Experiment of the Finger Post. Whenever the mind stands, as it were, at cross-roads, and knows not which way to select, it needs some decisive guide, and Bacon therefore assigned great importance and authority to in- stances or facts which serve in this capacity. The name given by Bacon has become exceedingly familiar ; it is perhaps almost the only one of Bacon's figurative expres- sions which has passed into common use. We even find Newton, as I have already mentioned, using the name (volii. p. 134). I do not think, indeed, that the common use of the word at all agrees with that intended by Bacon, Sir John Herschel says that ' we make an experiment of the crucial kind when we form combinations, and put in action causes from which some particular one shall be deliberately excluded, and some other purposely admitted"".' This, however, seems to be the description of any special ex- periment not made at haphazard. Pascal's experiment of causing a barometer to be carried to the top of the Fuy-de-Bdme has often been conndered as a perfect experiTnentum cntcis, if not the first distinct one on record"; but if so, we must dignify the doctrine of Nature's abhorrence of a vacuum with the position of a rival theory. A crucial experiment must not simply confirm one theory, but must negative another ; it must decide a mind which is in equilibrium, as Bacon says", ■" ' DiBcourae on the Study of Natural Philosophy,' p. 151. n Ibid. p. 229, o ' Novum Organum,' bk. II. Aphorism 36. by Google 160 THE PRINCIPLES OF SCIENCE. between two equally plausible views. ' When in search of any nature, the understanding comes to an equilibrium, as it were, or stands suspended as to which of two or more natures the cause of nature inquired after should be attributed or assigned, by reason of the frequent and common occurrence of several natures, then these Crucial Instances show the true and inviolable association of one of these natures to the nature sought, and the imcertain and separable aUiance of the other, whereby the question is decided, the former nature admitted for the cause, and the other rejected. These instances, therefore, afford great light, and have a kind of overruling authority, so that the course of interpretation will sometimes terminate in them, or be finished by them.' The long continued strife between the Corpuscular and Undulatory theories of light forms the best possible illus- tration of the need of an Experimentum Crucis. It is highly remarkable in how complete and plausible a manner both these theories agreed with the ordinary laws of geometrical optics, relating to re6ection and refraction. A movitig particle, according to the first law of motion, proceeds in a perfectly straight line, when undisturbed by extraneous forces. If the particle, being perfectly elastic, strike a perfectly elastic plane, it will bound off in such apath that the angles of incidence and reflection will be equal. Now a ray of light proceeds in a perfectly straight line, or appears to do so, until it meets a reflecting body, when its path is altered in a manner exactly similar to that of the elastic particle. Here is a remarkable correspondence which probably suggested to Newton's mind that light consisted of minute elastic particles moving with excessive rapidity in straight lines. The correspondence was found to extend also to the law of simple refraction ; for if these particles of light be supposed capable of attracting matter, and being attracted by it at insensibly small distances. by Google THE USE OF HTPOTIIESIS. 151 then a ray of light, falling on the surface of a transparent medium, will suffer an increase in its velocity of motion perpendicular to the surface, and the familiar law of sines is the necessary consequence. This remarkable expla- nation of the law of refraction had doubtless a very strong effect in leading Newton to entertain the corpuscular theory, and he appears to have thought that the analogy between the propagation of the rays of light and the motion of bodies was perfectly exact, whatever might be the actual nature of lightP. It is highly remarkable, again, that Newton was able to give, by his corpuscular theory, a plausible explanation of the inflection of light as dis- covered by Grimaldi. The theory would indeed have been a very probable one could Newton's own law of gravity have been applied ; but this was excluded, be- cause the particles of light, in order that they may move in straight lines, must be assumed devoid of any influence upon each other. The Huyghenian or Undulatory theory of light was also able to explain the same phenomena, but with one remarkable difference. If the undulatory theory be true, light must move more slowly in a dense refracting medium than in a rarer one ; but the Newtonian theory assumed that the attraction of the dense medium caused the par- ticles of light to move more rapidly than in the rare medium. On this point, then, there was a complete discrepancy between the two theories, and observation was required to show which theory was to be preferred. Now by simply cutting a uniform plate of glass into two pieces, and slightly inclining one piece so as to increase the length of the path of a ray passing through it, experi- menters have been able to show that the light does move P ' Principle,' bk. I. Sect. xiv. Prop, 96. Scholinm, 'Opticks,' Prop. VI. 31-d edit. p. 70. by Google 152 THE PRINCIPLES OF SCIESCE. more slowly in glass than in ain. More recently, in 1850, Fizeau and Foucault independently measured the velocity of light in air and water by a revolving mirror, and found that the velocity is greater in aar*". There are indeed a number of other points at which experience decides against Newton, and in favour of Huyghena and Young. Euler rejected the Corpuscular theory because particles of matter moving with the immense velocity of light must possess great momentum, of which there is no evidence in fact'. Bennet concentrated the light and heat of the sun upon a body so delicately suspended that an exceedingly small amount of momentum must have been rendered apparent, but there was no such effect*. This experiment, indeed, is of a negative kind, and is not absolutely conclusive, unless we could estimate the mo- mentum which Newton's theory would require to be present (see voL ii. p. 45) ; but there are other diificultiea. Laplace pointed out that the attraction supposed to exist between matter and the corpuscular particles of light, would cause the velocity of light to vary with the size of the emitting body, so that if a star were 250 times as great in diameter as our sun, its attraction would prevent the emanation of light altogether ". But so far as experi- ence shows, the velocity of light is uniform, and inde- pendent of the magnitude of the emitting body, as it should be according to the undulatory theory. Lastly, Newton's explanation of difiraction or inflection fringes of colours was only plausible, and not true ; for Fresnel ascertaiaed that the dimensions of the fringes are not what they would be according to Newton's theory. 1 Aire's ' Mathematical Tracts,' 3rd edit. pp. 386-288. ' Jamin, 'Uours de PhfBique,' vol. iii. p. 37a. • Euler'B 'Letters/ vol. ii. Letter XIX. p. 69. t Balfour Stewart, ' Elementory TreatiBe on Heat,' p. 161. " Young's 'Lectures on Natural Philosopliy' (1845), vol. i. p. 361. by Google TUB USE OF HYPOTHESIS. 153 Although the Science of Light presents us with the most beautiful examples of crucial experiments and ob- servations, instances are not wanting in other branches of science. Copernicus asserted in opposition to the ancient Ptolemaic theory that the earth and planets moved round the sun, and he predicted that if ever the sense of sight could be rendered sufficiently acute and powerful, we should see phases in Mercury and Venus. Galileo with his telescope was able, in 1610, to verify the prediction as regards Venus, and subsequent observations of Mercury lead to a like conclusion. The discovery of the aberra- tion of light added a new proof, still fiirther strengthened by the more recent determination of the parallax of fixed stars. Hooke proposed to prove the existence of the earth's diurnal motion by observing the deviation of a falling body, an .experiment successfully accomplished by Benzenberg; and Foucault's pendulum has since fur- nished an additional indication of the same motion, which is indeed also apparent in the direction of the trade winds. All these are crucial facts in favour of the Copernican theory. Davy's discovery of potassium and sodium in 1807 was a good instance of a crucial experiment ; for it decisively confirmed Lavoisier's views, and at the same time nega* tived the ancient notions of phlogiston. Descriptive Hypotheses. There are some, or probably many, hypotheses which we may call descriptive hypotheses, and which serve for little else than to furnish convenient names. When a certain phenomenon is of an unusual and mysterious kind, we cannot even speak of it without using some analogy. Every word implies some resemblance between the thing to which it is applied, and some other thing, which fixes Digitized by Google 154 THE PRINCIPLES OF SCIENCE. the meaning of the word. Thus if we are to speak of what constituteB electricity, we must search for the nearest analogy, and as electricity is chiefly characterised by the rapidity and facility of its movements, the notion of a fluid of a very subtle character presented itself as most appropriate. There is the single fluid and the double fluid theory of electricity, and a great deal of discussion has been uselessly spent upon them. The fact is that if these theories be understood as more than con- venient modes of describing the phenomena, they are grossly invalid The analogy extends only to the rapidity of motion, and the fact that a phenomenon occurs suc- cessively at difierent points of the body. The so-called electric fluid adds nothing to the weight of the conductor, and to suppose that it really consists of particles of matter would be even more absurd than to reinstate the Corpus- cular theory of light. An infinitely closer analogy exists between electricity and light undulations, which are about equally rapid in propagation ; and while we shall probably continue for a long time to talk of the electric fluid, there can be no doubt that this expression merely represents some phase of molecular motion, some wave of disturbance propagating itself at one time through material con- ductors, at another time through the ethereal basis of light. The invalidity of these fluid theories is moreover shown in the fact that they have not led to the invention of a single new experiment. When we speak of heat as jiowing from one body to another, we likewise use a descriptive hypothesis merely ; for Lambert's theory of he Corpuscular heses I should ' Fits of Easy . been since ex- )iit even when ibyGoo^lc TBE USE OF HYPOTHESIS. 155 really entertained it did not do more than describe what took place. It involved no deep analogy to any other phe- nomena of nature, for Newton could not point to any other substance which went through these extraordinary changes. We now know that the true analogy would have been the waves of sound, of which Newton had acquired in other respects so complete a comprehension. But though the notion of interference of waves had dis- tinctly occurred to Hooke, Newton had failed to see how the periodic phenomena of light could be connected with the periodic character of waves. Hie hypothesis fell be- cause it was out of analogy with everything else in nature, and it therefore did not allow him, as in other cases, to descend by mathematical deduction to consequences which could be verified or refuted. We are always at freedom again to imagine the existence of a new agent or force, and give it an appropriate name, provided there are phenomena incapable of explanation from known causea We may speak of vital force as oc- casioning life, provided that we do not take it to be more than a name for an undefined something giving rise to inexplicable facts, just as the French chemists called Iodine the Substance X, while they were unaware of its real character and place in chemisfiy f. Encke was quite justified in speaking of the resisting medium in space so long as the retardation of his comet could not be other- wise accounted for. But such hypotheses will do much harm whenever they divert us fixim attempts to reconcile the facta with known laws, or when they lead us to mix up entirely discrete things. We have no right, for instance, to confuse Encke'a supposed resisting medium with the ethereal basis of light. The name protoplasm, now BO familiarly used by phy8iolog;ists, is doubtlefs legitimate so long as we do not mix up different sub- >* Paris, ' Life of Davy,' p. 274, Digitized by Google 156 TUB PRINCIPLES OF SCIENCE. stances under it, or imagine that the name gives us any knowledge of the ohscure origin of life. To name a substance protoplasm uo more explams the infinite variety of forms of life which spring out of the substance, than does the vital force which may be supposed to reside in the protoplasm. Both expressionB appear to me to be mere names for an unknown and inerplicable series of causes - which out of apparently similar conditions pro- duce the most diverse results. Hardly to be distinguished from descriptive hypotheses are certain imaginary objects or conditions which we often frame for the more ready investigation or comprehension of a subject The mathematician, in treating abstract questions of probability, finds it convenient, to represent the conditions to his own or other minds by a concrete analogy in the shape of a material ballot-box. The funda- mental principle of the inverse method of probabilities upon which depends the whole of our reasoning in in- ductive investigations is proved by Poisson, who images a number of ballot-boxes, of which the contents are after- wards supposed to be mixed in one great box (vol. i. p. 280). Muny other such devices are also used by mathematicians. When Newton investigated the nature of waves, he employed the pendulum as a convenient mode of representing the nature of the undulation. Centres of gravity, oscillation, &c., poles of the magnet, lines of force, are other imaginary existences solely em- ployed to assist our thoughte (vol. i. p. 422). All such creations of the mind may be called Representative Hypo' theses, and they are only permissible and useful so &r as they embody analogies. Their further consideration pro- perly belongs either to the subject of Analogy, or to that of language and representation, founded upon analogy. by Google CHAPTER XXIV. EMPTETCAL KNOWLEDGE, EXPLANATION, AND PREDICTION. The one great method of inductive investigation, as we have seen, consists in the union of hypothesis and experi- ment, deductive reasoning being the link by wliich the experimental results are made to confirm or confute the hypothesis. Now when we consider this relation between hypothesis and experiment, it is obvious that we may classify our knowledge under four heads. (i) We may be acquainted with facts or phenomena which have come under our notice accidentally or without reference to any special hypothesis, and which have not been brought into accordance as yet with any hypotheas. Such facts constitute what is called EmpiricaX Know- ledge. {2) Another very extensive portion of our knowledge consists of those facts which, having been first observed empirically, have afterwards been brought into accord- ance with other facts by an hypothesis concerning the general laws applying to them. This portion of our knowledge may be said to be explained, reasoned, or (3) In a third place comes the collection of facts, minor in number, hut most important as regards their scientific value and interest, which have been anticipated by theory and afterwards verified by experiment. Digitized by Google 168 THE PRINCIPLES OF SCIENCE. (4) Lastly, there may and does exist knowledge of phenomena accepted solely on the ground of theory, and which is incapable of experimental confirmation, at least with the instrumental means at the time in our pos- session. It is a work of much interest to compare and illustrate in some degree the relative extent and value of these four groups of knowledge. As a general rule we shall observe that every great branch of science originates in facts observed accidentally, or without any distinct consciousness of what is to be expected. But as science progresses, its power of foresight rapidly increases, until the mathematician in his study seems to acquire the power of anticipating nature, and predicting what will happen in stated circumstances before the eye of man has ever witnessed the event Empirical Knowledge. By empirical knowledge we mean such as is derived directly from the examination of certain detached facts, and rests entirely on those facts, without corroboration or connexion with other branches of knowledge. It is con- trasted to generalised and theoretical knowledge, which embraces many series of fact-s under a few simple and comprehengdve principles, so that each series serves to throw light upon each other series of facte. Just as, in the map of a half- explored country, we see detached portions of rivers, isolated mountains, and undefined plains, not connected into any general plan, so a new branch of knowledge often consistB of groups of fects, each group standing apart, so as not to allow us to reason from one part to another. Before the time of Descartes, and Newton, and Huy- ghens, there was much empirical knowledge of the by Google BMPIKWAL KNOWLEDOK, EXPLANATION, Digitized by Google 170 TBB PRINCIPLES OF SCIENCE. in 1824. Arago acutely inferred from Gambey's experi- ment that if the copper were set in rotation while tiie needle was stationary the motion would gradually be communicated to the needle. The phenomenon never- theless puzzled the whole scientific world, and it required the deductive genius of Faraday to show that it was a neceesaiy result of the principles of electro-magnetism''. By an act of deductive reasoning Faraday antitapated that a piece of copper rotating between the poles of a powerful magnet must experience a kind of resistance which will soon bring it to rest, and this effect he proved to esist in a decisive experiment'. Many other curious facts might be mentioned which when once noticed were explained as the effects of well- known natural laws. It was accidentally discovered that the navigation of canals of email depth could be greatly facilitated by increasing the speed of the boats, the resiat- ance being actually reduced by this increase of speed, which enables the boat to ride as it were upon its own forced wave. Now mathematical theory might have pre- dicted this result had the right application of the formulee occurred to any one™. Gifl^rd's injector for supplying steam boilers with water by the force of their own steam, was, I believe, accidentally discovered, but no new prin- ciples of mechanics are involved in it, so that it might have been theoretically invented. The same may be said of the curious experiment in which a stream of air or steam issuing from a pipe is made to bold a free disc upon the end of the pipe and thus apparently obstruct its own free outlet The possession then of a true theory does not by any means imply the. foreseeing of all the k 'Experimental Bfieearcbes in Electricity,' ist Series, pp. 34-44. Paragrapba 81-139. I Jamin, 'Conn de Iliyiique,' torn. iii. p. 197. " Airy, ' On Tided and WaTes," EncTcloptedia Metropolitana, p. 348 *. by Google EMPIRICAL KNOWIEDOB, EXPLANATION, ^e. 171 results. The effects of even a few simple laws may be infimtely diverse, and some of the most curious and ueelul efiecta may remain uudetected until accidental observation brings them to ovr notice. Predicted Discoveries. The most interesting of the four classes of facts or phenomena as specified in p. 157, is probably the third — containing those the occurrence of which has been first predicted by theory, and then verified by observation. There is no more convincing proof of the soundness of scientific knowledge than that it thus confers the gift of foresight Auguste Comte said that ' Prevision is the test of true theory ; ' I should say that it is only one test of true theory, but that which is most likely to strike the public attention. Coincidence with fact is the test of true theory, but when the result of theory is announced before- hand, there can be no possible doubt as to the unpre- judiced and confident spirit in which the theorist inter- prets the results of his own theory. The earliest instance of scientific prophecy is naturally furnished by the science of Astronomy, which was the earliest in development. Herodotus narrates" that, in the midst of a battle between the Medes and Lydians, the day was suddenly turned into night, and the event had been foretold by Thales, the Father of Philosophy. A cessation of the combat and a peace confirmed by mar- riages was the immediate consequence of this happy scientific efibrt. Much controversy has taken place con- cerning the exact date of this occurrence, Baily assign- ing the year 610 b.c., but Sir Gr. B. Airy has lately decided that the exact day was the 28th of May, 584 b.c. 1 Lib. i. cap. 74. Digitized by Google 172 TEE PRINCIPLES OF SCIESCE. There can be no doubt tbat this and other predictions of eclipses attributed to ancient philosophers were due to an obscure knowledge of the Metonic Cycle, a period of 6585 days, or 223 lunar months, or about 19 years in which a nearly perfect recurrence of the phases and eclipses of the moon takes place ; but if so, Thales must have bad access to a long series of astronomical records either those of the Egyptians or the Chaldeans. There is a well known story as to the happy use which Columbus made of the power of predicting eclipses in overawing the islanders of Jamaica who refused bim necessary supplies of food for his fleet. He threatened to deprive them of the moon's light. ' His threat was treated at first with indifference, but when the eclipse actually commenced, the barbarians vied with each other in the production of the necessary supplies for the Spanish fleet.' Exactly the same kind of interest imd awe which the ancients experienced at the prediction of eclipses, bas been felt in modem times concerning the return of comets. Seneca indeed asseri^ in most distinct and remarkable terms tbat comets would be found to revolve in periodic orbits and return to sight. The ancient Chaldeans and the Pythagoreans are also said to have entertained a like opinion. But it was not until the age of Newton and HaUey that it became possible to calculate the path of a comet in future years. A great comet appeared in 1682, a few years before the first publication of the ' Principia,' and Halley showed that its orbit corresponded with those of remarkable comets rudely recorded to have appeared in the years 1531 and 1607. The intervals of time indeed were not quite equal, but Halley conceived the bold idea tbat this diflerence might be due to the disturbing power of Jupiter, near which great planet the comet bad passed in the interval 1607-1682. He predicted that the comet would return about the end of 1758 or the beginning of by Google EMPIRICAL KNOWLEDGE, EXPLANATION, Ac. 173 1759, and though Halley did not live to enjoy the sight, it waa actually detected on the night of Christmaa-day, 1758. A second return of the comet was witnessed in 1835 nearly at the time anticipated. In recent times the discovery of Neptune has heen the most remarkable instance of prevision in astronomical science. A fiall account of this discovery may he found in several works, as for instance Herscbel's ' Outlines of Astronomy' and 'Grant's History of Physical Astronomy,' Chapters xii and xni. Predictions in the Science of Light. Next after astronomy the science of physical optics has furnished the most beautifid and early instances of the prophetic power of correct theory. These cases are the more striking because they proceed from the profound application of mathematical analysis, and show an insight into the mysterious workings of matter which is sur- prising to all, but especially to the great majority of men who are unable to comprehend the methods of research employed. By its power of prevision the truth of the undulatory theory of light has been conspicuously proved, and it is especially to be remarked that even Newton received no assistance from his Corpuscular theory in the detection of new experiments. To his followers who embraced that theory we owe little or nothing in the science of light, and even the lofty genius of Laplace did not derive from it a single discovery. As Fresnel himself remarks" : — 'The assistance to be derived from a good theory is not to be confined to the calculation of the forces when the laws of the phenomena are known. There are certain laws so complicated and so singular, that observation o Taylor's ' Scientific ITeinmrB,' vol. v. p, 34 1. by Google 174 TBE PRINCIPLES OP SCIENCE. alone, aided by analogy, could never lead to their dis- coveiy. To divine these enigmas we must he guided by theoretical ideas founded on a true hypothesis. The theory of luminous vibrations presents this character, and these precious advantages ; for to it we owe the discovery of optical laws the most complicated and most difficult to divine.' Physicists who embraced the barren emission theory had nothing but their own native capacity and quickness of observation to rely upon. Fresnel having once seized the conditions of the true undulatory theory, as previously stated by Young, was enabled by the mere manipulation of his mathematical symbols to foresee many of the com- plicated phenomena of light. Who could possibly suppose, or even believe on the ground of mere common sense, that by stopping a large portion of the rays passing through a circular aperture, the illumination of a point upon a screen behind the aperture might be many times multi- plied. Yet this paradoxical effect was predicted by Fresnel, and verified both by himself, and in a careful repetition of the experiment in later years, by BiUet. Comparatively few persons even now are aware that in the very middle point of the shadow of an opaque circular disc is a point of Hght sensibly as bright as if no disc had been inter- posed. This startling fact was deduced fi^m Fresnel's theory by Poisson, and was then verified experimentally by Arago. Airy, again, was led by pure theory to pre- dict that Newton's rings would present a modified appear- ance if produced between a lens of glass and a plate of metal. This effect happened to have been observed fifteen years before by Arago, unknown to Airy ; but another prediction of Airy, that there would be a further modificar tion of the rings when made between two substances of very different refiractive indices, was verified by subsequent trial with a diamond. A reversal of the rings takes place by Google EMPIRICAL KNOWLEDGE, EXPLANATION, &c. 175 when the Bpace interreuiug between the plates ia filled with a substance of IntermedLate refractive power, another phenomenon predicted by theory and verified by experi- ment as Sir John Herschel has described. There is hardly a limit to. the number of other complicated effects of the interference of rays of light under different circum- stances which might be deduced from the mathematical expressions, if it were worth while, or which, being previously observed can be explained, as in an interesting ease observed by Sir John Herschel and explained by AiryP. By a somewhat different effort of scientific forraight, Presnel discovered that any solid transparent medium might be endowed with the power of double refraction by mere compressioiL For as he attributed the peculiar re- fi'acting power of crystals to the unequal elasticity in different directions, he inferred that unequal elasticity, if artificially produced, would give similar phenomena. With a powerful screw and a piece of glass, he then pro- duced not only the colours due to double refraction, but the actual duplication of images. Thus, by a great scien- tific generalisation, are the apparently unique properties of Iceland apar shown to belong to all transparent sub- stances under certain conditions'. All other predictions in optical science are, however, thrown into the shade by the theoretical discovery of conical refraction by the late Sir W. R. Hamilton, of Dublin. In investigating the passage of light through certain crystals, Hamilton found that Fresnel had slightly misinterpreted his own formuUe, and that, when rightly understood, they indicated a phenomenon of a kind never witnessed. A small ray of light sent into a crystal of arragonite in a particular direction, becomes spread out P Ahya, 'Mathematical Tracts,' 3rd edit. p. 312. ■ Tonng's 'WwkB,' vol. i. p. 4i». by Google 176 THE PRINCIPLES OF SCIENCE. into an infinite number of rays, which fonn a hollow cone within the crystal, and a hollow cylinder when emerging from the opposite side. In another case, a somewhat different, but equally strange, effect is pro- duced. These phenomena are peculiarly interesting, because cones and cylinders of light are not produced in any other casea They are, in fact, wholly opposed to all analogy, and constitute singular, or exceptional cases, of a kind which we shall afterwards have to consider more fully. Thar very strangeness rendered them peculiarly fitted to test the truth of the theory by which they were discovered; and when Professor Lloyd, at Hamilton's request, succeeded, after considerable diificulty, in wit- nessing l^e new appearances, no further doubt could remain of the validity of the great theory of waves, which we owe to Huyghens, Young, and Fresnel'. Predictions from the Theory of Undulations. It is curious to reflect that the undulations of light, although so inconceivably rapid and small, admit of more accurate observation and measurement than the waves of any other medium. But so far as we can carry out exact experiments on other kinds of waves, we find the phe- nomena of interference repeated, and analogy gives con- siderable powers of prediction. Sir John Herschel was perhaps the fij^t to suggest that two sounds might be made to destroy each other by 'interference ". For if one- half of a wave travelling through a tube could be sepa- rated, and conducted by a somewhat longer passage, so as, on rejoining the other half, to be one-quarter of a vibra- r Lloyd's 'Wave Theory,' Part il pp. 52-58. Babboge, 'NinUi Bridgwater Treatise,' p. 104, quoting Lloyd, ' Trans, of the Boyal Irish Academy,' toI. xvii. Clifton, ' Quarterly Journal of Pure and Applied Mathematics,' January, i860. * ' Ent^dopiedia Metropolitana,' art Sound, p. 753. by Google EMPIRICAL KNOWLEDGE, EXPLANATION, £c. 177 tion behindhand, the two portions would exactly neutralise each other. This experiment has recently been performed with success by Quincke and KOnig* The mterference arising between the waves from the two prongs of a tuning-fork was also predicted by theory, and proved to exist by Weber ; indeed it may be observed by merely turning round a vibrating fork close to the ear". It is a plain result of the theory of sound that, if we move rapidly towards a sounding body, or if it move rapidly towards us, the pitch of the sound will be a little more acute ; and, vice versd, when the relative motion is in the opposite direction, the pitch will be more grave. It arises from the less or greater intervals of time between the successive strokes of waves upon the auditory nerve, according as the ear moves towards or from the source of sound relatively speaking. This effect was predicted by theory, and afterwards verified by the experimente of M. Buys Ballot, on Dutch railways, and of Mr. Scott Russell, in England". Whenever, indeed, one railway train passes another, on the. locomotive of which the whistle is being sounded, the drop in the acuteneFS of the sound may be noticed at the moment of passing. This change gives the sound a peculiar howling character, which many persons must have noticed. I have calculated diat, with two trains travelling thirty miles an hour, the effect would amount to rather more than half a tone, and it would often amount to a tone. A corresponding effect is produced in the case of light undulations, when the eye and the luminous body rapidly approach or recede from each other. It is shown by a shght change in the refrangi- bility of the rays of light, and a consequent change in the place of the lines of the spectrum, which has been made to give roost important and unexpected information con- ' TyodallB ' Sound,' p. 261. " Ibid. p. 273. ' Ibid. p. 78. VOL. H. N Digitized by Google 178 THE PRINCIPLES OP SCIENCE. cerning the relative approach or recession of many stars as r^ards the earth. Tides are vast waveB, and were the earth's surface entirely covered by an ocean of uniform depth, they would admit of very exact theoretical investigation. The wholly irregular form of the several seas introduces unknown quantities and complexities with which theory cannot cope. Nevertheless, Whewell, observing that the tides of the German Ocean consist of interfering waves, which arrive partly round the north of Scotland and partly through the British Channel, was enabled to predict that at a point about midway between Lowestoft and Brill on the coast of Holland, in latitude 52" 27' N, and longitude 3 h. 14 m. E, no tides would be found to exist. At that point the two waves would be of exactly the same amount, but in opposite phases, so as to neutralise each other. This assertion was verified by a surveying vessel of the British navy?. Prediction in other Sciences. Generations, or even centuries, will probably elapse before mankind are in possession of a mathematical theory of the constitution of matter as complete and satisfactory as the theory of gravitation. Nevertheless, mathema- tical physicists have in recent years acquired a fair hold of some of the simple relations of the physical forces to matter, and the proof is found in some remarkable anti- cipations of curious phenomena which had never been observed. Professor James Thomson deduced from Car- net's theory of heat that the application of pressure would lower the melting-point of ice. He even ventured to assign the amount of this effect, and his statement was T Whewell'fl 'History of the Inductive Sciences,' vol, ii. p. 471. Henchel's 'Physical Geography,' J 77. by Google EMPIRICAL KNOWLEDGE, EXPLANATION, &e. 179 afterwards verified almost exactly by Sir "W. Thomson*. ' In this very remarkable speculation, an entirely novel physical phenomenon was predicted, in anticipation of any direct experiments on the subject ; and the actual observation of the phenomenon was pointed out as a highly interesting object for experimental research.' Just as liquids which expand in solidifying wUl have the tem- perature of solidification lowered by pressure, so liquids which contract in solidifying wiU exhibit the reverse effect. They will be assisted in soUdifying, as it were, by pressure, so as to become solid at a higher temperature, as the pressure is greater. This latter result was verified by BuDsen and Hopkins, in the case of paraffin, spermaceti, wax, and stearin. The effect upon water has more recently been carried to such an extent by Mousson, that under the vast pressure of 1300 atmospheres, water did not freeze until cooled down to -iS" Cent. Another remark- able prediction of Professor Thomson was to the effect that, if a metallic spring be weakened by a rise of tem- perature, work done against the spring, by bending it, must cause a cooling effect. Although the amount of effect to be expected in a ceitain apparatus was only about four-thousandths of a degree Centigrade, Dr. Joule' succeeded in detecting and measuring the effect to the extent of three-thousandths of a degree, such is the deli- cacy of modem methods of measurement. I cannot refrain from quoting Dr. Joule's refiections upon this fact*". 'Thus even in the above delicate case,' he says, • is the formula of Professor Thompson completely verified. The mathematical investigation of the thermo-elastic qualities of metals bos enabled my illustrious friend to ' Maxwell's "Theory of Heat,' p. 174. ' PhiloBophical Magaune,' Angnst, 1850. Third Series, vol. xxxrii. p. 133. ■ ' Philosophical Transactious,' 1858, vol. cxlviii. p. 127. '' Ibid. p. 1 30. by Google 180 THE PRINCIPLES OF SCIENCE. predict with certainty a whole class of highly interesting phenomena. To him especially do we owe the important advance which has been recently made to a new era in the history of science, when the iamous philosophical system of Bacon will be to a great extent superseded, and when, instead of arriving at discovery by induction from experiment, we shall obtain our largest accessions of new facta by reasoning deductively from fundamental principles.' The theory of electricity is a necessary part of the general theory of matter, and is rapidly acquiring the power of prevision. As soon as "Wheatstone had proved experimentally that the conduction of electricity occupied time, Faraday renuirked in 1838, with wonderful sagacity, that if the conducting wires were connected with the coatings of a large Leyden jar, the rapidity of conduction would be lessened. This prediction remained unverified for sixteen years, until the submarine cable was laid be- neath the Channel. A considerable retardation of the electric spark was then detected by Siemens and Latimer Clark, and Faraday at once pointed out that the wire surrounded by water resembles a Leyden jar on a large scale, so that each message sent through the cable verified his remark of 1838^ The joint relations of heat and electricity to the metals constitute almost a new science of thermo-electricity. Sir W. Thompson was enabled by theory to anticipate the following curious e£fect, namely, that an electric current passing in an iron bar from a hot to a cold part produces a cooling effect, but in a copper bar the effect is exactly opposite in character, that is the bar becomes heated*'. The action of crystals with regard to heat and electricity was partly foreseen on the groimds of theory by PoisBon. " TyndaH's 'Faraday,' pp. 73, 74 ; 'Life of Faraday,' toI. n. pp. 83, 83, * Twt's ' Therm odynamicB,' p. 77. by Google EMPIRICAL KNOWLEDGE, EXPLANATION, £c. 181 Chemistry, although to a great extent an empirical science, has not been without prophetical triumphs. The existence of the metals potassium and sodium was fore- seen by Lavoisier, and their elimination by Davy was one of the chief expertmenta crucis wbich established Lavoi- sier's system. The existence of many other metals which eye had never seen was almost a necessary inference, and theory has not been found at &ult. No sooner, too, had a theory of organic compounds been conceived by Pro- fessor A, W. Williamson than he foretold the formation of a complex substance conasting of water in which both atoms of hydrogen are replaced by atoms of acetyle. This substance, known as the acetic anhydride, was afterwards produced by Gerhardt. In the subsequent progress of organic chemistry occurrences of this kind have been mul- tiplied almost indefinitely. The theoretical chemist by the classification of bis specimens and the manipulation of his formulie can plan out as it were the creation of whole series of unknown oils, acids, alcohols, and such like products, just as a designer might draw out a multi- tude of patterns. The formation of many such substances is a matter of course, but there is 'an interesting predic- tion ^ven by Hofmann, concerning the possible existence of new compounds of sulphur and selenium, and even oxides of ammonium, which it remains for the future to verify e. Prediction by Inversion of Cause and Effect. There is one process of experiment which has so often led to important discoveries as to deserve separate de- scription and illustration — I mean the inversion of Cause and Effect. Thus if A and B in one experiment produce C as a consequent, then antecedents of the nature of B ' Hofnuum'a 'Introduction to ChemiBtiy,' pp. 234, 335. Digitized by Google 182 THE PRINCIPLES OF SCIENCE. and C may usually be made to produce a consequent of the nature of A inverted in direction. When we apply heat to a gas it tends to expand ; hence if we allow the gas to expand by its own elastic force, cold is the result ; that is B (air) and {expansion) produce the negative of A (heat). Or again, B (air) and compression, the nega- tive of C, produce A (heat). Similar results may be ex- pected in a multitude of cases. It is a most familiar law that heat expands iron and nearly all solid bodies. What may be expected, then, if instead of increasing the length of an iron bar by heat we use mechanical force and stretch the bar 1 Having the bar and the former consequent, ex- pansion, we should expect the negative of the former antecedent, namely cold. The truth of this inference was ■proved by Dr. Joule, who investigated the amount of the effect with his usual skills This inversion of cause and effect in the case of heat may be itself again inverted in a highly curious manner. It happens that there are a few substances which are un- explained exceptions to the general law of expansion by heat. India-rubber especially is remarkable for contracting when heated. Since, then, iron and india-rubber are oppo- sitely related to heat, we may expect that as distension of the iron produced cold, distension of the india-rubber ■will produce heat. This is actually found to be the case, and any one may detect the effect by suddenly stretching an india-rubber band while the middle part is in the mouth. Whenever stretched it will be found to grow slightly warm, and when relaxed cold. The reader will readily see that many of the sdentific predictions mentioned in preceding sections were due to the principle of inversion ; for instance. Professor Thomp- son's speculations on the relation of pressure and the melting-point. But many other illustrations could be f ' Philosopliical Transactions,' (1855) Tol. cxIt. pp. 100, &c. Digitized by Google EMPIRICAL KNOWLEDGE, EXPLANATION, Ae. 183 adduced. The usual agent by which we melt or liquefy a aubstance is heat ; but if we can melt a substance without heat, then we may expect the negative of heat as an effect. This is the foundation of all freezing mix- tures. The affinity of salt for. water causes it to melt snow or ioe, and may thus ■ reduce the temperature to Fahrenheit's zero. Calcium chloride has so much higher an attraction for water that a temperature of — 50° Fahr. may thus be attained. Even the solution of a certain alloy of lead, tin, and bismuth in mercury, may be made to reduce the temperature from 63° to 14° Fahr. AJl the other modes of producing cold are inversions of more familiar uses of heat. Carry's freezing machine is an inverted distilling apparatus, the distillation being occasioned by chemical affinity instead of heat Another kind of ireezing machine is the exact inverse of the steam engine. A very paradoxJcal efiect is due to another inversion. It is hard to believe at the first moment that a current of steam at 212° could raise a body of liquid to a higher temperatnre than the steam itself possesses. But Mr. Spence has pointed out that if the boiling-point of a saline solution be above 212°, it will continue, on account of its affinity for water, to condense steam when above 212° in temperature. It will condense the steam until heated to the point at which the tension of its vapour is equal to that of the atmosphere, that is, its own boiling-points. Since heat, again, melts ice, we might expect to produce heat by the inverse change from water into ice. Now this is accomplished in the phenomenon of suspended freezing. Water may be cooled in a clean glass vessel many degrees below the freezing-point, and yet retained in the liquid condition. But if disturbed, and especially if brought into contact with a small particle of ice, it immediately B * Proceedings of the Manchester PhHoBophical Society.' Feb. 1870. Digitized by Google 184 TUB PRINCIPLES OF SCIENCE. solidifies and rises in temperature to 32° Fahr. A like effect 18 still more beautifully displayed in the well known lecture-room experiment, of the suspended crystaUization of a solution of sodium sulphate, in which a sudden rise of temperature of 30° or even 40° Fahr. is often manifested. The science of electricity is full of the most -varied and interesting cases of inversion. As Professor TyndaU has remarked, Faraday had a profound belief in the reciprocal relations of the physical forces. The great starting-point of his' researches, the discovery of electro-magnetism, was clearly an inversion. Oersted and Ampere had proved that with an electric current and a magnet in a particular position as ante- cedents, motion is the consequent. If then a magnet, a wire and motion be the antecedents, aji opposite electric current will be the consequent. It would be an endless task to trace out the resiilts of this fertile relationship when once fully understood. No small part of Faraday's researches was occupied in ascertaining the direct and inverse relations of magnetic and diamagnetic, amorphous and crystalline substances in various circimistances. In all other relations of electricity the principle of inversion holds. The voltameter or the electro-plating cell is the inverse of the galvanic battery. As heat applied to a junction of antimony and bismuth bars produces electricity, it necessarily follows that an electric current passed through such a junction will produce cold. Thus it is apparent that inversion of cause and effect is a most fertile ground of prediction and discovery. The reader should carefiilly notice, however, that the inversion of natural phenomena is exactly true only of. the character of the effect, not the amount. There is always a waste of energy in every work, because a certain part of it is dissipated in the form of conducted or radiated heat, and escapes beyond our nee. Theoretically speaking. by Google EMPIRICAL KNOWLEDGE, EXPLANATION, ' Elementaty LeBsoas in Logic,' p. 163. by Google 196 THE PRIHCJPLBS OF SCIEXCE. in a succesfflon of predictions grounded on the theory of gravitation as to the inequalities of the planetary move- ment caused by mutual perturbations. These inequalities are so numerous, so small, and so complicated in character, that it would be an almost hopeless task to attempt to discover them empirically or tentatively by the compari- son and dassification of observations. But theory pretty easUy indicates the period and general nature of the inequality to be detected, and by elaborate calculations even the amount of the effect may be assigned. Thus the inequality arirang from the attraction of Venus and the earth was estimated by Sir George Airy to amount to no more than a few seconds at its maximum, white the period is no less than 240 years. Nevertheless, the in- direct effects of this inequality upon the moon's motion are considerable, and are entirely verified in the lunar theory. Although prediction by theory is the general rule in physical astronomy, yet the empirical investiga- tion of divei^nces from theory sometimes discloses effects which had been overlooked, or points out residual effects of unknown origin. Quantities determined hy Theory and not verijied. It will continually happen that we are able, from certain measured phenomena and a correct theory, to determine the amount of some other phenomenon which we may either be unable to measure at all, or to measure with an accuracy corresponding to that required to verify the prediction. Thus Laplace having worked out an almost complete theory of the motions of Jupiter's satel- lites on the hypothesis of gravitation, found that these motions were greatly affected by the spheroidal form of Jupiter. Hence from the motions of the satellites, which can be observed with great accuracy owing to the frequent ,d by Google ACCORDANCE OF QUANTITATIVE THEORIES, Ac. 197 eclipses and traDsits, he was able to argue mverBely, and assign the ellipticitj of the planet's section by theory. The ratio of the polar and ecLuatorial axes thus det^- mined was very nearly that of 13 to 14; and it agrees well with anch direct micrometrical measurements of the planet as have been made ; but Laplace believed that the theory gave a more accurate result than direct observation could yield, bo that the theory could hardly be said to admit of direct verification. The specific heat of air was believed on the grounds of direct experiment to amount to o"2669, the specific heat of water beiog taken as unity ; but the methods of expe- riment were open to considerable causes of error. The late Professor Bankioe showed in 1850 that it was possible to calculate from the mechanical equivalent of heat, and from other thermodynamic data, what this number should be, and he found for it 0*2378. This determination was at the time accepted }yy him and others as the most satisfactory result, although not verified ; subsequently in 1853 Hegnault obtained by direct experiment the number o'2377, proving that the prediction had been well grounded. It will be readily seen that in purely quantitative questions verification will be a matter of degree and probability. A less accurate method of measurement can- not verify the results of a more accurate method, so that if we arrive at a determination of the same physical quantity in several distinct modes it will oflen become -a delicate matter of investigation to decide which result is , most reliable, and should be used for the indirect deter- mination of other quantities. For instance, Joule's and Thomson's ingenious experiments upon the thermal phe- , nomena of fluids in motion*= involved, as one physical constant, the mechimical equivalent of heat; if requisite, c ' I^iloaophical Transactions' (J854), vol. cxiir. p. 364. Digitized by Google 198 THE PRINCIPLES OF SCIENCE. theD, they might have been used to predict or to correct that most important constant. But if other more direct methods of experiment give the mechanical equivalent oi heat with superior accuracy, then the experiments on fluids will be turned to a better use in detecting and assigning various quantities relating to the theory of fluids. We will further consider questions of iJiis kind in succeeding sections. There are of course many quantities aasigned on theo- retical grounds which we are quite imable to verify with corresponding accuracy. The thickness of a film of gold leaf, the average depths of the oceans, the velocity of a star's approach to or regression from the earth as inferred &om spectroscopic data, or other quantities indirectly determined (see vol. i. pp. 345-349), might be cases in point ; but many others might be quoted where direct verification seems impossible. Newton and many sub- sequent physicists have accurately measured the lengths of light undulations, and by several distinct methods we learn the velocity with which light travels. Since an undulation of the middle green is about five ten-millionths of a metre in length, and travels at the rate of nearly 300,ooo,cxx) of metres per second, it necessarily follows that about 600,000,000,000,000 undulations must strike in one second the retina of an eye which perceives such light. But how are we to verify such an astounding calculation 1^ directly counting pulses which recur sis hundred billions of times in a second 1 Discordance of Theory and Experiment. When a distinct want of accordance is found to exist between the results of theory and direct measiu'ement, several interesting questions may arise as to the mode in which we can account for this discordance. The ultimate by Google AGOORDANCE OF QUANTITATIVE THEORIES, &e. 199 explanation of the discrepancy may be accomplished in any one of at least four distinct ways, as follows ; — (i) The direct measurement may be erroneous owing to various sources of casual error. {2) The theory may be correct so far as regards the general form of the supposed laws, but some of the con- stant numbers or other quantitative data employed in the theoretical calculations may be inaccurate. {3) The theory may be false, in the sense that the forms of the mathematical equations assumed to expi^sa the laws of nature are incorrect. {4) The theory and the involved quantities may be approximately accurate, but some regular unknown cause may have interfered, so that the divergence may be re- garded as a residual effect representing possibly a new and interesting phenomenon. No precise rules can be laid down as to the best mode of proceeding to explain the divergence, and the experi- mentalist will have to depend upon his own insight and knowledge; but the foUowing general recommendations may perhaps be made. In the first place, if the experimental measiirements are not numerous, repeat them and take a more extensive mean result, the probable accuracy of which, as regards fireedom from casual enrors of experiment, will increase as the square root of the number of experiments. Supposing that no considerable modification of the result is thus effected, we may suspect the existence of some more deep- seated and constant source of error in our method of measurement. The next resource will be to change the size and form of the apparatus employed, and to introduce various modifications in the materials employed or in the course of procedure, in the hope, as before explained (vol. i. p. 462), that some cause of constant error may thus be removed. If the inconsistency with theory still re- by Google 200 THE PRINCIPLES OF SCIENCE. mains unreduced we may attempt to invent some widely different mode of arriving at the same physical quantity, so that we may be almost sure that the same cause of error will not affect both the new and old results. In some cases it is possible to find five or six essentially different modes of arriving at the same determination. Supposing that the discrepancy still exists we may well be^ to suspect that our direct measurements are correct, but that the data employed in the theoretical calculations are inaccurate. We must now review the grounds on which these data depend, consisting as they must ulti- mately do of direct measurements. A comparison of the various recorded results will show the degree of proba- bility attaching to the mean result employed ; and if there is any ground for imagining the existence of error, we should repeat the observations, and vary the forms of experiment just as in the case of the previous direct measurements. The continued existence of the discre- pancy must show that we have not really attained to a complete acquaintance with the theory of the causes in action, but two different cases still remain. We may have misunderstood the action of those causes which do exist, or we may have overlooked the existence of one or more other causes. In the first case our hypothesis appears to be wrongly chosen and inapplicable ; but whether we are to reject it will depend upon whether we can form any other hypothesis which yields a more accurate accordance. The probability of an hypothesis, it will be remembered (vol. i. p. 279), is to be judged entirely by the probability that if the supposed causes exist the observed residt follows ; but as there is now very little probability of reconciliug the original hypothesis with our direct measurements the field is open for new hypotheses, and any one which gives a closer accordance with measurement will so &r have claims to attention. Of course we must never estimate by Google ACCORDANCE OF QOANTITATIVE THEORIES, Ae. 201 the probability of an hypotheeifl merely by its accordance with a few results only. Its general analogy and accord- ance with other known laws of nature, and the fact that it does not conflict with any other probable theories, must be taken into account, as we shall see in the next book. The requisite condition of a good hypothesis, that it must admit of the deduction of facts verified in observation, must be interpreted in the widest possible manner, as including all ways in which there may be accordance or discordance. All our attempts at reconciliatioa having failed, the only conclusion we can come to is that some unknown cause of a new character exists. If the measurements be accurate and the theory probable, then there remains a rendttcU phenomenon, which, being devoid of theoretical explanation, must be set down as a new empirical fact worthy of deliberate investigation. As a matter of fact these outstanding residual discrepancies have often been found to involve new discoveries of the greatest im- portance. Accordance of Measurements of Astronomical Distances. One of the most instructive instances which we could meet, as regards the manner in which different measure- ments confirm or check each other, is furnished by the determination of the velocity of light, and the dimensions of the planetary system. Roemer first discovered that light requires time in travelling, by observing that the eclipses of Jupiter's satelhtes, although they of course occur at fixed moments of absolute time, are visible at different moments in different parts of the earth's orbit, according to the distance of the earth and Jupiter. The time occupied by light in traversing the mean semi- diameter of the earth's orbit is found to be about eight minutes. The mean distance o^ the sun and earth was DigitzedbyGOOgie 202 THE PRINCIPLES OF SCIENCE. loQg assumed by astroDomers aa being about 95,274,000 miles, this result being deduced by Bessel from the ob- servations of the transit of Venus, which occurred in 1 7 69, and which were found to give the solar parallax, or what is the same thing, the apparent size of the earth as seen from the sun, as equal to 8""578, Now, dividing the mean distance of the sun and earth by the number of seconds in 8™. 1 3^.3 we find the velocity of light to be about 192,000 miles per second. Nearly the same result was obtained in an apparently very diflFerent manner. The aberration of light is the apparent change in the direction of a ray of light owing to the composition of its motion with that of the earth's motion round the sun. If we know the amount of aber- ration and the mean velocity of the earth we can very simply estimate that of light which is thus found to be 191,102 miles {166,072 geographical miles) per second. Now this determination depends upon an entirely new physical quantity, that of aberration, which is ascertained by direct observation of the stars, so that tha close accord- ance of the estimates of the velocity of light as thus arrived at by different methods might seem to leave little room for doubt, the difference being less than one per cent. Nevertheless, experimentalists were not satisfied until- they had succeeded in actually measuring the velocity of light by direct experiments performed upon the earth's surface. Fizeau, by a rapidly revolving toothed wheel, estimated the velocity at 195,920 miles per second. As this result differed by about one part in sixty from esti- mates previously accepted, there was thought to be room for further investigation. The revolving mirror, previously used by Mr. Wheatstone in measuring the velocity of elec- tricity, was now applied in a more refined manner by Fizeau and by Foucault to determine the velocity of light. The latter physicist finally came to the startling Digitized by Google ACCORDANCE OF QUANTITATIVE THEORIES, ■ Clsnsius, ' PhiloBophicftl Magazine,' 4th Series, to), ii. p. 119. 1 Watta' 'Dictionary of Chemistry,' vol. iii. p. 139. by Google THE PRINCIPLES OF SCIEXCE. Residual Phenomena. Even when all the experimental data employed in the verification of a theory are sufficiently accurate, and the theory itself is sound, there may still exist discrepancies demanding further investigation. Sir John Herschel waa perhaps the first who pointed out the importance of such outstanding quantities, and called them residual pheno- menai. Now if the observations and the theory be really correct, such discrepancies must be due to the incomplete- ness of our knowledge of the causes in action, and the - ultimate explanation must consist in showing that there is in action (i) Some agent of known nature whose presence was not suspected. (2) Some new agent of unknown nature. In the first case we cannot be said to make any new discovery, for our ultimate success consists merely in reconciling the theory with known facts when our in- vestigation is more comprehensive. But in the second case we meet with a totally new fact, which may lead us to whole realms of new discovery. Take the instance adduced by Sir John Herschel. The theory of Newton and Halley concerning cometary motions was that they were gravitating bodies revolving round the sun in oblique orbits, and the actual return of Halley's Comet, in 1758, sufficiently verified this theory. But, when accurate observations of Encke's Comet came to be made, the verification was not found to be complete. Elach time Encke's Comet returned a little sooner than it ought, the period having regularly decreased from 121279 days, between 1786 and 1789, to i2io'44 be- i ' Fraliminaiy Diacouree on the study of Natural Philosophy,' §§ 158, 174. 'Oatlinee of Astronomy,' 4th. edit. § 856. by Google ACCORDANCE OF QUANTITATIVE THEORIES, &e. 213 tween 1855 and 1858. The theory of gravitation alone cannot account for such a continued decrease of period ; hence the hypothesis has been started that there is a resisiing medium filling the space through which the comet passes. This hypothesis is a deus ex machind for explaining this solitary phenomenon, and cannot pos- sess any validity or probability unless it can be shown that other phenomena are deducible from it. Many per- sons have identified this medium with that througb which heat undulations pass, but I am not aware that there is anything in the undulatory theory of light to show that the medium woxdd offer resistance to a moving body. If Professor Balfour Stewart can prove that a rotating disc experiences resistance even in a perfectly vacuous receiver, here J8 an experimental fact which distinctly supports the hypothesis. But in the mean time it is open to question whether other known agents, for injstance electricity, may not be brought in, and I have tried to show that if, aa seems highly probable, on other grounds, the tail of a comet is an electrical phenomenon, it is almost a neces- Fary result of the theory of the conservation of energy that the comet shall exhibit a loss of energy manifested in a diminution of its mean distance from the sun and its period of revolution K If so, the residual phenomenon seems to confirm an hypothesis as to the nature of the comet itself, rather than that of the medium througb which it moves. In other cases residual phenomena have involved im- portant inferences not recognised at the time. Newton showed how the velocity of sound in the atmosphere could be calculated by a theory of pulses or undulations from the observed tension and density of the air. He inferred that the velocity in the ordinary state of the !■ ' pFoceedingB of the HancheBter Literary and Philosophical Society,' 38th November 1371, vol. xi. p. 33. Digitized by Google 214 THE PRINCIPLES OF SCIENCE. atmosphere at the earth's surface would be 968 feet per second, and very rude experiments made by him in the cloisters of Trinity College seemed to show that this was: not tar from the truth. Subsequently it waa ascertained by other experimentalists that the velocity of sound was more nearly 1 142 feet, and the discrepancy being no less than one sixth part of the whole was far too much to attribute to casual errors in the numerical data, Newton attempted to explain away this discrepancy by hypotheses as to the relations of the molecules of air, but without success. Many new investigations having been made from time to time concerning the velocity of sound, both as observed experimentally and as calculated from theory, it was found that each of Newton's results was inaccurate, the theo- retical velocity being 916 feet per second, and the real velocity about 1090 feet. The discrepancy therefore re- mained as serious as ever, and it was not until Uie year 1816 that Laplace showed it to be due to the heat developed by the sudden compression of the air in the passage of the wave, this heat having the effect of in- creasing the elasticity of the air and accelerating the motion of the impulse. It is now perceived that this discrepancy really involved the whole doctrine of the equivalence of heat and energy, and the discrepancy was applied by Mayer, at least by implication, to give an estimate of the mechanical equivalent of heat. The esti- mate thus derived agrees satisfectorily with independent and more direct determinations by Dr. Joule and other physicists, so that fhe explanation of the residual dis- crepancy which so exercised Newton's ingenuity is now complete. As Sir John Herechel obseived, almost all the great astronomical discoveries have been first disclosed in the form of residual differences. It is the practice at well- Digitized by Google ACCORDANCE OF QUANTITATIVE THEORIES, ^c 215 conducted observatories to compare the position of the principal heavenly bodies as actually observed with what might have been expected theoretically. This practice was introduced by Halley when Astronomer Royal, and his reduction of the lunar observations gave a series of residual errors firom 1722 to 1739, by the examination of which the lunar theory was improved. Most of the greater astronomical variations arising from nutation, aberration, planetary perturbation were in like manner disclosed. The precession of the equinox was perhaps the earliest residual difference observed ; the systematic divergence of Uranus trom its calculated places was one of the latest, and was the foundation of the remarkable discovery of Neptune by anticipation. We may also class under residual phenomena all the so-called proper motions of the stars. A complete star catalogue, such as that of the British Association, gives a greater or less amount of proper motion for almost every star, consisting in the apparent difference of position of the star as derived from the earliest and latest good observations. But these apparent motions are often due, as is expressly explained by Baily', the author of the catalogue, to errors of obser- vation and reduction. In many cases the best astronomi- cal authorities have differed as to the very direction of the supposed proper motion of stars, and as regards the amount of the motion, for instance of a Polaris, the most different estimates have been formed. Residual quantities will of necessity be often so small that their very existence will be doubtful. Only the gradual progress both of theory and of accurate measurement will clearly show whether a discrepancy is to be referred to previous errors of obser- vation and theory or to some new phenomenon. But nothing is more requisite for the progress of science than ' ' British Association Catalogue of Stars,' p. 49. Digitized by Google 216 THE PRINCIPLES OF SCIENCE. the carefiil recording and investigation of all such discre- pancies. In no part of physical science can we be free from exceptions and outstanding facts, differences and discrepancies of which our present knowledge can give no account. It is among such anomalies that we must look for the key to wholly new realms of facts worthy of discovery. They are like the floating waifs which led Columbus to suspect the existence of the new world. by Google CHAPTER XXVr. CHARACTER OF THE EXPBHIMENTAU8T. There seems to be a tendency to believe that, in the present ^e, the importance of individual genius is lees than it formerly was. 'The individual withers, and the world is more and more.' Society, it seems to be supposed, has now assumed so highly developed a form, that what was accomplished in past times by the solitary exertions of a single great intellect, may now be gradually worked out by the united labours of an army of investigators. Just as the combi- nation of well-organized power in a modem army entirely supersedes the single-handed bravery of the mediaeval knight, so we are to believe that the combination of intel- lectual labour has superseded the genius of an Archimedec', a Roger Bacon, or a Newton. So-called original research is now regarded almost as a recognised profession, adopted by hundreds of men, and communicated by a regular system of training. All that we need to secure great additions to our knowledge of nature is the erection of great laboratories, museums, and observatories, and the offering of sufficiently great pecuniary rewards to those who can invent new chemical compounds, or detect new species, or discover new comets. Doubtless this is not the real meaning of the eminent men who are now urging upon Government the elaborate endowment of physical Digitized by Google 218 THE PRINCIPLES OF SCIESCE. research. They can only mean that the greater the pecu- niary and material aasistance given to men of science, the greater is the result which the available genius of the country may be expected to produce. . Money and oppor- tunities of study can no more produce genius than sun- Bhine and moisture can generate living beings ; the inex- plicable germ is wanting in both cases. But^ just as when the germ is present, the plant will grow more or less vigorously according to the circumstances in which it is placed, so it may be allowed that pecuniary assist- ance may favour the development of intellect. Public opinion however is not discriminating, and is Ukely to interpret the agitation for the endowment of science as meaning that science can be evolved trom money or labour. All such notions are, I believe, radically erroneous. In no branch of human affairs, neither in politics, war, literature, industry, nor science, is the influence of genius less considerable than it used to be. It is quite possible that the extension and organization of scientific study, assisted by the printing press and the accelerated means of communication, has increased the rapidity with which new discoveries are made known, and their details worked out by many heads and hands. A Darwin now no sooner propounds original ideas concerning the evolution of ani- ■ mated creatures, than those ideas are discussed and illus- trated, and applied by other naturalists in every part of the civilized world. In former days his labours and dis- coveries would have been hidden for decades of years in scarce manuscripts, and generations would have passed away before his theory had enjoyed the same amount of criticism and corroboration as it has already received in fifteen years. But the general result is that the genius of Dai win is more valuable, not less valuable, than it would foimerly have been. The advance of military Digitized by Google CHARACTER OF THE EXPERIMENTALIST. 219 science and the organization of enormous and well dis- ciplined armies has not decreased the value of a skilful general ; on the contrary, the rank and file are still more in need than they used to be of the guiding power of an ingenious and far-seeing intellect. The swift destruction of the French military power was not due alone to the perfection of the German array, nor to the genius of Moltke ; it was due to the combination of a well-disci- plined multitude, with a leader of the highest intellectual powers. So in every branch of human affiiirs the influence of the individual is not withering, but is growing with the extent of the material resources which are at his command. Nature of Genius. Turning to our own particular subject, it is a work of undiminished interest to reflect upon those qualities of mind which lead to great advances in natural knowledge. Nothing, indeed, is less amenable than genius to scientific analysis and explanation. Even precise definition is out of the question. Buffon said that ' genius is patience,' and certainly patience is one of its most constant and requisite components. But no one can suppose that patient labour alone will invariably lead to those con- spicuous results which we attribute to genius. In every branch of science, literature, art, or industry, there are thousands of men and women who work with unceasing patience, and thereby ensure at least a moderate success ; but it would be absurd to assent for a moment to crude notions of human equality, and to allow that equal amounts of intellectual labour yield equal results. A Newton may modestly and sincerely attribute bis dis- coveries to industry and patient thought, and there is much reason to believe that genius is essentially uncon- scious and unable to account for its own peculiar powers. Digitized by Google 220 THB PRINCIPLES OF SCIEUfCE. If genius, indeed, be that by which intellect diverges from what is common, it must necessarily be a phenomenon be- yond the domain of the ordinary laws of nature. Never- theless, it is always an interesting and instructive work to trace out, as fer as possible, the characteristics of miDd by which great discoveries have been achieved, and we shall find in the analysis much to illustrate the principles of scientific method. Error of the Baconian Method. Hundreds of investigators may be constantly engaged in experimental inquiry ; they may compile nimiberlesa notebooks full of scientific facts, and may frame endless tables full of numerical results ; but if the views of the nature of induction here maintained be true they can never by such work alone rise to new and great dis- coveries. By an organized system of research they may work out deductively the detailed results of a previous discovery, but to arrive at a new principle of nature is another matter. Francis Bacon contributed to spread abroad the hurtful notion that to advance science we must begin by accumulating facts, and then draw from them, by a process of patient digestion, successive laws of higher and higher generality. In protesting against the false method of the scholastic logicians, he exaggerated a partially true philosophy, until it became almost as false as that which preceded it. His notion of scientific method was that of a kind of scientific bookkeeping. Facts were to be indiscriminately gathered from every source, and posted in a kind of ledger, from which would emerge in time a clear balance of truth. It is difficult to imagine a less likely way of arriving at great discoveries. The greater the array of facts, the less is the probability that they will by any routine system of classification or Digit zed by Google CHASACTER OF THE EXPERIMENTALIST. 221 research disclose the lawB of nature they embody. Ex- haustive classiBcation in all possible orders is out of the question, because the possible orders are practically in- finite in number. It is before the glance of the philoso- phic mind that facts must display their meaning, and fall into logical order. The natural philosopher must there- fore have, in the first place, a mind of impression- able character, which is readily affected by the slightest exceptional phenomenon. Hisasaociating and identitying powers must be great, that is, a single strange fact must suggest to his miad whatever of like nature has pre- viously come within his experience. His imagination must be active, and bring before his mind multitudes of relations in which tJie unexplained facts may possibly stand with regard to each other, or to more common facts. Sure and vigorous powers of deductive reasoning must then come into play, and enable him to infer what will happen under each supposed condition. Lastly, and above all, there must be the love of certainty leading him dihgently and with perfect candour, to compare his spectdations with the test of fact and experiment. Freedom, of Theorizing. It would be a complete error to suppose that the great discoverer is one who seizes at once unerringly upon the truth, or has any special method of divining it. In all probability the errors of the great mind far exceed in number those of the less vigorous one. Fertility of imagination and abundance of guesses at truth are among - the first requisites of discovery ; but the erroneous guesses must almost of necessity be many times as numerous as those which prove well founded. The weakest analogies, the most whimsical notions, the most apparently absurd theories, may pass through the teeming brain, and no Digitized by Google 222 THE PRINCIPLES OF SCIENCE. record may remain of more than the hundredth part. There is nothing intrinsically absurd except that which proves contrary to logic and experience. The truest theories involve suppositions which are most inconceiv- able, and no limit can really be placed to the freedom of framing hypotheses. Kepler is an extraordinary instance to this effect. No minor laws of nature are more firmly established than those which he detected concerning the orbits and motions of planetary masses, and on these empirical laws the theory of gravitation was founded. Did we not know by his own writings the multitude of errors into which he fell, we might have imagined that he had some special faciJty of seizing on the truth. But, as is well known, he was full of chimerical notions ; his most favourite and long entertained theory was founded on a fanciful analogy between the planetary orbits and the regular solids. His celebrated laws were the outcome of a lifetime of speculation, for the most part vain and groundless. We know this with certainty, because he had a curious pleasure in dwelling upon erroneous and futile trains of reasoning, which most other persons care- fully consign to oblivion. But Kepler's name was des- tined to immortality, on account of the patience with which he submitted his hypotheses to comparison with observation, the candour with which he acknowledged failure after failure, and the perseverance and ingenuity with which he renewed his attack upon the riddles of nature. Next after Kepler perhaps Faraday is the phyacal philosopher who has afforded us the most important mate- rials for gaining an insight into the progress of discovery, by recording erroneous as well as successful speculations. The recorded notions, indeed, are probably at the most a tithe of the fancies which arose in his active brain. As Faraday himself said — ' The world little knows how by Google CHARACTER OF THE EXPERIMENTALIST. 223 many of the thoughts and theories which have passed through the mind of a scientific investigator, have been crushed in silence and secresy by his own Bevere criticism and adverse examination ; that in the most successful instances not a tenth of the suggestions, the hopes, the wishes, the preliminary conclusions have been realized.' Nevertheless, in Faraday's researches published either in the ' Philosophical Transactions' or in minor papers, in bis manuscript note-books, or in various other materials, fortunately made known in his interesting life by Dr. Bence Jones, we find invaluable lessons for the experi- mentalist. These writings are full of speculations which we must not judge by the light of subsequent discovery. It may even be said that Faraday sometimes committed to the printing press crude ideas which a cautious friend would have counselled him to keep back or suppress. There was occasionally even a wildness and vagueness in his notions, which in a less careful experimentalist might have been fatal to the attainment of trutlL This is especially apparent in a curious paper coDcemiDg Ray-vibrations ; but fortunately Faraday was fully aware of the shadowy character of his speculations, and expressed the feeling in words which must be quoted. * I think it likely,' he says ", * that I have made many mistakes in the preceding pages, for even to myself my ideas on this point appear only as the shadow of a speculation, or as one of those impressions upon the mind, which are allowable for a time as guides to thought and research. He who labours in experimental inquiries knows how numerous these are, and how often their apparent fitness and beauty vanish before the progress and development of real natural truth.' If, then, the experimentalist has no royal road to the discovery of the truth, it is an interesting matter *• ' Experimental Researches in Chemistry and Fliybica,' p. 371. Philosophical Magazine, 3rd Series, May 1846, vol, JtxviiL p. 350. by Google THE PRINCIPLES OF SCIENCE. to consider by what logical procedure he attains the truth. If I have taken a correct view of logical method, there is really no such thing as a distinct process of induction. The probability is infinitely small that a collection of complicated facts will fall into an arrangement capable of exhibiting directly the laws obeyed by them. The mathematician might as well expect to integrate his functions by a ballot-box, as the experimentalist to draw deep truths from haphazard trials. All induction is but the inverse application of deduction, and it is by the inexplicable mental action of a gifted mind that a multi- tude of heterogeneous facts are caused to range them- Belves in luminous order as the results of some uniformly acting law. So different, indeed, are the qualities of mind required in different branches of science that it would be absurd to attempt to give an exhaustive description of the character of mind which leads to discovery. The labours of Newton could not have been accomplished except by a mind of the utmost mathematical genius ; Faraday, on the other hand, has made the most extensive and undoubted additions to human knowledge without ever passing beyond common aritlimetic. I do not re- member meeting in Faraday's writings with a single algebraic formula or mathematical problem of any com- plexity. Professor Clerk Maxwell, indeed, in the preface to his new ' Treatise on Electricity,' has strongly re- commended the reading of Faraday's researches by all students of science, and has given his opinion that though Faraday seldom or never employed mathematical formulae, his methods and conceptions were not the less mathe- matical in their nature *". I have myself protested against the prevailing confusion between a mathematical and an b See alao 'Nature,' Sept. i8, 1873 ; vol. viii, p. 398, Digitized by Google CHARACTER OF THE EXPERIMENTALIST. 225 exact science'', yet I certaiuly think that Faraday's expe- riments were for the most part purely qualitative, and that his mathematical ideas were of a rudimentary cha- racter. It is true that he could not possibly investigate such a subject as magne-ciystallic action without involv- ing himself in geometrical relations of considerable com- plexity. I nevertheless think that he was deficient in purely mathematical deductive power, that power which is so exclusively developed hy the modem system of mathematical training at Cambridge. Faraday, for in- stance, was perfectly acquainted with the forms of his celebrated lines of force, but I am not aware that he ever entered into the subject of the algebraic nature of those curves, and I feel sure that he could not have explained their form as depending on the resultant attraction of all the magnetic particles acting according to general mathe- matical lawa There are even occasional indications that he did not understand some of the simpler mathematical doctrines of modem physical science. Although he so clearly foresaw the establishment of the tmity of the physical forces, and laboured so hard with his own hands to connect gravity with the other forces, it is very doubt- fol whether he understood the fundamental doctrine of the conservation of energy as applied to gravitation. Thus, while Faraday was probably equal to Newton in experimental skill and deductive power as regards the invention of simple qualitative experiments, he was con- trasted to him in mathematical power. These two in- stances are sufficient to show that minds of widely dif- ferent conformation may meet with suitable regions of research. Nevertheless, there are certain common traits which we may discover in all the highest scientific minds. " ' Principlea of Science,' vol. i. p. 317, and ' Theory of Foliticii Economy,' pp. 3-14. by Google THE PRINCIPLES OF SCIENCE. The Newtonian Method, the True Organum. Laplace was of opinion that the * Principia ' and the ' Opticks ' of Newton furnished the best models then available of the delicate art of experimental and theo- retical investigation. In these, as he says, we meet with the most happy illustrations of the way in which, from a series of inductions, we may rise to the causes of phenomena, and thence descend again to all the resulting details. The popular notion concerning Newton's discoveries is that in early life, while driven into the country by the Great Plague, a falling apple accidentally suggested to him the existence of gravitation, and that, availing him- self of this hint, he was led to the discovery of the law of gravitation, the explanation of which constitut-es the ' I^cipia.' It is difficult to imagine a more ludicrous and inadequate picture of Newton's labom^ and position. No originality, or at least priorityj could be or was claimed by Newton as regards the discovery of the celebrated law of the inverse square, so closely associated with his name. In a well-known SchoHmn^ he acknowledges that Sir Christopher Wren, Dr. Hooke, and Dr. Halley, had severally observed the accordance of Kepler's third law of motion of the planets with the principle of the inverse square. Newton's work was really that of developing the methods of deductive reasoning and experimental verifica- tion, by which alone great hypotheses can be brought to the touch-stone of fact. Archimedes was the greatest of ancient philosophers, for he showed how mathematical theory could be wedded to physical experiments ; and his works are the first true Organum. Newton is the modem d ' Principia,' bk. 1, Prop, iv. Digitized by Google CHABACTER OF THE EXPERIMENTALIST. 227 Archimedes, and the ' Principia ' forms the true Novum Organum of scientific method. The laws which he actually established are great, but bis example of the manner of establishing them is greater still. There is hardly a progressive branch of physical and mathe- matical science, excepting perhaps chemistty and eleo- tricity. which has not been developed from the germs of true scientific procedure which he disclosed in the * Prin- cipia' or the ' Opticks.' Overcome by the success of his theory of upiversal gravitation, we are apt to forget that in his theory of sound he originated the mathematical investigation of waves and the mutual action of particles ; that in his Corpuscular theory of light, however mistaken, he first ventured to apply mathematical considerations to molecular attractions and repulsions ; that in his prinmatic experiments he showed how far experimental verification could be pushed ; that in his examination of the coloured rings named after him, he accomplished the most remark- able instance of minute measurement yet known, a mere practical application of which by M. Fizeau was recently deemed worthy of a medal by the Eoyal Society. We only learn by degrees how complete was hia scientific infflght ; a few words in his third law of motion display his acquaintance with the fijndamental principles of modem thermodynamics and the conservation of energy, while manuscripts long overlooked prove that in his inquiries concerning atmospheric refraction he had overcome the main difficulties of applying theory to one of the most complex of physical problems. After all, it is only by examining the way in which he ^ected discoveries, that ve can rightly appreciate bis greatness. The ' Principia ' treats not of gravity so much as of forces in general, and the methods of reasoning about them. He investigates not one hypothesis only, but mechanical hypotheses in generaL NotMng so much Q2 Digitized by Google 228 THE FRINCIPLES OF SCIENCE. strikes t&e reader of the work as the eshaustiveness of his treatment, and the almost infinite power of his insight. If he treats of central forces, it is not any one law of force which he discusses, but many, or almost all imaginable Cases, the laws and results of each of which he sketch^ out in a few pregnant words. If his subject is a resisting medium, it is not air or water alone, nor any one resisting medium, but resisting media in general. We have a good example of his method in the SchoKum to the twenty- second proposition of the second book, in which he runs rapidly over many possible suppositions as to the laws of the compressing forces which might conceivably act in an atmosphere of gas, a consequence being drawn from each Case, and that one hypothesis ultimately selected which yidds results agreeing with experiments upon the pressure and density of the terrestrial atmosphere. Newton said that he did not frame hypotheses, but, in teality, the greater part of the 'Principia' is purely hypo- thetical, endless varieties of causes and laws being ima- gined which have no counterpart in nature. The most grotesque hypotheses of Kepler or Descartes were not more imaginary. But Newton's comprehension of logical method was perfect ; no hypothesis was entertained unless it was definite in conditions, and admitted of unquestion- able deductive reasoning ; and the value of each hypo- thesis was entirely decided by the comparison of its conse- quences with facts. I do not entertain a doubt that the general couree of his procedure is identical with that view of the nature of induction, as the inverse application of deduction, which I have advocated throughout these volumea Francis Bacon held that science should be founded on experience, but he wholly mistook the true mode of using experience, and in attempting to apply his method he ludicrously failed. Newton did not less found his method on experience, but he seized the true method Digitized by Google CHARACTER QF TUE EXPERIMENTALIST. 229 of treating it, and applied it with a power and Success- never since equaUed. It is wholly a mistake to say that modem science is the result of the Baconian philosophy ; it is the Newtonian philosophy and the Newtonian method ■which have led to all the great triumphs of physical science, and I repeat that the ' Principia ' forms the true * Novum Organum.' In bringing his theories to a decisive experimental veri- fication, Newton showed, as a general rule, an exquisite skill and ingenuity. In his hands a few simple pieces of apparatus were made to give results involving an unsus- pected depth of meaning. His most beautiful experimental inquiry was that by which he proved the differing refran- gibility of rays of light. To suppose that he originally discovered the power of a prism to break up a beam of white light would be a great mistake, for he speaks of procuring a glass prism to try the celebrated phenomena of colours. But we certainly owe to him the theory that white light is a mixture of rays differing in refran- gibility, and that lights which differ in colour, differ also in refrangibUity. Other persons might have conceived this theory ; in fact, any person regarding refraction as a quantitative effect, must see that different parts of the spectrum have suffered different amounts of refraction. But the power of Newton is shown in the tenacity with which he followed his theory into every consequence, and tested each result by a simple but conclusive experi- ment. He first shows that different coloiu'cd spots are displaced by different amoimts when viewed through a prism, and that their images come to a focus at different distances from the lense, as they should do, if the refran- -gibility differed. After excluding by various experiments a variety of indifferent circumstances, he fixes his atten- ■tion upon the question whether the rays are merely sliattered, disturbed, and spread out in a chance manner. Digitized by Google 230 THE PRINCIPLES OF SCIENCE. as Grimaldi supposed, or whether there is a constant relation between the colour and the refran^bility. If Grimaldi was right, it might be expected that any part of the spectrum taken separately, and subjected to a second refiraction, would suffer a new breaking up, and produce some new spectrum. Newton inferred from his own theory that a particular ray of the spectrum would have a constant retrangibility, so that a second prism would merely bend it more or less, but not further dis- perse it in any considerable degree. By simply cutting off most of the rays of the spectrum by a screen, and allowing the remaining narrow ray to fell on a second prism, he proved the truth of this conclusion ; and then slowly turning the first prism, go as to vary the colour of the ray falling on the second one, he found that the spot of light formed by the twice-refracted ray travelled up and down, a palpable proof that the amount of refran- gibility varied with the colour. For his further satisfac- tion, he sometimes refracted the light a third or fourth time, and he found that it might be refracted upwards or downwards or sideways, and yet for each coloured light there was a definite amount of refraction through each prism. He completes the proof by showing that the separated rays may again be gathered together into white light by an inverted prism. So that no number of refrac- tions alters the character of the light. The conclusion thus obtained serves to explain the confusion arising in the use of a common lense ; with homogeneous light he shows that there is one distinct focus, with mixed light an infinite number of foci, which prevent a clear view from being obtained at any one point. What astonishes the reader of the 'Opticks' is the persistence with which Newton follows out the conse- quences of a preconceived theory, and tests the one notion by a wonderful variety of simple comparisons with fact. Digitized by Google CUARAGTER OF THE EXPERIMENTALIST. 231 It is certainly the theory which leads him to the experi- ments, and most of these could hardly he devised by accident. The fertility with which he invents new comhi* nations, and foresees the results, subsequently verified* produces an invincible conviction in the reader that he has possession of the truth. Newton actually remarks that it was by mathematically determining all kinds of phenomena of colours which could be produced by refrac- tion that he had ' invented ' ahnost all the experiments in the book, and he promises that others who shall ' argue truly,' and tiy the experiments with care, will not be disappointed in the results. The philosophic method of Huyghens was almost ex- actly the same as that of Newton, and Huyghens' investi- gation of the laws of double refraction furnishes almost equally beautiful instances of theory guiding experiment. Double refraction was first discovered by accident, so far as we know, and was described by Erasmus Bartbolinus in 1 669. The phenomenon then appetu-ed to be entirely ex- ceptional, and the laws governing the two separate paths of the refixicted rays were so unapparent and compHcated, that even Newton altogether misimderstood the pheno- menon, and it was only at the latter end of the last century that scientific men generally began to comprehend its laws. Nevertheless, Huyghens had, with rare genius, arrived at the true theory as early as 1678. He regarded light as an undulatory motion of some medium, and in his ' Traite de la Lumi^re,' he pointed out that, in ordinary refraction, the velocity of propagation of the wave is equal in all directions, so that the front of an advancing wave is spherical, and reaches equal distances in equal times. But in crystals, as he supposed, the medium would be of unequal elasticity in different directions, so that a dis- turbance would reach unequal distances in equal times, and the wave produced would have a spheroidal form. Huy- Digitized by Google 232 TBE PSmCIPLES OF SCIENCE. ghens was not satisfied with an unverified theory. He calculated whgit might be expected to happen when a crystal of calc-epar was cut in various directions, and he says, ' I have examined in detail the properties of the extraordinary refraction of this ciystal, to see if each phenomenon which is deduced from theory would agree with what is really observed. And this being so, it is no slight proof of the truth of our suppositions and prin- ciples ; but what I am going to add here confirms them still more wonderfully ; that is, the different modes of cutting this crystal, in which the surfaces produced give rise to refraction exactly such as they ought to be, and as I had foreseen them, according to the preceding theory.' The supremacy of Newton's mistaken corpuaciilar doc- trine of light caused the theories and experiments of Huyghens to be disregarded for more than a century ; but it is not easy to imagine a more beautiful or successful application of the true method of inductive investigation, theory guiding experiment, and yet wholly relying on experiment for confirmation. Candour and Courage of the Philosophic Mind. Perfect readiness to reject a theory inconsistent with fact is, then, a primary requisite of the philosophical mind. But it would be a mistake to suppose that this candour has anything akin ^o fickleness; on the contrary, readiness to reject a false theory may be combined with a peculiar pertinacity and courage in maintaining anhypothesis as long as its falsily is not actually apparent. There must, indeed, be no prejudice or bias distorting the mind, and causing it to under-estimate or pass over the unwelcome results of experiment. There must he that scrupulous honesty and flexibility of mind, which assigns an adequate value to all by Google CBARACTSR OF THE MXPERIMEKTALIST. 233 (evidence ; indeed tiie more a man loves his theory, the more scrupulous should be his attention to its faults. Nothing is more common in life than to meet with some theorist, who, by long cogitation over a single theory, has allowed, it to mould his mind, and render him incapable of receiving anything but as a contribution to the truth of his one theory. A narrow and intense course of thought may sometimes lead to great results, but the adoption of a wrong theory at the outset is in such a mind irretriev- able. The man of one idea has but a single ch^mce of truth. The fertile discoverer, on the contrary, chooses between many theories, and is never wedded to any one, unless impartial and repeated comparison has convinced him of its validity. He does not choose and then compare ; but he compares time after time, and then choosea Having once deliberately chosen, the philosopher may rightly entertain his theory with the strongest love and fidelity. He will neglect no objection ; for he may chance at any time to meet a fatal one ; but he will bear in mind the inconsiderable powers of the human mind compared with the tasks it has to undertake. He will see that do theory can at first be reconciled with all possible, objec- tions, simply because tiiere may be many interfering causes, or the very consequences of the theory may have a com- plexity which prolonged investigation by successive gene- rations of men may not exhaust If then, a theory exhibit a number of very striking coincidences with fact, it must not be thrown aside until at least one conclusive dis- cordance is proved, regard being had to possible error in establishing that discordance. In science and philosophy something must be risked. He who quails at the least difficulty will never establish a new truth, and it was not unpluloEophic in Ledie to remark concerning his own experimental investigations into the nature of heat — Digitized by Google THE PBINCIPLES OF SCIENCE. ' In the course of inveatigation, I have found myself compelled to relmquish some preconceived notions ; but I have not abandoned them hastily, nor, till after a ■warm and obstinate defence, I waa driven from every postV Faraday's life, again, fiimiahes most interesting illustra- tions of this tenacity of the philosophical mind. Though so candid in rejecting some of his theories, there were others to which he clung through everything. One of his mo'st favourite notions was finally realised in a brilliant dis- covery ; another remains in doubt to the present day. The Philosophic Character of Faraday. In Faraday's researches concerning the conuexion of magnetism and light, we find an excellent instance of the pertinacity with which a favourite theory may be held and pursued, so long as the results of experiment are simply nugatory and do not clearly negative the notions entertained. In purely quantitative questions, as we have seen, the absence of apparent effect can seldom be regarded as proving the absence of all effect. Now Faraday waa convinced that some mutual relation must exist between magnetism and light. As early as 1822 he attempted to produce an effect upon a ray of polarized light, by passing it through water placed between the poles of a voltaic battery ; but he was obliged to record that not the slight- est effect was observable. Diying forty subsequent years the subject, we are told', rose again and again to his mind, and no failure could make him relinquish his search alter this unknown relation. It was in the year 1 845 that he * ' Experimeatal Inquiry into the Nature of Heat.' Pre&ce, p. xv. f Bence Jones, ' Life of Faradfty,' vol. i. p. 361. by Google CHARACTER OF THE EXPERIMENTALIST. 235 gained the first success ; on August 30th he began to work with common electricity, vainly trying glass, quartz, Iceland spar, &c. Several days of labour gave no result, yet he did not desist. Heavy glass, a transparent medium of great refiiictive powers, composed of borate of lead, was now tried, by being placed between the poles of a powerful electro-magnet, while a ray of polarized light was trans- mitted through it. When the poles of the electro-mE^et ■were arranged in certain positions with regard to the substance under trial, no effects were apparent; but at last !Faraday happened fortunately to place a piece of heavy glass so that contrary magnetic poles were on the eame side, and now an effect was witnessed. The glass was found to have the power of twisting the plane of polarization of the ray of light. All Faraday's recorded thoughts upon this great experi- ment are replete with curious interest. He attributes his success to the opinion, almost amounting to a conviction, that the various forme, under which the forces of matter are made manifest, have one common ori^n, and are so directly related and mutually dependent that they are convertible. ' This strong persuasion,' he saysK, ' extended to the powers of light, and led to many exertions having for their object the discovery of the direct relation of light and electricity. These ineffectual exertions could not remove my strong persuasion, and I have at last suc- ceeded.' He describes the phenomenon in somewhat figu- rative language as the Truxgnedzation of a ray of light, and also as ike illumination of a magnetic curve or line of force. He has no sooner got the effect in one case, than he proceeds, with his characteristic comprehensive- ness of research, to test the existence of a like phenomenon in all the substances available. He finds that not only 8 ' Life of Faraday,' vol. 11. p. 199. Digitized by Google THE PRINCIPLES OF SCIENCE. heavy glass, but solids and liquids, acids and alkalis, oils, water, alcohol, ether, all possess this power ; hut he was not ahle to detect its existence in any gaseous suh- stance. His thoughts cannot he restrained from running into curious speculations as to the possihle results of the power in certain easea ' What effect,' he says, ' does this force have in the earth where the magnetic curves of the earth traverse its substance ? Also what effect in a mag- net?' And then he falls upon the wholly original notion that perhaps this force tends to make iron and oxide of iron transparent, a phenomenon never previously or since observed. We can meet with nothing more instructive as to the course of mind by which great discoveries are made, than these records of Faraday's patient labours, and his varied success and failure. Nor are his unsuccess- ful labours upon the relation of gravity and electricity less interesting, and worthy of study. Throughout a large part of his life, Faraday was pos- sessed by the idea that gravity cannot be unconnected with the other forces of nature. On March 19th, 1849, he wrote in his laboratory book — ' Gravity. Surely this force must be capable of an experimental relation to elec- tricity, magnetism, and the other forces, so as to bind it up with them in reciprocal action and equivalent effect''.' He filled twenty paragraphs or more with reflections and suggestions, as to the mode of approaching the subject by experiment. He anticipated that the approach of one body to another would develope electricity in them, or that a body falling through a conducting helix would excite a current changing in direction as the motion was reversed. 'All this is a dream' he remarks; 'still ex- amine it by a few experiments. Nothing is too wonderful to be true, if it be consistent with the laws of nature ; b See also hiB more formal statement id the ' Experimental RoBearchea in Electricity,' 34tli Series, § 3702, vol. iii. p. i6t. by Google CHARACTER OF TEE EXPERTMENTALIST. 237 and in such things as these, experiment is the best test of such consistency.' He executed many difficult aud tedious experiments, vhich are described in the 24th Series of Experimental Researches; but the result was nil. And yet he con- cludes, ' Here end my trials for the present. The results are negative ; they do not shake my strong feeling of the existence of a relation between gravity and elec- tricity, though they give no proof that such a relation exists.* He returned to the work when he was ten years older, and in 1858-9 recorded many remarkable reflections and experimenta. He was much struck by the fact that elec- tricity is essentially a dual force, and it had always been a pecuUar conviction of Faraday that no body could be electrified positively without some other body becoming electrified negatively ; some of his researches bad been simple developments of this necessaiy relation. But ob- serving that between two mutually gravitating bodies there was no apparent circumstance to determine which shall be poritive and which negative, he does not hesitate to call in question an old opinion. ' The evolution of one electricity would be a new and very remarkable thing. The idea throws a doubt on the whole ; but still try, for who knows what is possible in dealing with gravity.' We cannot but notice the candour with which he thus in his laboratory book acknowledges the doubtfulness of the whole thing, and is yet prepared as a forlorn hope to frame experiments in opposition to all his pre- "vious experience of the course of nature. For a time his thoughts flow on as if the strange detection were already made, and he had only to trace out its conse- quences throughout the universe. ' Let us encourage our- selves by a little more imagination prior to experiment/ he says, and then he reflects upon the infinity of actions Digit zed by Google 238 THE PRINCIPLES OP SCIENCE. in nature, in which the matual relations of electricity and gravity would come into play ; he pictures to himself the plaoeta and the comets charging themselves as they ap< proach the sun ; cascades, rain, rising vapour, circulating currents of the atmosphere, the fumes of a volcano, the smoke in a chimney become so many electrical machines. A multitude of events and changes in the atmosphere eeem to be at once elucidated by such actions ; for a moment his reveries have the vividness of fact ' I think we have been dull and blind not to have suspected some such results,' and he sums up rapidly the consequences of bis great but ima^nary theory ; an entirely new mode of exciting heat or electricity, an entirely new relation of the natural forces, an analysis of gravitation, and a justifica- tion of the conservation of force. Such were Faraday's fondest dreams of what might be, and to many another philosopher they would have been a sufficient bafds for the writing of a great book. But Faraday's imagination was within his full control ; as he himself says, * Let the imagination go, guarding it by judgment and principle, and holding it in and directing it by experiment.' His dreams soon took a very practical form, and for many subsequent days he laboured with ceaseless energy, on the staircase of the Boyal Institution, in the clock tower of the Houses of Parliament, or in the Shot Tower at Southwark, raising ^id lowering heavy weights, and com- bining electrical helices and wires in every conceivable way. His skill and long experience in experiment were severely taxed to eliminate the effects of the earth's mag- netism, and time after time he saved himself from accept- ing mistaken indications, which to another man might have seemed conclusive verifications of his theory. When all was done there remained absolutely no results. ' The experiments,' he says, ' were well made, but the results are n^;ative;' and yet he adds, 'I cannot accept them as DigitizedbyGOOgle CHARACTER OF THE SXPERIMENTALIST. 3S9 coDcloedTe.' In this position the question remains to the present day ; it may be that the effect was too slight to be detected, or it may be that the arrangments adopted ■were not suited to develope the particular relation which exists, just as Oersted could not detect electro-magnetism, so long as his wire was perpendicular to the plane of motion of his needle. But these are not matters which concern us further here. We have only to notice the pro- found conTictioD in the unity of natural laws, the active powers of inference and imagination, the unboimded licence of theorizing, combined above all with the utmost dili- gence in experimental verification which this remarkable research manifests. Reservation of Judgment. There is yet another characteristic needed in the philosophic mind; it is that of suspending judgment when the data are insuffiaent. Many people will express a confident opinion on almost any question which is put before them, but they thereby manifest not strength, but weakness and narrowness of mind. To see all sides of a complicated subject, and to weigh all the different facts and probabilities correctly, may require no ordinary powers of comprehension. Hence it is most frequentiy the philosophic mind which is in doubt, and the ignorant mind which is ready with a positive decision. Faraday has himself said, in a very interesting lecture', ' Occa- sionally and frequently the exercise of the judgment ought to end in ahsolute reservation. It may be very distasteful, and great fatigue, to suspend a conclusion; but as we are not infalhble, so we ought to be cautious ; we shall eventually find our advantage, for the man who t Frinted in 'Modem Cnlture,' edited by Tounums, p. 219. Digitized by Google 240 THE PRINCIPLES OF SCIENCE. rests in his position is not bo fer from right as he who, proceeding in a wrong direction, is ever increasing his distance.' Arago presented a conspicuous example of this high quality of mind, as Faraday remarks ; for when he made tnown his curious discovery of the relation of a magnetic needle to a revolving copper plate, a number of supposed men of science in different countries gave immediate and confident explanations of it, which were all wrong. But Arago, who had both discovered the phenomenon and personally investigated its conditions, decUned to put forward publicly any theory at all At the same time we must not suppose that the truly philosophic mind can tolerate a state of doubt, while a chance of decision remains open. In science nothing like compromise is possible, and truth must' be one. Hence, doubt is the confession of ignorance, and must involve a painful feeling of incapacity. But doubt lies between error and truth, so that if we choose wrongly we are further away than ever from our goal. Summing up, then, it would seem as if the mind of the great discoverer must combine almost contradictory attributes. He must be fertile in theories and hypotheses, and yet full of facts and precise results of experience. He must entertain the feeblest analogies, and the merest guesses at truth, and yet he must hold them as worthless till they are verified in experiment. When there are any grounds of probability he must hold tenaciously to an old opinion, and yet he must be prepared at any moment to relinquish it when a single clearly contradictory feet is encountered. 'The philosopher,' says Faraday*, 'should be a man willing to listen to every suggestion, but deter- mined to judge tor himselfl He should not be biassed by ^ Bence Jones, ' Life of Faraday,' vol. i. p. 225- by Google CHARACTER OF THE EXPERIMENTALIST. 241 appearances ; have no favourite hypothesis ; be of no echool ; and in doctrine have no master. He should not be a respecter of persons, but of things. Truth should be his primary object. If to these qualities be added in- dustry, he may indeed hope to walk within the veil of the temple of nature.' by Google BOOK V. GENERALIZATION, ANALOGY, AND CLASSIFICATION. CHAPTER XXVII. GENERALIZATION. I HATE endeavoured to show in preceding chapters that all inductive reasoning is an inverse application of de- ductive ]%asoning, and consists in demonstrating that the consequences of certain assumed propositions or laws agree with facta of nature gathered by active or passive observation. The fundamental process of reasoning, as stated in the outset, consists in inferring of any thing what we know of similar objects, and it is on this prin- ciple that the whole of deductive reasoning, whether simply logical or mathematico-logical, is founded. All inductive reasoning must therefore be foimded on the same principle. Now it might seem that by a very plain use of this principle we might avoid the complicated pro- cesses of induction and deduction, and argue directly from one particular case to another, as the late Mr. J. S. Mill proposed. If the Earth, Venue, Mars, Jupiter, and other planets move in elliptic orbits, cannot we dispense with all elaborate precautions, and assert that Neptune, Ceres, or the last discovered planet must do so likewise? Do Digitized by Google 0£XERA LIZA TION. 24 3 we not know that Mr. Gladstone must die, because he is like other men ? May we not argue that because some men die therefore he must ? Is it requisite to ascend by induction to the general proposition ' all men must die,' and then descend by deduction from that gen^^ proposition to the case of Mr. Gladstone \ My answer will be undoubtedly that it is necessary to ascend to general propositions. The fundamental principle of the substitution of similars gives us no warrant in affirming of Mr. Gladstone what we know of other men, simply because we cannot be sure that Mr. Gladstone is exactly similar to other men. Until his death we cannot be perfectly sure that he possesses precisely all the attributes of other men ; it is a question of probability, and I have endeavoured to explain the mode in which the theory of probability is applied to calculate the probability that from a series of similar events we may infer the recurrence of like events under identical circumstances. There is then no such process as that of inferring from particulars to par- ticulars. A careful analyss of the conditions under which such an inference appears to be made, shows that the proceEs is really a general one, and that what is inferred of a particular case might be inferred of all Eomilar cases. All reasoning is essentiaUy general, and all science implies generalization. In the very birth-time of philosophy this was held to be so : ' Nulla scientia est de individiis, sed de solis universalibus,' was the doctrine of Plato, delivered by Porphyry. And Aristotle* held a like opinion — OvStfiia Se Tejfvii vKoirtt to Ka$' Stao-rov . , . to Si KaO' Ikuittov a-TTfipov, Koi ovK hrtmrrov. 'No art treats of particular* cases ; for particulars are infinite and cannot be known.' No one who holds the doctrine that reasoning may be from particulars to particulars, can be supposed to have ■ Aristotle's 'Rhetoric,' Liber I. a. ii. B 2 Digitized by Google 244 TUB PRIXCIPLBS OF SCIEXCB. the moat rudimentary notion of what constitutes reasoning and science. At the same time there can be no doubt that practi- cally what we find to be true of many similar objects will probably be true of the next similar object. This is the result to which an analysis of the Inverse Method of Probabilities leads us, and, in the absence of any precise data from which we may calculate probabilities, we are usually obliged to make a rough assumption that similars in some respects are similars in other respects. Thus it comes to pass that a very large part of the reasoning processes in which scientific men are engaged, seems to consist in detecting similarities between objects, and then rudely assuming that the like similarities will be detected in other cases. Distinction of Generalization and Analogy. There is no distinction but that of degree between what is known as reasoning by generalization and reasoning by analogy. In both cases from certain observed resemblances we infer, with more or less probability, the existence of other resemblances. In generalization the resemblances have great extension and usually little intension, whereas in analogy we rely upon the great intension, the extension being of small amount (vol J. p. 3 1 ). If we find that the qualities A and B are associated together in a great many instances, and have never been found separate, it is highly probable that on the next occasion when we meet •with A, B will also be found to be present, and vice versd. Thus wherever we meet with an object possessing gravity, it is found to possess inertia also, nor have we met with any material objects possessing inertia without discovering that they also possess gravity. The probability has there- fore become very great, as indicated by the rules founded by Google Q EN BRA LIZA TIOS 245 on the Inverse Method of Probabilities (vol. i. pp. 276- 312), that whenever in the future we meet an object pos- sessing either one of the properties of gravity and inertia, it will be found on examination to possess the other of these propertiee. This is a clear instance of the employ- ment of generalization. In analogy, on the other hand, we reason from likeness in many points to likeness in other points. The qualities or points of resemblance are now numerous, not the objects. At the poles of Mars are two white spots which resemble in many respecta the white regions of ice and snow at the poles of the earth. There probably exist no other similar objects with which to compare these, yet the exactness of the resemblance enables us to infer, with high probabiUty, that the spots on Mars would be found to consist of ice and snow, if we could examine them. In short, many points of resemblance imply many more. From the appearance and behaviour of those white spots we infer that they have all the chemical and physical properties of frozen water. The inference is of course only probable, and based upon the improbability that aggr^ates of many qualities should be formed in a like manner in two or more cases, without being due to some single uniform condition or causa In reasoning by analogy, then, we observe that two objects ABODE and A' B' C ly E' have many like qualities, as indicated by the identity of the letters, and we infer that, since the first has another quality, X, we shall also discover this quality in the second case by sufficiently close examina- tion. As Laplace says, — ' Analogy is founded on the probabiUty that similar things have causes of the same kind, and produce the same effects. The more perfect this similarity, the greater is this probability'''. The nature l> ' Eaeai PhiloBophique Bur lea Probabiliti^' p. 66. Digitized by Google 246 THE PRINCIPLES OF SCIENCE. of analogical inference is also very correctly described in the Logic attributed to Kant, where the rule of ordinary induction is stated in the words ' Eines in vielen, aiso in aUen,' one quality in laxDj things, therefore in all ; and the rule of analogy is ' Vieles in einem, also auch das Uhrige in demselhen'°, many (quaUties) in one, therefore also the remainder in the same. It is evident that there may be intermediate caaes in which, from the resemblance of a moderate number of objects in several properties, we may infer to other objects. Probability must rest either upon the number of instances or the depth of resethblance, or upon the occurrence of both in sufficient degrees. What there is wanting in extenwon must be made up by intension, and vice versd. Two Meanings of Generalization. The term generalization, as commonly used, includes two processes which are of different character, but are often closely associated together. In the first place, we generalize whenever we recc^pise even in two fiicts or objects a certain common nature. We cannot detect the slightest similarity without opening the way to inference from one case to the other. If we compare a cubical with a regular octa- hedral crystal, there is little apparent similarity ; but, so soon as we perceive that either can be produced by the symmetrical modification of the other, we discover a groundwork of similarity in the constitution of the crystals, which enables us to infer many things of one, because they are true of the other. Our knowledge of ozone took its rise from the time when the similarity of smell, attending electric sparks, strokes of lightning, and the slow combustion of phosphorus, was noticed by c Kant's 'Logik,' § 84, Konigsberg, tSoo, p. 307. Digitized by Google GBNERALIZA TlOIf. 2i 7 Scbonbein. There was a time when the rainbow was an entirely inexplicable phenomenon, a portent, like a comet, and a cause of superstitious hopes and fears. But we find the true spirit of science in Boger Bacon, who desires ns to consider &e objects which present the same colours as the rainbow ; he mentions hexagonal crystals from Ireland and India, but he bids us not suppose that the hexagonal form is essential, for similar colours may be detected in many other transparent stones. Drops of water scattered by the oar in the sun, the spray from a water-wheel, the dew-drops lying on the grass in the summer morning, all display a similar phenomenon''. No sooner have we grouped together these apparently diverse instances, tiian we have begun to generalize, and have acquired a power of applying to one instance what we can detect of others. Even when we do not apply the knowledge gained to new objects and phenomena, our comprehension of those already observed is vastly strengthened and deepened by thus learning to view them as particular cases of one more general property. A second process, to which the name of generalization is equally given, consists in passing from a g^ven feet or partial law to a multitude of imexamined cases, which we believe to be subject to the same conditions. Instead of merely recognising similarity as it is brought before us, we predict its existence before our senses can detect it, so that generalization of this kind endows us with a pro- phetic power of more or less probaHIity. Having ob- served that many substances assume, like water and mercury, the three states of solid, liquid, and gas, and having assured ourselves by frequent trial that the greater the means we possess of heating or cooling, the more sub- stances we can vapourize and freeze, we pass confidently ^ Whewell's ' PhiloBophy of the loductive Sciences,' and edit. vol. ii. p. 171, qnotiDg the 'OpuB MajuB,' p. 473. by Google 248 THE PRINCIPLES OF SCIENCE. in advance of fact, and assume that all substances are capable of these three forms. Such a generalization was accepted by men of the high intellect of Lavoisier" and Laplace' before many of the corroborative facta now in our possession were known. The reduction of a single comet beneath the sway of gravity was at once conBidered suffi- cient indication that all comets must obey the same power. Few persons doubted that the same great law extended over tlie whole heavens ; certainly the fiiet that a few stars out of many millions make manifest the action of gravity, is now held to be sufficient evidence to establish the general extension of the laws of Newton over the sphere of the visible universe. Value of Generalization. It might seem that if we know particular facts, there can be little use in connecting them together by a general law. The particulars must be more full of useful informa- tion than an abstract general statement. If we know, for instance, the properties of an ellipse, a circle, a parabola, and hyperbola, what is the use of learning all these pro- perties over again in the general theoiy of curves of the second degree? If we understand the phenomena of sound and light and water-waves separately, what is the need of erecting a general theory of waves, which, after all, is in- applicable to practice until resolved into particular cases again ? But, in reality, we never do obtain an adequate knowledge of particulars until we regard them as cases of the general. Not only is there a singular delight in dis- covering the many in the one, and the one in the many, but there is a constant interchange of light and knowledge. " * Cbembtiy,' translated hy Kerr, 3rd edit. pp. 63, 77. ^ 'Syetem of the World,' ditto vol. i. p. 201. by Google GENERA LIZA TION. 240 Properties which are unapparent in the hyperbola may readily be discovered in the ellipse. Most of the complex relations which the old geometers discovered in the circle will be reproduced mutcUis mutandis in the other conic sections. The undulatory theory of Ught might have been unknown at the present day, had not the theory of sound supplied hints by analogy. The study of light has made known many phenomena of interference and polarization, the existence of which had hardly been suspected in the case of sound, but which may now be Bought out, and per- haps found to possess unexpected interest and importance. The careful study of water-waves shows how waves may alter in form and velocity with varying depth of water. Analogous changes may sometimes be detected in sound waves. Thus them is a mutual interchange of aid. ' Every study of a generalization or extension,' as De Morgan has well saids, 'gives additional power over the par- ticular form by which the generalization is suggested. No- body who has ever returned to quadratic equations after the study of equations of all degrees, or who has done the like, will deny my assertion that ov ^Xivti fiXevwv may be pre- dicated of any one who studies a branch or a case, without afterwards making it part of a larger whole. Accordingly it is always worth whUe to generalize, were it only to give power over the partictdar. This principle, of daily fami- liarity to the mathematician, is almost unknown to the Comparative Generality of Physical Properties. Much of the value of science depends upon the know- ledge which we gradually acquire of the different degrees of generality of properties and phenomena of various kinds. e ' Syllabus of a proposed System of Logic,' p. j*. Digitized by Google 250 ■ THE PRINCIPLES OF SCIENCE. • The very iise of science consiets in enabling as to act with confidence, hecause we can foresee the result. Now this foresight must rest upon the knowledge of the powers which will come into play- That knowledge, indeed, can never be certain, because it rests upon imperfect induc- tion, and the most confident beliefs and predictions of the physiciBt may be falsified. Nevertheless, if we always estimate the probability of each belief according to the due teaching of the data, and bear in mind that probability when forming our anticipations, we shall ensure the mini- mum of disappointment. Even when he cannot exactly apply the theory of probabilities, the physicist may acquire the habit of making judgments in general agreement with its principles and results. Such is the constitution of nature, that the physicist soon leams to distinguish those properties which have wide and uniform extension, from those which vary between case and case. Not only are certain laws dis- tinctly laid down, with their extension carefiilly defined, but a scientific training gives a kind of tact in judging how far other laws are likely to apply under any parti- cular circumstances. We learn by degrees that crystals exhibit phenomena depending upon the directions of the axes of elasticity, which we must not expect in uniform solids. Liquids, compared even with non-crystalHne solids, exhibit laws of far less complexity and variety ; and gases assume, in many respects, an aspect of nearly complete uniformity. To trace out the branches of science in which varying degrees of generality prevail, would be found to be an inquiry of great interest and importance ; but want of space, if there were no other reason, would forbid me to attempt it, except in a very slight manner. Gases, so far as they are really gaseous, not only have ex- actly the same properties in all directions of space, but one gas exactly resembles other gases in a great many qualities. by Google GENERA LIE A TION. 251 All gases expand by beat, according to tbe one same law, and by nearly the same amoant; the specific heats of equivalent weights are equal, and the denaties, though not the same, are exactly proportional to the atomic weights. All such gases obey tbe general law, that the volume multiplied by. the pressure, and divided by the absolute temperature, is constant or nearly so. The laws of diSusion and transpiration are the same in all cases, and, generally speaking, all physical laws, as distinguished from chemical laws, which apply to one gas apply equally to all other gasea Even when gases differ in chemical or physical properties, the differences are minor in degree or number. Thus the differences of viscosity are far less marked than in the liquid and solid states. Nearly all gases, again, are colourless, the exceptions being chlorine, the vapours of iodine, bromine, and some other sub- Only in one single point, so far as I am aware, do gases present distinguishing marks unknown, or nearly so, in tbe solid and liquid states. I mean as r^ards the light given off when incandescent. Each gas, when suf- ficiently heated, yields its own peculiar series of rays, arising from the free vibrations of tbe constituent parts of the molecules when pursuing separate paths. Hence tbe possibility of distinguishing gases by the spectro- scope. But the molecules of solids and liquids appear to be continually in conflict with each other, so that only a ooniused noise of atoms is produced, instead of a definite series of luminous chords. At tbe same tempera- ture, accordingly, all sd-ids and liquids give off nearly the same rays when strongly heated, and we have in this case a single exception to the general rule of the greater generality of properties in gases. Liquids are in many ways intermediate in character between gases and solids. While incapable of possessing Digitized by GOO^Iv 252 TMB PRINCIPLES OF SCIEXCE. different elasticity in different directions, and thus de- nuded of the rich geometrical complexity of solidB, they retain the variety of density, colour, degrees of trans- parency, great diyersity in surface tension, viscosity, co- efficients of expansion, compressibility, and many other properties which we observe in solids, but not for the most part in gases. Though our knowledge of the phy- sical properties of liquids is thus much wanting in generality at present, there is ground to hope that by degrees laws connecting and explaining the varieties of character may be traced out. Liquids ought to be compared together, not at uniform temperatures, but at points of temperature similarly related to the points of fusion and ebullition. Solids are in every way contrasted to gases. Each solid substance has its own peculiar density, hardness, com- pressibility, degree of transparency, tenacity, elasticity, power of conducting heat and electricity, magnetic pro- perties, capability of producing frictional electricity, and so forth. . Even different specimens of the same kind of substance will be widely different, according to the acci- dental treatment it has received. And not only has each substance its own 8peci6c properties, but, when crystallized, ita own properties peculiar to each direc- tion, regard being had to the axes of crystallization. The velocity of radiation, the rate of conduction of heat, the coeificients of eipanability and compressibility, the thermo-electric properties, all vary in different crystallo- graphic directions. It is highly probable that many apparent differences among liquids, and even among solids, will be resolved and explained, when we learn to regard them under ex- actly corresponding circumstances. The extreme gene- rality of the properties of gases is really only true at an infinitely high temperature, when they are all equally by Google GESERA LIZA TION. 253 remote from their condensing points. Now, it is found that if we compare liquids — for instance, different kinds of alcohols — not at equal temperatures, but at points equally distant from their respective boiling-points, the laws and coefficients of expansion are nearly equal. The vapour-tensions of liquids also are much more nearly equal, when thus compared at corresponding points, and the boiling-points themselves appear to be simply related to the chemical composition in many cases. No doubt the progress of investigation will often enable vs to discover generality, where we at present only see variety and puzzling complexity. Id some cases substances exhibit the same physical pro- perties in the liquid as in the solid state. Lead has a high refractive power, whether in solution, or in solid salts, crystallized, or vitreous. The magnetic power of iron is conspicuous, whatever be its chemical condition ; indeed, the magnetic properties of substances, though varying with temperature, seem not to be greatly affected by physical changes. Colour, absorptive power for heat or light rays, and a few other properties are also often the same both in liquids and gases. Iodine and bromine possess a deep colour whenever they are chemically un- combined. Nevertheless, we can seldom argue safely from the properties of a substance in one condition to that in another condition. Ice is an insulator, water a con- ductor of electricity, and the same contrast exists in most other substances. The conducting power of a liquid for electricity increases with the temperature, while .that of a solid decreases. By degrees we may letim to distinguish between those properties of matter which depend upon the intimate construction of the chemical molecule, and those which depend upon the contact, conflict, mutual attraction, or other relations of distinct molecules. Tlie properties of a substance with respect to light seem gene- Digitized by Google 254 THE PRINCIPLES OF SCIENCE. rally to depend upon the molecule ; thus, the power of certain substances to cause the plane of polarization of a ray of light to rotate, is exactly the same whatever be its degree of density, or the diluteness of the solution in which it is contained. Taken as a whole, the physical properties of substances and their quantitative laws, pre- sent a problem of infinite complexity, and centuries must elapse before any moderately complete generalizations on the subject become possible. Uniform Properties of all Mattei: Some laws are held to be true of all matter in the universe absolutely, without exception, no instance to the contrary having ever been noticed. This is the case with the laws of motion, as laid down by Galileo and Newton. It is also cojispicuously true of the law of universal gravitation. The rise of modem physical science may perhaps be considered as beginning at the time when Galileo showed, in opposition to the Aristotelians, that matter is equally affected by gravity, irrespective of its form, magnitude, or texture. All objects &11 with equal rapidity, when disturbing causes, such as the resistance of the air, are removed or allowed for. That which was rudely demonstrated by Galileo from the leaning tower of Pisa, was proved by Newton to a high degree of approxi- mation, in an experiment which has already been referred to (vol. ii, p. 55). Newton formed two pendulums of as nearly as possible similar outward shape, by taking two equal round wooden boxes, and suspending them by equal threads, eleven feet long. The motion of each pendulum was therefore equally subject to the resistance of the air. He filled one box with wood, and in the centre of oscillation of the by Google GENERA LIZA TION. 255 other placed an equal weight of gold. The peDdulums were then equal in weight and in Bize ; and, on setting them Bimultaneously in motion, Newton found that they vibrated for a great length of time with exactly equal vibrations. He tried the same experiment with silver, lead, glass, sand, common salt, water, and wheat, instead of gold, and ascertained that the rapidity of motion of his pendulum was exactly the same whatever was the kind of matter inside them *>. He considered that a difference of a thousandth part would have been apparent. The reader must observe that the pendulimis were made of equal weight only in order that they might suffer equal retardation from the air. The meaning of the experiment is that all the substances manifest exactly equal accelera- tion from the force of gravity, and that therefore the inertia or resistance of matter to force, which is the only independent measure of mass in our possession, is always proportional to gravity. These experiments of Newton were consideied conclu- sive up to vety recent times, when certain discordances between the theory and observations of the movements of planets led Nicolai, in 1826, to suggest that the equal gravitation of different kinds of matter might not be absolutely exact. It is perfecUy philosophical and desir- able thus to call in question, from time to time, some of the best accepted laws. On this occasion Bessel carefully repeated the experiments of Newton with pendulums composed of ivory, glass, marble, quartz, meteoric stones, Ac, but was unable to detect the least difference. This conclusion is also confirmed by the ultimate agreement of all the calculations of physical astronomy based upon it. Thus, whether the mass of Jupiter be calculated from the motion of its own satellites, frxim the effect upon the small 1> ' Frincipia,' bk. III. Prop. VI, Motte's traDslBtion, vol. iL p. 230. Digitized by Google 256 THE PRINCIPLES OF SCIENCE. planets, Vesta, Juno, Ac, or from the perturbation of Encke's Comet, the results are closely accordant, showing that precisely the same law of gravity applies to the most different bodies which we can observe. The gravity or weight of a body, again, appears to be entirely inde- pendent of its other physical conditions, being totally unaffec-ted by any alteration in the temperature, density, electric or magnetic condition, or other phyacal proper- ties of the substance. One almost paradoxical residt of the law of equal gravitation is the theorem of Torricelli, to the effect that all liquids of whatever density fall or flow with equal rapidity. If there be two equal cisterns respectively filled with mercury and water, the mercury, though thirteen times as heavy, woidd flow from an aperture neither more rapidly nor more slowly than the water, and the same would be true of ether, alcohol, or any other liquids, allowance being made for the resistance of the air, and the differing viscosities of the liquids. In its exact equality and its perfect independence of every circumstance, except mass and distance, the force of gravity stands apart from all the other forces and pheno- mena of nature, and has not yet been brought into any relation with them except through the general principle of the conservation of energy. Magnetic attraction, as remarked by Newton, follows a wholly different law as depending upon the chemical quality and molecvdar struc- ture of each particular substance. We must remember that in saying ' all matter gravi- tates,' we exclude from the term matter the basis of light- undulations, which is almost infinitely more extensive in amount, and obeys in many other respects the laws of mechanica This adamantine basis of undulations appears, so far as can be ascertained, to be perfectly uniform in its properties when existing in epace unoccupied by matter. Digitized by Google GENERALIZATION. 257 Light and heat are conveyed by it with equal velocity in all directions, and in all parts of spice so far as observar tion informs us. But the presence of gravitating matter modifies the density and mechanical properties of the so-called ether in a way which is yet quite unexplained. Leaving gravity, it is somewhat difficult to discover other laws which are equally true of all matter. Boer- liaave was considered to have established that all bodies expand by heat, but not only is the expansion very dif- ferent in different substances, but we now know positive exceptions. Many liquids and a few solids contract by heat at certain temperatures. There axe indeed other relations of heat to matter which seem to be universal and uniform ; thus all eubstances begin to give off rays of heat or light at the eame temperature, according to Ihe law of Draper ; and gases will not be an exception if sufficiently condensed, as in the experiments of Frank- land. Grove considers it to be univereally Irue that all bodies in combining produce heat ; all solids, with the doubtful exception of sulphur and selenium, in becoming liquid, and all liquids in becoming gases, absorb a certain quantity of heat ; but the quantities of heat absorbed vary with the chemical qualities of the matter. On the other band, Camot's Thermodynamic Law is held to be exactly true of all matter without distinction ; it ex- presses the fact that the amount of mechanical energy which might be theoretically obtained from a certain amount of heat energy depends only upon the tempera- tures between which a substance is made to change, so that whether an engine be worked by water, air, alcohol, ammonia, or any other substance, the result would theo- retically be the same, if the boiler and condenser were employed at similar temperatures. by Google THE PRINCIPLES OF SCIENCE. Variable Properties of Matter. I have enumerated some of the few properties of matter, which are manifested in exactly the same maimer by all substances, whatever be their differences of chemical or physical constitution. But by far the greater number of qualities vary in degree ; substances are more or less dense, more or less transparent, more or less compressible, more or less magnetic, and so on. One very common result of the progress of science is to show that qualities supposed to be entirely absent from many substances are present only in so low a degree of intensity that the means of detection were insufiScient. Newton believed that most bodies were not affected by the magnet at all ; Faraday and Tyndall have rendered it very doubtful whether any substance whatever is wholly non-magnetic, including under that term diamagnetic properties. We are rapidly learning to believe that there are no sub- stances absolutely opaque, or non-conducting, non-electric, non-elaatic, non-viscous, non-compressible, insoluble, in- fusible, or non-volatile. All tends to become a matter of degree, or sometimes of direction. There may be some substances oppositely affected to others, as ferro-magnetic substances are oppositely affected to diamagnetics, or as substances which contract by heat are opposed to those which expand; but the tendency is certainly for e-vtry affection of one kind of matter to be represented by some- thing similar in other kinds. On this account one of Newton's rules of philosophizing seems quite to lose all validity; he said, ' Those qualities of bodies which are not capable of being heightened and remitted, and which are found in all bodies on which experiment can be made, must be considered as universal qualities of all bodies.' As far as I can see, the contrary is more probable, namely. by Google GENERALIZA TION. 259 that qualities variable in degree will be found in eveiy substance in a greater or less degree. It is bighlj remarkable that Newton, whose method of investigation was logically perfect, seemed incapable of generalizing and describing his own procedure. Hia celebrated ' Rides of reasoning in Philosophy,' described at the commencement of the third book of the ' Principia,' are of very questionable truth, and still more questionable value. Extreme Instances of Properties. Although, as we have seen, substances usually differ only in degree as regards their physical properties, great interest may attach to particular substances which mani- fest a property in a conspicuous and intense manner. Every branch of physical science has usually been de- veloped from the attention forcibly drawn to some sin- gular substance. Just as the loadstone disclosed mag- netism and amber frictional electricity, so did Iceland spar point out the existence of double refraction, and sulphate of quinine the phenomenon of fluorescence. When one such startling instance has dravra the attention of the scientific world, numerous less remarkable cases of the phenomenon will soon be detected, and it will pro- bably prove that the property in question ia actually univerwd to all matter. Nevertheless, the extreme in- stances retain their interest, partly in a historical point of view, partly because they fumJsh the most convenient substances for experiment. Francis Bacon was fully aware of the value of such examples, which he called Osie^isive Instances or Light- giving, Free or Predominant Instances'. 'They are those,' he says, * which show the nature under investigation naked, in an exalted condition, or in the highest degree » % Digitized by Google 260 THE PRINCIPLES OF SCIENCE. of power ; freed from impcdinients, or at least by its strength predominating over and suppressing them '.' He mentions quicksilver aa an ostensive instance of weight or densHj, thinking it not much less dense than gold, and more remarkable than gold as joining density to liquidity. The magnet is mentioned as an ostenBive instance of attraction^. It would not be very easy to distinguish clearly between these ostensive instances and those which he calls Instanttae Monodicae, or Irregulares, or Hetero- clitae, under which he places whatever is extravagant in its properties or magnitude, or exhibits least similarity to other things, such as the sun and moon among the heavenly bodies, the elephant among animals, the letter a among letters, or the magnet among stones '. Id optical science great use has been made of the high dispersive power of the transparent compounds of lead, that is, the power of giving a long spectrum (vol. i. p. 32). DoUand having noticed the peculiar dispersive power of lenses made of flin1>-glass employed them to produce an achromatic arrangement. The element strontium presents a contrast to lead in this respect, being characterized by a remarkably low dispersive power ; but I am not aware that this property has yet been turned to account. Compounds of lead have both a high dispersive and a high refractive index, and in the latter respect they proved very useful to Faraday. Having spent much labour in preparing various kinds of optical glass, Fara- day happened to form a compoimd of lead, silica, and boracic acid, now known as heavy glass, which possessed an intensely high refracting power. Many years after- wards in attempting to discover the action of magnetism upon light he failed to detect any effect, as has been I ' Novum Organum,' bk. II. Aphorism 34. k Ibid. AphoriBin 25, ' Ibid, AphoriBm z8. by Google GENERALIZA TION. 261 already mentioned (vol. ii. p. 235), until he happened to test a piece of the heavy glaea. The peculiar refractive power of this medium caused the magaetie strain to be apparent, and the rotation of the plane of polarization was discovered. In almost every other part of physical science there is some substance of powers pre-eminent for the special pur- pose to which it is put. Bock-salt is invaluable for its extreme diathermancy or transparency to the least re- frangible rays of the spectrum. Quartz is equally valu- able for its transparency, as regards the ultrarviolet or most refrangible rays. Diamond is the most highly refrac- ting substance whicb is at the same time transparent ; were it more abundant and easily worked it would be of great optical importance. Cinnabar is distinguished by possessing a power of rotating the plane of polarization of light, from 1 5 to 1 7 times that of quartz. In electric experiments copper is employed for its high conducting powers and exceedingly iow magnetic properties ; iron is of course essential for its enormous and almost ano- malous magnetic powers ; while bismuth holds a like place as regards its diamagnetic powers, and was of much im- portance in Tvndall's decisive researches upon the polar character of the diamagnetic force™. In regard to magne- crystallic action the mineral cyanite is highly remark- able, being so powerfully affected by the earth's magnetism, that when delicately suspended, it will assume a constant position with regard to the magnetic meridian, and may almost be used like the compass needle. Sodium is dis- tinguished by its luiique light-giving powers, which are so extreme that probably one half of the whole number of stars in the heavens have a yellow tinge in consequence. It is highly remarkable that water, though the most common of all fluids, is distinguished in almost every °» ' Philosophical TraUBactions,' (1856), wol. cxlvL p. 246. Digitized by Google 262 THE PRINCIPLES OF SCIENCE. respect by the most marked qualities. Of all known Bubstancea water has the highest specific heat, being thus peculiarly fitted for the purpose of warming and cooling, to which it is often put. It rises by capillary attraction to a height more than twice that of any other liquid. In the state of ice it is nearly twice as dilatable by heat as any other known solid substance". In proportion to its density it has a far higher surface tension than any other substance, being in fact surpassed in absolute tension only by mercury, and it would not be diflScult to extend con- siderably the hst of its remarkable and useful properties. Under extreme instances we may include cases of re- markably low powers or qualities, equally with those of the opposite extreme. Such cases seem to correspond to what Bacon calls Clandestine Iiistances, which exhibit a given nature in the least intensity, and as it were in a rudimentary state". They may often be important, he thinks, as allowing the detection of the cause of the pro- perty by difierence, I may add that in some cases they may be of use in experiments. Thxis hydrogen is at once the least dense of all known substances, and has the least atomic weight. Liquefied nitrous oxide has the lowest refractive index of all known fluids'*. The compounds of strontium have the lowest dispersive powers on light. It will be obvious that a property of very low degree may prove as curious and valuable a phenomenon as a property of very high degree. The Detection of Continuity. We should always bear in mind that phenomena which are in reality of a closely similar or even identical nature, ° ' FhiloBOpbical Magazine,' 4th Series, Januaiy 1870, vol. xxxix. p. 2. " 'Novum Organum,' bk. II. AphoriBin 35. P Faraday's ' ExperimenUl Bcsearches in Cliemistiy and Phpice,' P- 93- by Google GBSBUALIZA TWN. 2C3 may present to the Benses very different appearance?. Without a careful analysis of the changes ■which take place, we may often be in danger of -widely separating facts and processes, which are actually instances of the same law. Extreme difference of degree or magnitude is a very frequent cause of error. It is truly diflScult at the first moment to recognise any similarity between the gradual rusting of a piece of iron, and the rapid combustion of a heap of straw. Yet Lavoisier's chemical theory was founded upon the close similarity of the oxy- dizing process in one case and the other. We have only indeed to divide the iron into excessively small particles to discover that it is really the more combustible of the two, so that it actually takes fire spontaneously and bums like tinder. It is the excessive slowness of the process in the case of a massive piece of iron which disguises its real character. If Xenophon reports truly, Socrates was seriously mis- led by not making sufficient allowance for extreme differ- ences of degree and quantity. He rejected the acute opinion of Anaxagoras that the sun is a fire, on the ground that we can look at a fire, but not at the sun, and that plants grow by sunshine while they are killed by fire. He also pointed out that a stone heated in a fire is not luminouSj and soon cools, whereas the sun ever remains equally luminous and hof. All such mistakes evidently arise fiitim not perceiving that difference of quantity may be so extreme as to assume the appearance of difference of quality. It is the least creditable tJdng we know of Socrates, that when pointing out these supposed mistakes of earlier philosophers, he advised his followers not to study astronomy. Masses of matter of very different size may always be 1 ' Memorabiliii,' iv. 7 ; quoted by Whewell, ' History of Inductive Scieuces,' vol. i. p. 340. by Google 264 THE PRINCIPLES OF SCIEXCE. expected to exhibit great apparent differences of conduct, ariang simply from the very various intensity of the forces brought into play. Many persons have thought it requi- site to ima^ne occult forces producing the suspension of the clouds, and there have even been absurd theories representing cloud particles as minute water-balloons buoyed up by the warm air within them. But we have only to take proper account of the enormous comparative resistance which the air opposes to the fall of minute particles, to see that all cloud particles are probably con- stantly falling through the air, but so slowly that there is no apparent efiect. Mineral matter again is always regarded as inert and incapable of spontaneous movement- We are struck by astonishment on observing in a power- ful microscope, that every kind of solid matter suspended in extremely minute particles in pure water, acquires an oscillatory movement, often so marked as to resemble dancing or skipping. I conceive that this movement is entirely due to the vast comparative intensity of chemical actions when exerted upon minute particles, the effect being 5000 or 10,000 greater in proportion to the mass than in fragments of an inch diameter {vol. iL p. 9). Much that was formerly obscure in the science of elec- tricity, arose from the extreme differences of intensity and quantity in which this form of energy manifests itself. Between the instantaneous and brilliant discharge of a thimder-cloud and the gentle continuous current pro- duced by two pieces of metal and some dilute acid, there was no apparent analogy whatever. It was therefore a work of great importance when Faraday demonstrated the identity of the forces in action, showing that common frictional electricity would decompose water like that from the voltaic battery. The relation of the phenomena be- came plain when he succeeded in showing that it would require 800^000 discharges of his large Leyden battery to by Google GENERALIZE TIOS. _ 265 decompose one Bingle grain of water. Lightning was dow seen to be electricity of excessively high tension, but extremely small quantity, the difference beiiig somewhat analogous to that between the force of one million gallons of water falling through one foot, and one gallon of water falling through one million feet. Faraday estimated that one grain of water acting on four grains of zinc, would yield electricity enough for a great thunderstorm. It was long believed that electrical conductors and in- sulators belonged to two opposed classes of substances. Between the inconceivable rapidity witii which the cur- rent passes through pure copper wire, and the apparently complete manner in which it is stopped by a thin parti- tion of guttarpercha or gum-lac, there seemed to be no resemblance. Faraday, again, laboured successfully to show that these were but the extreme cases of a chain of 6ul> stances varying in all degrees in their powers of conduc- tion. Even the best conductors, such as pure copper or silver ofier some resistance to the electric current. The other metals have considerably higher powers of resist- ance, and we pass gradually down through oxides and sulpbidea The best insulators, on the other hand, allow of an atomic induction which is the necessary antecedent of conduction. Hence Faraday inferred that whether we can measure the effect or not, all substances discharge electricity more or les^^ One consequence of this doctrine must be, that every discharge of electricity produces an induced current. In the case of the common galvanic current we can readily detect the induced current in any parallel wire or other neighbouring conductor, and can separate the opposite currents which arise at the moments when the original currents begin and end. But a dis- charge of high tension electricity like lightning, though it certainly occupies time and has a beginning and an end, ' 'Experimental Reswrclics id Electricity,' Series xii. vol. i, p. 430. Digitized by Google 266 TUB PRINCIPLES OF SCIENCE. jet lasts so minute a fraction of a second, that it would be hopeless to attempt to detect and separate the two opposite induced currents, which are nearly simultaneous and exactly neutralise each other. Thus an apparent failure of analogy is explained away, and we are furnished with another instance of a phenomenon incapable of obser- vation and yet theoretically known to exist". Perhaps the most extraordinary and fundamental case of the detection of unsuspected continuity is found in the discovery of Cagniard de la Tour and Professor Andrews, that the liquid and gaseous conditions of matter are only remote points in a continuous course of change. Nothing is at first sight more apparently distinct than the physical condition of water and aqueous vapour. At the boiling- point there is an entire breach of continuity, and the gas produced is subject to laws incomparably more simple than the liquid &om which it arose. But Cagniard de la Toiu: showed that if we maintain a liquid under sufficient pressure its boiling point may be indefinitely raised, and yet the Uquid will ultimately assume the gaseous con- dition with but a small increase of volxune. Professor Andrews, recently following out a similar course of in- quiry, has shown that liquid carbonic acid may, at a par- ticular temperature {3o'*-92 C), and under the pressure of 74° atmosphere, be at once in -a state indistinguishable from that of liquid and gas. At higher pressures carbonic acid may be made to pass from a palpably liquid state to a truly gaseous state without any abrupt change whatever. The subject is one of some complexity, hecause as the pressure is greater the abruptness of the change from liquid to gas gradually decreases, and finally vanishes. As similar phenomena or an approximation to them have been observed in various other Uquids, there is little doubt that we may make a flide generalization, " ' Lire of Faraday,' vol ii. p. 7. Digitized by Google GMSERALIZA TION. 267 and assert that, under adequate pressure, every liquid might be made to pass into a gaa without any breach of continuity*. The liquid state, again, is considered by Professor Andrews to be but an intermediate step between the solid and gaseous conditions. There are various indica- tions that the process of melting is not perfectly abrupt ; and could the experiments be made under adequate pressures, it is believed that every solid could be made to pass by insensible degrees into the state of liquid, and subsequently into that of gas. These discoveries appear to open the way to most im- portant and fundamental generalizations, but it is probable that in many other cases phenomena now regarded as dis- crete may be shown to be different degrees of the same process. The late Profe^or Graham was of opinion that chemical affinity differed but in degree from the ordinary attraction which holds different particles of a body together. He found that sulphuric acid continued to evolve heat when mixed even with the fiftieth equivalent of water that is added to it, so that there seemed to be no distinct limit to chemical affinity. He concludes — ' There is reason to believe that chemical affinity passes in its lowest degree into the attraction of a^regation'". The atomic theory is well established, but its limits are not marked out. As Mr. Justice Grove suggests, we may by selecting sufficiently high multipliers express any com- bination or mixture whatever of elements in terms of their equivalent weights". Sir W. Thompson has suggested that the power which vegetable fibre, oatmeal, and many other substances possess of attracting and condensing aqueous vapour ia probably continuous, or, in fact, iden- ' ' Nature,' vol. ii, p. zj8. " 'Journal of the Chemical Society,' toL viii. p- Si- * 'Correlation of Phyeical Forces,' 3rd edit. p. 184. , by Google 268 TUB PRINCIPLES OF SCIENCE. tical with capillary attraction, which is capable of inter- fering with the pressure of aqueous vapour and aiding its condensationy. There are many cases of so-called catalytic or siu&ce action, such as the extraordinary power of animal charcoal for attracting organic matter, or of spongy pla^ tinum for condensing hydrogen, which can only be con- sidered as exalted cases of a much more general power of attraction. The number of substances which are decom- posed by light in a striking manner is very limited ; but many other substances, such as vegetable colours, are affected by long exposure ; on the principle of continuity we might well expect to find that all kinds of matter are more or leas susceptible of chMige by the incidence of light rays*. It is the opinion of Mr. Justice Grove that wherever an electric current passes there is a tendency to decom- position, a strain on the molecules, which when suflSciently intense leads to disruption. Even a metallic conducting wire may be regarded as tending to decomposition. Davy was probably correct in describing electricity as chemical affinity acting on masses, or rather, as Grove suggests, creating a distxirbance through a chain of particles'. Laplace went so far as to suggest that all chemical phe- nomena may be regarded as the results of the Newtonian law of attraction, applied to atoms of various mass and position ; but the time is probably long distant when the progress of molecular philosophy and of mathematical methods will enable such a generalization to be verified or refuted. The Law of Continuity. Under the title Law of Continuity we may place many applications of the general principle of reasoning, that r 'Philosophical Uagaztne,' 4th Series, vol. xlii. p. 451. * Grove, 'Correlation of Fhysicat Forces,' 3rd edit. p. 118. • Ibid. pp. i6fi, 199, &.C. by Google GEXEBA LIZA TIOX. 269 what is true of one case will be true of similar cases, and probably true of what are probably eimilar. Whenever we find that a law or amilarity is rigorously fulfilled up to a certain point in time or space, we expect with a very high degree of probability that it will continue to be ful- filled at least a little longer. If we see part of a circle, we naturally expect that the' form of the line wUl be maintained in the part hidden fi^m us. If a body has moved uniformly over a certain space, we expect that it will continue to move uniformly. The ground of such inference is doubtless identical with that of all other in- ductive inferences. In continuous motion every infinitely small space passed over constitutes a separate constituent fact, and had we perfect powers of observation the smallest finite motion would include an infinity of information, which, by the principles of the inverse method of probahilities, would enable us to infer with actual certainty to the next infinitely small portion of the path. But when we attempt to infer from one finite portion of a path to another finite part, the inference will be only more or less probable, according to the comparative lengths of the parts and the accuracy of the oheervations ; the longer our expe- rience is, the more probable our inferences will be ; the greater the length of time or space over which the in- ference extends, the less probable. This principle of continuity presents itself in nature in a great variety of forms and cases. It is familiarly expressed in the dictum Natura non agit per saltum, in other words, no change in a natural phenomenon comes on with perfect suddenness or abruptness. There is always some notice — some forewarning of every phenomenon, and eveiy change begone by insensible degrees, could we observe it with perfect accuracy. The cannon ball, indeed, is forced from the cannon in an inappreciable portion of time ; the trigger is pulled, the fuze fired, the powder inflamed, the by Google 2-0 THE PRINCIPLES OF SCIENCE. ball expelled, all simultaneously to our senses. But there is no doubt that time is occupied by every part of the process, and that the ball begins to move at first with indefinite slowness. Captain Noble Is able to measure by his chronoscope the progress of the shot in a 300- pounder gun, and finds that the whole motion within the barrel takes something less than — ■ of a second. It is an invariable principle of nature that no finite force can produce motion, except in a finite space of time. The amount of momentum communicated to a body is pro- portional to the accelerating force multiplied by the time through which it acts uniformly. Thus a slight force produces a great velocity only by long continued action. In a powerful shock, like that of a railway collision, the stroke of a hammer on a hard anvil, or the discharge of a gun, the time is very short, and therefore the accelerating forces brought into play are exceedingly great, but never infinite. In the case of a large gun the powder in ex- ploding is said to exert for a moment a force equivalent to at least 2,800,000 horses. Our belief in some of the most fundamental laws of nature rests upon the principles of continuity. Galileo ia held to be the first philosopher who consciously employed this principle In liis arguments concerning the nature of motion, and it is certain that we can never by pure ex- perience assure ourselves of the truth even of the first law of motion. A material particle, we are told, when not acted on hy extraneous forces vnll continue in the same staie of rest or motion. This' may be true, but as we can find no body which is free irom the action of extraneous causes, how are we to prove it ? Only by observing that the less the amount of those forces the more nearly is the law found to be true. A ball rolled along rougb ground is soon stopped; along a smooth pavement it continues Digitized by Google GENERA LIZA TION. 2 7 1 longer in movenieiit. A delicately suspended pendulum is almost free from friction against its supports, but it is ' gradually stopped by the resistance of the air ; place it in the vacuous receiver of an air-pump and we find the motion immensely prolonged. A large planet like Jupiter experiences almost infinitely less friction, in comparison to its vast momentum, than we can produce experimentally, and we find through centuries that there is not the least evidence of the falsity of the law. Expe- rience, then, informs us that we may approximate indefi- nitely to a uniform motion by sufiBciently decreasing the dieturhlng forces. It is a pure act of inference which enables us to travel on beyond experience, and assert that, in the total absence of any extraneous force, motion would be absolutely uniform. The state of rest, again, is but a singular case in which motion is infinitely small or zero, to which we may attain, on the principle of continuity, by considering successively cases of slower and slower motion. There are many interesting cases of physical pheno- mena, in which, by gradually passing from the apparent to the obscure, we can aasiure ourselves of the nature of phenomena which would otherwise be a matter of great doubt. Thus we can sufficiently prove, in the manner of Galileo, that a musical sound consists of rapid uniform pulses, by causing strokes to be made at intej^als which we gradually diminish until the separate strokes coalesce into a uniform hum or note. With great advantage we approach, as Tyndall says, the sonorous through the grossly mechanical. In listening to a great oigan we cannot fe,il to perceive that the longest pipes, or their partial tones, produce a tremor and fluttering of the building. At the other extremity of the scale, there is no fixed limit to the acuteness of sounds which we can hear ; some individuals can hear sounds too shrill for other ears, and as there is nothing in the nature of the Digitized by Google 272 THE PRISCIPLF.S OF SCIENCE. atmoBphere to prevent the existence of undulations in- comparably more rapid than any of which we are con- scious, we may infer, by the principle of continuity, that such undulations probably exist. There are many habitual actions which we perform we know not how. So rapidly are many acts of mind ac- complished that analysis seems impossible. We can only investigate them when in process of formation, observing that the best formed habit or instinct is slowly and con- tinuously acquired, and it is in the early stages that we can perceive the rationale of the process. Let it be observed that this principle of continuity must be held of much weight only in exact physical laws, those which doubtless repose ultimately upon the wmple laws of motion. If we fearlessly apply the prin- ciple to all kinds of phenomena, we may often be right in our inference, but also oflen wrong. Thus, before the development of spectrum analysis, astronomers had ob- served that the more they increased the powers of their telescopes the more nebulse they could resolve into dis- tinct stars. This result had been so often found true that they almost irresistibly assumed that all the nebuloe would be ultimately resolved by telescopes of sufficient power ; yet Mr. Huggins has in recent years proved by the spectroscope, that certain nebulse are actually gaseous, and in a truly nebulous state. Even one such observation is a real exception sufficient to invalidate previous in- ferences as to the constitution of the universe. The principle of continuity must have been continually employed in the inquiries of Galileo, Newton, and other experimental philosophers, but it appears to have been distinctly formulated for the first time by Leibnitz. He at least claims to have first spoken 'of the law of con- tinuity' in a letter to Bayle, printed in the 'Nouvelles de la Republique des Lettres,' an extract from which is Digitized by Google GENERALIZATION. 273 given in Erdmann's edition of Leibnitz' works, p. 104, under the title ' Sur un Principe G^n^ral utile \ I'expli- cation des Lois de la Nature^.' It has indeed been asserted that the doctrine of the latevs processus of Francis Bacon involves the principle of continuity", but I think that this doctrine, like that of the natures of substances is merely a vague statement of the principle of causation. Failure of the Law of Continuity. There are certain requisite cautions which must be given as to the application of the principle of continuity. In the first place, where this principle really holds true, it may seem to fail owing to our imperfect means of observation. Though a physical law may never admit of perfectly abrupt change, there is no limit to the approach which it may make to abruptness. When we warm a piece of very cold ice, the absorption of heat, the tem- perature, and the dilatation of the ice vary according to apparently simple laws until we come to the zero of the Centigrade scale. Everything is then changed ; an enor- mous absorption of heat takes place without any rise of temperature, and the volume of the ice decreases as it changes into water. Unless most carefuUy investigated, this change appears perfectly abrupt ; hut accui-ate ob- servation seems to show that there is a certain forewarn- ing; the ice does not turn into water all at once, but through a small fi'action of a degree the change ie gradual. All the phenomena concerned, if measured very esactly, would be represented not by angular lines, but con- tinuous curves, undergoing rapid flexures; and we may •> ' Life of Sir W. Hamilton,' p. 439. ' Powell'a'Hiatoryof Natural Philosophy,' p. 201. 'Novum Organum,' bk. II. AphoriBmB 5-7. VOL. rr. T by Google 274 THE PRINCIPLES OF SCIENCE. probably assert with safety that between whatever points of temperature we examined ice, there would be found some indication, doubtless almost in£nitesimally small, of the apparently abrupt change which was to occur at a higher temperature. It might also be pointed out that aU the most important and apparently simple physical laws, such as those of Boyle and Marriotte, Dalton and Gay-Lussac, &c., are only approximately true, and the divergences from observation are forewarnings of abrupt changes, which would otherwise break the law of con- tinuity. Secondly, it must be remembered that mathematical laws of any complexity will probably present singular cases or negative results, which may present the appear- ance of discontinuity, as when the law of refraction sud- denly yields us with perfect abruptness the entirely different phenomenon of total internal reflection. In the undulatoiy theory there is no real change of law between the phenomenon of refraction and that of reflection. Faraday in the earlier part of his career found so many substances possessing more or less magnetic power, that he ventured on a great generalization, and asserted that all bodies shared in the magnetic property of iron. His mistake, as he afterwards himself discovered, consisted in overlooking the feet that though magnetic in a certain sense, some substances might have negative magnetism, and be repelled instead of attracted by the magnet Between magnetism and diamagnetism there must be a zero near or even at which some substances may be classed, but otherwise magnetic properties appear to be universally present in matter. Thirdly, where we might expect to find a uniform mathematical law prevailing, the law may undergo abrupt change at singular points, and actual discontinuity may arise. We may sometimes be in danger of treating under Digitized by Google GENERA LIZA TIOS. 275 one law phenomena which really belong to difierent laws. It is generally known, for instance, that a spherical shell of unifonn matter attracts an external particle of matter with a force varying inversely aa the square of the distance from the centre of the sphere. But this law only holds true so long as the particle is external to the shell. Within the shell the law is wholly different, and the aggregate gravity of the sphere becomes zero, because the force in every direction is neutralized by an exactly equal force. If an infinitely small particle be in the superficies of a sphere, the law is again difierent, and the attractive power of the shell is half what it would lie on particles infinitely close to the surface of the shell. Thus in approaching the centre of a shell from a distance, the force of gravity evinces a double discontinuity in passing through the shell^. It may well admit of question, too, whether discontinuity is really unknown in nature. We perpetually do meet with events which are real breaks upon the jirevious law, though the discontinuity may then be a sign that some wholly independent cause has come into operation. If the ordinary course of the tides is interrupted by an enormous and irregular wave, we attribute it to an earth- quake, or some gigantic natural disturbance. If a meteoric stone falls upon a person and kills him, it is clearly a discontinuity in his life, of which he could have had no anticipation. A sudden sound may prfb through the air neither preceded nor followed by any continuous effect. Although, then, we may regard the Law of Con- tinuity as a principle of nature holding rigoroxisly true in many of the relations of natural forces, it seems to be a matter of difficulty to assign the limits within which the •> Thomson and Toit, 'Treatise on Natural Philosophy,' vol. i. pp. 346-35 >■ T 2 by Google 276 THE PRINCIPLES OF SCIENCE. law is verified. Much caution, therefore, is desirable in its application. Negative Arguments on the Principle of Continuity. Upon the principle of continuity we may often found arguments of great force which prove ui hypotheas to be imposaible, because it would involve a continual repetition of a process ad infinitum, or else a purely arbitrary breach at some point. Bonnet's famous theory of reproduction represented every living creature as containing germs which were perfect representatives of the next generation, so that on the same principle they necessarily included germs of the next generation, and so on indefinitely. The theory was sufficiently refuted when once clearly stated, as in the following poem called the Universe^, by Henry Baker: ' Each seed includee a plant : that plant, again, Has other Beeds, which other phuite contain : Those other plants have all their seeds, and those Hore planta again, BucceBBiTely iucloBC. ' Thus, ev'ry single berry that we find, Has, really, in itself whole forests of its kind. Empire and wealth one acorn may dispense, By fleets to sail a thousand ages hence.' The general principle of inference, that what we know of one case must be true of similar cases, if they really are identical in the essential conditions, prevents our asserting anything which we cannot apply time after time xmder the sfune circumstances. On this principle Stevinus beautifully demonstrated that weights resting on two inclined planes and balancing each other must be proportional to the lengths of the planes between their apex and a horizontal plane. He imagined an uniform » ' Fhilosopliical Transactions' (1740), vol. sli. p. 454. Digitized by Google GSNERALIZATION. 277 endless chain to be hung over the planes, and to hang below in a symmetrical festoon. If the chain were ever to move by gravity, there would be the same reason for its moving on for ever, and thus producing a perpetual motion. As this is absurd, the portions of the chain lying on the planes, and eqiial in length to the planes, must balance each other. On similar grounds we may disprove the existence of any self-moving maddne, for if it could once alter its own state of motion or rest, in how- ever small a degree, there is no reason why it should not do the like time after time ad infinitum. Even Newton's proof of hia third law of motion, in the case of gravity, is of this character. For he remarks that if two gravitating bodies do not exert exactly equal forces in opposite direc- tions, the one exerting the strongest pull will carry both itself and the other away, and will move with constantly increasing velocity ad infinitum. But though the argu- ment might seem sufficiently convincing, Newton in his characteristic way made an experiment with a loadstone and iron floated upon the surface of water^. In recent years the very foundation of the principle of conservation of energy has been placed on the assumption that it is impossible by any combination whatever of natural bodies to produce force continually trom nothingtt. The principle admits of frequent application in various subtle forms. Lucretius attempted to prove, by a most ingenious argu- ment of this kind, that matter must be indestructible. For if a definite quantity, however small, were to fall out of existence in any finite time, an equal quantity might be supposed to lapse in every equal interval of time, so that in the infinity of past time the universe must have ceased to exisf*. But the argument, however ingenious, ' ' Principia,' bk. I. Law iii. Corollary 6. s Helmholtz, Taylor's ' Scteutific Memoirs' (1853), vol. vi. p. 118. )■ ' Lucretius,' bk. L linca 231-264. by Google 278 THE FSINCIPLES OF SCIENCE. seems to fail at several points. If past time be infinite, why may not matter have been created infinite also ? It woald be most reasonable, again, to suppose the matter destroyed in any time to be proportional to the matter then remainiDg, and not to the original quantity ; under this hypothesis even a finite quantity of original matter could never wholly disappear from the universe. For like reasons we cannot hold that the doctrine of the Conserva- tion of Energy is really proved, or can ever be proved to be absolutely true, however probable it may be regarded on many grounds. Tendency to Hasty Generalization. In spite of all the powers and advantages of generali- zation, men require no incitement to generalize ; they are too apt to draw hasty and ill-considered inferences. As Francis Bacon said, our intellects want not wings, but rather weights of lead to moderate their course*. The process is inevitable to the human mind ; it begins with childhood and lasts through the second childhood. The child that has once been hurt fears the like result on all similar occasions, and can with difficulty be made to dis- tinguish between case and case. It is caution and dis- crimination in the adoption of general conclusions that we chiefly have to learn, and the whole experience of life is one continued lesson to this effect. Baden Powell has excellently described this strong natural propensity to hasty inference, and the fondness of the human mind for tracing resemblances real or fanciful. 'Our first induc- tions,' he says'', 'are always imperfect and inconclusive ; we advance towards real evidence by successive approxi- mations : and accordingly we find false generalization the ' 'NoTuin Organum,' bk. I. Aphorietn 104. ^ ' The Unity of Worlds and of Nature,' and edit. p. 16. by Google GENERALIZATION. 279 besetting error of most first attempts at scientiGc research. The faculty to generalize accurately and philosophically requires large caution and long trainiDg ; and lb not fully attained, especially in reference to more general views, even by some who may properly claim the title of very accurate scientific observers in a more limited field. It is an intellectual habit which acquires inmiense and accumulating force firom the contemplation of wider analogies/ Hasty and superficial generalizations have alwayB been the bane of science, and there would be no difficulty in finding endless illustrations. Between things which are the same in number there is a certain resemblance, namely in number, but in the infeuicy of science men could not be persuaded that there was not a deeper resemblance im- plied in that of number. Pythagoras was not the inventor of a mystical science of number. In the ancient Orimtal rehgions the seven metals were connected with the seven planetSj and in the seven days of the week we still have, and probably always shall have, a relic of the septiform system ascribed by Dio Caasius to the ancient Egyptians. The disciples of Pythagoras carried the doctrine of the number seven into great detail. Seven days are men- tioned in Genesis ; infents acquire their teeth at the end of seven months ; they change them at the end of seven years; seven feet was the Hmit of man's height; every seventh year was a climacteric or critical year, at which a change of disposition took place. Then again there were the seven sages of Greece, the seven wonders of the world, the seven rites of the Grecian games, the seven gates of Thebes, and the seven generals destined to conquer that city. In natural science there were not only the seven planets, and the seven metals, but also the seven primi- tive colours, and the seven tones of munic. So deep a by Google 280 TUE PRINCIPLES OF SCIENCE. hold did this doctrine take that we still have its results in many customs, not only in the seven days of the week, but the seven years' apprenticeship, puberty at fourteen years, the second climacteric, and legal majority at twenty- one yeaxa, the third climacteric. The system was repro- duced in the seven sacraments of the Roman Catholic Church, and the seven year periods of Comte's grotesque system of domestic worship. Even in scientific matters the loftiest intellects have occasionally yielded, as when Newton was misled by the analogy between the seven tones of music and the seven colours of his spectrum. Other numerical analogies, though rejected by Galileo, held Kepler in thraldom ; no small part of Kepler's labours diuing seventeen years was spent upon nu- merical and geometrical analogies of tbe most baseless character; and he gravely held that there could not be more than six planets, because there were not more than five regular solids. Even the acute genius of Huyghens did not prevent him from inferring that but one satellite could belong to Saturn, because, with those of Jupiter and the Earth, it completed the perfect number of six. A whole series of other superstitions and fallacies attach to the numbers six and nine'. It is by false generalization, again, that the laws of nature have been supposed always to possess that sim- plicity and perfection which we attribute to particular forms and relations. 1^6 heavenly bodies, it was held, must move in circles, for the circle was the perfect figure. Even Newton seemed to adopt the questionable axiom that nature must always proceed in the simplest way ; in stating his first rule of philosophizing, he adds™ : ' To this purpose the philosophers say, that nature does nothing in 1 Baring-Gould, ' On the Fatalities of Number,' iu ' Curious Mjilie of the Middle Ages' {i866), p. 209. * ' Principia,' bk. Ill, ad initium. by Google GENERALIZATION. 281 vain, when less will serve ; for Nature is pleased with BimpUcity, mid affects not the pomp of superfluous causes.' Keill, again, lays down" as an axiom that 'The causes of natural things are such, as are the most simple, and are sufficient to explain the phenomena ; for nature always proceeds in the simplest and most expeditioua method ; because by this manner of operating the Divine Wisdom displays itself the more.' If this axiom had any clear grounds of truth, it would not apply to proximate laws; for even when the ultimate law may appear simple the results may be infinitely diverse, as in the various elliptic, hyperbolic, parabolic, or circular orbits of the heavenly bodies. Simplicity is naturally agreeable to a mind of ■very finite powers, but to an Infinite Mind everything is simple. Every great advance in science consists in a great gene- ralization, pointing out deep and subtle resemblances. Tlie Copemican system was a generalization, in that it classed the earth among the planets ; it was, aa Bishop Wilkins expressed it, ' the discovery of a new planet,' but it was opposed by a more shallow generalization. Those who argued from the condition of things upon the earth's surface, thought that every object must be attached to and rest upon something else. Shall the earth, they said, atone be free ^ Accustomed to certain special results of gravity they could not conceive ita action under widely different circumstances". No hasty thinker coxdd seize the deep analogy pointed out by Horrocks between a pen- dulum and a planet, true in substance though mistaken in some details. All the advances of modern science rise from the conception of Galileo, that in the heavenly bodies, however apparently different their condition, we n Keill, ' IntroductioQ to Natural Philosophy,' p. 89. " .reiPiiiia; Hoiroccii ' Opera Postliuma' {1673), pp. 26, 17. by Google 282 THE PRINCIPLES OF SCIENCE. shall ultimately recognise the same fundamental principles of mechanical science which are true on earth. Generalization is the great prerogative of the intellect, but it is a power only to be exercised safely with much caution and after long training. Every mind must gene- ralize, but there are the widest differences in the depth of the resemblances discovered and the care with which the discovery is verified. There seems to be an innate power of insight which a few men have possessed pre-eminently, and which enabled them, with no exemption indeed from labour or temporary error, to discover the one in the many. Minds of excessive acuteness may exist, which have yet only the powers of minute discrimination, and of storing up, in the treasure-house of memory, vast accumu- lations of words and incidents. But the power of dis- covery belongs to a more restricted class of minda Xor place said that, of all inventors who had contributed the most to the advancement of human knowledge, Newton and Lagrange appeared to possess in the highest d^ee the happy tact of distinguishing general principles among a multitude of objects enveloping them, and this tact he conceived to be the true characteristic of scientific genius **. o YouDg'a ' Works,' voL ii. p. 564, by Google CHAPTER XXVIII. ANALOGY. As we have seen in the previous chapter, generaliza- tion passee insensibly into reasoDing hy analogy, and the difference is but one of degree. We are said to generalize when we view many objects as agreeing in a few pro- perties, 80 that the resemblance is extensive rather than deep. When we have two or only a few objects of thought, but are able to discover many points of resem- blance, we argue by analogy that the correspondence will be even deeper than appears. It may not be true that the words are always used in these distinct senses, and there is no doubt great vagueness in the employment of these and many other logical terms ; but, if there is any clear discrimination to be drawn between generaliza- tion and analogy, it is indicated above. It has been often said, indeed, that analogy denotes not a resemblance between things, but between tiie relations of things. A pilot is a very different man from a Prime Minister, but he bears the same relation to a ship that the minister does to the state, so that we may analogi- cally describe the Prime Minister as the pilot of the state. A man diiFers still more from a horse, nevertheless four men bear to three men the same relation as four horses bear io three horses ; there is the analogy. Four men : Three men : : Four horses : Three horses, or Four men : Four horses : : Three men : Three horses. There is a real analogy between the tones of the Mono- Digitized by Google 284 THE PRINCIPLES OF SCIENCE. chord, the Sages of Thebes, and the Gates of Thebes, but it does not extend beyond the fact that they were all seven in number. Between the most discrete notions, as, for instanM, those of time and space, analogy may exist, arising from the fact that the mathematical conditions of the lapse of time and of motion along a line are similar. There is no identity of nature between a word and the thing it signifies ; the substance iron is a heavy solid, the word iron is either a momentary disturbance of the air, or a film of black pigment on white paper ; but there is analogy between words and their significates. The substance iron is to the substance iron-carbonate, as tbe name iron is to the name iron-carbonate, when these names are used according to their correct scientific definitions. The whole structure of language and the whole utility of signs, marks, symbols, pictures, and representations of various kinds, rest upon analogy. I may, perhaps, hope to enter fully upon this important subject at some future time, and to attempt to show how the invention of signs enables us to express, guide, and register our thoughts. It will be sufficient to observe here that tbe use of words constantly involves analogies of a subtie kind ; we should often be at a loss how to describe a notion, were we not at liberty to employ in a metaphorical sense the name of anything sufficiently resembling it. There would be no expression for the sweetness of a melody, or the brilliance of an harangue, unless it were furnished by the taste of honey and the brightness of a torch. A very cursory examination of the cases in which we popularly use the word analogy, shows that it includes all degrees of resemblance or similarity. The analogy may consist only in similarity of number or ratio ; or in like relations of timo or space. It may also consist in more simple resemblance between physical properties. We should not be using the word inconsistently with custom. by Google if we said that there was an analogy between iron, nickel, and cobalt, manifested in the strength of their magnetic powers. There is a still more perfect analogy between iodine and chlorine ; not that every property of iodine is identical with the corresponding property of chlorine ; for then they would be one and the same kind of sub- stance, and not two substances ; but every property Of iodine resembles in all but d^ee some property of chlo- rine. For almost every substance in which iodine forma a component, a corresponding substance may be dis- covered containing chlorine, so that we may confidently infer from the compounds of the one to the compounds of the other substance. Potassium iodide crystallizes in cubes ; therefore it is to be expected that potassium chlo- ride will also crystallize in cubes. The science of chemistry, as now developed, rests almost entirely upon a careful and most extensive comparison of the properties of substances, bringing to light deep-lying analogies. When any ap- parently exceptional or new substance is encountered, the chemist is guided in his treatment of it entirely by the analogies which it seems to present with previously known In this chapter I cannot hope to illustrate the all- pervading influence of analogy in human thought and science. All science, it has been said, at the outset, arises from the discovery of identity, and analogy is but one name by which we denote the deeper-lying cases of re- semblance. I shall only try to point out at present how analogy between apparently diverse classes of phenomena often serves as an all-important guide in discovery. We thus conamonly gain the first insight into the nature of an apparently unique object, and we thus, in the progress of a science, often discover that we are treating over again, in a new form, phenomena which were well known to us in another form. by Google THE PRINCIPLES OF SCIENCE. Analogy as a Guide in Discovery. There can be no doubt that discovery is most frequently accomplished by following up hints received &om analogy, as Jeremy Bentham remarked'. Whenever a phenomenon is perceived, the first impulse of the mind is to connect it with the most nearly similar phenomenon. If we could ever meet a thing wholly sui generis, presenting no analog to anything else, we should be incapable of investigating its nature, except by purely haphazard trial. The probability of success by such a process is so slight, that it is preferable to follow up the slightest clue. As I have pointed out already (vol. ii. p. 24), the possible modifications of condition in experiments are usually in- finite in number, and intinitely numerous also are the hypotheses upon which we may proceed. Now it is self- evident that, however slightly superior the probability of success by one course of procedure may be over another, the most probable one should always be adopted first. The chemist having discovered what he believes to be a new element, will have an infinite variety of modes of treating and investigating it. If in any one of its qualities the substance displays a resemblance to an alkaline metal, for instance, he will naturally proceed to try whether it possesses other properties common to the alkaline metals. Even the apparently simplest phenomenon presents so many points for notice that we have a choice at each moment from among many hypotheses. It would be difScolt to find a more instructive instance of the way in which the mind is guided by analogy than in the description by Sir John Herechel of the course of thought by which he was led to anticipate in theory one of Faraday's greatest experimental discoveries. Sir John ■ ' EBBay on Logic,' ' Works,' vol. viii, p. 176. by Google Herschel noticed that in three physical phenomeiia, a screw-like form, technically called helicoidal die^Tnmetry, was presented, namely in electrical helices, plagihedral quartz crystals, and the rotation of the j)lane of polariza- tion of light. As he himself has said *>, ' I reasoned thus : Here are three phenomena agreeing in a very grange pecu- liarity. Probably, this peculiarity is a connecting link, physically speaking, among them. Now, in the case of the crystals and the light, this probability has been turned into certainty by my own experiments. Therefore, induc- tion led me to conclude that a similar connexion exists, and must turn ap, somehow or other, between the electric current and polarized light, and that the plane of polariza- tion would be deflected by magneto-electricity.' By this course of analogical thought Sir John Herschel had actu- ally been led to anticipate Faraday's great discovery of the influence of magnetic strain upon polarized light He had ti'ied as long ago as 1822-25 ^ discover the influence of electricity on light, by sending a ray of polarized light through a helix, or near a long wire conveying an electric current. Such a course of inquiry, followed up with the persistency of Faraday, and with his experimental re- sources, would doubtless have effected the strange dis- covery. Herschel also suggests that the plagihedral form of quartz crystals must be due to a screw-like strain during the progress of crystallization ; but the notion, although probable) remjuns unverified by experiment to the present day. If ever men approach the investigation of the me- chanism of thoizght, they must be guided by analogy. Already many philosophers have drawn analogies between nerve influence and the transmission of vibrations. Dr. Briggs, Newton in his 24th Query, and Hartley, have vaguely specidated concerning such vibrations, and some ^ ' Life of Faraday,' by Benc« Jones, vol. ii. p. 206. Digitized by Google 288 THE PRINCIPLES OF SCIENCE. countenaDce is now given to the notion by the aomewhat similar rate of propagation of nerve pulses and sound- waves in soft bodies. But the phenomena of memory are far more difficult to reduce to any material mechanism, and I know of no material analogy but the interesting one suggested by Hooke'', who likens memory to ' those bells or vases which Vitruvius mentions to be placed in the ancient theatre, which did receive and return the soimd more vigorous and strong; or like the unison- toned strings, bells, or glass^, which receive impressions from sounds without, and retain the impressions for some time, answering the tone by the same tone of their own.' Analogy in the Mathematical Sciences. Whoever wishes to acquire a deep acquaintance with the constitution of Nature must observe that there are deep analogies which connect whole branches of science in a parallel manner, and enable ua to infer of one class of phenomena what we know of the other. It has thus happened on several occasions that the discovery of an unsuspected analogy between two hitherto distinct branches of knowledge has been the starting-point for a rapid course of discovery. The truths readily observed in the one may be of a different character from those which present themselves in the other. The analogy, when once pointed out, leads us easily to discover regions of one science yet undeveloped, but to which the key is furnished by the corresponding truths in the other science. An interchange of aid most wonderful in its results may thus take place, and at the same time the mind rises to a higher generalization, and a more com- prehensive view of mind and nature. 'Posthumous Works,' p. 141. Digitized by Google No two scieDces might seem at 6rst sight more entirely discrete and divergent in their subject matter than geometry and arithmetic, or algebra. The first deals with circles, squares, parallelograuns, and various other forms in space ; the latter with mere symbols of number, the symbols having form indeed, but bearing a meaning independent of shape or size. Prior to the time of Des- cartes, too, the sciences actually were developed in a slow and painful manner in almost entire independence of each other. The Greek philosophers indeed could not avoid noticing occasional analogies, as when Flato in the Thseetetus describes a square number as equally equal, and a number produced by multiplying two unequal factors as oblong. Euclid, in the 7th and 8th books of his Elements, continually usee expressions displaying a consciousness of the same analogies, as when he calls a number of two factors a plane number, e-n-iVeiof apt6/*ot, and distinguishes a square number of which the two factors are equal as an equal-sided or plane number, t(TOT\evpoi Ka'i iviireSos aptOftot. He also calls the root of a cubic number its side, -rXtvpa, In the Diophantine algebra many problems of a geometrical character were solved by algebraic or numerical processes ; but there was no general system, so that the solutions were of an isolated character. In general the ancients were far more advanced in geometric than symbolic methods ; thus Euclid in his 4th book gives us the means of dividing a circle by purely geometric or mechanical means into 2, 3. 4. 5. 6, 8, 10, 12, 15, 20, 24, 30 parts, but he was totally unacquainted with the theory of the roots of unity exactly corresponding to this division of the circle. During the middle ages, on the other hand, algebra ad- vanced beyond geometry, and modes of solving equations were painfully discovered by those who had no notion that at every step they were implicitly solving important VOL. II. u Digitized by Google 2ft0 THE PRINCIPLES OF SCIENCE. geometric problems. It is true that Begiomontanus, Tar- taglia, Bombeili, and possibly other early algebraists, solved isolated geometrical problems by the aid of algebra, but particular numbers were always used, and no consciousueBS of a general method was displayed. Vieta in some degree anticipated the final discovery, and occasionally represented the roots of an equation geometrically, but it was reserved for Descartes to show, in the most general manner, that eveiy equation may be represented by some curve or figure in space, and that every bend, point, cusp, or other peculiarity in the curve indicates some peculiarity in the values of the algebraic symbola It is impossible to describe in any adequate manner the importance of this discovery. The advantage was twofold : algebra aided geometry, and geometry gave reciprocal aid to algebra. Curves such as tiie long described sections of the cone were found to correspond to quadratic equations of no great difficulty ; and it was impossible to manipulate the symbolic equa- tions without discovering properties of those all important curves. The way was thus opened for the algebraic treat- ment of motions and forces, vrithout which Newton's ' Principia ' could never have been worked out. Newton indeed was possessed by a strange and, to some extent, unfortunate infatuation in favour of the ancient geome- trical methods; but it is well known that he employed symbolic methods to discover his profound truths, and he every now and then, by some accidentia use of algebraic expressions, confessed its greater powers and generahty. Geometiy, on the other hand, gave the greatest assist- ance to algebra, by a£fording concrete representations of relations which would otherwise be too abstract for easy comprehension, A curve of no great complexity may give the whole history of the variations of value of a troublesome mathematical expression. As soon as we know, too, that every regular geometrical curve repre- Digitized by Google sents some algebraic equation, we are presented by simple observation of many mechanical movements with abun- dant suggestions towards the discovery of mathematical problems. Every particle of a carriage-wheel when mov- ing on a level road, is constantly describing a cydoidal curve, the curious properties of which exercised the in- genuity of all the most sHlful mathematiciaos of the seventeenth century, and led to important advancements in algebraic power. It may well be held even that the discovery of the Differential Calculus is mainly due to geometrical analogy, because mathematicians, in attempt- ing to treat algebraically the tangent of a continuously varying curve, were obliged to entertain the notion of infinitely small quantities*'. There can be no doubt that Newton's fluxional, or in fact geometrical mode of stating the differential calculus, however much it sub- sequently retarded its progress in England, facilitated its apprehension at first, and I should think it almost certain that Newton discovered the calculus geometrically. We may accordingly look upon this discovery of analogy, this happy alliance, as Bossut calls it^, between geometry and algebra, as the chief source of discoveries which have been made for three centuries past in mathe- matical methods. This is certainly the opinion of no lees an authority than Lagrange, who has said, ' So long as algebra and geometry have been separate, their progress was slow, and their employment limited ; but since these two sciences have been united, they have lent each other mutual strength, and have marched together with a rapid step towards perfection.' The advancement of mechanical science has also been greatly aided by analogy. An abstract and intangible d locroiz, ' Traits BHmeDtaire de Calcul Diff^restwl et de Calcul Integral,' 5 ™ &lit p. 699. " ' Histuire deB Math^mfttiqacs,' vol. i. p. igS. U 2 Digitized by Google 293 THE PRINCIPLES OF SCIENCE. existence like force demands much power of conception, but it has a perfect concrete representative in a line, the end of which may denote the point of application, and the direction the line of action of the force, while the length can be made arbitrarily to denote the amount of the force. Nor does the analogy end here; for the moment of the force about any point, or its product into the perpen- dicular distance of its line of action from the point, is found to be correctly represented by an area, namely twice the area of the triangle contained between the point and the ends of the line representing the force. Of late years a great generalization has been effected ; the Double Algebra of De Morgan is true not only of space relations, but of forces, so that the triangle of forces is reduced te a case of pure geometrical addition. Nay, the triangle of lines, the triangle of velocitif®, the triangle of forces, the triangle of couples, and perhaps other cognate theorems, are reduced by analogy to one ample theorem, which amounts merely to this, that there are two ways of getting from one angular point of a triangle to another, which ways, though different in length, are identical in their final results '. In the wonderful system of quaternions of the late Sir W. R. Hamilton, these analogies are embodied and carried out in the most genetul manner, so that whatever problem involves the threefold dimensions of space, or relations analogous to those of space, is treated by a symbolic method of the most comprehensive simplicity. Since nearly all physical problems do involve space relations, or those analogous to them, it is difficult to imagine any limits to the work which may be ultimately achieved by this calculus. It ought to be added that to the discovery of analogy f See Goodwin, ' Cambridge Philoeophical Transactione ' (1845), vol. viii. p. 269. O'Brien, ' On Symbolical StaticB,' Pliilosuphical Magazine, (th Series, vol. i. pp. 491 &c. by Google between the forms of mathematical and logical expressionB, we undoubtedly owe the greatest recent advance in logical science. Boole based his exteneion of logical proceBses entirely upon tbe notion that logic was an algebra of two quantities, o and i. His profound genius for the investi- gation of symbolic methods led him to perceive by analogy that there must exist a general system of logical deduc- tion, of which the old logicians had seized only a few stray fragments. Much mistaken as be was in placing algebra as a higher science than logic, no one can deny that the development of the m,ore complex and dependent science had advanced far beyond that of the simpler science, and that Boole, in drawing attention to the connexion, made one of the most important discoveries in tbe history of science. As Descartes had wedded algebra and geometry, so did Boole substantially accomplish the marriage of logic and algebra. Analogy in the Theory of Undulations. There is no class of phenomena which more thoroughly illustrates alike the power and weakness of analogy than the waves which a^tate every kind of medium. All waves, whatsoever be the matter through which they pass, obey certain common principlesof rhythmical orharmonic motion, and the subject therefore presents a vast field for mathema- tical generalization. At the same time each kind of medium may allow of waves pecidiar in their conditions, so that it is a beautiful exercise in analogical reasoning to observe how, in making inferences from one kind of medium to another, we must make allowance for difference of circum- stances. Tbe waves of the ocean are large and visible, and there are the yet greater tidal waves which extend around the globe. From such palpable cases of rhythmical by Google 294 THE PRINCIPLES OF SCIENCE. movement we pass by analogy to waves of sound, varying in length from about 32 feet to a Bmall fraction of an inch. We have but to imagine, if we can, the fortieth octave of the middle C of a piano, and we reach the undulations of yellow light, the ultra-violet being about the forty-first octave. Thus we pass gradually from the palpable and evi- dent to that which is obscure, if not incomprehensible. Yet the very same phenomena of reflection, iaterferenee, and refraction, which we find in the one case, may be expected to occur mutatis mutandis in the other cases. From the great to the small, from the evident to the obscure, ia not only the natural order in which inference proceeds, but it is the historical order of discovery. The physical science of the Greek philosophers must have re- mained incomplete, and their theories groundless, because they do not seem ever to have understood the nature and importance of undulations. All their systems were there- fore based upon the entirely different notion of continuous movement of translation from place to place. Modem Science tends more and more to the opposite conclusion that all motion is alternating or rhythmical, energy flowing onwards but matter remaining comparatively fixed in position. Diogenes Laertius indeed correctly compared the propagation of sound with the spreading of waves on the surface of water when disturbed by a stone, and Vitruvius displayed a more complete comprehension of the same analogy. It remained for Newton to create the theory of undulatory motion in showing by mathe- matical deductive reasoning that the particles of an elastic fluid, by vibrating backwards and forwards, might carry forward a pulse or wave moving onwards from the source of disturbance, while the disturbed particles return to their place of rest He was even able to make a first approxi- mation by theoretical calculation to the velocity of sound- waves in the atmosphere. His theory of sound formed a by Google hardly lees important epoch in science than hia far more celebrated theory of gravitation. It opened the way to all the subsequent applications of mechanical principles to the inseuEdble motion of molecules. He seemed to have been frequently, too, upon the brink of another appli- cation of the same principles which would have advanced science by at least a centuiy of progress, and made him the undisputed founder of all the theories of matter. He expressed opinions at various times that light might be due to uudulatory movements of a medium occupying space, and in one intensely interesting sentence remarks^ that colours are probably vibrations of different lengths, ' much after the manner that, in the sense of hearing, nature makes use of aenal vibrations of several big- nesses to generate sounds of divers tones, for the analogy of nature is to be observed '. He correctly foresaw that red and yellow hght would consist of the longer undula- tions, and blue and violet of the shorter, while white light would be composed of an indiscriminate mixture of waves of various lengths. Newton almost overcame one of the strongest apparent difBculties of the undulatory theory of light, namely, the propagation of light in straight lines. For he observed that though waves of sound bend round an obstacle to some extent, they do not do so in the same degree as water-waves'*. He had but to extend the ana- logy proportionally to light-waves, and not only would the difficulty have vanished, but the true theory of dif- fraction woiJd have been open to him. Unfortunately he had a preconceived theory that rays of hght are bent &om and not towards the shadow of a body, a theory which for once he did not sufficiently compare with ob- servation to detect its falsity. I am not aware, too, that « Bircb, ' History of the Boyal Society," vol. iii. j). a6i, quoted by Young, ' Works,' toI. i. p. 146. *> 'Opticka,' Query 28, srti edit. p. 33 J. by Google 296 THE PRINCIPLES OF SCIENCE. Newton has, in any of his works, displayed an under- standing of the phenomena of interference inseparable from the notion of waves. While the general principles of undulatory or harmonic motion will be the same in whatever medium the motion takes place, the circumstances must often be excessively different. Between light travelling 186,000 miles per second and sound travelling in air only about 1 100 feet in the same unit of time, or almost 900,000 times as slowly, we cannot expect a close outward resemblance. There are great differences, too, in the character of the vibrations. Gases scarcely admit of transverse vibration, so that sound travelling in air is a longitudinal wave, the particles of sir moving backwards and forwards in the same line in which the wave moves onwards. Light, on the other hand, appears to .consist entirely in the movement of points of force transversely to the direction of propaga- tion of the ray. The light-wave is partially analogous to the bending of a rod or of a stretched cord agitated at one end. Now this bending motion may take place in any one of an infinite number of planes, and waves of which the planes are perpendicular to each other cannot interfere any more than two perpendicular forces can interfere. Now the whole of the complicated phenomena of polar- ized light arise out of this transverse character of the luminous wave, and we must not expect to meet any analogous phenomena in atmospheric sound-waves. It is conceivable that in solids we might produce transverse sound undulations, in which many of the phenomena of polarization might be reproduced. But it would appear that even between transverse sound and light-waves the anal(^ holds true rather of the principles of harmonic motion than the circumstances of the vibrating medium ; from experiment and theory it is inferred that the plane of polarization in plane polarized light is perpendicular to by Google instead of being coincident with the direction of vibration, as it would be in the case of transverse sound undulations. Thus the laws of elastic forces appear to be essentially different in application to the luminiferous ether and to ordinary solid bodies'. Between light and heat, forms of energy, which at first sight appear so different, a perfect analogy has gradually been established. Not only do rays of hght and heat obey exactly the same laws of reflection and refraction, but they are subject to exactly the same laws of absorp- tion and polarization. Wherever a light-ray is deficient in the solar spectrum, a heat-ray is also missing. It is now considered that light is but the influence of heat-rays of certain wave-lengths upon the eye, so that we may in fact cease to distinguish radiant heat and rays of light. Heat in the radiant condition is, of course, to be distin- guished from the molecular vibration also called heat, and from the potential energy which it produces when absorbed by substances, and rendered latent. Use of Analogy in Agronomy. We shall be much assisted in gaining a true apprecia- tion of the value of analogy in its feebler degrees, by con- sidering how much it has contributed to the progress of astronomical science. Our point of obeervabion is so fixed with regard to the universe, and our means of examining distant bodies is so restricted, that we are obliged in many cases to be guided by limited and apparently feeble resemblances. In many cases the result has been con- firmed by subsequent direct evidence of the most forcible character. While the scientific worid was divided in opinion ' Bunkine, ' Philosophical Traosactions' (1856}, vol. cxlvi. p. 382. Digitized by Google 298 TUB PRINCIPLES OF SCIENCE. between the Copernican, and Ptolemaic Byetems, it was analogy which furnished the most satis&ctory arguments. Galileo discovered, by the use of his new telescope, the four small satellites which circulate round Jupiter, and make a miniature planetary world. These four Medi- cean Stare, as they were called, were plainly seen to re- volve round Jupiter in various periods, but approximately in one plane, and astronomers inesiBtibly inferred that what might happen on the smaller scale might also be found true of the greater planetary system. This dis- covery gave the holding turn, as Sir John Herschel has expressed it, to the opinions of mankind. Even Francis Bacon, who bad, in a manner little to the credit of his scientific eagacity, previously opposed ^e Copernican views, now became partially convinced, saying ' We affirm the Bolisequium of Venus and Mercury ; since it has been found hj Galileo that Jupiter also has attendants.' Nor did Huyghens think it superfluous to adopt the analogy as a valid argument''. Even in an advanced stage of the science of physical astronomy, the Jovian system has not lost its analo^cal interest ; for the mutual perturbations of the four satellites pass through all their phases within a few centuries, and thus enable us to verify in a miniar ture case the principles of stability, which Laplace has established for the great planetary system. Oscillations or disturbances which in the motions of the planets appear to be secular, because their periods extend over millions of years, can be watched, in the case of Jupiter's satellites, through complete revolutions within the historical periods of astronomy!. In obtEuning a knowledge of the stellar universe we mXist depend much upon somewhat precarious analogies. We must start with the opinion, entertained by Bruno as ^ ' CosmotlieoroB' (1699), p. 16. ' Lftplace, 'System of tbe World,' vol. ii. p. 316. by Google long ugo aa 1591, that the stars may be buds attended perhaps by planets like our earth. This is the most probable first assumption, eupported in some degree by very recent spectrum observations, which show the simi- larity of light derived from many stars with that of the sun. But at the same time we learn by the prism that there are nebulse and stars in coDditions widely different from an^'thing known in our system. In the course of time the analogy may perhaps be restored to comparative completeness by the discovery of many suns in various stages of nebulous condensation. The history of the evo- lution of our own world may, as it were, be traced back in bodies less developed, or traced forwards in systems more advanced towards the dissipation of energy, and the extinction of life. As in a great workshop, we may per- haps see the material work of Creation as it has variously progressed through tbouaands of millions of years. By the careful delineation ^id classification of the nebulfe and stellar systems, we may hope in time to find some parallel even to that apparently space-filling system of the Milky Way. Michell pointed out that the Pleiades form a remarkable group of worlds, and he thought that it might present an analogy to the sun and its immediate neighbours. The observations of the Herschels and other more recent astronomers, show that we really belong to a vast stratum of worlds of a peculiar split form, in- cluding countless myriads of stars of various sizes. The beUef in analogy is irresistible, and astrouomers have already looked into the depths of space, hoping to find distant nebulous specks which might resemble the sup- posed form of the Milky Wiiy, and extend our know- ledge to a higher order of universes. Such expectations are probably premature, or even unfoimded ; neverthe- less in the forms of the nebulse we may find much in- struction. The spiral form disclosed in many bodies Digitized by Google 300 THE PRINCIPLES OF SCIENCE. by Lord Rosse's telescope possesses some analogy to what would happen in a system revolving in a dense retard- ing medium. Let us once ascert^n by the spectroscope that there is a dense envelope of gas, and the forms of those bodies are at once bronght into harmony with the laws of matter on this globe. Viewing such worlds aa we do from a fixed distant point, they appear variously distorted according to the laws of perspective ; but when we find in many objects forms which might have pro- ceeded from the same object variously inclined to the line of vision, analogy will aid us in determining the real form. Thus when we see an apparent nebulous ring, we may be unable to decide whether it is really a ring of matter or a spherical shell, of which the ob- liquely seen edges are alone apparent. But if elsewhere we discover, as did Lord Rosse, another nebula present- ing the distinct appearance of a ring seen edgeways, we may infer with some probability from one case to the other. By similar processes of comparison and analogical reasoning, we may in time assign with much confidence the absolute forms of many classes of celestial objects'". In speculations concerning the physical condition of other planets and heavenly bodies, we must often depend upon analogies of a very weak character. We may be said to know that the moon has mountains and valleys, plains and ridges, volcanoes, and streams of lava, and, in spite of the absence of air and water, the rocky sur- face of the moon presents so many familiar appearances that we do not hesitate to compare them with the features of our own globe. We infer with high probability that Mars has polar snow and an atmosphere absorbing blue rays like our own ; Jupiter undoubtedly possesses a cloudy atmosphere, possibly not unlike a magnified copy of that surrounding the earth, but our tendency .to adopt an- ■" Grant's 'History of Physical Astronomy,' pp. 570-371. by Google alogiea receives a salutary correction in the recently dis- covered fact that the atmosphere of Uranxis contains hydrogen. Philosophers of the highest grade have not stopped at these comparatively safe inferences, but have speculated on the existence of living creatures in other planeta. Huyghens remarked that as we infer by analogy from the dissected body of a dog to that of a pig and ox or other animal of the same general form, and as we expect to find the same viscera, the heart, stomach, lungs, intestines, Sec, in corresponding positions, so when we notice the similarity of ike planets in many respects, we must expect to find them alike in other points". He even enters into an inquiry whether the inhabitants of other planets would possess reason and knowledge of the same sort as ours, concluding in the affirmative. Although the power of intellect might be different, he considers that they would have the same geometry if they had any at all,' and that what is true with us would be true with themo. As regards the sun, he wisely observes that every conjecture fails. Laplace entertained a strong belief iu the existence of inhabitants on other planets. The benign influence of the sun gives birth to animals and plants upon the surface of the earth, and analogy induces us to believe that his rays would tend to have a similar effect elsewhere. It is not probable that matter which xs here so fruitful of life, would be sterile upon bo great a globe as Jupiter, which, like the earth, has its days and nights and years, and changes which indicate active forces. Man indeed is formed for the temperature and atmosphere in which he lives, and, so far as appears, could not live upon the other planets. But there might be an infinity of organizations relative to the diverse constitution of the bodies of the universe. The most active imagination can- n 'Cosmotheoros' (169(1], p. '7- " Ibid p. 36. by Google 302 TBS PlilNCIPLES OF SCIENCE. not form any idea of such various creatures, but their existence is not unlikelyP. We now know that many metals and other elements never found in organic structures are yet capable of form- ing compoxmds, with substances of vegetable or animal origin. It ia therefore just possible that at different tem- peratures creatures formed of different but analogous com- pounds might exist, but it would seem indispensable that carbon should still form the basis of organic structures ; for we have no analogies to lead us to suppose that in the absence of that complex element, life can exist. Could we find globes surrounded by atmospheres resembUng our own in temperature and composition, we should be almost forced to believe them inhabited, but the probability of any analogical argument decreases rapidly as the condi- tion of a globe diverges from that of our own. The Cardi- nal Nicholas de Cusa held long ago that the moon was inhabited, but the absence of any appreciable atmosphere renders the existence of inhabitants highly improbable. Speculations resting upon weak analogies hardly belong to the scope of true science, and can only be tolerated as an antidote to the far worse dogmatism which would assert that the thousand million of persons on earth, or rather a small fraction of them, are the sole objects of care of the Power which designed this limitless Universe. Failures of Analogy. So constant is the aid which we derive from the use of analogy in all attempts at discovery or explanation, that it is most important to observe in what cases it may lead us into difficulties. That which we expect by analogy to exist may — • V ' System of the World,' vol. iL p. 336. ' Ebbsi PhiloBopbiqne,' p. 87. Digitized by Google ANALOGY. 3tf3 (i) Be found to exist ; (2) May seem not to exist, but nevertheless may really exist ; (3) May actually be non-existent. In the second case the failure is only apparent, and arises from our obtuseness of perception, the smallness of the phenomenon to be noticed, or the disguised cha- racter in which it appears. I have already pointed out that the analogy of sound and light seems to fail because light does not bend round a comer, the £ict being that it does so bend in the phenomena of difiraction, which present the effect, however, in such an unexpected and minute form, that even Newton was misled, and turned from the correct hypothesis of imdulations which he had partially entertwned. In tJie third class of cases analogy fails us altogether, and we expect that to exist which really does not exist. Thus we fail to discover the phenomena of polarization in sound travelling through the atmosphere, since air is not capable of any appreciable transverse undtdations. These failures of analogy are of peculiar interest, because they make the mind aware of its superior powers. There have been many philosophers who said that we can conceive nothing in the intellect which we have not previously received through the senses: This is true in the sense that we cannot image them to the mind in the concrete form of a shape or a colour ; but we can speak of them and reason concerning them ; in short, we often know them in everything but a sensuous manner. Accurate investi- gation shows that all material substances retard the motion of bodies through t^em by subetracting energy by impact. By the law of continuity we can frame the notion of a vacuous space in which tha% is no resistance whatever, nor need we stop there ; for we have only to proceed by analogy to the case where a medium should by Google 304 THE PRINdPLBS OF SCIESCB. accelerate the motion of bodies passiDg through it, s(Hne- what in the mode which Aristotelians attributed falsely ■ to the air. Thus we can frame the notion of negative density, and Newton could reason exactly concerning it, although no such thing exists i. In every direction of thought we may meet ultimately with similar £iilures of analogy. A moving point gene- rates a line, a moving line generates a surface, a moving surface generates a solid, but what does a moving solid generate? When we compare a polyhedron, or many- * sided solid, with a polygon, or pltme figure of many sides, the volume of the first is aDalogous to the area o the second ; the face of the solid answers to the side of the polygon ; the edge of the solid to the point of the figure ; but the comer, or junction of edges in the polyhedron, is left wholly imrepresented in the plane of the polygon. Even if we attempted to draw the analogies in some other manner, we should still find a geometrical notion embodied in the solid which has no representative in the plain figure "■. Faraday was able to frame some notion of matter in a fourth condition, which should be to gas what gas is to liquid '. Such substance, he thought, would not Ml far short of radiant matter, by which apparently he meant the supposed caloric or matter assumed to constitute heat, according to the Corpuscular Theory. Even if we could frame the notion, matter in such a state cannot be known to exist, and recent discoveries concerning the continuity of the solid, liquid, and gaseous states remove the basis of the speculation. From these and many other instances which might be 1 ' Principia,' bk. II. Section IL Prop. X. ' De Morgui, 'Ctunbridge Fhiloaophical Tnuisactioiu,' vol. xi. Part ii. p. 346. • ' Life of Faraday,' vol. i. p. z 1 6. by Google adduced, we learn that analogical reasoning leads us to the conception of many things which, so far as we can aecertajn, do not exist. In this way great perplexities have arisen in the use of language and mathematical symbols. AH language depends upon analogy; for we join and arrange words eo that they may represent the corresponding junctions or arrangements of things and their qualities. But in the use of language we are obviously capable of forming many combinations of words to which no corresponding meaning apparently exists. The same difficulty arises in the use of mathematical signs, and mathematicians have needlessly puzzled them- selves about the square root of a negative quantity, which is, in many applications of algebraic calculation, simply a sign without any analogous meaning, there being a failure of analogy. by Google CHAPTER XXIX. EXCEPTIONAL PHENOMENA. Ip srienee consieta in the detection of identity and the recognition of one uniformity existing in many objects, it follows almost of necessity that the progress of science depends apon the study of exceptional phenomena. Such new phenomena are the raw material upon which we are to exert our faculties of observation and reasoning, in order to reduce the new facts beneath the sway of the laws of nature, either those laws already well known, or those to be discovered. Not only are strange and inex- plicable facts those which are on the whole most likely to lead us to some novel and important discovery, but they are also best fitted to arouse our attention. So long aa events happen in accordance with our anticipations, and the routine of every-day observation is unvaried, there is nothing to impress upon the mind the smallness of its knowledge, and the depth of mystery, which may be hidden in the commonest sights and objects. In early times the myriads of stars which remained in apparently fixed relative positions upon the heavenly sphere, re- ceived far less notice from astronomers than those few planets whose wandering and inexplicable motions formed an unsolved riddle. Hipparchus was induced to prepare the first catalogue of stars, because a single new star had been added to those nightly visible ; and in the middle Digitized by Google EXCEPTIONAL PHENOMENA. ages two brilliant but temporary stars caused more popular interest in astronomy than any other events, and to one of them we owe all the obaervations of Tycho Brahe, the mediaval Hipparchua In other sciences, as well as in that of the heavens, exceptional events are commonly the points from which we start to explore new regions of knowledge. It has been beautifully said that Wonder is the daughter of Ignorance, but the mother of Invention ; and though the most familiar and slight events, if ftdly examined, will afiord endless food for wonder and for wisdom, yet it is the few peculiar and unlooked-for events which most often lead a scientific mind into a course of discoverv. It is true, indeed, that it requires much philosophy to observe things which are too near to ua. The high scientific importance attaching, then, to ex- ceptions, renders it desirable that we should carefully consider the various modes in which an exception may be disposed of; while some new facts will be found to confirm the very laws to which at first sight they seem clearly opposed, others will cause us to limit the generality of our previous statements. In some cases the exception may be proved to be no exception ; occasionally it will prove fatal to our previous most confident speculations; and there are some new phenomena which, without really destroying any of our former theories, open to ua wholly new fields of scientific investigation. The study of this subject is especially interesting and important, because, as I have before said (vol. ii. p. 233), no important theory- can be built up complete and perfect all at once. When unexplained phenomena present themselves as objections to the theory, it will often demand the utmost judgment and sagacity to assign to them their proper place and force. The acceptation or rejection of a theory will entirely depend upon discriminating the one insuperable contra- by Google 308 THE PRINCIPLES OF SCIENCE. dietiory feet from many, which, however singular and inexplicable at firet sight, may afterwards be shown to be results of wholly different causes, or possibly the most striking results of the very law with which they stand in apparent conflict. I can enumerate at least eight different classes or kinds of exceptional phenomena, to one or other of which any supposed exception to the known laws of nature will ultimately be referred ; they may be briefly described as below, and will be sufiSciently illustrated in the succeeding sections. (i) Imaginary, or false exceptions, that is, facts, ob- jects, or events which are not really what they are sup- posed to be. (2) Apparent, but congruent exceptions, which, though apparently in conflict with a law of nature, are really in agreement with it {3) Singular exceptions, which really agree with a law of nature, but exhibit remarkable and unique results of it. {4) Divergent exceptions, which really pi-oceed from the ordinary action of known processes of nature, but which are excessive in amount or monstrous in character. (5) Accidental exceptions, arising from the interferenM of some entirely distinct but known law of nature. (6) Novel and unexplained exceptions, wliich lead to the discovery of a new series of laws and phenomena, modifying or disguising the effects of previously known laws, without being inconsistent with them. (7) Limiting exceptions, showing the falsity of a sup- posed law in cases to which it had been extended, but not affecting its truth in other cases. (8) Contradictory or real exceptions which lead us to the conclusion that a supposed hypothesis or theory is in opposition to the phenomena of nature, and must therefore be abandoned. by Google EXCEPTIONAL PHENOMENA. ' 309 It ought to be clearly understood that in no case is a law of nature really thwarted or prevented from being fulfilled. The effects of a law may be disguised and hidden from our view in some instances — in others the law itself may be rendered inapplicable altogether — but if a law is applicable it must be carried out. Every law of nature must therefore be stated with the utmost generality of all the instances really coming under it. Babbage proposed to distinguish between universal prin- ciples, which do not admit of a single exception, such as that every number ending in 5 is divisible by five, and general principles which are more frequently obeyed than violated, as that 'men will be governed by what they believe to be their interest'.' But in a scientific point of view general principles must be universal as regards some distinct class of objects, or they are not principles at all. If a law to which exceptions exist is stated without allusion to those exceptions, the state- ment is erroneous. I have no right to say that 'AH liquids expand by heat,' if I know that water below 4° C. does not ; I ought to say, ' All liquids, except water below 4° C, expand by heat;' and every new exception discovered will falsify the statement until inserted in it. To speak of some taws as being generally true, meaning not universally but in the majority of cases, is a hurt- ful abuse of the word, but is quite usual. General should mean that which is true of a whole genus or class, and every true statement must be true of some assigned or assignable class. Imaginary or False Exceptions. When a supposed exception to a law of nature is brought to our notice, the first inquiry ought properly * Babbage, "The EipoBition of 1851,' p. i. Digitized by Google 310 THE PRINCIPLES OF SCIENCE. to be — Is there any breach of the law at all ? It may be that the supposed exceptional fact is not a fact at all, that it is a mere figment of the imagination. When King Charles requested the Eoyal Society to investigate the curious fact that a live fish put into a bucket of water does not increase the weight of the budiet and its contents, the Royal Society wisely commenced their deliberations by inquiring whether the fact was so or not. Every statement, however false, must have some cause or prior condition, and the real question for the Royal Society to investigate was, how the King came to think that the fact was BO. Mental conditions, as we have seen (voL ii. p. 4), enter into all acts of observation, and are often a worthy subject of inquiry. But there are many instances in the history of physical sciencej in which much trouble and temporary error have been caused by false assertions carelessly made, and carelessly accepted without experi- mental verification. The reception of the Copemican theory was much im- peded by the objection, that if the earth were perpetually moving, a stone dropped from the top of a high tower should be left behind, and should appear to move towards the west, just as a stone dropped &om the mast-head of a moving ship would fell behind, owing to the motion of the ship. The Copemicans attempted to meet this grave objection in eveiy way but the true one, namely, that of showing by trial that the asserted facts are not correct ones. In the first place, if a stone had been dropped with suitable precautions from the mast-head of a moving ship, it would have fallen close to the foot of the mast, because by the first law of motion it would remain in the same state of horizontal motion communicated to it by the mast. As the anti-Copemicans had assumed the contrary result as certain to ensue, their argument would of course have fellen through at once. Had the Copemicans next Digit zed by Google EXCEPTIONAL PHENOMENA. 311 proceeded to test with great care the other assertion in- volved, they would have become still better convinced of the truth of their own theory. A stone dropped from the top of a high tower, or into a deep well, would certainly not have been deflected from the vertical direc- tion in the considerable d^ree required to support the anti-Copemican views ; but, with very accurate obser- vation, they might have discovered, as Benzenberg sub- sequently did, a very small deflection towards the west (voL L p. 453). At the moment when a body begins to fall freely, it begins to resemble a very small satellite moving under the force of gravity, as exerted from the earth's centre of attraction, and it therefore describes> like other satellites, a portion of an elliptic orbits Had the Copemicans then been able to detect and interpret the meaning of this smaU divergence, they would have found in it a conclusive proof of their own views. Multitudes of cases might be cited in which laws of nature seem to be evidently broken, but in which the apparent breach entirely arises from a misapprehension of the &rcts of the case. It is a genend law, absolutely true of all crystals yet submitted to examination, Uiat no crystal has a re-entrant angle, that is an angle which towards the axis of the crystal is greater than two right angles. Wherever the feces of a crystal meet they pro- duce a projecting edge, and wherever edges meet they produce a comer. Many crystals, however, when care- lessly examined, present exceptions to this law, but closer observation always shows that the apparently re-entrant angle really arises from the oblique union of two distinct crystals. Other crystals seem to possess faces contradict- ing all the principles of crystallography ; but again careful examination shows that the supposed &ces are not t> 'Cambridge and Dublin Mathematical Joonul' (1848), vol. iii- by Google 312 THE PRINCIPLES OF SCIENCE. true faces, but surfaces produced by the orderly junction of an immense number of distinct thin crystalline plates, each plate being in fact a separate crystal, in which the laws of crystallography are strictly observed. The rough- ness of the supposed face, the strise detected by the microscope, or inference by continuity from other speci- mens where the true faces of the plates are clearly seen, prove the purely mistaken character of the supposed exception. In tracing out the isomorphic relations of the elements, great perplexity has often been caused by mistaking one substance for another. It was pointed out that though arsenic was supposed to be isomorphous with phosphorus, the arseniate of soda crystallized in a form distinctly different from that of the corresponding phos- phate. Some chemists held this to be a fatal objection to the doctrine of isomorphism ; but it was afterwards pointed out by Clarke, that the arseniate and phosphate in question were not corresponding compounds, as they differed in regard to the water of cryetalli2ation<=. Vana- dium again appeared to be an exception to the laws of isomorphism, until it was proved by Professor Eoscoe, that what Berzelius supposed to be metallic vanadium was really an oxide of vanadium d. In the science of crystallography many other ^parent exceptions present themselves, and sometimes cause con- siderable perplexity. Four of the faces of a regular octa- hedron may become so enlarged in the crystallization of iron pyrites and some other substances, that the other four faces become altogether imperceptible and a regular tetrahedron appears to be produced, contrary to the laws of crystallographic symmetry. Many other crystalline c Daubeny's ' Atomic Theoij,' p. 76. ■■ 'Bakerian Lecture,' ' PhiloBophical TransectioitB,' (1868) vol. civiii. by Google EXCEPTIONAL PHENOMENA. 313 forms are similarly modified, so as to produce a series of what are called kemihedral forms. Apparent but Congruent Exertions. Not unfrequently a law of nature will present results in cert^n circumstances which appear to be entirely in conflict with the law itself. Not only may the action of the law be much complicated and disguised, but it may in various ways be reversed or inverted, so that all care- less observers are misled. Ancient philosophers gene- rally believed that while some bodies were heavy by nature, others, such as flame, smoke, bubbles, clouds, &c., were essentially light, or possessed a tendency to move upwards. So acute and learned an inquirer as Aristotle entirely failed to perceive the true nature of buoyancy or apparent lightness, and the doctrine of intrinsic lightness, being expounded in bis works, became the accepted view for many centuries. It is true that Lucretius was fully aware why flame tends to rise, holding tbat — ' The flame has weight, though highly rare, Nor mounta but when compelled hy heavier air.' Arcbimedes also was so perfectly acquainted with the buoyancy of bodies immersed in water, that he could not fail to perceive the existence of a parallel eflect in air. Yet throughout the early, middle ages the light of true science, clear though feeble, could not contend with the powerful but confused glare of the false Peripatetic doc- trine. The genius of Galileo and Newton waa required to convince people of the simple truth that all matter is heavy, but that the gravity of one substance may be overborne by tbat of another, as one scale of a balance is carried up by the preponderating weight in the oppo- site scala It is curious to find Newton gravely explaining by Google 311 THE PRINCIPLES OF SCIENCE. the difference of abflolute and relative gravity, as if it were a new discovery proceeding from his theory^. More than a century elapsed before other apparent exceptions to the Newtonian philosophy were explained away. Newton himself allowed that the motion of the apsides of the moon's orbit appeared irreconcilable with the law of gravity, and it remained for Clairaut to remove the reproach by more complete mathematical analysis. There must always indeed remain, in the motions of the tides or of the heavenly bodies, discrepandes of some amount between theory and observation; but like discrepanciea have so often yielded in past times to prolonged investi- gation that all physicists have come to regard them as merely apparent exceptions, which will afterwards be found to be new confirmations of the law with which they now seem to conflict. The most beautiful instance, perhaps, which can be adduced of an apparent exception, is found in the total reflection of light, which occurs when a beam of light within a medium falls very obliquely upon the boundary separating it from a rarer medium. It is the general law that when a ray strikes the limit between two media of different refractive indices, part of the light is reflected and part is refracted, but when the obliquity of the ray within the denser medium passes beyond a certain point there is a sudden apparent breach of continuity, and the whole of the light is reflected. A very clear reason can be given for this exceptional conduct of the light ; for according to the law of refraction the sine of the angle of incidence always bears a fixed ratio to the sine of the angle of refraction, bo that the greater of the two angles, which is always that in the less dense medium, may increase up to a right angle, but when the media diSer in refractive power, the less angle cannot become a right angle, as this * ' Principia,' bk. II. Prop. 20. Corollaries, 5 and 6. by Google BXCSPTIONAL PHENOMENA. 315 would require the rane of an angle to be greater than the radiuB. It might seem, perhaps, that this was an exception of the kind elsewhere described as a limiting exception, in which a law is shown to be inapplicable beyond certain de- Bned limits ; but in the explanation of the exception according to the undulatoiy theoiy, we find that there is really no breach or exception to the general law. Whenever an undulation strikes any point in a bounding surface, spherical waves are produced and spread from the point The refracted ray is the resultant of an infi- nite number of such spherical waves, and the bending of tbe ray at the common surface of two media depends upon the comparative velocities of propagation of the undula- tions in those media. But if a ray falls very obliquely upon the Burfafie of a rarer medium, the waves arising fi:«m successive points of the surface may spread so rapidly as never to intersect, and no resultant wave will then be produced. We thus perceive that from general mathe- . matical conditions may arise very distinct apparent effects. There may occur from time to time distinct failures in our most well-grounded predictions. A comet, of which the orbit has been well determined, may fail, like JjexeU'e Comet, to appear at the appointed time and place in the heavens. In the present day we should not bold such an exception to our successful predictions to weigh against our belief in the theory of gravitation, but should assume that some unknown body had through the action of gravi- tation itself deflected tbe comet As Clairaut remarked, in publishing his calculations concerning the expected re- appearance of Halley's Comet, a body which passes into regions so remote, and which is hidden from our view during such long periods, might be exposed to the influ- ence of forces totally unknown to us, such as the action of other comets, or even of some planet too far removed from the sun to be ever perceived by us. In the case of Digitized by Google 316 TUE PRINCIPLES OF SCIENCE. Lexell's Comet it was afterwards shown, curiously enough, that its appearance was not one of a regular series of periodical returns within the sphere of our vision, but a single exceptional visit never to be repeated, and probably due to the perturbing powers of Jupiter. Tet this soli- tary visit was a strong confirmation of the law of gravity with which it seemed to be in conflict. The division of Biela's Comet into two companion comets was at the time when it occurred one of those unlooked- for and inexplicable events which awaken the attention and interest of observers in the highest degree. Comets indeed have alt(:^;ether the character of eccentric strangers intruding into our planetary system, and in almost every point they are yet inexplicable ; but there is a possibility tliat the separation of Biela's Comet may prove to be a comparatively ordinary event of cometary history. For if, as is now believed, comets be aggregates of small me- teoric stones or particles, forming the denser parts of con- tinuous streams of such bodies circulating round the sun, then it is not unlikely that these aggregates may at times be increased or diminished by the meeting or separation of meteoric streams. Singular Exceptions. Among the most interesting of apparent exceptions are those which I propose to call singular exceptions, because they are more or less closely analogous to the singular cases, or solutions which occur in mathematical science. A general mathematical law embraces an infinite multi- tude of cases which have a perfect agreement with each other in a certain respect. It may nevertheless happen that a single case, while obe3dng the general law, stands out as apparently different from all the rest. The daily rotation of the earth upon its axis gives to all the stars by Google EXCEPTIONAL PHENOMENA. 317 in the heavens an apparent relative motion of rotation from east to west ; but out of countless thousands which obey the rule the Pole Star alone seems to break it Exact observations indeed show that it also revolves in a small circle, but it might happen for a short time that a star existed so close to the pole tliat no appreciable change of place would be caused by the daily rotation. It would then constitute a perfect singular exception ; for, while really obeying the law, it would break the terms - in which it is usually stated. In the same way the poles of every revolving body are singular points. Whenever the laws of nature are reduced to a mathe- matical form we may expect to meet with singular cases, and, as all the physical sciences will meet in the mathema* tical principles of mechanics, there is no part of nature where we may not probably encounter them. In me- chanical science itself the circular motion of rotation may be considered a single exception to the rectilineal motion of translation. It is a general law that any number of parallel forces, whether acting in the same or opposite directions, will have a resultant which may be substituted for them with like effect This resultant will be equal to the algebraic sum of the forces, or the difference of those acting in one direction and the other ; it will pass through a point which is determined by a simple formula, and which may be described as the mean point of all the points of application of the parallel forces (vol i. p. 422). Thus we readily determine the resultant of parallel forces, except in one peculiar case, namely, when two forces are equal and opposite but not in the same straight line. Being equal and opposite the amount of the resultant is nothing, yet, as the forces are not in the same straight line, they do not balance and destroy each other. Exami- ning the formula for the point of application of the re- sultant, we 6nd that it gives an infinitely great magnitude, DigitizedbyGOOgle 318 THE PRINCIPLES OF SCIENCE. BO that the resultant is nothing at all, and acts at an infi- nite dietance, which is practically the same as to say that there is no possible single resultant Two such forces constitute what is known in mechanical science as a couple, which occasions rotatory instead of rectilineal motion, and can only be neutralized by an equal and opposite couple or pair of forces. The most beautiful instances of singnlar exceptious are furnished by the science of optica It is a general law, for instance, that in passing through transparent media the plane of vibration of polarized light remains un- changed. But in certain cases, to which reference has already been frequently made, namely, certain liquids, some peculiar crystals of quartz, and tr^isparent solid media suhjected to a magnetic strain, as in Faraday's ex- periment (vol ii. pp. 234, 287), the plane of polarization is rotated in a screw-like manner. This efiect is so entirely mi generis, so unlike any other phenomena in nature, as to appear truly exceptional; yet mathematical analysis shows it to be only a single case of much more general laws. A^ stated by Thomson and Tait', it arises from the composition of two uniform circular motions. If while a point is moving round a circle, the centre of that circle move upon another circle, a great variety of curious curves will be produced aojording as we vary the dimen- sions of the circles or the rapidity of the motions. In one case where the two circles are exactly equal, the point will be found to move gradually round the centre of the stationaiy circle, and describe a curious star-like figure connected with the molecular motions out of which the rotational power of the media arises. Among other sin- gular exceptions in optics may be placed the conical refrac- tion of light, already noticed (vol. ii. p. 175), connected with the peculiar form assumed by a wave of light when f ' Treatiae on Nataral Philosophy,' vol. i. p. 50, Digitized by Google EXCEPTIONAL PHENOMENA. 319 passing through certain double-refracting crystals. The laws obeyed by the wave are exactly the same as in other cases, yet the results are entirely sui generis. So far are such cases from conttadicting the theory of ordinary cases, that they afford the supreme opportunities for verification. In astronomy singular exceptions might occur, and in an approximate manner they do occur. We might point to the rings of Saturn as objects which, though undoubt- edly obeying the law of gravity, are yet entirely unique, as far as our observation of the universe has gone. They agree, indeed, with the other bodies of the planetary system in the stability of their movements, which never diverge far from the mean position. But a truly singular event might happen, or might have happened, under slightly different circumstances. Had the rings been exactly uniform all round, and with a centre' of gravity coinciding for a moment with that of Saturn, a singular case of unstable equilibrium would have arisen, necessarily re- sulting in the sudden collapse of the rings, and the fall of their debris upon the surface of the planet. Thus in one single case the theory of gravity would give a result wholly unlike anything else known in the mechanism of the heavens. It is possible that we might meet with Angular excep- tions in crystallography. If a crystal of the second or dimetric system, in which the third axis is usually unequal to either of the other two, happened to have the three axes equal, it might be mistaken at first sight for a crystal of the cubic system, but would in many ways exhibit different faces and dissimilar properties. There is, again, a possible class of diclinic crystals in which two axes are at right angles and the third axis inclined to the other two. This class is chiefly remarkable for its non- existence in a material point of view, since no crystals Digitized by Google 320 THE FRWCIFLES OF SGIBA'CE. have yet been proved to have such axes. It seems likely that the class would constitute only a singular case of the more general triclinic system, in which all three axes are inclined to each o^er at various angles. Now if the di- clinic form were merely accidental, and not necessitated by any general law of molecular constitution, its actual occurrence would be infinitely improbable, just as it is infinitely improbable that any star should indicate the North Pole with perfect exactness. In the curves denoting the relation between the temper- ature and pressure of water there is one very remarkable point entirely single and unique, at which alone water can remain in the three conditions of gas, liquid, and solid in the same vessel It is the point at which three curves intersect, namely, the steam line showing at what temper- atures and pressures water is just upon the point of be- coming gaseous, and other similar lines which show when ice is just on the point of melting, and when ice is just about to assume the gaseous state directly. Divergent Exceptions. Closely analogous to singular exceptions are those diver- gent exceptions, in which a phenomenon manifests itself in very unusual magnitude or character, without however in any degree becoming subject to peculiar laws. Thus in throwing ten coins, it happened in four cases out of 2048 throws, that all the coins fell with Leads uppermost (vol. i, p. 238) ; these woidd usually be regarded as very singular events, and, according to the theory of proba- bilities, they would be comparatively very rare ; yet they proceed only from an unusual conjunction of accidental events, and from no really exceptional causes. In all classes of natural phenomena we may expect to meet with similar divergencies from the average. Sometimes due merely to by Google EXCEPTIONAL PHENOMENA. 321 the principles of probability, Bometimes to deeper reasons. Among every large collection of persons, we shall probably find some persons who are remarkably large or remark- ably small, giants or dwarfs, whether in bodily or mental conformation. Such cases appear to be not mere lusus naturcB, since they usually occur with a frequency clcBely accordant with the law of error or divergence from an average, as shown by M. Quetelet and Mr. Galton (vol, i. p. 446). The rise of genius, or tJie occurrence of extra- ordinary musical or mathematical faculties, are attributed by M, Galton to the same principle of divergence. Under this class of exceptions I am inclined to place all kinds of remarkable events arising fixtm an unusual conjunction of many ordinary tendencies. When several distinct forces happen to concur together, we may have surprising or alarming results. Great storms, floods, droughts and other extreme deviations from the average condition of the atmosphere thus arisa They must be expected to happen from time to time, and will yet be very unfrequent compared with minor disturbances. They are not anomalous but only extreme events, exactly analogous to extreme runs of ludc. There seems, indeed, to be a fallacious impression in the minds of many persons, that the theory of probabiHties necessitates uni- formity in the happening of events, so that in the same space of time there will always be closely the same nimaber, for instance, of railway accidents and murders. Buckle has superficially remarked upon the comparative constancy of many such events as ascertained by Quetelet, and some of his readers acquire the false notion that there is a kind of mysterious inexorable law producing uniformity in natural and human affeirs. But nothing can be more opposed to the teachings of the theory of probability, which always contemplates the occurrence of extreme and unusual runs of luck. That theory shows VOL. n. Y Digitized by Google 322 THE PRINCIPLES OF SCIENCE. the great improbability that the number of railway acci- dents per month should be always equal, or nearly so. The public attention is atrongly attracted to any unusual conjunction of events, but there is a fallacioua tendency to suppose that every such conjunction must be due to a peculiar new cause coming into operation. Unless it can be clearly shown that such unusual conjtmctions occur more fiequently than they shoxild do according to the theory of probabilities, we should regard them as merely divergent exceptions. Eclipses and remarkable conjunctions of the heavenly bodies may also be regarded as results of ordinary laws, which nevertheless appear to break the reg^ar course of nature, and never fe.il to excite surprise or even fear. Such conjunctions of bodies vary greatly in frequency. One or other of the satellites of Jupiter is eclipsed almost every day, but the simultaneous eclipse of three satellites can only take place, according to the calculations of War- gentin, after the lapse of 1,317,900 years. The relations of the four satellites aie so remarkable, that it is actually im- possible, according to the theory of gravity, that they should alt suffer edipse simultaneously. But it may happen occa- sionally that while some of the satellites are really edipsed by entering Jupiter's shadow, the others are either occulted or rendered invisible by passing over his disk, as seen by us. Thus on four occasions, in 1681, 1S02, 1826, and 1843, Jupiter has been witnessed in the singular condition of being apparently deprived of satellites. A close conjunc- tion of two planets always excites surprise and admira- tion, though conjunctions must naturally occur at intervals in the ordinary course of their motions. We cannot wonder, then, that when three or four planets approach each other closely, the event is long remembered. A most exceptional conjxmction of Hars, Jupiter. Satum, and Mer- cury, which took place in the year 2446 B.O., was adopted by Google EXCEPTIONAL PHESOMSNA. by the Chmeee Emperor, Chuen Hio, as a new epoch for the chronology of that Empire, though there is some doubt ■whether the conjunction was really observed or was calculated from the supposed laws of motion of the planeta It is certain that on the nth November, 1524, the planets Yenus, Jupiter, Mars, and Saturn were seen veiy doae together, while Mercury was only distant 1^ about 16° or thirty apparent diameters of the smi, this conjunction being probably the most remarkable which has occurred in his- torical timea Among the perturbations of the planetary motions we may find divergent exceptions arising from the peculiar accumulation or intensification of effects, as in the case of the long inequaUty of Jupiter and Saturn (vol ii. p. 70). Leverrier has shown that there is one place between the orbits of Mercury and Venus, and another between those of Mara and Jupiter, in either of which, if a small planet happened to exist, it would suffer comparatively immense disturbance in the elements of its orbit. Now between Mars and Jupiter there do occur the minor planets, the orbits of which are in many cases exceptionally divergent?. It is worthy of notice that even in such a subject as formal logic, divei^nt exceptions seem to occur, not of course due to chance, but exhibiting in an tmnsual decree a phenomenon which is more or less manifested in all other cases. I pointed out in p. 163 of the first volume, that propositions of the general type A= BC + he are capable of expression in six equivalent logical forms, so that they manifest la a higher d^^e than any other proposition yet discovered, the phenomenon of logical equivalency. Under the head of divergent exceptions we might doubtless place all or nearly all of the instances of sub- stances possesmng physical properties in a very high or low degree, which were described in the chapter on « Chntnt's ' HiBtoi7 of Phjncftl Ajrtronomy,' p. 1 16. Y 2 by Google 324 THE PRINCIPLES OF SCIENCE. Generalization, (vol. ii. p. 259). Quicksilver is divergent amoDg metals as regards ita melting point, and potasBium and sodium as regards their specific gravity. Monstrous productions and variations, whether in the animal or vegetable kingdoms, should probably be assigned to this dass of exceptions. Accidental Exceptions. The third and largest class of exceptions contains those which arise from the casual interference of extraneous causes. A law may be in operation, and, if so, must be perfectly fulfilled, but, while we conceive that we are examining its results, we may have before us the efiecte of a totally different cause, possessing no connexion with the subject of our inquiry. The law is not really broken, but at the same time the supposed exception is not illusory. It may be a phenomenon which cannot occur but under the con- dition of the law in question, yet there has been such subsequent interference and modification of the result, that there is an apparent failure of science. There is, for instance, no subject in which more rigorous and in- variable laws have been established than in crystallo- graphy. As a general rule, each chemical substance pos- sesses its own definite form, by which it can be infallibly recognised ; but the mineralogist has to be on his guard against what are called 'pseudomorphic crystals. In some circumstances a substance, having perfectly assumed its proper crystalline form, may afterwards undergo chemi- cal change ; a new ingredient may be added, a former one removed, or one element may be substituted for another. In carbonate of hme the carbonic acid is some- times replaced by sulphuric acid, so that we find gypsum in the form of calcite ; other cases are known where the change is inverted and calcite is found in the form of gypsum. Mica, talc, steatite, hematite, are other minerals Digitized by Google EXCEPTIONAL PHENOMENA. 326 subject to these curious transmutations. Sometimes a crystal embedded in a matrix is entirely dissolved away, and subsequently a new kind of mineral is gradually deposited in the cavity as in a mould. Quartz is thus found cast in many forms wholly tmnatural to it. A still more perplexing case sometimes occurs. Carbonate of lime is one of the substances capable of assuming two distinct forms of crystallization, in which it bears respectively the names of calcite and arragonite. Now arragonite, while retaining its outward form imcbanged, may undergo an internal molecular change into calcite, as indicated by the altered cleavage. Thus we may come across ciystals apparently of arragonite, which seem to break all the laws of crystallography, by possessing the cleavage of an enlirely different system of crystallization. Some of the most invariable and certain laws of nature are disguised by interference of imlooked-for causes. While the barometer was yet a new and curious subject of investigation, its theory, as stated by TorriceUi and Pascal, seemed to be contradicted by the fact that in a well-constructed instrument the mercury would often stand far above 31 inches in height Boyle showed'' that the mercury coxild be made to rise as much as 75 inches in a perfectly cleansed tube, or about two and a half times as high as could be due to the pressure of the atmosphere. Many absurd theories about the pres- sure of imaginary fluids were in consequence put forth', and the subject was involved in much confusion until the adhesive or cohesive force between glass and mercury, when brought into perfect contact, was pointed out as the real interfering cause. Guy-Lussac, again, observed that the temperature of boiling water was very difierent in some kinds of v h 'Digcourae to the Royal Society,' aSth May, 1684. > Robert Hooke's ' Posthumoiu Works,' p. 365. by Google 326 THE PRINCIPLES OF SCIENCE. from what it was in others. It is only in contact with metallic surfaces or sharply broken edges that the tem- perature is at all fixed at 312° Fahr. The suspended freezing of liqmds is another case where the action of a law of nature appears to be interrupted. Spheroidal ebullition seemed at first sight a most anomaloos phe- nomenon ; it was almost incredible that water should not boil in a red-hot vessel, or that ice could actually be produced in a red-hot crucible. These paradoxical results are now fully explained as due to the interposition of a non-conducting film of vapour between the globule of liquid and the sides of the vesseL The feats of con- jurors who handle Uquid metals are readily accounted for in the same manner. At one time the passive state of steel was regarded as entirely anomalous. It may be assumed as a general law that when two pieces re- spectively of electro-negative and electro-positive metal are placed in nitric acid, and made to touch each other, the electro-negative metal will tmdergo rapid solution. But when iron is the electro-negative and platinum the electro-positive, the solution of the iron entirely and abruptly ceases. Faraday ingeniously proved that this eflect was due to a thin film of oxide of iron, which forma upon the surface of the iron and protects it*^. The law of gravity is of so simple and general a cha- racter, and is apparently so disconnected from the other laws of nature, that it never sufiers any disturbance, and is in no way disguised, but by the complication of its own effects. It is otherwise, however, with those entirely secondary laws of the planetary system, which have only an empirical basis. The fact that all the long known planets and satellites have a similar motion from west to east is not necessitated by any principles of science, but points merely to some common condition existing in the k 'Experimental Besearchea in Electricity,' vol. ii. pp. a4i>-345. by Google EXCEPTIONAL PHENOMENA. 327 nebulous mass from which our system has doubtless been evolved. The retrog^rade motions of the satellites of Uranus constituted a distinct breach in this law of uniform direction, which became all the more interesting when the single satellite of Neptune was also found to be retro- grade. It now became probable, as Baden Powell well observed, that the anomaly would cease to be singular, and become a case of another law, pointing to some general interference, which has taken place on the bounds of the planetary system. Not only have the satellites suffered from this perturbance, but Uranus is also anomalous in having an axis of rotation lying nearly in the echptic ; and Neptune constitutes a distinct exception to the empirical law of Bode concerning the distances of the planets, which exceptional circumstance may pos- sibly be due to the same disturbance. Geology is a science in which accidental exceptions are very likely to occur. Only when we find strata in their original relative positions, can we surely infer that the order of succession is the order of time. But it not uncommonly happens that strata are inverted by the bending and doubling action of extreme pressure. Land- sUps may carry one body of rock into proximity with an unrelated series, and produce results apparently inex- plicable i. Floods, streams, icebergs, and other casual agents, may occasionally lodge remains in places where they would be wholly unexpected. Though such interfering causes may have been often wrongly supposed to explain important discoveries, the geologist must of course always bear the possibility of interference in mind. Scarcely more than a century ago it was yet held by many persons that fossils were acci- dental productions of nature, mere forms into which minerals had been shaped by no peculiar cause. Voltaire 1 Unrchisoa's 'Silurian SyBtem,' vol. ii. p. 733, Ac. by Google 328 THE PmSGIPLES OF SCIESCE. appears not to have been able to accept Buch an ex- planation ; but fearing that the occurrence of fossil fishes on tbe Alps would support the Mosaic account of the deluge, he did not hesitate to attribute them to the remains of fishes accidentally brought there by travellers or pilgrims. In archxological investigations the greatest caution is requisite in allowing for secondary burials in ancient tombs and tumuli, for imitations, casual coin- cidences, disturbance by subsequent races, or even by other archeeologists, in feet, for a multitude of interfering circumstances. In common liie extraordinary events must happen from time to time, as when a shepherdess in France was astonished at an iron chain falling out of the sky near to her feet, the &ct being that G-uy-Lussac had thrown it out of his balloon, which was passing over her head unseen at the time. To this class of accidental exceptions I would refer the innumerable breaches of the rules of inflexion in grammar. These rules woidd be invariable were it not that the forms derived from distinct roots sometimes get mixed together, that mistaken analogies sometimes occasion con- fusion, and a variety of such disturbing causes produce irregularity. Philology already presents beautiful in- stances of tbe manner, in which a comprehensive law may be traced out in a thoroughly scientific manner, iu spite of apparently inexplicable exceptions. Ntyvd and Unexplained Exceptions. When a law of nature appears to fail because some other law has interfered with its action, two cases may obviously present themselves ; — the interfering law may be a known and familiar one, or it may have been pre- viously undetected. In the first case, which we have Btifficiently considered in the preceding section, we have by Google EXCEPTIOiTAL PHENOMENA. 329 notliing to do but calculate as exactly as possible fbe amount of interference, and make allowance for it ; the apparent failure of the law under examination should then disappear. But in the second case the results may be much more important. A phenomenon which entirely fails to be explained by any known laws may indicate the interference of some wholly new series of natural forces. The ancients could not help perceiving that the general tendency of bodies downwards failed in the case of the loadstone, nor would the doctrine of essential lightness explain the exception, since the substance drawn upwards by the loadstone is a heavy metal. We now see clearly that there was no breach in the perfect generality of the law of gravity, but that a new form of energy manifested itself in a conspicuous form in the loadstone for the first time. In this case the forces concerned, those of gravity and electrical attraction, have never yet been brought into correlation with each other. Other sciences show us that laws of nature, rigorously true and exact, may often be developed by those who are ignorant of far more complex phenomena involved in their appUcation. Newton's comprehension of geometrical optics was sufficient to explain all the ordinary refractions and reflections of light. The simple laws of the bending of rays apply to all rays, whatever the character of the undulations composing them. Newton suspected the existence of other classes of phenomena when he spoke of rays as having sides; but it remained for later experi- mentalists to show that light is a transverse undulation, like the bending of a rod or cord. Dalton's atomic theory is doubtless true of all chemical compounds, and the essence of it is that the same com- pound will always be found to contain the same elements in certain definite proportions. Pure calcium carbonate contains 48 parts by weight of oxygen to 40 of calcium. Digitized by Google 330 THE PRINCIPLES OF SCIENCE. and 1 2 of carbon. But when careful analyses were made of a great many minerals, this law often appeared to laiL What was unquestionably the same mineral, judging by its crystalline form and physical propeirties, would often give vaiying proportions of its compouents, and would sometimcB conttun unusual elements which yet could not be set down as mere impuritiea Dolomite, for instance, is a compound of the carbonates of magoeoa and lime, but speci- mens from different places do not exhibit any fixed ratio between the lime and magnesia, and carbonate of iron occasionally forms a real constituent of the mineral. Such facts could be reconciled with the laws of Dalton only by supposing the interference of a new law, that of Isomorphism. It is now sufficiently established that certain elements are closely related to each other, so that they can, as it were, step into each other's places without apparently altering the form of the compound molecules, or the shape of the crystals which they constitute. The car^ bonates of iron, calcium, and magnesium, are nearly identical in their crystalline forms, hence they may crystallize t<^ther in harmony, producing mixed minerals of considerable complexity, which nevertheless perfectly verify the laws of equivalent proportions. This principle of isomorphism once established, not only explains what was formerly a stumbling-block, but gives most valuable aid to chemists in deciding upon the real constitution of new salts, since those compounds of ieomorphous elements which have identical crystalline forms must possess cor- responding chemical formulEe. We may always expect that from time to time new and extraordinary phenomena will be discovered, and will lead to new views of the laws of nature. The recent observa- tion, for instance, that the resistance of a bar of seleniiuu to a current of electritaty is affected in an extraordinary by Google BZOEPTIONAL PHENOMENA. 331 degree by rays of light falling upon the selenium, points to a wholly new relation between light and electricity. The peculiar so-called allotropic changes which sulphur, Belenium, and phosphorus undergo by an alteration in the amount of latent heat which they contain, will pro- bably lead at some future time to important inferences concerning the molecular constitution of solids and liquids. The curious substance ozone has perplexed many chemists, and Andrews and Tait thought that it afforded evidence of the decomposition of oxygen by the electric dischai^e. The researches of Sir B. C. Brodie negative this notion, and afford evidence of the real constitution of the sub- stance "", which still, however, remains exceptional in its properties and relations, and affords a hope of important discoveries in chemical theory. Limiting Exceptions. We may pass to cases where exceptional phenomena are actually irreconcilable with a law of nature previously regarded as true by philosophers. Error must now be allowed to have been committed, but it is obvious that the error may be more or less extensive. It may be that a law holding rigorously trutf of the facts actually under notice had been extended by generalization to other series of facts then unexamined. Subsequent investiga- tion may show the falsity of this generalization, and the result must be to limit the law for the future to those objects of which it is really true, while we bring the other classes of objects under distinct generalizations. The contradiction to our previous opinions is partial and not total. Newton laid down as a result of experiment that every ray of homc^eneous light has a definite refrangibility, which >° ' PhiloBophicol TranractioDa '(187 3), vol. clxii. do. 13. Digitized by Google 332 THE PRINCIPLES OF SCIENCE. it preserves throughout its course until extinguished. This is indeed but one case of the general principle of undula- torj movement, which Sir John Herschel has stated in the most complete manner under the title, 'Principle of Forced Vibrations' (vol. ii. p. 65), and has asserted to be absolutely universal and without exception. But Sir John Herschel himself described in the ' Philosophical Transac- tions' for 1845 a curious appearance in a solution of qui- nine ; as viewed by transmitted light the solution appeared colourless, but in certain aspects it possessed a beautiiul celestial blue tint. Curiously enough the coloured light comes only from the first portion of liquid which the light entera Similar phenomena in fluor-spar had been described by Sir D. Brewster in 1838. Professor Stokes, having minutely investigated the phenomena, discovered that they were more or less present in almost all vegetable injFusions, and in a number of mineral substances. He came to the conclusion that this phenomenon, called by him Fluorescence, could only be explained by a degrada- tion or alteration in the refrangibility of the rays of light ; he asserts, in fact, that light-rays of very short length of vibration in falling upon certain atoms excite undulations of greater length, in total opposition to the principle of forced vibrations. No complete explanation of the mode of change is yet possible, because it evidently depends upon the intimate constitution of the atoms of the sub- stances concerned ; but Professor Stokes believes that the principle of forced vibrations is true only so long as the excursions of an atom are very small compared with the magnitude of the complex molecules". It is now also well known that in Calorescence the refrangibility of rays may be increased and the wave-length diminished. Bays of obscure heat and low refrangibility may he concentrated so as to heat a solid substance, and make it give out rays ■> ' Philwophical Transactions' (1852), vol. cxlii. pp. 465, 548, &c. by Google EXCEPTIONAL PHENOMENA. 333 belonging to any part of the spectrum, and it seems pro- bable that this effect arises from the impact of distinct but confiicting atoms. Nor is it in light only that we discover limiting exceptions to the law of forced vibrations ; for if we closely observe gentle waves lapping upon the stones at the edge of a lake or other piece of water, we shall notice that each larger wave in breaking upon a stone gives rise to a series of waves of a smaller order. Thus there must be constantly in progress a degradation in the magnitude of water-waves. The principle of forced vibra- tions seems then to be too generally stated by Sir John Herschel, but it must be a very difficult question of me- chanical theory to discriminate the circumstances in which it does and does not hold true. We may sometimes foresee the possible existence of exceptions yet unknown by experience, and limit the statement of our discoveries accordingly. Very extensive inquiries have diown that all substances yet examined fall into one of two classes ; they are all either ferro- magnetic, that b, magnetic in the same way as iron, or they are diamagnetic like bismuth. But it does not thence follow that every substance must be ferro-magnetic or diamagnetic. The magnetic properties are shown by Sir W. Thomson" to depend upon the specific inductive capacities of the substance in three rectangular directions. If these inductive capacities are all positive, we have a ferro-magnetic substance ; if negative, a diamagnetic sub- stance ; but if the specific inductive capacity were posi- tive in one direction and negative in the others, we should have an exception to previous experience, and could not place the substance under either of the present recognised So many gases have been reduced to the liquid state, and so many solids fused, that scientific men rather hastily ■> ' PliiloBOphicAl Hagazine,' 4th Series, vol. i. p. 181. Digitized by Google 334 THS PRINCIPLES OF SCIENOB. adopted the generalization that all suhstances could exist in all three states. A certain number of gases, such as oxygen, hydrogen, and nitrogen, have resisted all efforts to liquefy them, and it now seems probable from the ex- periments of Dr. Andrews that they are limiting excep- tions. Dr. Andrews finds that above 88° Fahr. carbonic atad cannot be liquefied by any pressure he could apply, whereas below this temperature lique&ction is always possible. By analogy it becomes highly probable that even hydrogen might be liquefied if cooled to a sufficiently low temperature. We must modify our previous views, and either assert that helow a certain critical temperature every gas may be liquefied, or else we must assume that a highly condensed gas is, when above the critical temper- ature, undistinguishable from a liquid. At the same time we receive an explanation of a remarkable exception pre- sented by liquid carbonic acid to the general rule that gases expand more by heat than liquids. This liquid carbonic acid was found by Thilorier in 1835 to expand more than four times as much as air ; but by the light of Dr. Andrews' experiments we may learn to regard the liquid as rather a highly condensed gas than an ordinary liquid, and it is actually possible to reduce the gas to the apparently liquid condition without any abrupt conden- sationP. ' It is an empirical law of the planetary system that all the bodies composing it revolve fit)m west to east ; that law is broken, as we have seen, in the cases of one planet and several satellites, probably by the interference of an accidental disturbing force. The law also fails to be true of comets, which, taken as a whole, appear to move according to no single uniform law. This exception, how- ever, is one of limitation only, for in all probability comets, although at present members of our system, have not P Maxwell, 'Theoij of Heat," p. IJ3. Digitized by Google EXCEPTIONAL PHENOMENA. 335 always been bo, but have, in wandering through space, been entangled in our ff^stem and retained hy the attrac- tive influence of Jupiter, or one of the other larger planets. We must then limit the statement of the law of uniform direction to bodies which are derived from the original constituents of the nebulous mass. Limiting exceptioDs occur most frequently in the natural sciences of Botany, Zoology, Geology, Ac, the laws of which are almost wholly empirical. In innumerable in- stances the confident belief of one generation haa been falsified by the wider observation of a succeeding one. Aristotle confidently held that all swans are white*', and the proposition seemed true until not a hundred years ^o black swans were discovered in Western Australia. At one time all the animal remains discovered in the Scottish Old Red Sandstone were fishes or shells, imtil at last a single small air-breathing reptile occurred oppor- tunely to prevent any haaty conclusions'. In zoology and physiology we may expect a fundamental identity to exist in the vital processes, but continual discoveries show that there is no limit to the apparently anomalous expedients by which life is reproduced. Alternate generation, fer- tilization for several successive generations, hermaphro- ditism, are opposed to alt we should expect from induction founded upon the higher animals. But such phenomena are only limiting exceptions showing that what is true of one class is not true of another. In certain of the cephalopoda we meet the extraordinary fact that an arm of the male is cast off and lives inde- pendently until it encounters the female. Q ' Prior Analytics,' ii i, 8, and eleewbere. r Murcbieon's 'Siluria' (1854), p. 154, by Google TUB PRINCIPLES OF SCIENCE. Real Exceptions to Supposed Laws. The exceptions which we have lastly to consider, are perhaps the most important of all, since they lead to the entire rejection of a law or theory before accepted. No law of nature can fail ; there are no such things as real exceptions. Where contradiction exists it must be in the mind of the experimentalist. Either the law is imaginaiy or the phenomena which conflict with it ; ii^ then, by our senses we can satisfy ourselves of the actual occurrence of the phenomena, the law must be rejected as illusory. The followers of Aristotle held that nature abhorred a vacuum, and thus accounted for the rise of water in a pump. When Torricelli pointed out the visible fact that water woidd not rise more than 33 feet in a pump, nor mercury more than about 30 inches In a glass tube, they attempted to represent these facte as limiting exceptions, saying that nature abhorred a vacuum to a certain extent and no further. But the Academicians del Cimento completed their discomfiture by showing that if we remove the pressure of the surroiinding air, and in proportion as we remove it, nature's feelings of abhor- rence decrease and finally disappear altogether. Even Aristotelian doctrines could not stand such direct contra- dictioD. Lavoisder's ideas concerning the constitution of acids received complete refutation. He named oxygen the add generator, because he believed that all acids were com- pounds of oxygen, a generalization based on insufficient data. Berthollet, as early as 1789, proved by analysis that hydrogen sulphide and prusslc acid, both clearly acting the part of acids, were devoid of oxygen; the former might perhaps have been interpreted as a Umiting excep- tion, but when so powerful an acid as hydrogen chloride by Google BSCBPTIOSAL PHENOMENA. 337 (muriatic acid) was found to contain no oxygen the theory- had to be relinquished. Berzelius' theory of the dual formation of chemical compounds has met a similar fete. It is obvious that all conclusive experimenta cruets constitute real exceptions to the supposed laws of the theory which is overthrown. Newton's corpxiscular theory of light was not rejected on account of its absurdity or inconceivability, for in these respects it is, as we have seen, far superior to the undulatory theory. It was re- jected because certain small diffraction fringes of colour did not appear in the exact place and of the exact size which calculation showed that they ought to appear according to the conditions of the theory {vol ii. pp. 145- 151). One single feet clearly irreconcilable with a theory involves its total rejection. In the greater number of cases, what appears to be a fatal exception, may be after- wards explained away as a singular or disguised result of the very laws with which it seems to conflict, or as due to the interference of extraneous causes ; but if we fail thus to reduce the fact to congniity, it remains more powerful than any theories or any dogmas. Of late years not a few of the favourite doctrines of geologists have been rudely destroyed. It was the general belief that human remains were to be found only in those deposits which are actxially in progress at the present day, so that the creation of man appeared to have taken place at the beginning, as it were, of this geological age. The discovery of a single worked flint in older strata and in connexion with the remains of extinct mammals was suf- ficient to explode such a doctrine. Similarly, the opinions of geologists have been altered by the discovery of the Eozoon in the Laurentiaa rocks of Canada ; it was pre- viously held that no remains of life occurred in any older strata than those of the Silurian system. As the exami- voL. n. z Digitized by Google 338 TEE PRINCIPLES OF SCIENCE. nation of the strata of the globe becomes more and more complete, our views of the origin and succession of life upon the globe must undergo many changes and ex- tensions. Unclasaed Exceptions. At every period of scientific progress there will neces- sarily exist a multitude of exceptional and unexplained phenomena which we know not how to regard. They are the outstanding facts upon which the labours of investi- gators must be exerted, — the ore from which the gold of future discovery is to be extracted. It might be thought that, as om- knowledge of the laws of nature increases, the number of such exceptions should decrease ; but, on the contrary, the more we know the more there is yet to learn and explain. This arises from several reasons ; in the first place the principal laws and forces in nature are numerous, so that he who bears in mind the wonderfully large numbers developed in the doctrine of combinations, will anticipate the existence of almost infinitely nume- rous relations of one law to another. When we are once in possession of a law, we are potentially in possession of all its consequences ; but it does not follow that tJie mind of man, so limited in its powers and capacities, can actu- ally work them all out in detail. Just as the aberration of light was discovered empirically, though it should have been foreseen, so there are doubtless multitudes of unex- plained facts, the connexion of which with laws of nature already known to us, we should perceive, were we not hindered by the imperfection of our deductive powers. But, in the second place, as will be more fully pointed out, it is not to be supposed that w^ have in any degree approximated to an exhaustion of nature's powers. The by Google EXCEPTIONAL PHENOMENA . 339 most familiar facts may teem with indicationB of forces, now secrets bidden fix)m ub, because we have Dot mind- directed eyes to discriminate them. The progress of science will conaflt in the discovery from time to time of new exceptional phenomena, and their assignment by degrees to one or other of the heads already described. When a new fact proves to be merely a false, apparent, singular, divergent, or accidental exception, we may gain a more minute and accurate acquaintance with the effects of certain laws already known to exist. We have indeed no addition to what was implicitly in our possession, but, as already explained, there is much difference between knowing the laws of nature and percei\'ing all their com- plicated effects. Should a new fact prove to be a limiting or real exception, we have to alter, in part or in whole, our views of nature and are saved from eiTors into which we had fallen. Ljistly, the new fact may cdme under the sixth class, and may eventually prove to be a novel and unexplained phenomenon, indicating the existence of new laws and forces, complicating but not otherwise interfering with the effects of laws and forces previously known. The best instance which I can find of an unresolved ex- ceptional phenomenon, consists in the anomalous vapour- densities of phosphorus, arsenic, mercury, and cadmium. It is one of the most important laws uf chemistry, dis- covered by Gay-Lu8sac, that equal volumes of gases exactly correspond to equivalent weights of the substances, and this holds generally true of any elements which we can convert into gas or vapour. Unfortunately phosphorus and arsenic give vapours exactly twice as dense as they should do by analogy, and mercury and cadmium diverge in the other direction, giving vapours half as dense as we should expect. We cannot treat these anomalies as limit- ing exceptions, and say that the law holds true of sub- z 2 Digitized by Google 340 THE PRINCIPLES OF SCIENCE. stances generally but not of these ; for the propertieB of gaaee, aa previouslv noticed (vol. ii. p. 250), usually admit of the surest and widest generalizations. Besides, the preciseness of the ratio of divergence pointa to the real observance of the law in a modified manner. We might endeavour to reduce the exceptions by doubling the atomic weights of phosphorus and arsenic, and halving those of mercury and cadmium. But this step has of course been maturely considered by chemists, and is found to conflict with all the other analogies of the substances and the principles of isomorphism. One of the most probable ex- planations is that phosphorus and arsenic produce vapour in an aUotropic condition, which might perhaps by intense heat be resolved into a simple gas of half the density ; but facta are wholly wanting to support this hypothesis, and it cannot be applied to the other two exceptions without supposing that gases and vapours generally are capable of resolution into something simpler. In short, chemists can at present make nothing of these anomalies. As Hofmann distinctly says, ' Their philosophical inter- pretation belongs to the future . . . They may turn out to be typical facts, round which many others of the like kind may come hereafter to be grouped ; and they may prove to be allied with special properties, or dependent on particular conditions as yet unsuspected".' The expansion of solids and liquids by heat is also a general law, in which we cannot expect to find any real anomaUes, any facts indicating too wide generalization, or even any accidental disturbing causes. The con- traction of water and several other liquids, even of fusible metal, by heat, t<^ther with the few cases in which a solid contracts by heat, must therefore be probably re- garded as results of the veiy law of exptmsion acting in a ■ HofmanD'a, 'Introduction to Chemistry,' p. 198. Digitized by Google EXCEPTIONAL PHENOMENA. 341 complicated and disguised manner. It would be easy to point out an almost infinite number of other unex- plained anomalies. Physicists assert, as an absolutely universal law, that in liquefaction heat is absorbed ^ yet sulphur is at least an apparent exception. The two substances, Sulphur and Selenium, are re- markable for their relations to heat. Sulphur may almost be said to have two melting points, for, though liquid like water at 120' C, it becomes quite thick and tenacious between 221° and 249°, melting once again at higher temperatures. As well as the other element named, it may be thrown into several curious states, which chemists conveniently dispose of by calling them aXlotropic, a term freely used when they are puzzled to know what has happened. The chemical and physical history of iron, again, is full of anomalies ; not only does it undergo inexplicable changes of hardness and texture in its alloys with carbon and other substances, but it is almost the only substance which conveys sound with greater velocity at a higher than at a lower tem- peratiu"e, the velocity increasing from 20° to 100° C, and then decreasing. Silver is also anomalous in regard to sound. These are all instances of inexplicable exceptions, the bearing of which must be ascertained in the future progress of science. When the discovery of new and peculiar phenomena conflicting with our theories of the constitution of nature is reported to us, it becomes no easy task to steer a philo- sophically correct course between credulity and scepticism. "We are not to assume, on the one hand, that there is any limit to the wonders which nature can present to us. Nothing except the contradictory is really impossible, and many things which we now regard as common-place were * Stewart's 'Elementary Treatise on He>t,' p. 80. Digitized by Google 342 THE PHINCIPLES OF SCIENCE. conradered ae little short of the miraculous when first perceived. The electric telegraph was a visionary dream among mediseval physicists ; it has hardly yet ceased to excite our wonder; to our descendants centuries hence it will probably appear inferior in ingenuity to some inventions which they will possess. Now every strange phenomenon may be a secret spring which, if rightly touched, will open the door to new chambers in the palace of nature. To refuse to believe, then, in the occurrence of anything new and strange would be to neglect the most precious chances of discovery. We may say with Hooke that ' the believing strange things possible may perhaps be an occasion of taking notice of such things as another would pass by without regard as useless.' We are not, therefore, to shut our ears even to such apparently absurd stories as those concerning second sight, clairvoyance, animal magnetism, ode force, table-turning, or any of the popular delusions which from time to time are current. The facts recorded concerning these matters are facts in some sense or other, and they demand explanation, either as new natural phenomena, or as the results of combined credulity and imposture. Most of the statements con- cerning the supposed phenomena referred to have been, or by careful investigation would doubtless be, referred to the latter head, and the absence of any appearance of scientific ability or care in many of those who describe them, is sufficient to cast a doubt upon their value. It is mainly upon this ground, and not on account merely of the strangeness and intrinsic improbability of the state- ments made that we shoidd hesitate to accept them. Cer- tainly in the obscure phenomena of mind, those relating to memory, dreams, somnambulism, and other peculiar actions or states of the nervous system, there are many inexplicable and almost incredible fects, and it is equally unphiloBophical to believe or to disbelieve without cletu- by Google EXCEPTIONAL PHENOMENA. 343 evidence. There are many facts, too, concerning the instincts of animals, and the mode in which they find their way from place to place, which are at present quite inex- plicable. We may always feel sure that there are many things not yet dreamt of in oxir philosophy. by Google CHAPTER XXX. CLASSIFICATION. The extensive subject of Classification has been deferred to a late part of this treatise, because it involves many questions of difficulty, and did not seem naturally to fall into any earlier place. But it must not be supposed that, in now formally taking up the subject, we are for the first time entertaining the notion of classification. All logical inference involves classificatioD, which is indeed the Deces- sary accompaniment of the action of judgment. It is impossible to detect a point of similarity between two or more objects without thereby joining them together in thought, and thus forming an incipient or potential class. Nor can we ever bestow a common name upon two or more objects without thereby equally implying the exis- tence of a class. Every common name is the name of a class, and every name of a class is a common name. It is evident also that every general notion, or concept is but another way of speaking of a class. Usage alone leads us to use the word classification in some cases and not in others. We are said to form the general notion paraUdo- gram when we regard an infinite number of possible four- sided rectilinear figures as resembling each other in the conmion property of possessing parallel sides. We should be said to form a class, Trilobite, when we place alongside of each other in a museum a number of hand specimens resembling each other in certain defined qualities. But by Google CLASSIFICA TION. ^ 346 the lo^cal nature of the operation is, or should be, exactly the same in both cases. We form a class of figures called parallelograms, and we form a general notion of Trilo- bites. Science, it has been said at the outset, is the detection of identity, and classification is the placing together, either in thought or in actual proximity of space, those notions or objects between which identity has been detected. Ac- cordingly the value of classification is co-extensive with the value of science and general reasoning. Whenever we form a class we reduce multiplicity to unity, and detect, as Plato said, the one in the many. The result of such classification is to yield generalized knowledge, as distinguished from the direct and sensuous knowledge of particular fects. Of every class, so far as it is correctly formed, the great principle of substitution is true, and whatever we know of one object in a class we also know of the other objects, so far as identity has been detected between them. The facilitation and abbreviation of mental labour is at the bottom of all mental progress. The reasoning faculties of Newton were not diflferent in quali- tative character from those of a ploughman ; the difference lay in the extent to which they were exerted, and the number of &cte which could be treated. Evety thinking being generalizes more or less, but it is the depth and extent of bis generalizations which distinguish the philo- sopher. Now it is the exertion of the classifying and generalizing powers which thus enables the intellect of man to cope in some degree with the infinite number and variety of natural phenomena and objecta In the chapters upon Combinations and Permutations it was rendered quite evident, that from a few elementary differences immense numbers of various combinations can be produced. The process of classification enables us to resolve these com- binations, and refer each one to its place according to by Google 346 THE PRINCIPLES OF SCIENCE. one or other of the elementary circumstances out of which it was produced. We restore nature, as it werej to the simple conditions oat of which its endless variety was developed. As Professor Bowen has excellently said*, ' The first necessity which is imposed upon us by the constitution of the mind itself, is to break up the infinite wealth of Nature into groups and classes of things, with reference to their resemblances and affinities, and thus to enlarge the grasp of our mental feculties, even at the expense of sacrificing the minuteness of information which can be acquired only by studying objects in detail. The first efibrts in the pursuit of knowledge, then, must be directed to the business of Classification. Perhaps it will be found in the sequel, that Classification is not only the beginning, but the culmination and the end, of human Classijicaiion Involving Induction. The purpose of classification must always be the detec- tion of resemblances and laws of nature. However much the process may in some cases be disguised, classification is not really distinct from the process of perfect induction, whereby we endeavour to ascertain the connexions which exist between the several properties of the objects under treatment. There can be no use in placing an object in a class unless something more than the fact of being in that class is thereby implied. If we arbitrarily formed a class of metals and placed therein a selection from the list of known metals made by the ballot — we should have no reason to expect that the metals in question would re- semble each other in any points except that they are • ' A Treatise on Logic, or, the Laws of Pure Thought,' by Francis Bowen, Professor of Moral Philosophy ia Harvard College, Cambridge, United States, 1866, p. 315. by Google CLAS8IFICA TION. 34 7 metals, and have been selected by the ballot But when chemists carefully selected from the list the five metaJs, Potassium, Sodium, Csesium, Rubidium, and Lithium, and called them the Alkaline metals, a great deal was implied in this classification. On comparing the qualities of these substances, they are all found to combine veiy energetically with oxygen, to decompose water at all temperatures, and to form strongly basic oxides, which are very soluble in water, yielding powerfully caustic and alkaline hydrates from which water cannot be expelled by heat. Their carbonates are also soluble in water, and each metal forms only one chloride. It may also be ex- pected as a general rule that each salt into which one of the five metals enters will correspond to salts into which the other metals enter, there being a general analogy between the properties and compounds of these metals. Now in forming this class of alkaline metals, we have done more than merely select a convenient order of statement. We have arrived at a diacoveiy of certain empirical laws of nature, the probability being very con- siderable that a metal which exhibits some of these pro- perties will also possess the others. If we discovered another metat whose carbonate was soluble in water, and which energetically combined with water at all tem- peratures, producing a strongly basic oxide, we should infer that it would form only a single chloride, and that, generally speaking, it would enter into a series of compounds corresponding to the salts of the other alkaline metals. The formation of this class of alkaline metals, then, is no mere matter of convenience ; it ls an important and highly successful act of inductive dis- covery, enabling us to register many undoubted propo- sitions as results of perfect induction, and to make an almost indefinite series of inferences depending upon the principles of imperfect induction. by Google 348 THE PRINCIPLES OF SCIENCE. ProfeBsor Huxley has defined the process of classifica- tion in the following terms \ ' By the classification of any series of objects, is meant the ^tual or ideal arrange- ment together of those which are like and the separation of those which are unlike ; the purpose of this arrange- ment being to facilitate the operations of the mind in clearly conceiving and retaining in the memory the cha- racters of the objects in question.' This statement is doubtless correct, so far as it goes, but it does not include all that Professor Huxley himself implicitly treats under classification. He is fully aware that deep correlatious, or in other terras deep imifonni- tiee or laws of nature, will be disclosed by any well chosen and profound system of dasfiification. I should therefore propose to modify the above statement, as fol- lows : — ' By the classification of any series of objects, is meant the actual or ideal arrangement together of those which are like and the separation of those which are unlike, the purpose of this arrangement being, primarily, to disclose the correlations or laws of union of proper- ties or circumstances, and, secondarily, to facilitate the operations of the mind in clearly conceiving and retain- ing in the memory the characters of the objects in question.' Multiplicity of Modes of Classijlcation. In approaching the question how any given group of objects may beat be classified, let it be remarked that there must generally be an unlimited number of modes of classifying any group of objects. Misled, as we shall see, by the problem of classification in the natural sciences, philosophers otlen seem to think that in each subject there must be one essentially natural classification which *> ' Lectnraa on the Elemeute of Compai&tire Anatomy,' 1S64, p. i. by Google CLASSIFICATION, 349 is to be selected, io the exclu^on of all others. This erroneous notion probably proceeds also in part from the limited powers of thought and the inconvenient mechani- cal conditions under which we labour. If we arrange the books in a library catalogue, we must arrange them in some one order ; if we compose a treatise on mineralogy, the minerals must be successively described in some one arrangement ; if we describe even such simple things as geometrical figures, they must be taken in some fixed order. We shall naturally therefore select that classification which appears to be most convenient and instructive for our principal purpose. Bui it does not follow that this system of classification possesses any exclusive excellence, and there will be usually many other possible arrange- ments, each valxiable in its own way. A perfect intel- lect would not confine itself to one order of thought, but would simultaneously regard a group of objects as classified in all the ways of which they are capable. Thus the elements may be classified according to their atomicity into the groups of Monads, Dyads, Triads, Tetrads, Pentads, and Hexads, and this is probably the most instructive clasfdfication ; but it does not prevent us from also classifying them according as they are metallic or non-metallic, solid, liquid or gaseous at ordi- nary temperatures, useful or useless, abuudant or scarce, ferro-magnetic or diamagnetic, and so on. Mineralogists have spent a great deal of labour in trying to discover a so-called natural system of classifi- cation for minerals. They have constantly encountered the difficulty that the chemical composition did not run together with the crystallographic form, and the various physical properties of the minend. Substances identical in the form of their crystals, especially those belonging to the first or cubical system of crystals, were often found to have no resemblance in chemical compo- Digitized by Google 350 THE PRINCIPLES OF SCIENCE. sition. The identically same substance, again, is occa- sionally found crystallized in two essentially different cryBtallographic forms ; calcium carbonate, for instance, appearing as calc-spar and arragonite. Now the simple truth is that if we are imable to discover any correspond- ence, or, as we shall call it, any correlation between the several properties of a mineral, we cannot make any one arrangement which will enable us to treat at any one time all these properties. We must reaJly classify mine- rals in as many different methods as there are different unrelated properties of sufficient importance. Even if, for the purpose of describing minerals successively in some one older in a treatise, we select one system, that, for instance, having regard to chemical composition, we ought mentally at least to regard the same minerals as classified in all other possible modea Exactly the same may be said of the classification of plants. An immense number of different modea of classi- fying plants have been proposed at one time or other, an exhaustive account of which wiU be found in Rees' ' Cyclopaedia,' article ' Classification,' or in the Introduc- tion to Lindley's ' Vegetable Kingdom.' There have been the Fmctistte, such as Cffisalpinue, Morison, Hermann, Boerhaave or Gaertner, who airauged plants according to the form of the fruit. The Corollistse, Eivinus, Lud- wig, and Tournefort, paid attention chiefly to the number or arrangement of the parts of the corolla. Magnol se- lected the calyx as the critical part, whUe Sauvage arranged plants according to their leaves ; nor are these instances more than a small selection from the actual variety of modes of classification which have been tried. Of such attempts it may be said that every proposed sys- tem will probably yield some information concerning the relations of plants, and it is only after trying many modes that it is possible to approximate to the best. by Google CLASSIFICATION. Natural and Artificial Systems of Classification. It haa been usual to distinguish systems of classifica- tion as natural and artificial, those being called natural which seemed to express the order of existing things as determined by nature. Artificial methods of classification, on the other hand, included those formed for the mere convenience of men in remembering or treating natural The difference, as it is commonly regarded, has been well described by Ampfere*, as follows : ' We can distinguish two kinds of classifications, the natural and the artificial. In the latter kind, some characters, arbitrarily chosen, serve to determine the place of each object ; we abstract all other characters, and the objects are thus found to be brought near to or to be separated from each other, often in the most bizarre manner. In nattural systems of classi- fication, on the contrary, we employ concurrently all the characters essential to the objects with which we are occupied, discussing the importance of each of them ; and the results of this labour are not adopted unless the objects which present the closest analogy are brought most near together, and the groups of the several orders which are formed from them are also approximated in pro- portion as they offer more similar characters. In this way it arises that there is always a kind of connexion, more or less marked, between each group and the group which follows it.' There is much, however, that is vague and logically false in this and many other definitions which have been proposed by naturalists to express their notion of a natural system. We are not informed how the import- " ' Essai sur la Philoaophle dea Sciences', p. 9. Digitized by Google 362 THE PRINCIPLES OF SCIENCE. once of a resemblance is to be determined, nor what iB the meaBTU* of the closeness of analogy. Until all the words employed in a definition are made clear in meaning, the definition itself is worse than useless. Now if the views concerning classification here upheld are true, there can be no sharp and precise distinction between natural and artificial systems. All arrangements which serve any purpose at all must be more or less natural, because, if closely enough scrutinized, they will involve more resem- blances than those whereby the class was defined. It is true that in the biological sciences there would be one arrangement of planta or animals which would be couspiaiously instructive, and in a certain sense natural, if it could be attained, and it is that after which natural- ists have been in reality striving for nearly two centuries, namely, that arrangement which would display the genea- logical descent of every form fcom the original life germ. Those morphological resemblances upon which the classi- fication of living beings is almost always based are in- herited resemblances, and it is evident that descendants will usually reeemble their parents and each other in a great many points. I have said that a natural is distinguished from an arbitrary or artificial system only in degree. It will be found almost impossible to arrange objecta according to any one circumstance without finding that some correla- tion of other circumstances is thus made apparent. No arrangement could seem more arbitrary than the common alphabetical arrangement according to the initial letters of the name. But we cannot scrutinize a list of names of persons without noticing a predominance of Evans's and Jones's, imder the letters E and J, and of names bt^nning with Mac under the letter M. The predomi- nance is so great that we could not attribute it to chance, and inquiry would of course show that it arose from im- Digitized by Google CLASSIFICATION. 353 portant facts concerning the nationality of the persons. It would appear that the Evans's and Jones's were of Welsh descent, and those whose names bear the prefix Mac of Scotch descent. With the nationality would be more or less strictly correlated many peculiarities of physical con- stitution, language, habits, or mental character. In other cases I have been interested in noticing the empirical inferences which are displayed in the most apparently arbitrary arrangements. If a large register of the names of ships be examined it will often be found that a number of ships bearing the same name were buUt about the same time, a correlation due to the occurrence of some striking incident shortly previous to the building of the ships. The age of ships or other structures is usually closely cor- related with their general form, nature of materials, &a It is impossible to examine the details of some of the most apparently artificial systems of classification of plants, without finding that many of the classes are natural in character. Thus in Tournefort's arrangement, depending almost entirely on the formation of the corolla, we find the natural orders of the Labiatse, Cruciferse, Rosaceee, XJmbellifene, LiliaceEe, and Papilionaceae, recognise J in his 4th, 5th, 6th, 7th, 9th, and lOth classes. Many of the classes in Linnaeus' celebrated sexual system also approxi- mate to natural classes. Correlation of Properties. Habits and usages of language are always apt to lead us into the error of imagining that when we employ different words we mean different things. In introducing the subject of classification nominally I was careful to draw the reader's attention to the feet that all reasoning and all operations of scientific method really involve classification, though we are accustomed to use the name VOL. 11. A a by Google 354 THE PHINCTPLES OF SCIENCE. in some cases and not in others. Now the name correla- tion requires to be used with the same qualification. Things are correlated {con, relata) when they are so re- lated or bound to each other that where one is the other is, and where one is not the other is not. Throughout this work we have then been dealing with correlations. In geometry the occurrence of three equal angles in a triangle is correlated with the existence of three equal sides; in physics gravity is correlated with inertia; in botany exogenous growth is correlated with the posses- sion of two cotyledons, or the production of flowers with that of spiral vessels. But it ia in the classificatory sciences especially that the word correlation has been em- ployed. We find it stated that in the class Mammalia IJie possession of two occipital condyles, with a well-ossified basi-occipital, is correlated with the possession of man- dibles, each ramxis of which is composed of a single piece of bone, articulated with the squamosal element of the skull, and also with the possession of mammte and non- nucleated red blood-corpuscles. Professor Huxley remarks'* that this statement of the character of the class mammalia is something more than an arbitrary definition ; it is a statement of a law of correlation or co-existence of animal structures, from which most important conclusions are deducible. It involves a generalization to the effect that in nature the structures mentioned are always found associated together. This simply amounts to saying that the formation of the class mammalia involves an act of inductive discovery, and residts in the establishment of certain empirical law^ of nature. Professor Huxley has excellently expressed the mode in which discoveries of this kind entible naturalists to make deductions or predictions ^ ' Lectures on the Elements of ComparatiTe Anotomy, and on the Classification of Aniaials,' 1864, |). 3. Digitized by Google CLA SSI PICA TION. 355 with considerable confidence, but he has also pointed out that such inferences are likely from time to time to prove mistaken. I will quote his own words : ' If a fragmentary fossil be discovered, consisting of no more than a ramus of a mandible, and that part of the ekxill with which it articulated, a knowledge of this law may enable the palasontologist to aflSrm, with great con- fidence, that the animal of which it formed a part suckled its young, and had non-nucleated red blood-cor- puscles ; and to predict that shoidd the back part of that skull be discovered, it will exhibit two occipital condyles and a well-ossified basi-occipital bone. 'Deductions of this kind, such as that made by Cuvier in the famous case of the fossil opossum of Montmartre, have often been verified, aud are well calculated to im- press the vulgar imagination ; so that they have taken rank as the triumphs of the anatomist. But it should carefully be borne in mind, that, like all merely empirical laws, which rest upon a comparatively narrow observa- tional basis, the reasoning from them may at any time break down. If CuA-ier, for example, had had to do with a fossil Thylacinus instead of a fossil Opossum, he would not have found the marsupial bones, though the inflected angle of the jaw would have been obvious enough. And 80, though, practically, any one who met with a character- istically mammalian jaw would be justified in expecting to find the characteristically mammalian occiput associ- ated with it ; yet, he would be a bold man indeed, who should strictly assert the belief which is implied in this expectation, viz., that at no period of the world's history did animals exist which combined a mammalian occiput with a reptilian jaw, or vice versd.' One of the most distinct and remarkable instances of correlation in the animal world is that which occurs in ruminating animals, and which could not be better stated A a 2 Digitized by Google 356 THE PRINCIPLES OF SCIENCE. than in the following extract from the classical work of Cuvier*: ' I doubt if any one would have divined, if imtaught by observation, that all t^iminants have the foot cleft, and that they alone have it. I doubt if any one would have divined that there are frontal horns only in this class : that those among them which have sharp canines for the meet part lack home. * However, since these relations are constant, they must have some sufficient cause ; but since we are ignorant of it, we must make good the defect of the theory by means of observation : it enables ub to establish empirical laws which become almost as certain as rational laws when they rest on suflSciently repeated observations; so that now whoso sees merely the print of a cleft foot may con- clude that the animal which left this impression rumi- nated, and this conclusion is as certain as any other in physics or morals. This footprint alone, then, yields to him who observes it, the form of the teeth, the form of the jaws, the form of the vertebrae, the form of all the bones of the legs, of the thighs, of the shoulders, and of the pelvis of the animal which has passed by : it is a surer mark than all those of Zadig,' We meet with a good instance of the purely empirical correlation of circumstances when we classify the planets of the solar system according to their densities or periods of axial rotation'. If we examine a table specifying the usual astronomical numbers of the solar system, we find that four planets resemble each other very closely in the period of axial rotation, and the same four planets are all found to have high densities, thus : — • 'Oraemenii Fosules,' 4tL edit vol. i. p. 164. Quoted by Huxlej, ' Lectures,' Ac, p. 5. f Chambers, ' Descriptive ABtronomy,' ist edit. p. 23, by Google CLASSIFICATION. 367 N»meof PUnet. Poriod of Aii&l RoUtioD. Dewilj. Mercary VenuB Earth Mars 24 honra 5 minutes 34 .. 37 .- ■ •■ 794 " ■■ 5-33 .. .. 5-61 .... S-84 Forming a as follows :— similar table for the other chief planeta, it is Name of PUoet. Period of AiUl BotfttioQ. DsoBity. Jnpiter 9 hours 55 minutes .. .. 1-35 Saturn .. 10 „ 29 „ .. .. -74 UranuB .- 9 „ 30 „ ., ., •97 Neptune .... — — .. ,. i-oa It will of course be observed that in neither group is the equality of the rotational period or of the density more than rudely approximate, nevertheless the difference of the numbers In the first and second group is so very marked, the periods of the first being at least double and the densdties four or five times those of the second, that the coincidence cannot be attributed to accident. The reader will also notice that the first group consists of the planets nearest to the sun, that with the exception of the earth none of them possess sateUites, and that they are all comparatively small ; the second group are furthest from the sun, and all of them possess several aatellites, and are comparatively great. - Therefore, with but slight exception, tlie following correlations hold true : — These coincidences certainly point with much proba- biUty to a difierence in the conditions of origin of the two groups, but no further explanation of the matter is yet possible. The classification of comets by Mr, Hind and Mr. A. S. Davis according to their periods tends to establish the conclusion that distinct groups of comets have been by Google 358 THE PRINCIPLES OF SCIENCE. brought into the solar system by the attractive powers of Jupiter, Uranus, or other planets P. The classification of nebulse as commenced by the two Herschels, and con- tinued by Lord Rosse, Mr. Hu^ins, and others, will probably lead at some future time to tlie discovery of important empirical laws concerning the constitution of the universe. The minute examination and classilication of meteorites, as carried on by Mr. Sorby and others, seems likely to afford us an insight into the constitution of the material universe. We should never fail to remember and record the slightest and most apparently inexplicable coincidences or correlations, for they may prove of importance in tlie future. Discoveries begin when we are least expecting them. It is a very significant fact that the greater number of variable stars are of a reddish colour. Not all variable stars are red, nor all red stars variable, but considering that only a small fraction of the observed stars are known to be variable, and only a small iraction are red, the number which fall into both classes is too great to be accidental •". It is also remarkable that the greater number of stars possessing great proper motion are double stars, the star 6i Cygni being especially noticeable in this respect'. The correlation in these cases is not perfect and without exception, but the preponderance is so great as to point to some natural correlation, the exact nature of which must be a matter for future investigation. Sir John Herschel has remarked that the two double stars 6 1 Cygni and a Centauri of which the orbits were well ascertained, evidently belonged to the same family or genus''. 8 ' Pliilosophical Magazine,' 4th Series, vol. xsjtix. p. 396; vol, xl. p. 183; vol. xli. p. 44. h Humboldt, ' CosmoH,' (Bolin) vol. iii. p. 124. ' Baily, ' British Aaaociation Catalogue,' p. 48. ^ ' Outlines of Astronomy,' § 850, 4th edit, p, 578. Digit zed by Google CLASSIFICATION. Classification in Crystallography. One of the most perfect and inatructive instances of classification which we can find is furnished by the science of cry8taUc^;raphy, already briefly noticed (vol i. p. 153). The system of arrangement now generally adopted is conspicuously natural, and is even mathematically perfect. A crystal consists in every part of similar molecules simi- larly related to the adjoining molecules, and connected with them by forces the nature of which we can only learn by their apparent effects. But these forces are exerted in space of three dimensions, so that there is a limited number of suppositions which can be entertained as to the relations of these forces. In one case each mole- cule will be similarly related to all those which are next to it ; in a second case, it will be similarly related to those in a certain plane, but differently related to those not in that plane. In the simpler cases the arrangement of molecules is rectangular ; in the remaining cases oblique either in one or two planes. In order to simplify the explanation and conception of the complicated phenomena which crystals exhibit, an hypothesis has been invented which is an excellent illus- tration of the class of Descriptive Hypotheses before men- tioned (vol. ii. p. 153). Crystallographers imagine that there are within each crystal certain axes, or lines of direction, by the comparative length and the mutual inclination of which the nature of the crystal is deter- mined and recorded. In one somewhat exceptional class of crystals there are three such axes lying in one plane, and a fourth perpendicular to that plane ; but In all the other classes there are imagined to be only three axes. Now these axes can be varied in three ways as regards by Google 360 THE PRINCIPLES OF SCIENCE. length : (i) they may be all equal, or (2) two equal and one unequal, or (3) all unequal. They may also be varied in four ways as regards direction : (i) they may be all at right angles to each other; (2) two axes may be at right angles and the third perpendicular to one of them and oblique to the other ; (3) two axes may be at right angles to each other and the third oblique to both ; (4) the three axes may be all oblique to each other. Now if all the variations as regards length were combined with those regarding direction, it would seem to be possible to have twelve classes of crystals in all, the enumeration being then logically and geometrically complete. But as a matter of empirical observation, many of these classes are not found to occur, oblique axes being seldom or never equal. There remain in all seven distinct classes of crystals, but even of these one class is not positively known to be represented in nature. The first class of crystals is defined by possessing three equal rectangular axes, and equal elasticity in all direc- tions. The primary or most simple form of the crystals is the cube , but by the modification or removal of the comers of the cube by planes variously inclined to the axes, we have the reguljir octohedron, the dodecahedron, or various combinations of these forms. Now it is a law of this class of crystals that as each axis is exactly like each of the other two, every modification of any corner of a crystal must be repeated symmetrically with regard to the other axes ; thus the forme produced are symmetri- cal or regular, and the class is called the Regular System of Crystals. It includes a great variety of substances, some of them being elements, such as carbon in the form of diamond, others more or less complex compounds, such as rock-salt, potassium iodide and bromide, the several kinds of alum, fluor-spar, iron bisulphide, garnet, spinelle, &C. No correlation then is apparent between the form of by Google CLASSIFICATION. 361 crystallization and the chemical composition. But what we have to notice is that the physical properties of the crystallized suhstances with regard to light, heat, elec- tricity, &c., are closely similar. Light and heat unduW tions, wherever they enter a crystal of the regular system, spread with equal rapidity in all directions, just as they would in a uniform liquid, gas, or amorphous solid, such as unstrained glass. Crystals of the regular systera accord- ingly do not in any case exhibit the phenomena of double refraction, unless by mechanical compression we alter the conditions of elasticity. These crystals, again, expand equally in all directions when heated, and if we could cut a sufficiently large plate from a cubical crystal, and ex- amine the sound vibrations of which it is capable, we should find that they indicated an equal elasticity in every direction. Thus we see that a great number of important properties are correlated with that of crys- tallizing in the regular system, and as soon hja we know that the primary form of crystallization of a substance is the cube, we are able to infer with approximate cer- tainty that it possesses all these properties. The class of cubical crystals is then an evidently natural class, one disclosing general laws connecting together the physical and mechanical properties of the substances so classified. In the second class of crystals, called the dimetric, square prismatic, or pyramidal system, there are also three axes at right angles to each other, two of which are equal, and the third or principal axis is unequal, being either greater or less than either of the other two. In such crystals accordingly the elasticity and other physical properties are alike in all directions perpendicular to the principal axis, but vary in all other directiona If a point within a crystal of this system be heated, the heat spreads with equal rapidity in planes perpendiciUar to the prin- Digitized by Google 362 THE PRINCIPLES OF SCIEAXE. cipal axis, but more or less rapidly in the direction of this axis, BO that the isothermal surface is an ellipsoid of revo- lution round that axis. Nearly the same statement may be made concerning the third or hexagonal or rbombohedral system of crystals, in which there are three axes lying in one plane and meeting at angles of 60°, while the fourth axis is perpendicular to the other three. The hexagonal prism and the rhombobedron are the two commonest forma assumed by crystals of this system, and in ice, quartz, and calc-spar, we have abundance of beautiful specimens of the various forms produced by the modification of the primitive form. Calc-spar alone is said to crystallize in at least 700 varieties of forms. Now of all the crystals belonging both to this and the dimetric class, we know that a ray of light passing in the direction of the prin- cipal axis will be refracted singly as in a crystal of the regular ejspim ; but in every other direction the light will suffer double reiraction being separated into two rays, one of which obeys the ordinary law of refraction, but the other a much more complicated law. The other physical properties- vary in an analogous manner. Thus calc-spar expands by heat in the direction of the principal eixis, but contracts by a small quantity in directions perpendicular to it. So closely indeed are these various physical pro- perties correlated that Mitscberlich, having observed the law of expansion in calc-spar, was enabled to predict that the double refracting power of the substance would be de- creased by a rise of temperature, as was proved by expe- riment to be the case. In the fourth system, called the trimetric, rhombic, or right prismatic system, there are three axes, at right angles, but all unequal in length. It may be asserted in general terms that the mechanical properties vary in such crystals in every direction, and heat spreads so that by Google CLASSIFICA TION. 363 the isothermal surface is au ellipsoid with three unequal axes. In the remaining three classes, called the monoclinic, diclintc, and triclinic, the axes are more or less oblique, as described above {vol, ii. p. 360), and at the same tlrae unequal. The complication of phenomena is therefore greatly increased, and it need only be stated that there are always two directions in which a ray is singly re- fracted, but that in all other directions double refraction takes place. The conduction of heat is unequal in all directions, the isothermal surface, being an ellipsoid of three unequal axes. The relations of such crystals to other phenomena are often very complicated, and hardly yet reduced to law. Thus some crystals, called pyro-electric, manifest vitreous electricity at some points of their sur- face, and resinous electricity at other points when rising in temperature, the character of the electricity being changed when the temperature sinks again. This pro- duction of electricity is believed indeed to be connected with the hemihedral character of the crystals exhibiting it. The crystalline structure of a substance again influ- ences its magnetic behaviour, the general law being that the direction in which the molecules of .a crystal are most closely approximated tends to place itself axially or equa- torially between the poles of a magnet, according as the body is magnetic or diamagnetic. .Further questions arise if we apply pressure to crystals. Thus doubly refracting crystals with one principal axis acquire two axes when the pressure is perpendicular in direction to the principal axis. All the phenomena peculiar to crystalHDe bodies are thus closely correlated with the formation of the crystal, or will almost certainly be found to be so as iovestigation proceeds. It is upon empirical observation indeed that the laws of connexion are in the first plivce founded, but by Google 364 THE PRINOIPLBS OF SCIENCE. the simple hypothesis that the elasticity and approxima- tion of the particles vary in the directions of the crystalline axes allows of the application of deductive reasoning. The whole of the phenomena are gradually being proved to be consistent with this hypothesis, so that we have in this subject of crystallography a beautiful instance of successful classification, connected with a nearly perfect physical hypothesis. Moreover this hypothesis was veri- fied experimentally as regards the mechanical vibrations of sound by Savart, who found that the vibrations in a plate of biaxial crystal indicated the existence of varying elasticity in varying directions. Classijication an Inverse and Tentative Operation. If all attempts at so-called natural classification be really attempts at perfect induction, it follows that they are all subject to the remarks which were made upon the inverse character of the inductive process, and upon the difficulty of every inverse operation (vol. i. pp. 14, 15, 140, Ac). There will of necessity be no royal road to the discovery of the best system, and it will even be im- possible to lay down any series of rules of procedure to assist those who are in search of a good arrangement. The only invariable logical rule which could be stated would be as follows : — Having given certain objects, group them in every way in which they can be grouped, and then observe in which method of grouping the coincidence of properties is most conspicuously manifested. But this method of exhaustive classification will in almost every case be impracticable, owing to the immensely great number of modes in which a comparatively small number of objects may be grouped together. About sixty-three elements have been classified by chemists in six principal groups as Monad, Dyad, Triad, &c. elements, the numbers Digitized by Google CLAS3IFICA TION. 365 in the classes varying from three to twenty elements. Now if we were to calculate the whole number of ways in which axty-three objects can be arranged in six groups, we should find the number to be Bo great that the life of the longest lived man would be wholly inadequate to enable him to go through these possible groupings. The rule of exhaustive arrangement, then, is absolutely impracticable. It follows also that mere haphazard trial cannot as a general rule give any useful result. If we were to write the names of the elements in succession upon sixty-three cards, throw them into a ballot-box, and draw them out haphazard m six handfuls time after time, the probability is excessively small that we take them out at any one trial in a specified order, for in- stance that at present adopted by chemiBts. The usual mode in which an investigator proceeds to form a classification of any new group of objects, seems to consist in tentatively arranging them according to their most obvious similarities. Any two objects which present a close resemblance to each other will be joined and formed into the rudiment of a class, the definition of which will at first include all the apparent points of resemblance. Other objects as they come to our notice will be gradually assigned to those groups with which they present the greatest number of points of resem- blance, and the definition of a class will often have to be altered in order to admit them. The early chemists, for instance, could hardly avoid classing together the common metals, gold, silver.copper, lead, and iron, which present such conspicuous points of similarity as regards density, metallic lustre, malleability, &c. With the pro- gress of discovery, however, difficulties begin to present themselves in such a grouping. Antimony, bismuth, and arsenic are distinctly metallic aa regards lustre, density, and some chemical properties, but are wanting in malle- by Google 366 THE PRINCIPLES OF SCIENCE. ability. The more recently discovered and rare tellurium presents greatei- difficulties, for it has many of the physical properties of metal, and yet all its chemical properties are analogous to those of sulphur and selenium which have never been regarded as metals. Great chemical differences again are by degrees discovered between the five metals just mentioned ; and the class, if it is to have any chemical validity, must be made to include other elements, having none of the original properties on which the class was founded. Hydrogen is a transparent colourless gas and the least dense of all substances, yet in its chemical ana- logies it is a metal, as suggested by Faraday™ in 1838, and almost proved by the late Profeeeor Graham ; it must be placed in the same class as silver. In this way it comes to pass that almost every classification which ia proposed in the early stages of a science will be foxmd to break down as the deeper similarities of the objects come to be detected. The most obvious points of differ- ence will have to be neglected. Chlorine is a gas, bromine a liquid, and iodine a solid, and at first sight these might have seemed formidable circumstances to overlook ; but in chemical analogy the substances are closely united. The progress of organic chemistry, too, has yielded wholly new ideas of the similarities of compounds. Who, for instance, would recognise without extensive research a close simi- larity between glycerine and alcohol, or between fatty sub- stances and ether. The class of paraffins contains three substances gaseous at ordinary temperatures, several Uquids, and some crystalline solids. It required much in- sight to detect the perfect affinity which exists between such apparently different substances. The science of chemistry now depends to a great extent on a correct classification of the elements, as will be learnt by consulting the able article on Classification by Pro- "> ' Life of Fumrfay,' vol. ji, p. 87. by Google CLASSIFICATION. 2G7 feasor G. C. Foster in Watts's 'Dictioiiaiy of Chemistry,' But the present theory of classification was not reached until at least three previous false Kj'stems had been long entertained. And though there is much reason to believe that the present system of classification according to atomicity is substantially correct, many errors may yet be discovered in the details of the grouping. Syntholic Statement of the I%eory of Classification. The whole theory of clasaification can be explained in the most complete and general manner, by reverting for a time to the use of the Logical Abecedarium, which was found to be of supreme importance in Formal Logic {vol i. p. 109). That form expresses in fact the necessary classi- fication of all objects and ideas as depending on the laws of thought, and there is no point concerning the purpose and methods of classification which may not be explained most precisely by the use of letter combinations, the only inconvenience being the somewhat abstract and repulsive form in which the subject is thus represented. If we pay regard only to three qualities or circum- stances in which things may resemble each other, namely the qualities A, B, C, then there are according to the laws of thought eight possible classes of objects. If there exist objects belonging to aU these eight classes, thus indicated, ABC oBC ABo oBe A6C oSC Ale ahe it follows that the qualities A, B, C are subject to no conditions except the primary laws of thought and nature (vol. i. p. 6). There is then no special law of nature to Digitized by Google 368 THE- PRINCIPLES OF SCIENCE. discover, and, if we arrange the classes in any one order rather than another, it must be for the purpose of showing that the combinations are logically complete. It will be obvious that there are three different possible arrange- ments which may be of some use ; firstly, that employed above in which all the combinations containing A stand first, and those devoid of it follow ; secondly, and thirdly, the similar arrangements in which the combinations con- taining B, and C, respectively stand first. Suppose now that there are but four kinds of objects possessing the qualities A, B, C, and that these kinds are represented by the combinations ABC, AiO, aBc, ahc. The order of arrangement will now be of importance ; for if we place them in the order JABO laBe rA6C Xahc plaang the B's first and those which are 6's last, we shall perhaps overlook the law of correlation of properties in- volved. But if we arrange the combinations as follows JABC lAJC raBc (.oJe it becomes apparent at once that where A is, and only where A is, the property C is to be found, B being in- differently present and absent. The second arrangement then would be called a natural one, as rendering mani- fest the conditions under which the combinations exist. As a further instance, let us suppose that eight objects "are presented to us for classification, which exhibit combi- nations of the five properties. A, B, C, D, E, in the follow- ing manner : — by Google CLASSIFICATION. 369 ABCdE aBCiffi ABcde oBcrfe A6CDE aiCDE AJbcDe oicDe. They are now classified, so that those containing A stand first, and those devoid of A second, but no other property seema to be correlated with A. Let us alter this arrange- ment and group the combinations as follows : — ABCrfE A6CDE ABcdc AftcDe aBCrfE aiCDE oBcde abcDe. It requires very little examination to discover that, in the first group, B is always present and D absent, whereas in the second group, B is always absent and D present. This is the result which follows from a law of the form B = d (see vol i, p. 157), so that in this mode of arrangement we readily discover a close correlation between two letters. Altering the groups again as follows : — kBCdE ABcde oBCdE oBcde A6CDE AftcDe oiCDE ahcDe, we discover another evident correlation between C and E. Between A and the other letters, or between the two pairs of letters B, D and C, E there is no logical connexion whatever. This example may perhaps seem tedious, but it will be found instructive in this way. We are claeafying only seven objects or combinations, in each of which only five qualities are considered. There are only two laws of cor- relation between four of those five qualities, and those laws are of the amplest logical character. Yet the reader would hardly discover what those laws were, and confi- VOL. II. B b Digitized by Google 370 THE PRINCIPLES OF SCIENCE. dently aseign them by mere contemplation of the combina- tions, as given in the Brst group. Several tentative classi- fications must probably be made before we can resolve the question. Let us now suppose that instead of seven objects and five qualities, we have, say, five hundred objects and fifty qualities. If we were to attempt the same method of exhaustive grouping which we before employed, we should have to arrange the five hundred objects in fifty different ways, before we could be sure that we had discovered even the simpler laws of correlation. But even the successive grouping of all those possessing each of the fifty properties would not necessarily give us all the laws. There might exist complicated relations between several pro}>erties simultaneously, for the detection of which no ride of pro- cedure whatever can be given. If the reader entertains any doubt as to the diflaculty of classifying combinations so as to disclose their rela- tions, let him test the matter practically upon the fol- lowing series of combinations. They involve only sis letters denoting six qualities, which are subject to four laws of correlation of no great complexity, ABCDEF ABcDe/ ABCDeF AhcdEf ABCD^/* aBcDEF ABCdE/ aBcDeF ABcDEF aBcDe/ ABcDeF abcdE./. I shall be happy to receive the solution of the above problem in classification from any reader who thinks he has solved it ; that is to say, I shall be glad to ascer- tain whether any reader succeeds in detecting the laws of correlation between the letters, which yield the above combinations, according to the principles of the Indirect Method described in Chapter VI. by Google CLASSIFICATION. Bifurcate Classification. Every system of clasBification. ought theoretically to be formed on the principles of the Logical Abecedarium. Each superior dasa should be divided into two inferior classes, distinguished by the possession and non-posaession of a single specified property. Each of these minor classes, again, is divisible by any other property whatever which can be sug^ated, and thus every classification logically consists of an infinitely extended series of subaltern genera and species. The classifications which we form are in reality very small fragments of those which would correctly and fully represent the relations of existing things. But if we take more than four or five qualities into account, the number of subdivisions grows imprac- ticably large. Our finite minds are unable to treat any complex group exhaustively, and we are obliged to simplify and generalize scientific problems, often at the risk of overlooking particular conditiona and exceptions. Every system of classes displayed in the manner of the Logical Abecedarium may be called htfurcate, because every class branches out at each step into two minor classes, existent or imaginary. It would be a great mistake to regard this arrangement as in any way a peculiar or special method ; it is not only a natural and important one, but it is the inevitable and only system which is lo^cally perfect, according to the fundamental laws of thought. AH other arrangements of classes correspond to the bifurcate arrangement, with the implication that some of the minor classes are not represeuted among existing things. If we take the genus A and divide it into the species AB and AC, we imply two propositions, namely that in the class A, the properties of B and G B b 2 DigitizedbyGOOgle 372 THE PRINCIPLBS OF SCIENCE, never occur together, and that they are never both absent ; ■these propositions are indeed logically equivalent to one, namely AB= Ac. Our classification is then identical with the following bifurcate one : — If, again, we divide the genus A into three species AB, AC, AD, we are either logically in error, or else we must be understood to imply the existence of three propositions excluding the union within the genus A of the properties of B. C and D, namely AB = ABcd, AO = A&Cd, and AD=A6cD. It comes to the same thing if we say that our classification is really a bifurcate one, as follows :— aIc" A^CDAicd a£d aL^ A^CD khCd khcD Abed = =o -o =o * =0 The lo^cal necessity of bifurcate classification has been clearly and correctly stated in the 'Outline of a New System of Logic' by George Bentham, a work of which the logical value has been quite overlooked until lately. Mr. fientham points out, in p. 113, that every classification must be essentially bifurcate and takes, as an example, the division of vertebrate animals into four subclasses, as follows : — Mammifera — endowed with mamms and lungs. Birds without mammse but with lungs and wings. Fish deprived of lungs. Reptiles deprived of mammse and wings but with lungs. Digit zed by Google CLASSIFICA TION. 373 We have, then, ae Mr. Bentham Bays, three hifid divi- sions, thus represented : — Vertebrata I ' T-l Eadowed with lungs deprired of lungs Endowed with deprived of Ush Mammifera with wings without wings Birds Beptilee It is however quite evident that according to the laws of thought even this arrangement is incomplete. The subclass mammifera must either have wings or be deprived of them ; we must subdivide this class, or assume that none of the mammifera have wings, which is, as a matter of fact, the case, the wings of bat-s not being true wings in the meaning of the term as applied to birds. Fish, again, ought to be considered with regard to the possession of mammae and wings ; and in leaving them undivided we really imply that they never have mammte nor wings, the wings of the flying-fish, again, being no exception. If we resort to the use of our letters and define them as follows — A = vertebrata, B = having lungs, C = having mammee, D = having wings, tlien there are four existent classes of vertebrata which appear to be thus described — ABC ABcD KRcd Ah. But in reality the combinations are implied to be by Google 374 THE PRINCIPLES OF SCIENCE. ABCd = Maromifera, ABcD= Birds, ABcrf = Reptiles, khcd =Fi8h, and wo imply at the same time that the other four con- ceivable combinations containing B, C, or D, namely ABCD, A6CD, AhCd, and KbcD, do not exist in nature. The bifurcate form of classification seems to be needless when the property according to which we classify any group of things admits of numerical discrimination. It would seem absurd to arrange things according as they have one degree of the property or not one degree, two degrees or not two degrees, and so on. The elements, for instance, are classified according as the atom of each satu- rates, one, two, three or more atoms of a monad element, such as chlorine, and they are called accordingly Monad, Dyad, Triad, Tetrad elements, and so on. It would be wholly useless to apply the bifid arrangement, thus : — not-Dyad not-Triad The reason of this is that, by the very nature of number as described in Chapter VIII, every number is logically discriminated from every other number. There can thus be no logical confusion in a numerical arrangement, and the series of numbers indefinitely extended is also exhaus- tive. Every thing admitting of a property expressible in numbers must find its place somewhere in the series of numbers. The chords in music correspond to the various simpler numerical ratios and must admit of complete by Google CLASSIFICA TION. 375 exhaustive classification in respect to the complexity of the ratios forming them. Plane rectilinear figures may also be classified according to the numher of their sides as triangles, quadrilateral figiures, pentagons, hexagons, heptagons, &c. The bifurcate arrangement is not false when applied to such series of objects; it is even neces- sarily involved in the arrangement which we do apply, so that its formal statement is needless and tedious. The same may be said of the division of portions of space. Keid and Eames endeavoured to cast ridicule on the bifurcate arrangement " by proposing to classify the parts of England into Middlesex and what is not Middlesex, dividing the latter again into Kent and what is not Kent, the latter again into Sussex and what is not Sussex ; and so on. This is so far, however, from being an absurd proceeding that it is requisite to assure us that we have made an exhaustive enumeration of the parta of England. The Five Predicahles. As a general rule it is highly desirable to consign to oblivion all the ancient logical names and expressions, which have infest«d the science for many centuries past. If logic is ever to be a useful and progressive science, logicians must distinguish between logic and the history of logic. As in the case of any other science it may be desirable to examine the course of thought by which logic has, before or since the time of Aristotle, been bronght to its present state ; the history of a science is always instructive aa giving instances of the mode in which dis- coveries take place. But at the same time we ought carefully to disencumber the statement of the science " George Bentham, ' Outline of a New System of Logic,' p- i >5- DigitizedbyGOOgle 576 THE PRINCIPLES OF SCIENCE. itself of all names and other vestiges of antiquity which are not actually useful at the present day. Among those ancient expressions which may well be excepted from such consideratioDs and ever retained in use, are the 'Five Words' or 'Five Predicables' which were described by Porphyry in his ' Introduction to Aris- totle's Organum.' Two of them indeed, namely Genus and Species, are the most venerable names in philosophy, having probably been first employed in their present logical meanings by Socrates. In the present day it requires some mental effort, as Mr. Georges Lewes has remarked V to see anything important in the invention of notions now so familiar as those of Genus and Species. But in reality the introduction of such terms showed the rise of the first germs of logic and scientific method ; it showed that men were beginning to analyse their pro- cesses of thought. The Five Predicables are Genus, Species, Difference, Property, and Accident, or in the original Greek ■j-fVos, tJSot, Statpopa, 'Siov, truftfit^^Kot. Of these, Genus may be taken to mean any class of objects which is regarded as broken up into two minor classes, which form Species of it. The Genus is defined by a certain number of qualities or circumstances which belong to all objects included in the class, and which are sufficient to mark out these objects from all others which we do not intend to include. Interpreted as regards intension, then, the Genus is a group of qualities ; interpreted as regards extension, it is a group of objects possessing those quidities. If now another quality be taken into account which is possessed by some of the objects and not by tlie others, this quality becomes a Difference which divides the Genus into two Species. We may interpret the Species " 'Biograpbical HiBtory of Philosophy,' (1857) voL i. p. ia6. Orote's ' History of Greece," vol, viii. p. 578. by Google CLASSIFICA TION. 377 either in intension or extension ; in the former respect it is more than the Genua ae containing one more quality, the Difference : in the latter respect it is less than the Genus as containing only a portion of the group consti- tuting the Genus. We may say then, with Aristotle, that in one sense the Genus is in the Species, namely in inten- sion, and in another sense the Species is in the Genus, namely in extension. The Difference, it is evident, can be interpreted in intension only. A Property is a quality which belongs to the whole of a class, but does not enter into the definition of that class. Thus if it be a generic property it belongs to every indi- vidual object contained in the genus. It is a property of the genus Parallelogram that the opposite angles are equal. If we regard a Rectangle as a species of parallel- ogram, the difference being that one angle is a right angle, it follows as a specific property that all the angles are right angles. Though a property in the strict logical sense must belong to each of the objects included in the class of which it is a property, it may or may not belong to other objects. The property of having the opposite angles equal may belong to many figures besides parallel- ' ograms, for instance, regular hexagons. It is a property of the circle that all triangles constructed upon the dia- meter with the apex upon the circumference are right angled triangles, and vice versd, all closed curves of which this is true must' be cirdea We might with ad- vantage distinguish properties which thus belong to a class, and only to that class, as peculiar properties. They enable us to make statements in the form of simple iden- tities (vol. i. p. 44). Thus we know it to be a peculiar property of the circle that for a given length of perimeter it encloses a greater area than any other possible curve ; hence we may say — Curve of equal curvature = curve of greatest area. Digitized by Google 378 THE PRINCIPLES OF SCIENCE. It is a peculiar property of equilateral triangles that they are equiangular, or, vice versd, it ia a peculiar pro- perty of equiangular trianglee that they are equilateral. It is a property of crystala of the regular system that they are devoid of the power of double refraction, but this Is not a property peculiar to them, because vitreous and other amorphous transparent solids, such as glaes. together with all liquids and gases, ore also devoid of the same property. An Accident, the fifth and last of the Predicables, is any quality, which may or may not belong to certain objects, and which has no-connexion with the classification adopted. The particular size of a crystal does not in the slightest degree affect the nature of the crystal, nor does the manner in which it may be grouped with other crystals ; these, then, are Accidents as regards a crystallographic classification. With respect to the chemical composition of a substance, again, it is an accident whether the sub- stance be crystallized or not, or whether it be organized or not. As regards botanical classification the absolute size of a plant is an accident, due to external circum- stances. Thus we see that a logical accident is any quality or circumstance which is not known to be cor- related with those qualities or circumstances forming the definition of the species. The use of the Predicables can be very concisely ex- plained by our symbols. Thus, let A be any definite group of qualities and B another quality ; then A will constitute a genus, and AB, A6 will be species of it, B being the difference. Let C, D and E be other qualities, and on examining the combinations in which A, B, C, D, E occur let them be as follows : — ABODE kbCdE ABCDe AJCrfc. by Google CLASSIFICATION. 379 Here we see that wherever A is C is also foxmd, so that C is a generic property ; D occurs always with B, so that it constitutes a specific property, while E is indifierentJy present and absent, so as not to be in any way correlated with any of the other letters ; it represents, therefore, an accident. It will now be seen that the Logical Abece- dariura leally represents an interminable series of subor- dinate genera and species ; it is but a concise symbolic statement of what was involved in the ancient doctrine of the Predicables. Summum Genus and Injima Species. As a genua means any class whatever which is re- garded as composed of minor classes or species, it follows that the same class will be a genus in one point of view and a species in another. Metal is a genus as r^ards alkaline metal, a species as regards dement, and any extensive system of classes consists of a series of subor- dinate, or as they are technically called, subaltern genera and species. The question, however, arises, whether any such chain of classes has a definite termination at either end. The doctrine of the old logicians was to the effect that it terminated upwards in a genus generaliasimum or surtimum genus, which was not a species of any wider class. Some very general notion, such as substance, object oi thing, yfSiB supposed to be so comprehensive as to in- clude all thinkable objects, and for all practical purposes this might be so. But as I have already explained (vol. i. p. 88), we cannot really think of any object or class without thereby separating it from what is not that object or class. All thinking is relative, and implies discrimina- tion, so that every class and every logical notion must have its negative , If so, there is no such tiling as a summum by Google 380 THE PRINCIPLES OF SCIENCE. genus, for we cannot frame tbe requisite notion of a class forming it without implying the existence of another class discriminated from it, but which with the supposed summum genus will form the speraes of a still higher genus, which is absurd. Although there is no absolute summum genus, neverthe- less relatively to any biunch of knowledge or any special argument, there is always some class or notion which bounds OUT horizon as it were. The chemist restricts liis view to material substances and the forces manifested in them ; the mathematician extends his view so as to com- prehend all notions capable of numerical discrimination. The biolo^t, on the other hand, has a narrower sphere containing only organized bodies, and of these the botanist and the zoologist take parts. In other subjects there may be a still narrower summum genus, as when the lawyer regards only living and reasoning beings of his own country. In the description of the Logical Abecedarium, it was pointed out (vol. i. p. io8) that every series of com- binations was really the development of some one single class, denoted by X, which letter indeed was accord- ingly placed in the first column of the table on p. 109. This is the formal acknowledgment of the principle clearly stated by De Morgan, that all reasoning pro- ceeds within some assumed summum genus. But at the same time the fact that X as a lexical term must have its negative x, shows that it cannot be an absolute summum genus. There arises, again, the question whether there be any such thing as an I'n^nia species, which cannot be divided into any smaller species. The ancient logicians were of opinion that there always was some assignable class which could only be divided into individuals, but this doctrine appears to me theoretically incorrect, as Mr. George Digitized by Google GLA SSIFWA TION. 38 1 Bentham indeed long ago stated p. We may always put an arbitrary limit to the subdivisions of oiir classification at any point convenient to our purpose. The crystallo- grapher would not generally consider as diflferent species of crystalline form those which differ only in the degree of development of the faces. The naturalist overlooks innu- merable slight differences between plants or animals which he refers to the same species. But in a strictly Ic^cal point of view classification might be carried on so long as there is a single point of difference, however minute, between two objects, and we might thus go on until we arrived at individual objects which are numerically distinct in the logical sense attributed to that expression in the chapter upon Number. We must either, then, call the individual the injima species or allow that there is no such species at all. 77ie Tree of Porphyry. The bifurcate method of classification, arising as it does from the primary laws of thought, is the very founda- tion of all strict scientific method, and its application in formal logic constitutes the method of Indirect Inference, of which the nature and importance were shown in Chap- ter VI. So slight, however, has been the attention paid to this all important subject, that I shall in this case break the rule which I have laid down for myself, not to mingle the subject of logic as a science with the history of logic. Both Plato and Aristotle were fully acquainted with the value of bifurcate division which they occasionally employed in an explicit manner. It is impossible, too, p * Outline of a New System of Logic,' 1827, p. ii7- Digitized by Google 382 THE PRINCIPLES OF SCIENCE. that Aristotle should state the laws of thought, and employ the predicables without implicitly recognising the logical necessity of that method. It is, however, in Por- phyry's remarkable and in many respects excellent ' Intro- duction to the Categories of Aristotle' that we find the most distinct account of it. Porphyry not only fully and accurately describes the Predicables, but incidently intro- duces an example for illustrating those predicables, which constitutes a good specimen of bifurcate classification. Translating his wordsi freely we may say that he takes Substance as the genus to be divided, under which are succeesively placed as Species — Body, Animated Body, Animal, Bational Animal, and Maa Under Man, again, come Socrates, Plato, and other particular men. Kow of these notions Substance is the genus generalissimum, and is a genus only, not a species. Man, on the other hand, is the species specialissima (infima species), and is a species only, not a genus. Body is a species of substance, but a genus of animated body, which, again, is a species of body but a genus of animal. Animal is a species of animated body, but a genus of rational animal, which, again, is a species of animal, but a genus of man. Finally, man is a species of rational animal, but is a species merely and not a genus, being divisible only into particular men. Porphyry proceeds at some length to employ his example in fiirther illustration of the predicables. We do not find in Porphyry's own work any scheme or diagram exhibiting this curious specimen of classifi- cation, but some of the earlier commentators and epitome writers drew what has long been called the iW of Porphyry, Thus in the 'Epitome Logica' of Nicephorus Blemmidae, 1 ' FoTphyrii Isagoge,' Caput ii. 34. by Google CLASSIFICATION. we find a diagram •■ of which the following is nearly a facsimile : — 17 ovtTia oiaipIJTat awfta aawfiarov A A^ aifffiijTiitoi' avalvQ>rrov Av _ fitrafiartKOV ifirrdfiaToy A XoytKoy aXoyov rov avOpimrov. In the above scheme we find the ^iiuicate principle accurately but not completely applied. Each genus is subdivided into two species, described by a pair of posi- tive and negative terms, so that the species are together equal in extent to the genus. But it will of course be observed that each negative branch is left without further subdivision, so that there is only a single iufima species, namely man, instead of thirty-two final branches, as there would be in a theoretically complete system. This tree was subsequently reproduced in the works of a multitude of logicians in a form which is more complicated and not so good as that of Nicephorus. Thus ' ' Epitome Logico, Angustce Viadel.' 1605, p. 118. by Google 384 THE PRmCIPLES OF SCIENCE. in the ' Opuscula' of Aquinas, as quoted by Man&el in his edition of Aldrich's 'Artia Logicffi Rudimenta,' second edition, p. 31, we find the Tree nearly in the following form: — Substantia Corporea Incorporea Corpus Animatuni Inanimatum Vivena Senaibile Inseusibile Animal Rntionale Imtttonale Homo Socrates Plato. This example of the bifurcate method, although re- peated in almost all compendiums and treatises on logic, attracted no particular attention until the time of Peter Kamus and bis followers, who are commonly said to have bestowed so much attention and praise upon it as to be by Google CLASSIFICA TION. 586 regarded by some persons as its inventors. The Eamean Tree is a name frequently employed instead of the Por- phyrian Tree, or the iirXi>af, that is, the Ladder of Por- P^ijry. *s it was sometimes called by the Greek logicians. Although I have looked through several commentaries upon the Dialectics of Eamus, I do not find that very much is said upon the subject In the Questions of Frederick Beurhusius"", the method of dichotomy is described as 'ilia naturalls et antiquissimorum philoso- phorum priestantissima Dichotomia,' but in none of the works do I find the Tree itself given. Among modem logicians Jeremy Bentham possesses the great merit of having drawn attention to the logical importance of bifurcate division. His remarks on the subject are contained in that extraordinary collection of digressive, and of^n almost incomprehensible papers, called Chrestomathia*, two of the formidable title-pages of which are given below. The fifth appendix in this work, fonning the larger and most important part of the book, consists of an Eesay on Nomenclature and ClasEnfi- cation'. Although written in his later and worse style, this essay is well worth reading, and full of forcible remarks. It may be regarded, I beUeve, as the first of ' In Petri Bami, R^i Profeasoria Clariss. IMalecticte LibroB daos Lutetue Anno LXXII, postremo sine Pnel«ctaonibiiB leditoa, explica- tionum Qiuesliones : qnte Ptedagogioe Lc^ck de Docenda Ducendaqne Dialectics. Auctore Frederico Beurbusio. Londoni, 1581, p. 110. ■ 'ChreBtomatliia 1 being a Collection of Papers, esplanatAry of the Design of an Institution proposed to be set on foot, under the name of the Chreetomatbic Day School, or Chrestomatbic School, for the extension of the New Sfstem of Instruction,' &o. By Jeremy Bentham, Esq., London, 1816. t 'An Essay on Nomenclature and Claeeificatiou: including a Critical Examination of the Encyclopaedical Table of Lord Bacon, as improved by D'Alembert : and the first lines of a new one grounded on the application of the Logical Principle of Exhaustively Bifiircate Analyda.' London, 1817. VOL. n. c c Digitized by Google THE PRINCIPLES OF SCIENCE. the series of Ecglish writings which have, in the present century, made log^c a new and progressive science. In Table IV. Bentham gives the Arbor Porphyriana, as exhibited in the course of a coll^;e lecture in 1761, call- ing it the original form. His reading of logic seems to have been restricted to the compendiums of Saxmderson and Watts, and it was only after the text was written that he obtained an opportunity of consulting the work of Porphyry, and was surprised to find no diagram therein. He attributes its invention to Peter Bamua, although he had never seen the writings of that logician, and had merely learnt their titles firom a dictionary. In this essay he states in the most powerful way the advantages of the bifurcate method of classification, which had been suggested to him by a chapter in Saundersou's logio and the diagram given in the college course. Although the Tree of Porphyry and the principles of bifurcation had been mentioned by almost all logicians, the utility and excellence of the method, he says (p. 287), had not made itself apparent. Indeed the method was mentioned but to be slighted, or to be made a subject of pleasantry by Keid and Kames. Bentham sufficiently states his own opinion when he speaks (p. 295) of 'the matchless beauty of the Eamean Tree.' After fully show- ing its logical value as an exhaustive method of classifi- cation, and refuting the objections of Keid and Kames, on a wrong ground, as I think, he proceeds to inquire to what length it may be carried. He correctly points out two objections to the extensive use of bifid arrangements, (i) because they soon become impracticably extensive and unwieldy, and (2) because they are uneconomical. In his day the recorded number of difierent species of plants was 40,000, and he leaves the reader to estimate the im- mense number of branches and the enormous area of a biftircate table which should exhibit all these species in by Google CLASSIFICATTOK 387 one scheme. He also points out the apparent loss of labour in making any large bifurcate classification ; but this he considers to be fuUy recompensed by the logical value of the result, and the logical training acquired in its execution. Jeremy Bentham, then, fully recognises, as I conceive, the value of the Logical Abecedarium under another name, though he apprehends tte limit to its use placed by the finiteness of our mental and manual powers. Mr, George Bentham has also fully recognised the value of bifurcate classification, both in his ' Outline of a New System of Logic'" {pp. 105-118), and in his 'Essai Bur la Nomenclature et la Classification.' This latter work consists of a free translation or improved version in French of Jeremy Bentham's 'Essay on Classification.' Further Illustrations of the value of the bifurcate method are adduced from the natural science, and Mr. Bentham points out that it is really this method which was employed by Lamark and DecandoUe in their so-ccdled analytical arrangement of the French Flora. The following table contains an excellent example of bifurcate division, con- sisting of the principal classes of Decandolle's system, as ^ven by Mr. Bentham in Table No. HI. p. 108 of hia Essay, the names, however, being translated : — ° ConceruiDg the cotmexion of this work with the great discorery of the quantification of the predicate, I may refer the reader to the remarks and articles of Hr. Herhert Spencer and Professor Thomas Spencer Baynea, in the 'Contemporary Review' of March, April, and July, 1873, vol. xxi. pp. 490, 796 ; vol. zxiL p. 318 ; as also to my own article in answer to Professor Baynes in the same Beriew for May, 1 873, vol. zxi. p. 8>i. Professor Baynes makes it evident that, when Sir W. Hamilton reviewed Mr. Bentham's work in 1833, he did not sufficiently acquaint himself with its contents. I must continue to hold that the principle of quantification is explicitly stated hy Mr. Bentham, and it must be re- garded as a remarkable fact in the history of logic that Hamilton, while vindicating, in 1847, his own claims to originality and priority against the scheme of De Morgan, should have overlooked the much earlier and more closely related discoveries of Bentham. C C 2 Digitized by Google TBE PXINCIPLBS OF SCIESCE. ill li -111 11 ll ■■i Sjfi |S-i Hi ■fis 3, 1| i-ISla b, Google CLASSIFICATION. Mt. Bentham also gives a bifurcate arrangement of animals after the method proposed by Dum^ril in hie ' Zoologie Analytique/ this naturalist being distinguished by his clear perception of the logical importance of the method. A more recent binary classification of the animal king- dom as re^rds the larger classes may be found in Pro- fessor Eeay Greene's ' Manual of the Ctelenterata,' p. i8. Does Abstraction imply Generalization ? Before we can acquire a sound comprehension of the subject of classification we must answer a very difficult question, namely, whether logical abstraction does or does not always imply generalization. It comea to exactly the same thing if we ask whether a species may be coexten- sive with its genus, or whether, on the other hand, the genus must contain more than the species. To abstract logically is, as we have seen (vol. L p. 33), to overlook or withdraw our notice firom some point of difference. When- ever we form a class we abstract, for the time being, the differences of the ol^ects so united in respect of some common quaUty. If, for instance, we class together a great number of objects as dwelling-houses, we overlook or abstract the feet that some dwelling-hoiises are con- structed of stone, others of brick, wood, iron, Ac. Veiy often at least the abstraction of a circumstance increases the number of objects included under a class according to the law of the inverse relation of the quantities of exten- sion and intension (vol. i. p. 32). Dwelling-house is a wider term than brick dwelling-house. House, or building, is more general still than dwelling-house. But the ques- tion before us is, whether abstraction always increases the number of objects included in a class, which amounts to asking whether the law of the inverse relation of logical quantities is always true. The interest of the question by Google 390 THE PRINCIPLES OF SCIENCE. partly arises firom the &ct, that so high a philosophical authority as Mr. Herbert Spencer has denied that gene- ralization is implied in abstraction'', making this doctrine the ground for rejecting previous methods of classifying the sciences, and for forming an ingenious but peculiar method of his own. The question is also a fundamental one of the highest logical importance, and involves subtle difficulties which have made me long hesitate in forming a decisive opinion. Let us attempt to answer the question by examination of a few examples. Compare the two classes gun and iron gun. It is certain that there are many guns which are not made of iron, so that abstraction of the circumstance ' made of iron* increases the extent of the notion. Next compare gun and metallic gun. All guns made at the present day consist, I believe, of metal, so that the two notions seem to be co-extensive ; but guns were at first made of pieces of wood bound together like a tub, and as the logical term gun takes no account of time, it must include alt guns that have ever existed. Here again extension increases as intension decreasea Compare once more ' stewn-locomotive engine' and 'locomotive engine.' In the present day so far as I am aware all locomotives are worked by steam, so that the omission of that qualifica- tion might seem not to widen the term ; hut it is quite possible that in some future age a different motive power may he used in locomotives ; and as there is no limitation of time in the use of logical terms, we must certainly assume that there is a class of locomotives not worked by steam, as well as a class that is worked by steam. When the natural class of Euphorbiaceie was origin- ally formed, all the plants known to belong to it were devoid of corollas ; it would have seemed therefore that the two classes ' Euphorbiaceae,' and 'Euphorbiaceae devoid * ' The ClaBBification of the Sciences,' &c., 3rd edii p. 7. by Google CLASSIFICATION. 391 of Corollas,' were of equal extent. Subsequently a number of plants plainly belonging to the same class were found in tropical countries, and they possessed bright coloured corollas. Naturalists believe with the utmost confidence that 'Ruminanta' and 'Ruminants with cleft feet' are identical terms, because no ruminant has yet been dis- covered without cleft feet. But we can see no impossibility in the conjunction of rumination with uncleft feet, and it would be too great an assumption to say that we are certain that an example of it will never be met with. Instances can be quoted, without end, of objects being ulti- mately discovered which combined properties or forms which had never before been seen together. In the animal kingdom the Black Swan, the Ornithorhyncus Paradoxus, and more recently the singular fish called Ceratodus For- steri, all discovered in Australia, have united characters never previously known to co-exist. At the present time deep-sea dredging is bringing to light many animaU of a new and unprecedented nature. Singular exceptional dis- coveries may certainly occur in other branches of science. When Davy first succeeded in eliminating metallic potas- mum, it was a well established empirical law that all metallic substances possessed a high specific gravity, the least dense of all metals then known being zinc, of which the specific gravity is fi. Yet, to the surprise of chemists, potassium was found to be an undoubted metal of less density than water, its specific gravity being 0*865. It is hardly requisite to prove by further examples that our knowledge of nature is incomplete, so that we cannot safely assume the non-existence of new combinations. Logically speaking, we ought to leave a place open for animals which ruminate but are without cleft feet, and for every other possible intermediate form of animal, plant, or minerid. A purely logical classification miist take account not only of what certainly does exist, but of what pay in after ages be found to exist. Digitized by Google 392 THE PRINCIPLES OF SCIESCB. I will go a step further, and say that we must have places in our scientific claseifications for purely imaginary existences. A very large proportion of the mathematical functions which are conceivable have no application to the circumstances of this world. Physicists certainly do in- vestigate the nature and consequences of forces which nowhere exist. Newton's ' Principia ' is full of such inves- tigations. In one chapter of his ' Mecanique Celeste ' Laplace indulges in a remarkable speculation as to what the laws of motion would have been if momentum instead of varying simply as the velocity had been a more com- plicated fimction of it. I have already mentioned {vol, i. p. 256) that Sir George Airy contemplated the existence of a world in which the laws of force should be such that a perpetual motion would be possible, and the Law of Conservation of Energy would not hold true. Thought is not bound down to the limits of what is mate- rially existent, but is circumscribed only by those Funda- mental Laws of Identity, Contradiction and Duality, which were laid down at the outset. This is the point at which I should differ from Mr. Herbert Spencer. He appears to suppose that a classification is complete if It has a place for every existing object, and this may perhaps seem to be practically sufficient ; but it is subject to two profound objections. Firstly, we do not know all that exists, and therefore in limiting our classes we are erroneously omitting multitudes of objects of unknown form and nature which may exist either on this earth or in other parts of space. Secondly, as I have explained, the powers of thought are not limited by material existences, and we may or, for some purposes, must imagine objects which probably do not exist, and if we imagine them we ought (strictly speak- ing) to find appropriate places for them in the classifi- cations of science. The chief difiSculty of this subject, however, consists in by Google CLASSIFICATION. 393 the fact that mathematical or other certain laws may en- tirely forhid the existence of eome combinationa The circle may be defined ae a plane curve of equal curvature, and it is a property of it that it contains the greatest area within the least possible perimeter. May we then eon- template mentally a circle not a figure of greatest possible area ? Or, to take a still simpler example, a parallelc^;ram possesses the property of having the opposite angles equal. May we then mentally divide parallelograms into two classes according as they do or do not have their opposite angles equal \ It might seem absurd to do so, because we know that one of the two species of parallelogram would be non-existent. But, then, what is the meaning of the thirty-fourth proposition of Euclid's first book, unless the student had previously contemplated the existence of both species as possible. We cannot even deny or dis- prove the existence of a certain combination without thereby in a certain way recognising that combination as an object of thought. The general conclusion, then, at which I arrive, is in opposition to that of Mr. Herbert Spencer. I think that whenever we abstract a quality or circumstance we do generalize or widen the notion from which we abstract. Whatever the terms A, B, and C may be, I hold that in strict logic AB is mentally a wider term than ABC, because AB includes the two species ABC and ABc. The term A is wider still, for it includes the four species ABC, ABc, A6C, Ahc. The Logical Abecedarium, in short, is the only limit of the classes of objects which we must contem- plate in a purely logical point of view. Whatever notions be brought before us, we must mentally combine them in all the ways sanctioned by the laws of thought and ex- hibited in the Abecedarium, and it is a matter for after consideration to determine how many of these combina- tions exist in outward nature, or how many are actually Digitized by Google 394 THE PRINCIPLBS OF SCIESGE. forbidden by the nature of space. A clasafication is essen- tially a mental not a material tbing. Discovery of Marks or Characteristics. Although the chief purpose of classification is to disclose the deepest and most general reeemblances of the objects classified, yet the practical value of any particular system will partly depend upon the ease with which we can refer an object to its proper dass, and thus infer concerning it all that is known generally of that class. This operation of discovering to which class of a system a certain speci- men or case belongs is generally called Diagnosis, a technical term very familiarly used by physicians, who constantly require to diagnose or determine the nature of the disease from which a patient ia suffering. Now every class is defined by certain specified qualities or cir- cumstances, the whole of which are present in every object contained in the class, and not all present in any object excluded from it. These defining circumstances ought to consist of the deepest and most important circum- stances, by which we vaguely mean those probably forming the conditions with which the minor circum- stances are correlated. But it will often happen that the so-called important points of an object are not those which can most readily be observed. Thus the two great classes of phanerogamous plants are defined respectively by the possession of two cotyledons or seed-leaves, and one coty- ledon. But when a plant comes to our notice and we want to refer it to the right class, it will often happen that we have no seed at all to examine, in order to dis- cover whether there be one seed-leaf or two in the germ. Even if we have a seed it will often be very small, and a careful dissection under the microscope will be requisite to ascertain the number of cotyledons. Occasionally the by Google CLA SSI FIG A TION. 89B examination of the germ wouJd mislead us, for the coty- ledons may be obsolete, as in Cuscuta, or united together, as in Clintonia. Botanists therefore seldom actually refer to the seed for such simple information. Certain other characters of a plant are closely correlated with the number of seed-leaves; thus monocotyledonons plants almost always possess leaves with parallel veins like those of grass, while dicotyledonous plants have leaves with reti- culated veins like those of an oak leaf. In monocotyle- donons plants, too, the parts of the flower are most often three or some multiple of three in niunber, while in dico- tyledonous plants the numbers four and five and tiieir multiples prevail Botanists, therefore, by a glance at the leaves and -flowers can almost certainly refer a plant to its right class, and can infer not only the number of coty- ledons which would be found in the seed or young plant, but also the structure of the stem and the other general characters and relations of a dicotyledon or a mono- cotyledon. Any conspicuous and easily discriminated property which we thus select for the purpose of deciding to which class an object belongs, may be called a characteristic. The logical conditions of a good characteristic mark are very simple, namely, that it should be possessed by all objects entering into a certain class, and by none othera The characteristic may consist either of a single quality or circumstance, or of a conjunction of such, provided that they aU be constant and easily detected. Thus in the classification of mammals the teeth are of the greatest assistance, not because a slight variation in the number and form of the teeth is of any great importance in the general economy of the animal, but because such variations are found by empirical observation to coincide with most important diflerences in the general affinities. It is found that the minor classes and genera of mammals can be Digitized by Google 396 THE PRINCIPLES OP SCIENCE. registered and discriminated atxjurately by their teeth, especially hy the foremost molars and the hindmost pre- molars. Some of the teeth, indeed, are occasionally missing, so that zoologists prefer to trust to those characteristic teeth which are most constant^, and to infer from them not only the arrangement of the other teeth, but the whole conformation of the animal. It is a very difficult matter to mark out any boundary- line between the animal and vegetable kingdoms, and it may even be doubted whether any rigorous division can be established. The most fundamental and important character of a vegetable structure probably consists in the absence of nitrogen from the constituent membranes. Supposing this to be the case, the difficulty arises that in examining minute organisms we cannot ascertain directly whether they contain nitrogen or not. Some minor but easily detected circumstance is therefore needed to dis- criminate between animals and vegetables, and this is furnished to some extent by the fact that the production of starch granules is restricted to the vegetable kingdom. Thus the Desmidiacese may be safely assigned to the vege- table kingdom, because they contain starch. But we must not employ this characteristic negatively ; the Diato- macese are probably vegetables, though they do not produce starch. Diagnostic Systems of Classijication. We have seen that diagnosis is the process of dis- covering the place in any system of classes, to which an ^|®5i?°9^^^^iA'^^t3B(^^ady been referred by some previous investi- ^^*»,s8^^S v5wfi' 'J^''^ ^^^^S *° avail ourselves of the informa- ■i^ij^^jpv''^^ ** Jig such an object which has been already j^^S^N 'ilj^ ^JCjb ^and recorded. It is obvious that this is a {bv ^^ ^^^^S^y <>'^ ^^ ClasBiiicatioii and Oeographical Distribution of Digitized by Google CLASSIFICA TION. 397 matter of the greatest importance, for, unlesB we can recognise, &om time to time, objects or substances which have been before investigated, all recorded discoveries would lose their value. Even a angle investigator must have some means of recording or systematizing his ob- servations of any large number of objects like those furnished by the vegetable and animal kingdoms. Now whenever a class has been properly formed, a definition must have been laid down, stating the qualities and circumstances possessed by all the objects which are intended to be included in the class, and not possessed completely by any other objects. Diagnosis, therefore, consists simply in comparing the qualities of a certain object with the definitions of a eeries of classes ; the absence in the object of any one quality stated in the definition excludes it from the class thus defined; whereas, if we find every point of a definition exactly fulfilled in the specimen, we may at once assign it to the class in question. It is of course by no means certain that every- thing which has been aflBrmed of a class is true of all objects afterwards referred to the class; for this would be a case of imperfect inference, which is never more than a matter of probability. A definition can only make known a finite number of the properties of an object, so that it always remains possible that objects agreeing in those assigned properties will differ in other ones. An individual cannot be defined, and can only be made known by the exhibition of the individual itself, or by a material specimen exactly representing it. But this and many other questions relating to definition must be treated if I am able to take up the general subject of language in another work. Diagnostic systems of classification should, aa a general rule, be arranged on the bifurcate method explicitly. Any property may be chosen which divides the whole group Digitized by Google 398 THE PRINCIPLES OF SCIENCE. of objects into two distinct parts, and each part may be Bub-divided succeesively by any prominent and well marked circumstance which is present in a large part of the genus and not in the other. To refer an object to its proper place in such an arrangement we have only to note whether it does or does not possess the successsive critical circumstances. Dana devised a classification of this kind^ by which to refer any crystal to its place in the series of BIX or seven classes already desciibed. If a crystal has all its edges modified alike or the angles replaced by three or six similar planes, it belongs to the monometric system ; if not, we observe whether the number of similar planes at the extremity of the crystal is three or some multiple of three, in which case it is a crystal of the hexagonal system ; and so we proceed with further successive dis- criminations. To ascertain the name of a mineral by examination with the blow-pipe, an arrangement more or less evidently on the bifurcate plan, has been laid down by Von Kobell*. Minerals are divided according as they possess or do not possess metallic lustre ; as they are fusible (including under fusible substances those which are volatile) or not fusible in a determinate degree, according as they do or do not on charcoal give a metallic bead, and so on. Perhaps the best example to be found of any anunge- ment simply devised for the purpose of diagnosis, is Mr. George Bentham's ' Analytical Key to the Natural Orders and Anamolous Genera of the British Flora,' g^ven in his 'Handbook of the British Flora ''.' In this ■ Dana's ' Mineralogy,' vol. i. p. 1 33. Quoted in Watts's ' Dictionary of ChemiBtry,' toI, ii. p. 166, " ' InatructioDB for the Discrimination of Minerals by Simple Chemical Experiments,' by Franz von Kobell, translated from the Qemian by R. C. Campbell, Glasgow, 1841. >• Edition of 1 866, p. Uiil by Google CLASSIFICA TION. 399 scheme, the great composite family of plants, together with the closely approximate genus Jasione, are first separated from all other flowering plants by the compound character of their flowers. The remaining plants are sub- divided according as the perianth is double or single. Since no plants are yet known in which the perianth can be said to have three or more distinct rings, this division becomes practically the same as one into double and not- double. Flowers with a double perianth are nest discrimi- nated according as the corolla does or does not consist of one piece, according as the ovary is free or not-free, as it is simple or not simple, as the corolla is regular or irre- gular, and so on. On looking over this arraDgement, it will be found that numerical discriminations often occur, the numbers of petals, stamens, capsules, or other parts being the criteria, in which cases, as already explained (vol. ii. p. 374), the actual exhibition of the bifid division would be tedious. Linnseus appears to have been perfectly acquainted with the nature and uses of diagnostic classification, which he describes under the name of Synopsis, saying'' : — ' Synopsis tradit Divisiones arbitrarias, longiores aut brevi- ores, plures aut pauciorea : a Botanicis in genere non agnoscenda. Synopsis est dicbotomia arbitraria, quie instar vise ad Botanicem duciL Limites autem non deter- minat.' The rules and tables drawn out by chemists to facilitate the discovery of the nature of a substance in qualitative analysis are usually arranged on the bifurcate method, and form excellent examples of diagnostic classification, the qualities of the substances employed in testing being in most cases merely characteristic properties of little importance in other respects. The chemist does not detect potassium by reducing it to the state of metallic potas- < ' PhiloBopbia Botanica' (1770), § 154. p- 98- Digitized by Google 400 THE PRINCIPLES OF SCIENCE. slum, and then- observing whether it has all the principal qualities belonging to potassium. He selecta from among the whole number of compounds of potassium that salt, namely the compound of platinum tetrarchloride and potassium chloride, which has the most distinctive ap- pearance, as it is comparatively insoluble and produces a peculiar yellow and highly crystalline precipitate. Ac- cordingly whenever this precipitate can be produced by adding platinum chloride to a solution potassium is pre- sent. The fine purple or violet colour which potassium salts usually communicate to the blowpipe flame, had long been used as a characteristic mark. Some other elements were readily detected by the colouring of the blowpipe flame, barium giving a pale yellowish green, and salte of strontium a bright red. By the use of the spectroscope the coloured light given off by any incan- descent vapour is made to give perfectly characteristic marks of the elements contained in the vapour. Diagnosis seems to be identical with the process termed by the ancient logicians abscissio infiniti, the cutting off of the infinite or negative part of a classification when we discover by observation that an object possesses a par- ticular property. At every step in a bifurcate division, some objects possessing the difference will fall into the affirmative part or species ; all tbe remaining objects in the world fall into the negative part which will be infinite in extent. Diagnosis consists in tbe successive rejection from further notice of those almost infinite classes with which the specimen in question does not agree. Index Classifications. Tinder the general subject of classification we may certainly include all arrangements of objects or names, which we make for the purpose of saving labour in the by Google CLA SSI PICA TION. 40 1 discovery of an object Even such apparently trivial and arbitrary arrangements as alphabetical or other indices, are really classifications subject to all the principles of the subject. No such arrangement can be of any use unless it involves some correlation of circumstances, so that knowing one thing we learn another. If we merely arrange letters in the pigeon-holes of a secretaire we establish a correlation, for all letters in the first hole will be written by persons, for instance, whose names begin with A, and so on. Knowing then the initial 'letter of the writer's name we know also the place of the letter, and the labour of search is thus reduced to one twenty-sixth part of what it would be without any arrangement. Now the purpose of a mere catalogue is to discover the place in which an object is to be found, but the art of cataloguing involves logical considerations of some interest and importance. We want to establish a correlation be* tween the place of an object and some circumstance about the object which shall enable us readily to refer to it ; this circumstance therefore should be that which will most readily dwell in the memory of the searcher. A piece of poetry, for instance, wiU be best remembered, in all probability, by the first line of the piece, according to the laws of the association of Ideas, and the name of the author will be the next most definite circumstance ; a catalogue of poetry should therefore be arranged alpha- betically according to the first word of the piece, or the name of the author, or, still better, in both ways. It would be wholly absurd and impossible to arrange poems according to their subjects, so vague and mixed are these found to be when the attempt is made. It is a matter of considerable literary importance to decide upon the best mode of cataloguing books, so that any required book in a library shall be most readily found. Books may be classified in a great number of VOL. II. D d Digitized by Google 402 THE PRINCIPLBS OF SCIENCE. waya, according to subject, language, date or place of publication, size, the initial words of the book itaeli^ of the title-page, the colophon, the author's name, the piiblisher's name, the printer's name, the character of the type, and so on. Every oue of these modes of arrange- ment may be useful, for we may happen to remember one circumstance about a book when we have forgotten all others ; but as we cannot usually go to the expense of forming more than two or three indices at the most, we must of course select those circumstances for the basis of arrangement which will he likely to lead to the dis- covery of a book most surely. Many of the criteria mentioned are evidently inapplicable. The language in which a book is written is no doubt definite enough, but would afford no criterion for the classification of any large group of English books, or of those written in any one language. Classification by subjects would be an exceed- ingly useful method if it were practicable, but experience, or indeed a little reflection, shows it to be a logical absurdity. It is a very difficult matter to classify the sciences, so close and complicated are in many cases the relations between them. But with books the complica- tion is infinitely greater, since the same book may treat successively of different sciences, or it may discuss a problem involving many entirely diverge principles and branches of knowledge. A good history of the steam engine will be antiquarian, so far as it traces out records of the earliest efforts at discovery ; purely scientific, as regards the principles of fhermodynamics involved ; technical, as regards the mechanical means of applying those principles ; economical, as regards the industrial results of the invention ; biographical, as regards the lives of the inventors. A history of Westminsta- Abbey might belong either to the history of architecture, the history of the church, or the histoiy of England. If we Digitized by Google CLASSIFICATION. 403 abandon the attempt to carry out an arrangement accord- ing to the natural classification of the sciences, and form comprehensive practical groups, we shall be continually perplexed by the occurrence of intermediate cases, and opinions will differ ad injinitum, as to the details. If, to avoid the difficulty about Westminster Abbey, we form a class of books devoted to the History of Buildings, the question will then arise whether Stonehenge is a building, and if so, whether, cromlechs, mounds, or even monoliths are so. At the other end of the scale we shall be uncer- tain whether to include under the class History of Build- ings, lighthouses, monuments, bridges, &c. In regard to purely literary works, rigorous classification is still less possible. The very same work may partake of the nature of poetry, biography, history, philosophy, or if we form a comprehensive class of Belles-Lettres, nobody can say exactly what does or does not come imder the term. My own experience entirely bears out the opinion of the late Professor De Morgan, that classification according to the name of the author is the only one practicable in a large library, and this method has been admirably carried out in the great Catalogue of the British Museum. The name of the author is the most precise circumstance con- cerning a book, which usually dwells in the memory. It is more nearly a characteristic of the book than anything else. In an alphabetical arrangement we have an exhaus- tive classification, including a place for every possible name. The following remarks'" of De Morgan seem there- fore to be entirely correct. ' From much, almost daily use, of catalogues for many years, I am perfectly satisfied that a classed catalogue is more difficult to use than to make. It is one man's theory of the subdivision of knowledge, and the chances are against its suiting any other man. Even if all doubtful works were entered under several ■I ' Philosopliical Magazine,' 3rd Series (1845), vol. xxvi, p. gaa. D d 2 by Google 404 THE PRINCIPLES OF SCIENCE. diflFerent heads, the frontier of the dubious region would itself he a mere matter of doubt. I never turn from a claseed catalogue to an alphabetical one without a feeling of relief and security. With the latter I can always, by taking proper pains, make a library yield its utmost ; with the former I can never be satisfied that I have taken proper pains, until I have made it, in fact, as many different catalogues as there are different headings, with separate trouble for each. Those to whom bibliographical research is familiar, know that they have much more frequently to hunt an author than a subject : tiiey know also that in searching for a subject, it is never safe to take another person's view, however good, of the limits of that subject with reference to their own particular purposes.' It is often very desirable, however, that an alphabetical name catalogue should be accompanied by a subordinate subject catalogue, but in this case no attempt should be made to devise a theoretically complete classification. Every principal subject treated in a book should be entered separately in an alphabetical list, under the name most likely to occur to the searcher, or under several names. This method was partially carried out in "Watta's valuable ' Bibliotheca Britannica,' but it was perfectly applied in the admirable subject index to the ' British Catalogue of Books,' and equally ^ell in the ' Catalogue of the Manchester Free Library at Campfield,' this latter being the most perfect model of a printed catalogue with which I am acquainted. The public Catalogue of the British Museum is arranged as far as possible according to the alphabetical order of the author's names, but in writing the titles for this catalogue several copies are simultaneously produced by a manifold writer, so that a catalogue according to the order of the books on the shelves, and another according to the first words of the title-page, are created by a mere re- Digitized by Google CLASSIFICATION. 405 arrangement of the spare copies. In the ' English Cyclo- peedia' it is soggested that twenty copies of the book titles might readily have been utilized in forming additional catalogues, arranged according to the place of publication, the language of the book, the general nature of the sub- ject, and so forth «. It will hardly be a digression to point out the enormous saving of labour, or, what comes to the same thing, the enormous increase in our available knowledge, both lite- rary and scientific, which arises from the formation of ex- tensive indices. The ' State Papers,' containing the whole history of the nation, were practically sealed to literary inquirers until the Government undertook the taafc of calendaring and indexing them. The British Museum Catalogue is another national work, of which the im- portance in advancing knowledge cannot be overrated. The Royal Society is accomplishing a work of world-wide importance, in publishing a complete catalogue of memoirs upon physical science. The time will perhaps come when our views upon this subject will be extended, and either Government or some public society will undertake the systematic cataloguing and indexing of masses of his- torical and scientific information which are now almost closed against inquiry. Clojtsificaiion in the Biological Sciences. The great generalizations established in the works of Herbert Spencer and Charles Darwin have thrown great light upon many other sciences, and, strange as it may seem to say so, they have removed several difficulties out of the way of the logician. The subject of classification has long been studied in almost exclusive reference to the e 'English Cyclopedia,' 'Arts and Sciences,' vol. V. p. 333. Digitized by Google 40e TUB PRINCItLES OF SCIMSCE. arrangement of the various kinds of animals and plants. Systematic Botany and Zoology have been commonly known as the Olassificatoiy Sciences, and scientific men Beemed to suppose that the methods of arrangement, which we e suitable for living creatures, must be the best for all other classes of objects. Several mineralogists, especially Mohsj have attempted to arrange minerals in genera find species, just as if they had been animals capable of reproducing their kind with variations, and thus having relatives like distant cousins. It is highly remarkable that this confusion of ideas between the relationship of living forms and the logical relationship of things in general prevailed from the earliest times, as manifested in the etymology of words. We familiarly speak of a hind of things meaning a class of things, and the kind consists of those things which are akin, or come of the same race. It is even believed by some etymologists that second means other kind, the Latin suffix cund being thus regarded as cognate with kind^. Similarly when Socrates and his followers wanted a name for a class regarded in a philosophical light, they again adopted the analogy in question, and called it a 7Ei'Dr, or race, the root yev- being distinctly connected with the notion of generation. So long as the species of plants and animals were believed to proceed from distinct and unconnected acts of Creation, the multitudinous points of resemblance and difference which they present, possessed a simply logical character, and might be treated as a guide to the classifi- cation of other objects generally. But when once we come to regard these resemblances as purely hereditary in their origin, we see that the sciences of systematic Botany and Zoology have a special character of their own. There is no reason whatever to suppose that the f VernoD, 'Anglo-Saxon Gukle,' p. 6B. by Google CLASSIFICA TION. 407 same kind of natural classification which is best in biology- will apply also in mineralogy, in chemistry, or in astronomy. The universal logical principles which underlie all classifi- cations are of course the same in natural history as in the sciences of brute matter, but the special logical resem- blances which arise from the relation of parent and ofifepring will not be found to prevail between different kinds of crystals or mineral bodies. The genealogical view of the mutual relations of ani- mals and plants leads us to discard all notions of any regular progression of Uving forms, or any theory as to their symmetrical relations. It was at one time a great question whether the ultimate scheme of natural claEsifi- cation would prove to be in a simple line, or a circle, or a combination of circles. Macleay's once celebrated system was a circular one, and each class-circle was composed of five order-circles, each of which was composed again of five tribe-circles, and so on, the subdivision being at each step into five minor circles. Thus he held that in the animal kmgdom there were five sub-kingdoms — the Ver- tebrata, Annulosa, Kadiata, Acrita, and Mollusca. Each of these was again divided into five — the Vertebrata con- sisting of Mammalia, Reptilia, Pisces, Amphibia, and AveaS. It is quite evident that in any such symmetrical system the animals were made to suit themselves to the classes instead of the classes being suited to the animals. We now perceive that the ultimate system will be an almost infinitely extended genealogical tree, which will be capable of representation by lines on a plane surface of sufl&cient extent. But there is not the least reason to suppose that this tree will have a symmetrical form. Some branches of it would be immensely developed com- pared with others. In some cases a form may have pro- K SwainsoD, ' Treatise on the Qeogrsphy and Classification of Aniinals, ' Cabinet Cydoptedift,' p. aoi. by Google TUE PRINCIPLES OF SCIENCE. pagated itself almost from primeval times with little vari- ation. In other ciiseB frequent differentiations will have oo- curred. Strictly speaking, this genealogical tree ought to represent the descent of each individual living form now existmg or which has existed. It should be as personal and minute in ita detail of relations, aa the Stemma of the Kings of England. We must not assume that any two forms are absolutely and exactly alike, and in any case they are numerically distinct. Every parent then must be represented at the apex of a series of divergent hues, representing the generation of so many childrea Any complete and perfect system of classification must regard individuals as the infimse species. But as in the lower races of animals and plants the differences between indi- viduals are usually very slight, and apparently imimportant, while the ntunbers of such individuals are immensely great, beyond all possibility of separate treatment, scientific men have always stopped at some convenient but arbitrary point, and have assumed that forms so closely resembling each other as to present no constant difference were all of one kind. They have, in short, fixed their attention entirely upon the main features of family difference. In the genealogical tree which they have been unconsciously aiming to construct, diverging lines meant races diverging in character, and the purpose of all efforts at so-called natural classification was to trace out the relationships between existing plants or animals. Now it is evident that hereditary descent may have in different cases pro- duced very different results as regards the problem of classification. In some cases the differentiation of charac- ters may have been very frequent, and specimens of all the characters produced may have been transmitted to the present time. A living form wUl then have, as it were, an almost infinite number of cousins of various degrees. Digit zed by Google GLASSIFIGATIOif. 409 and there -will be an immense number of forms finely graduated in their resemhkncee. Exact and distinct classification will then be almost impossible, and the wisest course will be not to attempt arbitrarily to distin- guish forms closely related in nature, but to allow that there exist transitional forms of every degree, to mark out if possible the extreme limits of the family relationship, and perhaps to select the most generalized form, or that which presents the greatest niimber of close resemblances to others of the family, as the type of the whole. Mr. Darwin, in his most interesting work upon Orchids, points out that the tribe of Malaxese are distinguished from Epidendreee by the absence of a caudicle to the pollinia, but aa some of the Malaxese have a minute cau- dicle the division really breaks down in the most essential point. ' This is a misfortune,' he remarks'", ' which every natu> ralist encounters in attempting to classify a largely de- veloped or so-called natural group, in which, relatively to other groups, there has been little extinction. In order that the natiuralist may be enabled to give precise and clear definitions of his divisions, whole ranks of interme- diate or gradational forms must have been utterly swept away : if here and there a member of the intermediate ranks has escaped annihilation, it puts an effectual bar to any absolutely distinct definition.' In other cases a particular plant or anioaal may perhaps have transmitted its form from generation to generation almost unchanged, or, what comes to the same result, those forms which diverged in character from the parent stock, may have proved unsuitable to their circumstances, and may have perished sooner or later. We shall then find a particular form standing apart from all others, and marked by various distinct characters. Occasionally we i> Darwin, ' Fertilization of Orchids,' p. 1 59. by Google 410 THE PRINCIPLES OF SCIENCE. may meet with specimens of a race which was formerly far more common but is now undergoing extinction, and is nearly the last of its kind. Thus we may explwn the occurrence of exceptional forms such as are found in the Amphioxus. The Equisetaceae perplex botanists by their want of affinity to other orders of Acrogenous plants. This doubtless indicates that their genealogical con- nexion with other plants roust be sought for in the most distant past ages of geological development. Constancy of character, as Mr. Darwin has said', is what is chiefly valued and sought after by naturalists ; that is to say naturalists wish to find some distinct family- mai'k, or group of characters by which they may clearly recognise the relationship of descent between a large group of living forms. It is accordingly a great relief to the mind of the naturalist when he comes upon a defi- nitely marked group, such as the DiatomaccEe, which are clearly separated from their nearest neighbouri t'le Des- midiaceffi by their siliceous framework and the absence of chlorophyll. But we must no longer think that because we fail in detecting constancy of character the fault is in oxur classificatory sciences. Where gradation of charac- ter really exists, we must devote ourselves to defining and registering the degrees and limits of that gradation. The ultimate natural arrangement will often be devoid of strong lines of demarcation. Let naturalists, too, form their systems of natural classification with all care they can, yet it will certainly happen &om time to time that new and exceptional forms of animals or vegetables will be discovered, and will require the modification of the system. A natural system is directed, as we have seen, to the discovery of empirical laws of correlation, but these laws being purely empirical will frequently be falsified by more extensive investiga- ■ 'Descent of Uaii,' vol. i. p. 314. by Google CLASSIFICA TION. 4 1 1 tion. From time to time the notioos of naturalists have been greatly widened, especially in the case of Australian animals and plants, by the diecovery of unexpected com- binatione of organs, and such events must often happen in the future. If indeed the time shall come when all the forms of plants are discovered and accurately de- Gcribed, the science of Systematic Botany will then be placed in a new and more favourable position, aa remarked by Alphonae Decandolle''. It ought, I think, to be allowed that though the genea- logical classification of plants or animals is doubtless the most natural and instructive of all, it is not necessarily the best for all purposes. There may be correlations of properties important for medicinal, or oUier practical purposes, which do not corret^nd to the correlations of descent. We must regard the bamboo as a tree rather than a grass, although it is botanically a grass. For legal purposes we may still with advantage continue to treat as fish, the whale, seal, and other cetace». We must class plants together according as they are Arctic, or Alpine, or belong to the temperate, sub-tropical or tropical regions. There may be some causes of likeness apart from hereditary relationship, and in a logical and practical point of view we must not attribute exclusive excellence to any one method of classification. Classification hy T^pes. Perplexed by the difficulties arising in natural history from the discoveiy of intermediate forms, naturalists have resorted to what they call classification by types. In- stead of forming one distinct class defined by the invari- able possession of certain assigned properties, and rigidly including or excluding objects according as they do or ■■ ' LuwB of Botanical Nomenelatuw,' p. i6. by Google 412 THE PRINCIPLES OF SCIENCE. do not poBsess all these properties, naturalists select a typical form or specimen, and they group around it all other forms or specimens which resemble this type more than any other selected type. ' The type of each genus,' we are told*, 'shoxild be that species in which the charac- ters of its group are best exhibited and most evenly balanced.' It would usually consist of those descendants of a form which had undergone little alteration, while other descendants had suffered slight differentiation in various directions. It would be a great mistake to suppose that this classi- fication by types is a logically distinct method. It is either not a real method of classification at all, or it is a merely abbreviated mode of representing a very com- plicated system of arrangement A class must be defined by the invariable presence of certain common properties. If, then, we venture to include an individual in which one of these properties does not appear, we either fall into Ic^cal contradiction, or else we form a new class with a new definition. Even a single exception consti- tutes a new class by itself, and by calling it an excep- tion we merely imply that this new class closely resembles that firom which it diverges in one or two points only. Thus if in the de6nition of the natural order of Kosacese, we find that the seeds are one or two in each carpel, but that in the genus Spiraea there are three or four, this must mean either that the number of seeds is not a part of the fixed definition of the class, or else that Spirsea does not belong to that class, though it may be closely ap- proximated to it. Naturalists continually find themselves between two horns of a dilemma ; if they restrict the number of marks specified in a definition so that every form intended to come within the dass shall possess all ' Wnterhouse, quoted by Woodward in his ' Rudimentai; Treatiae of Rewnt and Fosdl Shells,' p. 61. by Google CLA SSIFICA TJON. 4 1 3 those marks, it will then be iisually found to include too many forms ; if the definition be made more particular, the result is to produce so-called anomalous genera, which, while they are held to belong to the class, do not in all rejects conform to its definition. The practice has hence arisen of allowing considerable latitude in the definition of natural orders. The family of Cruciferae, for instance, forms an exceedingly well marked natural order, and among its characters we find it specified that the fruit is a pod, divided into two cells by a thin partition, from which the valves generally separate at maturity ; but we are also informed that, in a few genera, the pod is one-celled, or indehiscent, or separates transversely into several joints™. Now this must either mean that the formation of the pod is not an essential point in the definition, or that there are several closely associated families. The same holds true of typical classification. The type itself is an individual, not a class, and no other object can be exactly like the type. But so soon as we abstract the individual peculiarities of the type and thus specify a finite number of qualities in which other objects may resemble the type, we immediately constitute a class. If some objects resemble the type in some points and others in other points, then each definite collection of points of resemblance constitutes intensively a separate class. The very notion of classification by types is in fact erroneous in a strictly logical point of view. The naturalist is constantly occupied by endeavouring to mark out definite groups of living forms, where the forms them- selves do not in many cases admit of any such rigorous fines of demarcation. A certain laxity of logical method is thus apt to creep in, the only remedy for which will be m Bentliam's 'Handbook of the Bridsli Flora' (r866), p. 25. Digitized by Google 414 THE PRiyCIPLES OF SCIEXCS. the frank recognition of the fact that according to tlie theory of hereditary descent, the gradation of characters is probably the rule, and the precise demarcation between groups the exception. Natural Genera and Species. One important result of the establishment of the theory of evolution, is to explode all notions about the existence of natural groups constituting separate creations. Naturar lists have long held that every plant belongs to some species or group, marked out by invariable characters, which do not change by difference of soil, climate, crosa- breeding, or other circumstances. They were unable to deny the existence of such things as sub-species, varieties, or hybrids, so that a species of plants was often sub- divided and classified within itself. But then the dif- ferences upon which this sub-classification depended were supposed to be variable, and thus distinguished from the invariable characters imposed upon the whole species at its creation. Similarly a Natural Genus was a group of species, and was marked out from other genera by eternal differences of still greater importance. We now, however, perceive that the existence of any such groups as genera and species is an arbitrary creation of the naturalist's mind. All resemblances of plants, in- deed, are natural, so far as they express their hereditary affinities, but this applies as well to the variations within the species as to the species itself, or the larger natiural classes. All is a matter of degree. The deeper differences between plants have been produced by the differentiating action of circumstances during millions of years, so that it would naturally require millions of years to undo this result, and prove experimentally that the forms can be approximated together again. Sub-species may often have by Google OLA SSTFICA TION. 4 1 5 arisen within historical times, and varieties approaching to sub-species may often be produced by the horticul- turist in a few years. Such varieties can easily be brought back to their original form, or, if placed in the original circumstances, will themselves revert to that form ; but according to Darwin's views all forms are capable of un- Umited change, and, it might possibly be, unlimited re- version, if sufficient time and suitable circumstances be granted. Many fruitless and erroneous attempts have been made to establish some rigorous criterion of specific and generic difference, so that these classes might have a definite value or rank in all branches of biology. Linnsus adopted the view that the species was to be defined as a distinct Creation saying", 'Species tot numeramus, quot diversse forma? in principio sunt create,' or again, 'Species tot sunt, quot diversas formas ah initio produxit Infinitum Ens ; quffi formje, secundum generationis inditaa leges, pro- duxere plures, at slbi semper similes.' Of genera he also says°, 'Genus omne est naturale, in primordio tale crea- tum.' It was a common doctrine added to and essential to that of distinct creation that these species coidd not produce intermediate and variable forms, so that we find LinnsBus in another work obliged by the ascertained exis- tence of hybrids to take a different view ; he says'', 'Novas species immo et genera ex copula diversarum specierum in regno vegetabilium oriri prime intuitu paradoxum videtur ; interim ohservationes sic fieri non Ita dissuadent.' Even supposing in the present day that we could assent to the notion of a certain number of distinct creatlonal acts, this notion would not help us in the theory of classi- " ' Philosopliia Botaoica' (1770), $ 157, p- 99. " Ibid, § 159, p. ioo. 9 'AoHBnitates Academicie' (1744), vol. i. p. 70. Quoted in 'Edin- burgh Review/ October 1868, vol. cxxviii. pp. 416, 417. by Google 416 THE PRINCIPLES OF SCIENCE. ficatioD. Naturalists have never pointed out any separate method of deciding what are the resulte of distinct crea- tions, and what are not. Aa Darwin says^, * the definition must not include an element which cannot possibly be ascertained, such as an act of creation.' It is, m fact, by investigation of forms and classification that we should ascertain what were distinct creations and what were not ; this information would be a result and not a means of classification. The eminent naturalist Agassiz seems to consider that he has discovered an important principle, to the effect that general plan or structure is the true ground for the dis- crimination of the great classes of animals, which may be called branches of the animal kingdom'. He also thinks that genera are definite and natural groups. 'Genera,' he says", 'are most closely allied groups of animals, differ- ing neither in form, nor in complication of structure, but simply in the ultimate structural peculiarities of some of their parts ; and this is, I believe, the best definition which can be given of genera,' But it is surely apparent that there are endless degrees both of structural peculi- arity and of complication of structure. It is impossible to define the amount of structural peculiarity which consti- tutes the genus as distinguished from the speciea The form which any classification of plants or ammals tends to take is that of an imlimited series of Bubaltem classes. Originally botanists confined themselves for the most part to a limited number of such classes ; thus Linnaeus adopted Class, Order, Genus, Species, and Variety, and even seemed to think that there was some- thing essentially natural in a five-fold arrangement of groups *. 1 ' Descent of Man,' vol. i. p. 228, '' AgasaU, ' Essay on ClasBificfition,' p. 319. " Ibid. p. 349. ' ' PhiloBophia Botanies,' $ 155, p. g8. by Google CLASSIFICA TION. 417 With the progress of botany intermediate and ad- ditional divisions have gradually been introduced. Ac- cording to the Laws of Botanical Nomenclatm'e adopted by the International Botanical Congress, held at Paris " in August, 1867, no less than twenty-one names of classes are recognised — namely. Kingdom, Division, Sub-division, Class, Sub-class, Cohort, Sub-cohort, Order, Sub-order, Tribe, Sub-tribe, Genus, Sub-genus, Section, Sub-section, Species, Sub-species, Variety, Sub-variety, "Variation, Sub-variation. It is allowed by the authors of this scheme, that the. definition or degree of importance to be attributed to any of these terms may vary in a certain degree according to individual opinion. The only point on which botanists are not allowed discretion is as to the order of the successive sub-divisions ; the division of genera into tribes, or of tribes into orders ; any inversion, in short, of the arrangement being inadmissible. There is no reason to suppose that even the above list is complete and inextensible. The Botanical Congress itself recognised the distinction between variations according as they are Seedlings, Half-breeds, or Lusus Nalurce, The compli- cation of the inferior classes is increased agwn by the existence of hybrids, arising from the fertilization of one species by another deemed a distinct species, nor can we place any limit to the minuteness of discrimination of degrees of breeding short of an actual pedigree of descent. It will be evident to the reader that in the remarks upon classification as applied to the Natural Sciences, given in this and the preceding sections, I have not in the least attempted to treat the subject in a manner adequate to ita extent and importance. A volume would be insuf- ficient for tradng out the principles of sdentific method ■■ ' Iavb of Botanical NomencUtiire,' by AlphonM Decandolle, trans- lated IroDi the French, 1868, p. 19. VOL. n. Be by Google TEE PRINCIPLES OF SCIENCE. specially applicable to these branches of sdence. What raore I may be able to say upon the subject will be better said, if ever, when I am able to take up the closely- connected subject of Scientific Nomenclature, Terminology, and Descriptive Representation. In the meantime, 1 have wished to show, in a negative point of view, that natural classification in the animal and vegetable kingdoms is a special problem, and that the special methods and diffi- culties to which it gives rise are not those common to all cases of classification, as so many pbysiciata have sup- posed. Genealogical resemblances are only a special case of resemblances in general. Unique or Exceptional Ohj'ects. In framing a system of classification in almost any branch of science, we must expect to meet with unique or peculiar objects, which are so called because they seem to stand alone, having few analogies with other objects. They may also be said to be sui generis, each unique ob- ject forming, as it were, a class by itself; or they are called nondescript, because in thus standing apart it is difficult to find terms in which to explain their properties. The rings of Saturn, for instance, form a unique object among the celestial bodies. We have indeed considered this and many other instances of unique objects in the preceding chapter, on Exceptional Phenomena. Apparent, Singular, and Divergent Exceptions especially, are analo- gous in nature to unique objects. In the classification of the elements. Carbon stands apart as a substance entirely unique in its powers of producing compounds. It is considered to be a quadri- valent element, and it obeys all the ordinaiy laws of chemical combmation. Yet it manifests powers of affinity in such an exalted degree that the substances in which it by Google CLASSIFICA TION. 4 1 9 appears are more numerous than all the other compounds known to chemists. Almost the whole of the substances which have been called organic contain carbon, and are probably held together by the carbon atoms, so that many chemists are now inclined to abandon the name Organic Chemistry, and substitute the name Chemistry of the Carbon Compounds. It used to be believed that the production of the so-called organic compounds was due solely to the action of a vital force, or some inexplicable cause involved in the phenomena of life, but it is now found that chemists are able to commence with the elementary materials, pure carbon, hydrogen, and oxygen, and by strictly chemical operations, combine these together 80 as to form complicated organic compounds. So many compounds have already been thus formed that the proba- bility is very great that many others will be so formed in the course of time, and we might be inclined to generalize, and infer that all so-called organic compounds might ulti- mately be produced without the agency of living beings. Thus the distinction between the organic and the inorganic kingdoms seems to be breaking down, but our wonder at the peculiar powers of carbon must increase at the same time. In considering generalization, the law of continuity was applied chiefly to physical properties capable of mathe- matical treatment. But in the classificatory sciences, also, the same important principle is often beautifully ex- emplified. Many objects or events seem to be entirely exceptional and abnormal, and in regjird to degree or magnitude they may be so termed. We might adduce examples on the one hand of such extreme cases, but it is often easy to show, on the other hand, that they are connected by intermediate links with other apparently difierent cases. In the organic kingdoms of natxure there is a common E e 2 by Google 420 THE PRINCIPLES OF SCIENCE. groiondwork of similMity numing through all classes, but particular actions and processes present themselves conspicuously in particular families and classes. Tenacity of life is most marked in the Rotifera, and some other kinds of microscopic organisms, which can be dried and boiled without loss of life. Beptiles are distinguished by torpidity, and the length of time they can live without food. Birds, on the contrary, exhibit ceaseless activity and high muscular power. The ant is as conspicuous for intelligence and size of brain among insects as the quad- rumana and man among vertebrata. Among plants the Leguminosae are distinguished by a tendency to sleep, folding their leaves at the approach of night In the genus Mimosa, especially the Mimosa pudica, commonly called the sensitive plant, the same tendency is magnified into an extreme irritability, almost resembUng voluntary motion. More or less of the same irritability probably belongs to v^etable forms of every kind, but it is of course to be investigated with special ease in such an extreme case. In the Gymnotus and Torpedo, we find that organic structures can act like galvanic batteries. Are we to suppose that such animals are entirely anomalous ex- ceptions ; or may we not justly expect to find less intense manifestations of electric action in all animals and plants 1 In the animal world we find many phenomena which seem to be peculiar to certain classes, but are afterwards found to differ but in d^^e from what is always present. The lower animals, for instance, seem to difler entirely from the higher ones in the power of reproducing lost limbs. A kind of crab has the habit of casting portions of its claws when much frightened, but they soon grow again. There are multitudes of smaller animalfi which, like the Hydra, may be cut in two and yet live and develop into new complete individuala No mammalian animal can repro- Digitized by Google OLASSIFICA TION. 421 dace a limb, and in appearance there is no analogy. But it was suggested by Blumenbach that the healing of a wound in the higher animala really represents in a lower degree the power of reproducing a limb. That this is true may be shown by adducing a multitude of inter- mediate cases, each adjoining pair of which are clearly analogous, so that we pass gradually from one extreme to the other. Darwin holds, moreover, that any such re- storation of parts is closely connected with that perpetual replacement of the particles which causes every organized body to be after a time entirely new as regards its con- stituent substance. In short, we approach to a great generalization under which alt the phenomena of growth, restoration, and maintenance of organs are effects of one and the same power*. It is perhaps still more sur- prising to find that the complicated process of sexual reproduction in the higher animals may be gradually traced down to a simpler and simpler form, which at last becomes undistinguishable fmra the budding out of one plant from the stem of another. By a great generalization we may regard all the modes of reproduction of organic life as alike in their nature, and varying only in com- plexity of development?. Limits of Classification. Science can extend only so far as the power of accurate dassification extends. If we cannot detect resemblances, and asdgn their exact character and amount, we cannot have that generalized knowledge which constitutes science; we cannot infer frx>m case to case. It will readily be ' Darwin, ' The Yarialion of Animals and Plants,' vol. iL pp. 393, 359, ftc i quoting Paget, ' Lectures on Pathology,' 1853, PP- '6'> "^4- T Ibid. vol. ii p. 373. by Google 422 THE PRINCIPLES OF SCIENCE. observed that classification is the opposite process to dis- crimination. If we feel that two tastes differ, for instance, the tastes of two specimens of wine, the mere feet of difference existing prevents inference. The detection of the difference saves us, indeed, from false inference, be- cause so far as difference exists, all inference is impossible. But classification consists in detecting resemblances of all degrees of generality, and ascertaining exactly how far such resemblances extend, while assigning precisely at the same time the points at which difference begins. It enables iis, then, at once to generalize and make inferences where it is possible, and it saves us at the same time from going too far. FuU classifications constitute a complete record of all our knowledge of the objects or events classified, and the limits of exact or scientific knowledge are identical with the limits of classification. It must by no means be supposed that every group of natural objects will be found capable of rigorous classification. There may be substances which vary by- insensible degrees, consisting, for instance, in varying mixtures of simpler substances. Granite is a mixture of quartz, felspar, and mica, hut there are hardly two specimens in which the proportions of these three constituents are alike, and it would be impossible to lay down definitions of distinct species of granite without finding an infinite variety of intermediate species. The only true classification of granites, then, would be founded, on the proportions of the constituents present, and a chemical or microscopic analysis would be requisite, in order that we should assign any specimen to its true position in the series. Granites vary, again, by insensible degrees, as regards the magnitude of the crystals of fel- spar and mica. Precisely similar remarks might be made concerning the classification of other plutonic rocks, such as syenite, basalt, pumice-stone, lava, tuff, &c. by Google CLASSIFWA TION. 423 The nature of a ray of homogeneous light is strictly defined, either by its place in the spectrum or by the cor- responding wave-length, but a ray of mixed light adnuts of no simple claasification ; any of the infinitely numerous rays of the continuous spectrum may be present or absent, or present in various intensities, so that we can only class and define a mixed colour by defining the intensity and wave-length of each ray of homogeneous light which is present in it. Complete spectroscopic analjrsis and the determination of the intensity of every part of the spec- trum yielded by a mixed ray is requisite for its accurate classification. Nearly the same may be said of complex sounds. A simple sound undulation, if we could meet with such a sound, would admit of precise and exhaustive classification as regards pitch, the length of wave, or the number of waves reaching the ear per second being a suf- ficient criterion. But almost all ordinary sounds, even those of musical instruments, consist of complex aggregates of undulations of several difierent pitches, and in order to classify the sound we should have to measure the Inten- sities of each of the constituent sounds, a work which has been partially accomplished by Professor Helmholtz, as regards the vowel sounds. The difierent tones of voice distiuctive of difTerent individuals must also be due to the intermixture of minute waves of various pitch, which are at present quite beyond the range of experimental in- vestigation. We cannot, then, at present, attempt to classify the different kinds or timhres of sound. The difBculties of classification are even greater when a varying phenomenon cannot be shown to be a mixture of simpler phenomena. If we attempt, for- instance, to classify the tastes of natural and artificial substances, we may rudely group them according as they are sweet, hitter, saline, alkaline, acid, astringent, or fiery ; but it is evident that these groups are bounded by no sharp lines by Google 424 THE PRINCIPLES OF SCIENCE. of definition. Tastes of mixed or intermediate character may exist almost ad infinitum,, and, what is still more troublesome, the tastes clearly united within one claas may differ more or less from each other, without our being able to arrange them in subordinate genera and species. The same remarks may be made concerning the classifi- cation of odours, which may be roughly grouped according to the arrangement of Linnseus as. Aromatic, Fragrant, Ambrosiac, Alliaceous, Fetid, Virulent, Nauseous. Within each of these vague classes, however, there would be infinite shades of variety, and each class would graduate probably into each other class. The varieties of odour which can be discriminated by an acute olfactory organ are almost infinite ; every rock, stone, plant, or animal has some slight odour, and it is well known that dogs, or even blind human beings, can discriminate persons by a slight distinctive odour which usually passes unnoticed. Nearly similar remarks may be made concerning the higher feelings of the human mind, usually called emotions. We know what is anger, grieC fear, hatred, love ; and many systems for classifying these feelings have been proposed at one time or another. They may be roughly distinguished according as they are pleasurable or painful, prospective or retrospective, selfish or sympathetic, active or passive, and possibly in many other ways, but each mode of arrangement wiU be indefinite and unsatisfactory when followed into detaila As a general rule, the emo- tional state of the mind at any moment will be neither pure anger nor pure fear, nor any one pxu^ feeling, but an indefinite and complex aggregate of feelings. It may he that the state of mind is really a sum of several distinct modes of agitation, just as a mixed colour is the sum of the several distinct rays of the spectrum. In this case there may be more hope of some method of analysis being successfully applied at some future time. But it may by Google CLASSIFICATION. 425 be found that states of mind really graduate into each other, so that rigorous classification would prove to be hopeless. A little reflection will show that there are whole worlds of existences which in like manner are incapable of logical analysis and classification. One friend may be able to single out and identify another friend by his countenance among a million other countenances. Faces are capable of infinite discrimination, but who shaU classify and define them, or say by what particular shades of feature he does judge. There are of courBe certain distinct types of face, but each type is connected with each other type by infinite intermediate specimens. We may classify melodies according to the major or minor key, the character of the time, and some other distinct points ; but every melody has independently of such cir- ciunstances its own distinctive character and effect upon the ruind. Similar remarks might be made concerning a multitude of other circumstances. We can detect dif- ferences between the styles of literary, musical, or artistic compositions. We can even in some cases assign a picture to its painter, or a symphony to its composer, hj a subtle feeling of resemblances or differences of character and expression, which may be felt, but cannot be described. Finally, it is apparent that in human character there is un&thomable and inexhaustible diversity. Every mind is more or less like every other mind ; there is always a basis of fflmilarity, but there is a superstructure of feelings, impulses, or motives which is distinctive for each person. We can often, indeed, predict the general character of the feelings or actions which will be produced in a given individual well known to us, by a given external event, but we also know that we are oflen inexplicably at &,ult in all our inferences. No one can safely generalize upon the subtle variations of temper and emotion which may Digitized by Google 426 THE PRINCIPLES OF SCIENCE. arise even in a pereon of ordinary character. As human knowledge and civilization progress, these characteristic differences tend to develop and multiply themselves rather than decrease. Character grows more evidently many- sided. Two well educated Englishmen are far better dis- tinguished from each other than two common labourers, and these are better distinguished, again, than two Australian aborigines. Thus the complexities of exist- ing phenomena develop themselves more rapidly than scientific method can overtake them. In spite of all the boasted powers of science, we cannot really apply method to those existences, namely, our own minds and characters, which are more important to us than all the stars and nebulae. ■¥*^- DijiiiMb, Google BOOK YI. CHAPTER XXXI. KEFLECnONS ON THE RESULTS AMD LIMTTS OP SCIENTIFIC METHOD. Before concluding a work on the Principles of Science, it will not be inappropriate to add some remarks upon the limits and ultimate bearings of the knowledge which we may acquire by the constant employment of scientific method. All Bcience consists, it has several times been stated, in the detection of identities and uniformities in the action of natural agents. The purpose of inductive inquiry is to ascertain the apparent existence of necessary connexion between causes and effects, the establishment of natural laws. Now so far as we thus leam the in- variable course of natiure, the future becomes the neces- sary sequel of the present, and we are brought beneath the sway of powers with whidi nothing can interfere. By degrees it is found, too, that the chemistry of organized substances is not widely separated irom, but is rather continuous with, that of earth and stones. Life itself seems to be nothing but a special form of that energy which is manifested in heat and electricity and mechanical force. The time may come, it almost seems. by Google 428 THE PRINCIPLES OF SCIENCE. when the tender mechanism of the brain will be traced out, and every thought reduced to the expenditure of a determinate weight of nitrogen and phosphorus. No apparent limit exists to the success of scientific method in weighing and measuring, and reducing beneath the sway of law, the phenomena both of matter and of mind. And if mental phenomena be thus capable of treatment by the balance and the micrometer, can we any longer hold that mind is distinct irom matter ? Must not the same inexorable reign of law, which is apparent in the motions of brute matter, be extended to the most subtle feelings of the human heart ? Are not plants and animals and ultimately man himself, merely crystals, as it were, of a complicated form ? If so, oiu? boasted Free Will becomes a delusion, Moral Eeaponsibility a fiction. Spirit a mere name for the more curious manifestations of material energy. All that happens, whether right or wrong, plea- surable or painful, is but the outcome of the necessary relations of time and space and force, and of the laws of matter emerging from them, which are fixed in the very nature of things. Materialism seems, then, to be the coming reli^on, and resignation to the nonenity of human will the only duty. Such may not generally be the reflections of men of science, but I believe that we may thus describe the secret feelings of fear which the constant advance of scientific investigation excites in the minds of many who view it Jrom a distance. Is science, then, essentially atheistic and materialistic in its tendency \ Does the uniform action of material causes, which we leam with an ever increasing approach to certainty, preclude the hypothesis of an intelligent and benevolent Creator, who has not only designed the existing universe, but who still retains the power to alter its course &om time to time ? by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 429 To enter actually upon theological discussions would be evidently beyond the scope of this work. It is with the Bcientific method common to all the sciences, aod not with any of the separate sciences, that we are concerned. ■ Theology therefore would be at least as much beyond my scope as chemistry or geology. But I believe that grave misapprehensions exist as regards the very nature of this scientific method. There are scientific men who assert that the interposition of Providence is impossible, and prayer an absurdity, because the laws of nature are inductively proved to be invariable. Inferences are drawn not so much from particular sciences as from the lo^cal foundations of science itself, to n^ative the impulses and hopes of men. Now I may properly venture to state that my own studies in logic lead me to call in question all such negative inferences. Those so-called laws of nature are uniformities observed to exist in the action of certain material agents, biit it is logically impossible to show that ail other agents must behave as these do. The too exclusive study of particular branches of physical science seems in some cases to generate an over confident and dogmatic spirit. Rejoicing in the success with which a few groups of facts are brought beneath the apparent sway of laws, the investigator hastily assumes that he is dose upon the ultimate springs of being. A particle of gelatinous matter is foimd to obey the ordinary laws of chemistry ; yet it moves and lives. The world is therefore asked to believe that chemistry can resolve the m3r8teriea of existence. The Meaning of Natural Law. ' Pindar speaks of Law as the Buler of the Mortals and the Immortals, and it seems to be commonly supposed that the so-called Laws of Nature, in like manner, rule by Google 430 THE PRINCIPLES OF SCIENCE. man and his Creator. The course of nature is regained as being determined by invariable principles of mechanics ■which have acted since the world began, and will act for infinite ages to come. Even if the origin of all things be attributed to an intelligent creative mind, that Being ia regarded as having yielded up arbitrary power, and as being subject like a human legislator to the laws which he has himself enacted. Such notions I should describe as superficial and erroneous, being derived, as I think, from false views of the nature of scientific inference, and the degree of certainty of the knowledge which we acquire by inductive investigation. A law of nature, as I regard the meaning of the expression, is not a uniformity which must be obeyed by all objects, but merely a uniformity which is as a matter of fcu3t obeyed by those objects which have come beneath our observation. There is nothing whatever incompa- tible with logic in the discovery of objects which should prove exceptions to any law of nature. Perhaps the best established law is that which asserts an invariable cor- relation to exist between gravity and inertia, so that all gravitating bodies are found to possess inertia, and all bodies possessing inertia are found to gravitate. But it would be no reproach to our scientific method, if something were ultimately discovered to possess gravity without in- ertia. Strictly defined and correctly interpreted, the law itself would acknowledge the possibility ; for with the statement of every law we ought properly to join an esti- mate of the number of instances in which it has been observed to hold true, and the probability thence calcu- lated, that it will hold true in the next case. Now as we before found (vol. i. p. 299) no finite number of instances can warrant us in expecting with certainty that the next instance will be of like nature ; in the formulas yielded ■ by the inverse method of probabilities a unit always by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 431 appears to represent the probabilltj that our inference will be mistaken. I demur to the assumption that there is any necessary truiJi even in such fundamental laws of nature as the Indestructibility of Matter, the Conservation of Force, or the Laws of Motion. Certain it is that men of science have recognised the conceivability of other laws, or even investigated their mathematical conditions. Sir George Airy investigated the mathematical conditions of a perpetual motion (vol. i. p. 256), and Laplace and New- ton discussed various imaginary laws of forces incon- sistent with those so &r observed to operate in the universe (vol ii. pp. 304, 392). The laws of nature, aa I venture to r^;ard them, are simply general propositions concerning the correlation of properties which have been observed to hold true of bodies hitherto observed. On the assumption that our experience is of adequate extent, and that no arbitrary interference takes place, we are then able to assign the probability, always less than certainty, that the next object of the same apparent nature will confonn to the same law. Infiniteness of the Universe. We may safely accept as a satisfactoiy scientific hypo- thesis the doctrine so grandly put forth by Laplace, who asserted that a perfect knowledge of the universe, as it existed at any given moment, would give a perfect know- ledge of what was to happen thenceforth and for ever after. Scientific inference is impossible, unless we may regard the present as the necessary outcome of what is past, and the necessary cause of what is to come. To the view of Perfect Intelligence nothing is uncertwn. The astronomer can calculate the positions of the heavenly bodies when thousands of generations of men shall have by Google 432 THE PRINCIPLES OF SCIENCE. passed away, and in this fact we have some illustra- tion, as Laplace remarks, of the power which scientific prescience may attain. Douhtlese, too, all efforts in the investigation of nature tend to bring us nearer to the possession of that ideally perfect power of intelligence. Nevertheless, as Laplace with profound wisdom adds*, we must ever remain at an infinite distance from the goal of our aspirations. Let us assume, for a time at least, as a highly probable hypothesis, that whatever is to happen must be the out- come of what is ; there then arises the question, What is ? Now our knowledge of what exists must ever remain im- perfect and fallible in two respects. Firstly, we do not know all the matter that has been created, nor the exact manner in which it has been distributed through space. Secondly, assuming that we had that knowledge, we should still be wanting in a perfect knowledge of the way in which the particles of matter will act upon each other. The power of scientific prediction extends at the most to the limits of the data employed. Every con- clusion is purely hypothetical and conditional upon the non-interference of agencies previously undetected. The law of gravity asserts that every body tends to approach towards every other body, with a certain determinate force, but even supposing the law to hold true, it does not assert that the body will approach. No single law nor science can warrant us in making any one absolute prediction. We must know all the laws of nature and all the existing agents acting according to those laws before we can say what will occur. To assume, then, that scientific method can take everything within its cold embrace of uniformity, is to imply that the Creator cannot outstrip the intelligence of his creatures, and that the existing > ' Tb^rie Analytique des Probabilit^s,' quoted by Babbage, ' Ninth Bridgwater TretitiBe,' p. 1^3. by Google RESULTS AJfD LIMITS OF SCIENTIFIO METHOD. 433 universe is cot infinite in extent and complexity, an assumption for which I can see no logical basis whatever. The Indeterminate Prohlem of Creation. A second and very serious misapprehension concern- ing the import of a law of nature may now be pointed out. It is not uncommonly supposed that a law deter- mines the character of the results which shall take place, as, for instance, that the law of gravity determines what force of gravity shall act upon a given particle. Surely a little reflection must render it plain that a law by itself determines nothing. It is a law plm agents obeying that law which have results, and it is no part of the law to govern or define the number and place of its own agents. Whether a particle of matter shall gravi- tate, depends not upon the law of Newton only, but upon the distribution of surrounding particles. The theory of gravitation may perhaps be true throughout all time and in all parts of space, and even the Creator may never find occasion to create those possible excep- tions to it which I have asserted to he conceivable. Let this be as it may, and our science cannot certainly determine the question, yet the theory of gravitation itself gives no indication of the forces which may be brought to act at any point of space. The force of gravitation acting upon any particle depends, as we have seen, upon the number, mass, distance, and rela- tive position of all the other particles of matter within the hounds of space at the instant in question. Even assuming that all matter when once distributed through space at the Creation, was thenceforth to act in an in- variable manner without subsequent interference, yet the actual configuration of matter at any moment, and VOL. II. F f by Google 434 THE PRINCIPLES OF SCIENCE. the coDBequent results of the law of gravitation muBt have been entirely a matter of free choice. Chalmers has most distinctly pointed out that the existing collocations of the material -world are at least as important as the laws which the objects obey. He remarks that a certain class of writers entirely over- look the distinction, and forget that mere laws without collocations would have afforded no security against a ttu*bid and disorderly chaosl>. Mr. J. S. Mill has recog- nised " the truth of Chalmers' statement, without draw- ing the proper inferences from it. He says'* of the di&- tribution of matter through space, 'We can discover nothing regular in the distribution itself; we can reduce it to no uniformity, to no law.' More lately the Duke of Argyle in his well known work on the ' Eeign of law' has drawn attention to -the profoimd distinction between laws and collocations of causes. The original conformation of the material universe was, BO far as we can possibly tell, free from all restriction. There was unlimited space in which to frame it, and an unlimited number of material particles, each of which could be pleiced in any one of an infinite number of different positions. It mxist also be added that each particle might be endowed with any one of an infinite nxmiber of degrees of vis viva acting in any one of an nfinite number of different directions. The ' Creation was, then, what a mathematician an indeterminate problem, and it was inde- in an infinitely infinite number of waya In- imerouH and various imiverses might then fashioned by the various distribution of the idgwftter Treatise' (1834), pp. 16-34. )f Logic," sth edit. bk. III. chap. V. $ 7. Chap. XVI § 3. i. p. 384- by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 435 original nebulous matter, although all the partides of matter should obey the one law of gravity. Lucretius tells us bow in the original rain of atoms some of these little bodies diverged from the rectilineal direction, and coming into contact with other atoms gave rise to the various combinations of substances and phe- nomena which exist. He omitted, indeed, to tell us whence the atoms came, or by what force some of them were caused to diverge, but surely these omissions involve the whole question. I accept the Lucretian conception of creation when properly supplemented. Every atom which existed in any point of space must have existed there previously, or must have been created there by a previously existing Power. When placed there it must have had a definite mass and a definite energy, kinetic or potential as regards other existing atoms. Now, as before remarked, an unlimited number of atoms can be placed in unlimited space in an entirely imlimited number of modes of distribution. Out of infinitely infinite choices which were open to the Creator, that one choice must have been made which has yielded the universe as it now exists. It would indeed be a mistake to suppose that the law of gravity, when it holds true, is no restriction in the dis- tribution of force. That law is a geometrical law, and it would in many cases be mathematically impossible, as fer as we can see, that the force of gravity acting on one particle should be small while that on a neighbouring particle was great We cannot conceive that even Omni- potent Power should make the angles of a triangle less or greater than two right angles. The primary laws of thought and the fiindamental notions of the mathemati- cal sciences do not seem to us to admit of any alter- ation. Into the metaphysical origin and meaning of the apparent necessity attaching to such laws I have not P f 2 Digitized by Google 436 THE PRINCIPLES OF SCIENCE. attempted to inquire in this work, and it is act requisite for my present purpose. If the law of gravity were the only law of nature and the Creator had chosen to render all matter ohedient to that law, there would doubtleBs be restrictiona upon the effects derivable irom any one distribution of matter. Hierarchy of Natural Laws. A further consideration inevitably presenta itself. A natural law like that of gravitation expresses a certain uniformity in the mode of action of agents submitted to it, and this uniformity produces, as we have seen, certain geometrical restrictions upon the effects which those agents may produce. But there are other forces and laws besides those of gravity. One force may override another, and two laws may each be obeyed and may each disguise the action of the other. In the intimate constitution of matter there may be hidden springs of force which, while acting in accordance with their own fixed laws, may lead to sudden and xmexpected changes. So at least it has been found from time to time in the past, and so there is every reason to believe it will be found in the futura To the ancients it seemed incredible that one lifeless stone could make another leap towards it. A piece of iron while it obeys the magnetic forces of the loadstone does not the less obey the law of gravity. A plant also gravitates downwards as r^ards every con- stituent cell or fibre, and yet it persists in growing upwards. Life altogether is an exception to the ample phenomena of mineral substances, not in the sense of disproving those laws, but in that of superadding forces of new and inexpUcable character. Doubtless no law of chemistry is broken by the action of the nervous cells, and no law of physics by the pulses of the nervous Digitized by Google RESULTS AND LIMITS OP SCIENTIFIC METHOD. 437 fibres, but something requires to be added to our sciences in order that we even explain these subtle phenomena. Now there is absolutely nothing in science or in scien- tific method to warrant us in assigning any limit to this hierarchy of laws. When in many undoubted cases we find law overriding law, and at certain points in our experience producing unexpected results, we can never venture to affirm that we have exhausted the strange phenomena which may have been provided for in the original constitution of matter. The Universe might have been so designed that it should for long intervals go through the same round of almost unvaried existence, and yet so that events of exceptional character should from time to time be produced. Charles Babbage showed in that most profound and eloquent work, 'The Ninth Bridgwater Treatise,' that it was theoretically possible for human artists to design a machine, consisting of metallic wheels and levers, which should work invari- ably by one simple law of action during any finite number of steps, and yet at a fixed moment, however distant, should manifest a single breach of law. Such an engine might go on counting, for instance, the natural numbers until they might reach a number requiring for its expression a hundred million digits. * If every letter in the volume now before the reader's eyes,' says Babbage", ' were changed into a figure, and if all the figures con- tained in a thousand such volumes were arranged in order, the whole together would yet fall far short of the vast induction the observer would have had in favour of the truth of the law of natural numbers . . . Yet shall the engine, true to the prediction of its director, after the lapse of myriads of ages, iulfil its task, and give that one, the first and only exception to that time-sanctioned law. What would have been the chances against the appear- • 'Ninth Bridgwater Treatise,' p. 140. by Google 438 THE PRINCIPLES OF SCIENCE. ance of the excepted caae, immediately prior to its occur- rence V As Babbage further showed', a calculating engine, afler proceeding through any required number of motions according to a first law, may be made suddenly to suffer a change, so that it shall then commence to calculate according to a wholly new law. After giving the natural numbers for any finite time, it might suddenly begin to give triangular, or square, or cube numbers, and these changes might theoretically be conceived aa occurring time after time. Now if such occurrences can be deagned and foreseen by a human artist, it is surely within the capacity of the Divine Artist to provide for similar changes of law in the mechanism of the atom, or the construction of the heavens. Physical science, so far as its highest speculations can be trusted, gives some indication of a change of law in the past history of the Universe. According to Sir W. Thomson's deductions from Fourier's Theory of Heat, we can trace down the dissipation of heat by conduction and radiation to an infinitely distant time when all things will be uniformly cold. But we cannot similarly trace the beat-history of the Universe to an infinite distance in the past. For a certain negative value of the time the formulas g^ve impossible values, indicating that there was some initial distribution of heat which could not have re- sulted, according to known laws of nature, from any pre- vious distribution K. There are other cases in which a consideration of the dissipation of energy leads to the conception of a limit to the antiquity of the present order of things**. Human science, of course, is fallible, and f ' Ninth Bridgwater Treatise,' pp. 34-43. > Tait'e ' Thermodynamica,' p. 38, ' Cambridge Mathematical Jour- nal,' vol. iii. p. 174. •> Clerk Maxwell's ' Theory of Heat,' p. 345. by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 439 some oversight or erroneous siinplificatioti in these theo- retical calculatioDS may afterwards be discovered ; but as the present state of scientific knowledge is the only ground on which erroneous interpretations of the uniformity of nature and the reign of law are founded, I am right in appealing to the present state of science in opposition to these interpretations. Now the theory of heat places us in the dilemma either of believing in Creation at an assign- able date in the past, or else of supposing that some inexpUcable change in the working of natural laws then took place. Physical science gives no countenance to the notion of infinite duration of matter in one continuous course of existence. And if in time past there has been a discontinuity of law, why may there not be a similar event aw^ting the world in the future. Infinite ingentiity could have implanted some agency in matter so that it might never yet have made its tremendous powers mani- fest. We have a very good theory of the conservation of etiergj, but the foremost physiciate do not deny that there may possibly be forms of energy, neither kinetic nor poten- tial, and therefore of unknown nature'. We can imagine reasoning creatures dwelling in a world where the atmosphere was a mixture of oxygen and in- flammable gas like the fire-damp of coal mines. If devoid of fire, they might have lived on through long ages in complete unconsciousness of the tremendous forces which a single spark could call into play. In the twinkling of an eye new laws might have come into action, and the poor reasoning creatures who were so confident in their know- ledge of the uniform conditions of their world, might have had no time even to speculate upon the overthrow of all their theories. Can we with our finite knowledge be sure that such an overthrow of our theories is impossible ? f Maxwell's 'Theory of Haat,* p. 91. by Google THE PRINCIPLES OF SCIENCE. The Amhiguous Expression. — Uniformity of Nature, I have asserted that a seiious misconception arises from on ambiguouB interpretation of the expression Uniformity of Nature. Every law of nature is the statement of a certain uniformity observed to exist among phenomena, and since the laws of nature are supposed to be invariably obeyed it seems to follow that the course of nature itself is uniform, so that we can safely judge of the future by the present. This inference is supported by some of the most profound results of physical astronomy. Laplace proved that the planetary system was stable, so that no one of the perturbations which planet produces upon planet shall become so great as to cause a disruption, and a permanent alteration in the planetary orbits. A full comprehension of the law of gravity shows that all such disturbances are essentially periodic, ao that ^ter the lapse of millions of years the planets will all return to the same relative positions and a new cycle of disturbances will commence. As other branches of inquiry progress, we seem to gain assurance that no great alteration of the world's condition is to be expected. A conflict with a comet has long been a cause of fear to some persons, but now it is credibly asserted that we have passed through a comet's tail with- out the fact being known at the time, or manifested by any more serious a phenomenon than a slight luminosity of the heavens. More recently still the earth is said to have actually touched the comet Biela, and the only result was a beautiful and perfectly harmless display of radi- ating meteors, A decrease in the heating power of the sun seems to be the next most probable circumstance, from which we might fear an extinction of life on the earth. But calculations founded on reasonable physical by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. «1 data show that bo appreciable change can be going on, and experimental data to indicate any change are wholly wanting. Geological inveBtigations show indeed that there have been extensive variations of climate in past times; vast glaciers and icebergs have swept over the temperate regions at one time, and tropical vegetation has flourished near the poles at another time. But here again the vicis- situdes of climate assume a periodic character, so that the ultimate stability of the earth's condition does not seem to be aflected. All these statements may be reasonable, but they do not in the least establish the Uniformity of Nature in the sense that extensive alterations or sudden cataslrophea are impossible. In the first place Laplace's theory of the stability of the planetary system is of an abstract character, as paying regard to nothing but the mutual gravitation of the planetary bodies and the sun. It overlooks several physical causes of change and decay in the system which were not so well known in his day as at present, and it also presupposes the absence of any interruption of the course of things by conflict with foreign astronomical bodies. It is now commonly acknowledged by astronomers that there are at least two ways in which the vis viva of the planets and satellites may sufier loss. The friction of the tides upon the earth produces a small amount of heat which is radiated into space, and this loss of energy must result in a decrease of the rotational velocity, so that ultimately the terrestrial day will become identical with the year, just as the periods of revolution of the moon upon its own axis and around the earth have already become equal. Secondly, there can now be little doubt that various manifestations of electiicity upon the earth's surface depend upon the relative motions of the planets and the sun, which give rise to various periods of in- creased intensity. Such electrical phenomena must result by Google 442 THE PRINCIPLES OF SCIENCE. in the production and dissipation of heat, the energy of which must be drawn, partially at least, irom that of the moving bodies. This effect is probably identical, as I have suggested (vol. ii. p. 213), with the very evident loss of energy of comets attributed to a so-called resisting^ medium. But whatever be the theoretical explanation of these phenomena, it is almost certain that there exists a tendency to the dissipation of the energy of the planetary system, which will in the indefinite course of time result in the fall of the planets into the sun. It is hardly probable, however, that the planetary system will be left undisturbed throughout the enormous period of time required for the dissipation of its enei^y in this way. Conflict with other bodies is so far firom being improbable, that it becomes approximately certain when we take very long intervals of time into account. As regards cometary conflicts, I am by no means satisfied with the negative conclusions drawn from the remarkable display on the evening of the 27th of November, 1872. We may often have passed through the tails of comets, which are probably electrical manifestations no more substantial than the aurora borealis. Every remarkable shower of shooting stars may also be considered as pro- ceeding firom a cometary body, so that we may be said to have passed through the thinner parts of various comets. But the earth has probably never passed, in times of which we have any record, through the nucleus of a comet-, which consists perhaps of a dense swarm of small meteorites. We can only speculate upon the efiects which might be pro- duced by such a conflict, but it would probably be a much more serious event than any yet registered in history. The probability of its occurrence, too, can hardly be assigned ; for though the probability of conflict with any- one cometary nucleus is almost infinitesimal, yet the number of comets is immensely great (vol. ii. p. 1 1). by Google RESULTS AND LIMITS OF SCIENTIFIC METBOD. 443 It is far from impoBsible, again, that the planetary system may be invaded by bodies of greater mass than any comets. The sun seems to be placed in so extensive a portion of empty space, that its own proper motion would not bring it to the nearest known star (a Centauri) in less than 139,200 years. But in order to be sure that this long interval of undisturbed life is granted to our globe, we must prove that there are no stars moving so as to meet us, and no dark bodies of considerable size flying through intervening space unknown to us. The intrusion of comets into our system, and the fact that many of them have hyperbolic paths, is sufficient to show that the sur- rounding parts of apace are occupied by multitudes of dark bodies of some size. It is quite probable that small suns might have cooled sufficiently to become non- liuninous ; for even if we discredit the theory that the variation of brightness of periodic stars is due to the revolution of dark companion stars, yet there is oiu" own globe as an unquestionable example of a smaller body which has cooled below the luminous point. Altogether, then, it is a mere assumption that the Uniformity of Nature involves the imaltered existence of our own globe. There is no kind of catastrophe which is too great or too sudden to be theoretically consistent with the reign of law. For all that our science can tell, human history may be closed in the next instant of time. The world may be dashed to pieces against some intruding . body ; it may be involved in a nebulous atmosphere of hydrogen to be exploded a second afterwards ; it may be scorched up or dissipated into vapour by some great ex- plosion in the smi ; there might even be within the globe itself some secret cause of disruption, which only needs time for its manifestation. There are even some indications, as already noticed (vol. ii. p. 327), that some violent disturbances have by Google 444 THE PRINCIPLES OF SCIENCE. actually occurred in the history of the aolar system. Olbers sought for the minor planets or asteroids, on the sup- position that they were fragments of an exploded or fractured planet^ and he was rewarded with the discovery of some of them. The retrograde motion of the satellites of the more distant planets, the abnormal position of the poles of Uranus and the excessive distance of Neptune, are other indications of some violent event, of which we have no other evidence. I adduce all these facta and argu- ments, not to show that there is any appreciable proba- bility, so far as we can judge, of actual interruption within the scope of human history, but to prove that the Uniformity of Nature is theoretically consistent with the most unexpected events of which we can form any con- ception. Possible States of the Universe. When we give the rein to scientific imagination, it becomes apparent that conflict of body with body must not be regarded as the rare exception, but as the general rule and the inevitable fate of each star system. So far as we can trace out the results of the law of gravitation, and the dissipation of energy, the universe must be re- garded as undergoing gradual condensation into a single cold solid body of gigantic dimensions. Those who so frequently use the expression Uniformity of Nature, seem to forget that the universe might exist consistently with the laws of nature in the most diverse conditions. It - might consist, on the one hand, of a glowing nebulous mass of gaseous substances. The heat might be so intense that all elements, even carbon and silicon, would resemble permanent gases, and all atoms, of whatever nature, would be flying about in chemical independence, diffusing them- selves almost uniformly in the neighbouring parts of space. There would then be no life, unless we can by Google RESULTS AlfD LIMITS OF SCIENTIFIC METHOD. 446 apply that name to the passage through each part of space of similar average ^aiDS of atoms, the particular snccee^ons of atoms being gOTemed only by the theory of probability, and the law of divergence irom a mean exhibited in the Arithmetical Triangle. Such a universe would correspond partially to the Lucretian rain of atoms, and to that nebular hypothesis out of which Laplace proposed philosophically to explain the evolution of the planetary system. According to another extreme supposition, the intense heat energy of this nebulous mass might have been mostly radiated away into the unknown regions of outer space. The attraction of gravity would then have shown itfielf between each two particles, and the energy of motion thence arising would, by incessant conflicts, be resolved into beat and dissipated. Inconceivable agea might be required for the com- pletion of this process, but the dissipation of energy thus proceeding could end only in the production of a cold and motionless stone-like universe. The relation of cause and effect, as we see it manifested in life and growth, would then degenerate into the constant existence of ■ every particle in a fixed position relative to every other particle. Logical and geometrical resemblances would still exist between atoms, and between groups of atoms crys- tallized in their appropriate forma for ever more. But time, the great variable, would bring no variation, and as to human hopes and troubles, they would have come to eternal rest. Science is not really adequate to proving that such is the inevitable fate of the universe, for we can seldom trust our beat established theories and moat careftd in- ferences far from their data. Nevertheless, the most probable speculations which we can form as to the history, especially of our own planetary system, is that it origi- Digitized by Google 446 THE PRINCIPLES OF SCIENCE. nated in a heated revolving nebtilous mass of gas, and is in a state of almost infimtelj slow progress towards the cold and stony condition. Other speculative hypotheses might doubtless be entertEiiued. Eveiy hypotbems is pressed by difficultiea If the whole universe be cooling, where does the heat go to 1 If we are to get rid of it entirely, outer space must be infinite in extent, so that it shall never be stopped and reflected back. But not to speak of metaphysical difficulties, if the medium of heat undulations be infinite in extent, why should not the material bodies placed in it be infinite also in number and quantity. It i3 quite apparent that we are venturing into speculations which altogether surpass our powers of scientific inference. But then I am arguing nega- tively ; I wish only to show that those who speak of the uniformity of nature, and the reign of law, often misinterpret entirely the meaning involved in those expressions. Law is not inconsistent with Extreme di- versity, and, BO &r as we can read the history of this planetary system, it did most probably originate in heated nebulous matter, and man's history forms but a moment in its progress towards the cold and stony condition. It is by very doubtful and speculative hypotheses alone that we can avoid such a conclusion, and I depart least from undoubted facts and well-established laws, when I assert that, whatever uniformities may underlie the phenomena of natiuB, constant variety and ever-progressing change is the real outcome. Speculations on the Beconcentration of Energy. There are unequivocal indications, as I have said, that the material universe, as we at present see i^ is pro- gressing firom some act of creation, or some discon- tinuity of existence of which the date may be approxi- by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 447 mately fixed by scientific inference. It is progressing towards a etate in which the available energy of matter will be dissipated through infinite surrounding space, and all matter will become cold and lifeless. This constitutes, as it were, the historical period of physical science, that over which our scientific insight may more or less extend. But in this, as in other cases, we have no right to interpret our experience negatively, so as to infer that because the pt%sent state of things b^an at a particular time, there was no previous existence. It may be that the present period of material existence is but one of Ka indefinite series of like periods. All that we can see, and feel, and infer, and reason about may be, as it were, but a part of one single pulsation in the existence of the universe. After Sir W. Thomson had pointed out the prepon- derating tendency which now seems to exist towards the conversion of all eneigy into heat-energy, and its equal diffusion by radiation throughout space, the late Pro- fessor Rankine put forth a remarkable speculation^. He suggested that the ethereal, or rather, as I have called it, the adamantine mediimi in which all the stais exist, and all radiation takes place, may have bounds, beyond which only empty space may exist. All heat undulations reach- ing this boundary will be totally reflected, according to the theory of undulations, and will in all probability be recon- centrated into foci situated in many parts of the medium. Whenever a cold and extinct star happens to pass through one of these foci, it will be instantly ignited and resolved by intense heat into its constituent elements. A discon- tinuity will occur in the history of that portion of matter, and the star will begin its history afi-esh with a renewed store of energy. This is doubtless a mere speculation, incapable of veri- ^ ' Report of the British Aasociation' (1853), Report of Sections, p. 12. Digitized by Google 448 THB PRINCIPLES OF SCIENCE. fication by observatioD, and almost free &om any re- strictions afforded by present knowledge. We might attri- bute various shapes to the whole body of adamantine medium, and the consequences would be various. But there is this value in such specidations, that they draw attention to the finiteness of our knowledge. We cannot deny the possible truth of such an hypothesis, nor can we place a limit to the scientific imagination in the framing of other like hypotheses. It is impossible, indeed, to follow out our scientific inferences without falling into speculation. If heat be radiated into outward space it must either proceed ad injinitum, or it must be stopped somewhere. In the latter case we fall upon Rankine's hypothesis. But if the material universe consist of a finite collection of heated matter situated in a finite portion of an infinite adamantine medium, then either this universe must have existed for a finite time, or else it must have cooled down during the infinity of past time indefinitely near to the absolute zero of temperature. I objected to Lucretius' argimient against the destructihility of matter, that we have no knowledge whatever of the laws accord- ing to which it would undergo destruction. But we do know the laws according to which the dissipation of beat appears to proceed, and the conclusion inevitably is that a finite heated material body placed in a perfectly cold infinitely extended medium would in an infinite time become infinitely approximated to zero. Now our own world is not yet cooled down near to zero, so that physical science seems to place us in the dilemma of admitting either the finiteness of past duration of the world, or else the finiteness of the portion of medium in which we exist. In either case we become involved in metaphysical and mechanical difficulties surpassing our mental powers. by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 449 The Divergent Scope for New Discover^/. In the writings of some recent philosophers, especially of Auguste Comte, and in some degree John Stuart Mill, there is an erroneous and hurtful tendency to represent our knowledge as assuming an approximately complete character. At least these and many other writers fail to impress upon their readers a truth which I think cannot he too constantly home in mind, namely, that the utmost successes which our scientific method can accomplish will not enahle us to comprehend more than an infinitesimal fraction of what there doubtless is to comprehend. Pro- fessor TyndaU seems to me open to the same charge in a less degree. He remarks* that we can probably never bring natural phenomena completely under mathematical laws, because the approach of our sciences towards com- pleteness may perhaps be asymptotic, so that howevw far we may go, there may still remain some facts not subject to scientific explanation. He thus likens the supply of novel phenomena to a convergent series, the earlier and larger terms of which have been successfully disposed of, so that only comparatively minor groups of phenomena remain for future investigators to occupy themselves upon. On the contrary, aa it appears to me, the supply of new and imexplained facts is divergent in extent, so tJiat the more we have explained, the more there is to explain. The further we advance in any generalization, the more numerous and intricate are the exceptional cases still demanding further treatment. The experiments of Boyle, Mariotte, Dalton, Gay-Lussac, and others, upon the physical properties of gases might seem to have ex- hausted that subject by showing that all gases obey the ' 'Fragments of Science,' p. 363. VOL. n. G g by Google 450 THE PRINCIPLES OF SCIENCE. same laws as regards temperature, pressure, and volume. But in reality these laws are only approximately true, and the divergences have afforded a -wide and yet quite unexhausted field for further generalization. The more recent discoveries of Cagniard de la Tour and Professor Andrews might seem to have summed up many of these exceptional facts under a wider generalization, but in reality they have opened to us vast new regions of in- teresting inquiry, and they leave wholly untouched the question why one gas or one substance behaves differently Ax)m anolJier. The science of Crystallography is that perhaps in which the most precise and general laws have been detected, but it would be utterly untrue to assert that it haa lessened the area of future discovery. We can show that each one of the seven or eight hundred fonns of calcite is derivable by plain geometrical modifications from an hexagonal prism, but who has attempted to explain the molecular forces producing these modifications, or the chemical con- ditions in which they arise \ The law of isomorphism is an importjmt generalization, for it establishes a general resemblance between the forme of crystallization of natural classes of elements. But if we examine a little more closely we find that these forms are only approximately alike, and the divergence peculiar to each substance is an unexplained exception. By many similar iUustrations it might be readily shown that in whatever direction we extend our investigations and successAilly harmonize a few facts, the result is only to raise up a host of other unexplained facts. Can any scientific man venture to state that there is less opening now for new discoveries than there was three centuries ago % Is it not rather true that we have but to open a scientific book and read a page or two, and we shall in all probability come to some recorded phenomenon of which by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 451 no precJBe explanation can yet be given ? In every such fact there is a possible opening for new discoveries, and it can only be the fault of the investigator's mind if he can look around him and find no scope for the exercise of his faculties. The Infinite Incompleteness of the Mathematical Sciences. There is one privilege which a certain amount of know- ledge should confer ; it is that of becoming aware of the indefinite weakness of our powers compared with the tasks which they might undertake if stronger. To the poor savage who cannot count twenty, the arithmetical accom- plishments of the ordinary schoolboy are miraculously great in comparison. The schoolboy cannot comprehend the almost infinitely greater powers of the student, who has acquired facility with algebraic processes. The student can but look with feelings of surprise and reverence at the powers of a Newton or Laplace. But the question at once suggests itself, Do the powers of the highest human intellect bear any moderate ratio to the things which are to be understood and calculated ? How many further steps must we take in the rise of mental ability and the extension of mathematical method before we begin to exhaust the knowable % I am inclined to find fault with mathematical writers because they often exult in what they can accomplish, but omit to point out that what they do is but an indefinitely, nay an infinitely, small part of what might be done. They exhibit a general inclination, with few exceptions, not to do so much as mention the existence of problems of an impracticable character. This may be excusable so far as the immediate practical result of their researches is in question, but the custom has the effect of misleading the Qg 2 Digitized by Google 452 THE PRINCIPLBS OF SCIENCE. general public into the fallacious notion that mathematics is a perfect science, which accomplishes what it under- takes in a complete manner. On the contraiy, it may be said that if a mathematical problem were selected by pure chance out of the whole variety which might be proposed, the probability is infinitely slight that a human mathe- matician could solve it. Just as the numbers we can count or &ame to the mind are literally nothing compared with the numbers which might exist, so the whole accomplish- ments of a Laplace or a logrange are, as it were, the little comer of the multiplication table, which has really an indefinite extent. I have Bufficientty pointed out that the rude character of all our observations prevents us from being aware of the existence of the greater number of effects and actions of nature. It must be added that, if we perceived them, we should usually be incapable of including them in our theories from want of mathematical power. Some persons may be surprised that though nearly two centuries have elapsed since the time of Newton's discoveries, we have yet no general theory of molecular action. Some approxi- mations have been made towards such a theoiy. Joule and Clausius have measiu*ed the velocity of gaseous atoms, or even determined the distance between the col- lision of atom and atom. Sir W. Thomson has approxi- mated to the number of atoms in a given bulk of sub- stance. Rankine has formed some reasonable hypotheses as to the actual constitution of atoms, but it would be a _:.i_i„ X ^ ^^^ these ingenious results of theory brm any appreciable approach to a com- f molecular motions. There is every judging from the spectra of the elements, masons, that even chemical atoms are very tures. An atom of pure iron is probably aplicated system than that of the planets by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 453 Eoid their satellites. A compound atom may perhaps he compared with a stellar system, each star a minor system in itself. The smallest particle of solid substance will consist of a vast numher of such stellar systems united in regular order, each bounded by the other, communi- cating with it in some manner yet wholly imcomprehen- Bible. Now what are our mathematical powers in com- parison with this problem ? After two centuries of continuous labour, the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other, imder the simple hypothesis of the law of gravity. Concerning these calculations we must further remember that they are piuely approximate, and that the methods would not apply where four or more bodies are acting, and all pro- duce considerable effects each upon the other. There is every reason to believe that each constituent of a chemical atom must go through an orbit in the millionth part of the twinkling of an eye, in which it successively or simul- taneously is under the influence of many other consti- tuents, or possibly comes into colUsion with them. It is, I apprehend, no exaggeration to say that mathematicians have scarcely a notion of the way in which they could successfully attack so difScult a problem of foi'ces and motions. Each of these particles is for ever solving dif- ferential equations, which, if written out in full, might perhaps belt the earth, as Sir J. Herschel has beautifully remarked™. Some of the most extensive calculations ever made, were those required for the reduction of the measurements executed in the course of the Trigonometrical Survey of Great Britain. The calculations arising out of the prin- cipal triangulation alone occupied twenty calculators during three or four years, in the course of which the IK ' Faiailiar Lectures on Scientific Subjecte,' p. 43S, by Google 454 THE PRINCIPLES OF SCIENCE. computers bad to solve aimultaneonB equations involving seventy-Beven unknown quantities. The reduction of the levellingB again required the solution of a system of ninety-one equations. But these vast calculations present no approach whatever to what would be requisite for the complete treatment of any one physical problem. The motion of glaciers is supposed to be moderately well understood in the present day. A glacier is a viscid, slowly yielding mass, neither absolutely solid nor abso- lutely iTgid, but it is expressly remarked by Forbes", that not even an approximate solution of the mathe- matical conditions of such a moving mass can yet be pos- sible. ' Every one knows,' he says, ' that such problems are beyond the compass of exact mathematics;' but though mathematicians may knowthis, they do not often enough impress that knowledge on other people. The problems which are solved in our mathematical books consist of a small selection of those which happen from peculiar conditions to be practicable. But the very simplest problem in appearance will often give rise to impracticable calculations. Mr. Todhunter" seems to blame Condorcet, because in one of his memoirs he men- tions a problem to solve which would require ft + «.' + n" + n'" —2 successive integrations. Now if our mathematical sciences are to pretend to cope with the problems which await solu- tion, we must be prepared to effect an unlimited number of successive integrations ; yet at present, and almost beyond doubt for ever, the probability that even a single integration, taken haphazard, will be found to come within our powers is exceedingly small. In some passages of that most remarkable work, the ■> ' Fhilosophical Uagaztne,' 3rd Series, vol. xzvi. p. 406. o ' History of the Theory of Probability,' p. 398. by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 455 ' Ninth Bridgwater Treatise p,' Mr. Babbage has pointed out that if we bad power to follow and detect the minutest effects of any disturbance, each particle of existing matter must be a register of all that has happened. ' The track of every canoe — of every vessel that has yet disturbed the surface of the ocean, whether impelled by manual force or elemental power, remains for ever registered in the future movement of all succeeding particles which may occupy its place. The furrow which it left is, indeed, instantly filled up by the closing waters ; but they draw after them other and larger portions of the surrounding element, and these again, once moved, communicate mo- tion to others in endless succession.' We may even say that ' The air itself is one vast library, on whose pages are for ever written aU that man has ever said or even whispered. There, in their mutable but unerring charac- ters, mixed with the earliest, as well as the latest sighs of mortality, stand for ever recorded, vows unredeemed, promises unAilfilled, perpetuating in the united move- ments of each particle, the testimony of man's changeful will".' When we read truthful reflections such as these, we may congratulate ourselves that we have been endowed with minds which, rightly employed, can form some esti- mate of their incapacity, to trace out and account for all that proceeds in the simpler actions of material nature. It ought to be added that, wonderful as is the extent of physical phenomena open to our investigation, intel- lectual phenomena are yet vastly more extensive. Of this I might present one satis&ctory proof were space available by pointing out that the mathematical functions employed in the calculations of physical science, form an infinitely small fraction of the functions which may be p ' Ninth Bridgwater Treatise,' p. 1 15. 4 Ihid. p. T13. by Google 456 THE PRINCIPLES OP SCIENCE. inveDted Common trigonometry, for instance, conraate of a great series of useftil formulas, all of which arise out of the simple fundamental relation of the sine and cosine expressed in the one equation sin *a; + cos *»: = 1. But this is not the only trigonometry which may exist ; mathematicians also recognise the so-called hyperbolic trigonometry of which the fundamental equation is cos 'x— sin *x= I, De Morgan has pointed out that the symbols of ordinary algebra form but three of an interminable series of ood- ceivable systems*. As the logarithmic operation is to addi- tion or addition to multiplication, so is the latter Xa a higher operation, and bo on without limit. We may rely upon it that indefinite, and to us incon- ceivable, advances will be made by the human intellect, in the absence of any unforeseen catastrophe to the species or the globe. Almost within historical periods we can trace the rise of mathematical science Irom its simplest germs. We can prove our descent from ancestors who counted only on their fingers, but how almost infinitely is a Newton or a Laplace above those simple savages. Pythagoras is said to have sacrificed a hecatomb when he discovered the Forty-seventh Proposition of Euclid, and the occasion was worthy of the sacrifice. Archimedes was beside himself when he first perceived his beautiful mode of determining specific gravities. Yet these great dis- coveries are the simplest elements of our schoolboy-know- ledge. Step by step we can trace upwards the acquirement of new mental powers. What could be more wonderful and unexpected than Napier's discovery of logarithms, a wholly new mode of calculation which has multiplied perhaps a hundred-fold the working powers of every computer, and indeed has rendered easy calculations which ■ ' THgonometr^- aod Double Algebra,' chap. IX. by Google RESULTS AND LIMITS OF SCIENTIFIC METBOD. 457 were before almost impracticable. Since tbe time of Newton and Leibnitz whole worlds of problems have been solved which before were hardly conceived as matters of inquiry. In our own day extended methods of mathe- matical reasoning, such as the system of quaternions, have been brought into existence. What intelligent man will doubt that the recondite speculations of a Cayley or a Sylvester may possibly lead to some new methods, at the simplicity and power of which a fiiture age will wonder, and yet wonder more that to us they were so dark and difficult. May we not repeat the words of Seneca: ' Veniet tempus, quo ista quse nunc latent, in lucem dies extrahat, et longioris ebvI diligentia : ad inquisitionem tantorum tetaa una non sufficit Veniet tempus, quo pos- teri nostri tam aperta nos nesdsse mirentur.' The Reign of Law in Mental and Social Phenomena. After we pass from the so-called physical sciences to those which attempt to investigate mental and social phenomena, the same general conclusions will hold true. No one will be foimd to deny that there are certain uni- foimities of thinking and acting which can be detected in reasoning beings, and so f^ as we detect such laws we successfully apply scientific method. But those who attempt thus to establish social or moral sciences, soon become aware that tfaey are dealing with subjects of enormous perplexity. Take, for instance, the science of Political Economy. If a science at all, it must be a mathe- matical science, because it deals with quantities of com- modities. But so soon as we attempt to draw out the equations expressing the laws of variation of demand and supply, we discover that they must have a complexity entirely surpassing our powers of mathematical treatment. by Google 458 THE PRINGIPLES OF SCIENCE. We may lay down the general form of the equations, ex- presdng the demand and supply for two or three commo- dities among two or three trading bodies, but all the functions involved are of so complicated a character that there is not much fear of scientific method making a rapid progress in this direction. If such be the prospects of a comparatively formal science, like Political Economy, what shall we say of Moral Science 1 Any complete theory of morals must deal with quantities of pleasure and pain, as Bentham pointed out, and must sum up the general tendency of each kind of action upon the good of the community. If we are to apply scientific method to morals, we must have a calculus of moral effects, a kind of physical astronomy investigating the mutual per- turbatioDS of individuals. But as astronomers have not yet fuUy solved the problem of three gravitating bodies, when shall we have a solution of the problem of three moral bodies 1 Now the sciences of political economy and morality are, comparatively, abstract and general, treating mankind from simple points of view, and attempting to detect general grounds of action. They are to social phenomena what the general sciences of chemistry, heat, and electri- city, are to the concrete science of meteorology. Before we can investigate the actions of any aggregate of men, we must have fairly mastered all the more abstract sciences applying to them, somewhat in the way that we have acquired a fair comprehension of the simpler truths of chemistry and physics. But all our physical sciences do not enable us to predict the weather two days hence with any great probability, and the general problem of meteorology is almost imattempted as yet. What shall we say then of the general problem of social science, which shall enable us to predict the course of events in a nation % There have indeed been several writers who have pro- by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 469 posed to lay the foundations of the science of history. The late Mr. Buckle undertook to write the * History of Civilisation in England,' and showed how the character of a nation could be explained by the nature of the climate and the fertihty of the soil. He omitted to explain the contrast between the ancient Greek nation and the present one; either there must have been an extraordinary revolution in the climate and the soil, or some more complex causes must be imagined to have come into operation. Auguste Comte detected some very fundamental and simple laws of development through which nations pass. There are always three phases of intellectual condition, — the theological, the metaphysical, and the positive ; and applying this general law of progress to concrete cases, Comte was enabled to predict that in the hierarchy of European nations, Spain wo\dd necessarily hold the highest place. Such are the paro- dies of science offered to us by the so-called positive philosophers. A "science of history in the true sense of the term is an absurd notion. A nation is not a mere sum of in- dividuals whom we can treat by the method of averages ; it is an organic whole, held together by ties of infinite complexity. Each individual acts and re-acts upon his own smaller or greater circle of friends, and those who acquire a public position, exert an influence on much larger sections of the nation. There will always be a few great leaders of exceptional genius or opportunities, the unaccountable phases of whose opimons and incli- nations Bway the whole body, even when they are least aware of it. From time to time arise critical positions, battles, delicate negotiations, internal disturbances, in which the slightest incidents may profoundly change the course of history. A rainy day may hinder a forced march, and change the course of a campaign ; a few in- Digitized by Google 460 THE PRINCIPLES OF SCIENCE. judicious words in a despatch may irritate the national pride ; the accidental discharge of a gun may precipitate a collision, the effects of which will last for centuries. It is Baid that the history of Europe at one moment depended upon the question whether the look-out man upon Nelson's yessel would or would not descry a ship of Napoleon's expedition to Egypt which was passing not far o£ In human af&^irs, then, the smaUeBt effects may produce the greatest results, and in such circum- stances the real application of scientific method is out of the question. The Theory of Evolution. Very profound philosophers have lately generalized concerning the production of living forms and the mental and moral phenomena regarded as their highest develop- ment. Mr. Herbert Spencer's Theory of Evolution pur- ports to explain the origin of all specific differences, so that not even the rise of a Homer or a Beethoven would escape from bis broad theories. The homogeneous is un- stable and must differentiate itself, says Spencer, and hence comes the variety of human institutions and characters. In order that a living form shall continue to exist and propagate its kind, says Mr. Darwin, it must be suitable to its circumstances, and the most suitable forms will prevail over and extirpate those which are less suitable. From these fruitful ideas are developed theories of evo- lution and natural selection which go far towards ac- counting for the existence of immense numbers of living creatures — plants, and animals. Apparent adaptations of organs and limbs to useful purposes, which Paley and other theologicans regarded as distinct products of cre- ative intelligence, are now seen to follow as natural by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 461 effects of a constaDtly acting tendency. Even man, according to these theories, is no distinct creation, but rather an extreme specimen of brain development. TTig nearest cousins are the apes, and his pedigree extends backwards until it joins that of the lowliest zoophytes. The theories of Darwin and Spencer are doubtless not demonstrated ; they are to some extent hypothetical, just as all the theories of physical science are to some extent hypothetical, and open to doubt. But I venture to look upon the theories of evolution and natural selection in their main features as two of the most probable hypo- theses ever proposed, harmonizing and explaining as they do immense numbers of diverse facts. I question whether any scientific works whicb have appeared since tbe ' Prin- cipia' of Newton, are comparable in importance with those of Darwin and Spencer, revolutionizing as they do all our views of the origin of bodily, mental, moral, and social phenomena. Granting all this, I cannot for a moment admit that . the theory of evolution wiU alter oar theological views. That theory embraces several laws or uniformities which are observed to be true in the production of living forms ; but these laws do not determine the size and figure of living creatures, any more than the law of gravitation determines tbe magnitudes and distances of the planets. Suppose that Darwin is correct in saying that man is descended from the Ascidiana : yet the precise form of tbe human body must have been influenced by an infinite train of circumstances affecting the reproduction, growth, and health of tbe whole chain of intermediate beings. No doubt, the circumstances being what they were, man could not be otherwise than he is, and if in any other part of tbe universe an exactly similar earth, furnished, with exactly similar germs of life, existed, a race must have grown up tbere exactly similar to the human race. by Google 462 THE PRINCIPLES OF SCIENCE. By a difTerent distribution of atoms in the primseval world a different series of living forms on this earth must have been produced. From the same eauscB acting accord- ing to the same laws, the same results will follow ; bnt irom different causes acting according to the same lawB, different results will follow. So far as we can see, then, infinitely diverse living creatures might have been cre- ated consistently with the theoty of evolution, and the precise reason why we have a back-bone, two hands with opposable thumbs, an erect stature, a complex brain, about 223 bones, and many other peculiarities, is only to be found in the original act of creation. I do not, any less than Paley, believe that the eye of masi manifests design. I believe that the eye was gradually developed, and we can in fact trace ita gradual development from the first germ of a nerve affected by light rays in some simple zoophyte. In proportion as the eye became a more delicate and accurate instrument of vision, it enabled its possessor to escape destruction, but the ultimate result must have been contained in the aggregate of the causes, and these causes, so far as we can see, were subject to the arbitrary choice of the Creator. Although Professor Agassiz is clearly wrong In holding that every species of animals or plants has appeared on earth by the immediate intervention of the Creator, which would amount to saying that no laws of connexion be- tween forms are discoverable, yet he seems to be right in asserting that living forms are entirely distinct irom those produced bom purely physical causes. ' The products of what are commonly called physical agents," he says", 'are everywhere the same {i. e. upon the whole surface of the earth) and have always been the same ({. e. during all geo logical periods) ; while organized beings are everywhere dif- ferent and have differed in all ages. Between two such series '^ Agassiz, ' Essay on Clasaificntion,' p. 75. by Google RESULTS AND LIMITS OF SCIENTIFIC METSOD. 463 of phenomena there can be no causal or genetic connexion.' Living forme as we now regard them are essentially variable. Now from constant mechanical causes coostant eSects would ensue- If vegetable cells are formed on geometrical principles, being first spherical, and then by mutual compression dodecahedral, then all cells should have similar forms. In the Foraminifera and some other of the more lowly organisms, we do seem to observe the pro- duction of complex forms on pure geometrical principles. But from similar causes acting according to similar laws and princaples, only amilar results could be produced. If the original life-germ of each creature is a simple particle of protoplasm, unendowed with any distinctive forces, then the whole of the complex phenomena of animal and vege- table life are effects without causes. Protoplasm may be chemically the same substance, and the germ-cell of a man and of a fish may be apparently the same, so tar as the microscope can decide ; but if certain cells produce men and others as uniformly produce a given species of fish, there must be a hidden constitution determining the extremely different results. If this were not so, the generation of every living creature firom the uniform germ would have to be regarded as a distinct act of arbitrary creation. Theologians have dreaded the establishment of the theories of Darwin and Spencer, as if they thought that those theories could explain everything upon the purest mechanical and material principles, and exclude all notions of design. They do not see that those theories have opened up more questions than they have closed. The doctrine of evolution gives a complete explanation of no single living form. While showing the general principles which prevail in the variation of living creatures, it only points out the infinite complexity of the causes and cir- cumstances which have led to the present state of things. by Google 464 THE PRINCIPLES OF SCIENCE. Any one of Mr. Darwin's books, admirable though tbey'all are, consists but in the setting forth of a multitude of indeterminate problema He proves in the most beautiful manner that each flower of an orchid is adapted to some insect which firequents and fertilizes it, and these adapta- tions are hut a few cases of those imraeusely numerous ones which have occurred throughout the life of plants and animals. But why orchids should have been formed so differently from other plants, why anything, indeed, should be as it is, rather than in some of the other in- finitely numerous possible modes of existence, he can never show. The origin of everything that exists is wrapped up in the past history of the imiverse. At some one or more points in past time there must have been arbitrary determinations which led to the produc- tion of things as they are. PosdhUity of Divine Interference. I will now draw iiie reader's attention to pages 168-17 1 of the first voluma I there pointed out that all inductive inference involves the assumption that our knowledge of what exists is complete, and that the conditions of things remain unaltered between the time of our experience and the time to which our inferences refer. Recxirring to the illustration of a ballot-bos, employed in the Chapter on the Inverse Method of Probabilities, we assiuae when predicting the probable natm^ of the next drawing, that our previous drawings have been sufficiently numerous to give us nearly complete knowledge of the contents of the box ; and, secondly, that no interference with the ballot- box takes place between the previous and the next draw- ings. The results yielded by the theory of probabilities are quite plain. No finite number of casual drawings can g^ve us sure knowledge of the contents of the box, so that. by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD, 466 even in the absence of all disturbance, our inferences are merely the best whioh can be made, and do not approach to in&Uibility. If, however, interference be possible, even the theory of probability ceases to be applicable, for, the amount and nature of that interference being arbitrary and unknown, there ceases to be any connexion between premises and conclusion. Many years of reflection have not enabled me to see any way of avoiding this hiatus of scientific certainty. The conclusions of scientific inference appear to be always of an hypothetical and purely pro- visional nature. Given certain experience the theory of probability yields us the true interpretation of that ex- perience and is the surest guide open to us. But the beet calculated results which it can give are never absolute probabilities ; they are purely relative to the extent of our information. It seems to be imposnble for us to judge how far our experience gives us adequate information of the universe as a whole, and of all the forces and pheno- mena which can have place therein. I feel that I cannot in the space remaining at my com* mand in the present volume, sufficiently follow out the lin« of thought suggested, or define with precision my own conclufflons. This chapter contains merely Reflections upon subjects of so weighty a character that I should myself wish for many years — nay for more than a lifetime of further reflection. My purpose, as I have repeatedly said, is the purely negative one of showing that atheism and materialism are no necessaiy results of Scientific Uetiiod. From the preceding reviews of the value of our scientific knowledge, I draw one distinct conclusion, that we cannot disprove the possibility of Divine interference in the course of nature. Such interference might arise, so fiir as our knowledge extends, in two ways. It might consist in the disclosure of the existence of some agent or spring of enei^ previously unknown, but which effects a VOL. n. H h Digitized by Google TBS PRINCIPLBS OF SCIENCE. given purpose at a given moment. Like the pie-arranged change of law in Babbage's Imaginary Calculating Machine, there may exist pre-arranged surprises in the order of nature, as it presents itself to us. Secondly, the same Power, which created material nature, might, so far as I can see, create additions to it, or annihilate portions which do exist Such events are doubtless inconceivable to us in a certain sense ; yet they are no more inconceiv- able than the existence of the world as it is. The in- destructibility of matter, and the conservation of energy, are very probable scientific hypotheses, which accord very satisfactorily with experiments of sdeutific men during a few years past, but it would be a gross misconception of scientific inference to suppose that they are certain in the sense that a proposition in geometry is certain, or that any fiwt of direct consciousness is certtun in it- self. Philosophers no doubt hold that de nihilo nihil fit, that is to say, their senses give them no means of imagining to the mind how creation can take place. But we are on the horns of a trilemma ; we must either deny that anything exists, or we must allow that it was created out of nothing at some determinate date, or that it existed firom past eternity. The first alternative is absurd ; the other two seem to me equally conoeivable. Conclusion. It may seem that there is one point where our specu- lations must end, namely, where contradiction be^na. The laws of Identity and Difference and Duality were the foundations from which we started, and they are, so 8 I can see, the foundation which we can never quit, itific Method must begin and end with the laws of ght, but it does not follow that it will save us from untering inexplicable, and at least apparently contra- by Google RESULTS AND LIMITS OF SCIENTIFIC METHOD. 467 dictory results. The very nature of continuous quantity leads us into extreme diflScultiea. Any finite length is composed of an infinite numher of infinitely small spaces, each of which, again, is composed of an infinite number of spaces of a second order of infinite smallness ; these spaces of the second order are composed, again, of infinitely small spaces of the third order. Even these spaces of the third order are not absolute geometrical points answering to EucUd's definition of a point, as position without mag- nitude. Go on as far as we will, in the subdivision of continuous quantity, yet we never get down to the ab- solute point. Thus Scientific Method leads us to the inevitable conception of an infinite series of successive orders of infinitely small quantities. If so, there is nothing impossible in the existence of a myriad universes within the compass of a needle's point, each with its stellar sys- tems, and its suns and planets, in number and variety unlimited. Science does nothing to reduce the number of strange things that we may believe. When fairly pursued it makes large drafts upon our powers of com- prehension and belief. Some of the most precise and beautiful theorenu in mathematical science seem to me to involve apparent con- tradiction. Can we imagine that a point moving along a perfectly straight line towards the west, would ever get round to the east and come back again, having performed a circuit through infinite space, as it were, yet without ever diverging from a perfectly straij^t direction ? Yet this ia what happens to the intersecting point of two straight lines, when, being in the same plane, one line revolves about a fixed point. The same principle is exhibited in the hyperbola, which may be regarded as an infinite ellipse, one extremity of which has passed to an infinite distance and come back in the opposite direction. A varying quantity may change H h 2 DigitizedbyGOOgle 468 THE PRINCIPLES OF SCIENCE. its BigQ by passmg, as niathemati pose. Let us be faithful to our scientific method, and investigate also those instincts of the human mind, by which man is led to work as if the approval of a Higher Being were the aim of life. by Google INDEX. Abacds, the logical, L 119; Brigg^, Abeceduioin, the lexical, I. 109, 114, 334; ii- 367.380,387. AbemtioD of light, iL iGg, i8j. Abadaaio infimti, i. 94 ; ii. 400. Abstract tenua, I. 33 ; nnmben, 357. AbstractiDii, logical, i. 30; it. 3891 numerical, i. 177, iij ; of iDdiCTerent Amderaie ttel Cimento, ii. 36, 43, 46. Aooident, logical, ii. 378. Accidental diKnTeiy, ii. t63- Achroioatic lansea, li. 43. Actinometer, L 389. Agaaaz, il. 416. Ally, Sir George Blddel, il, 171, 190, 1 - '*. acddeatol tfiscorarjr, 167 ; deoiit; of earth, 340 ; ii. 309 ; pendnlniD, f. 355; emira ofobaerTatioD, 459; tide wave, ii. 1 1 1 ; extrapolation, 1 10, AlchamistB, ii. 37, 133. Algebra, i. 141, 174.185. Algebruo geometrj, ii, 190. Allotropic oondition, it, 331, 340, 341. Alloja, number of. i. ilS; pTopertiee of, ii, 161. Alphabet, permatationB of, i. 196, M3. Alphabetic iadeies, ii, 401. Alternative relation, i. So. Amp^ electricitj, li. 184 ; olaariBea- Hon, 351. Anagrams, i. 146. Analogjr, 11.944,183: in dgn of equality, i. iB ; iufl(tf,ii. 140; nse in diicoTeiy, 186: in mathematiaa, 188 ; tn theoiy of nndolattoiu, 193 ; in attranomy , 197: &iliirM in. 301. An^jsii, logical, L 14a. Andrews, Dr., ii. 366. 334. Antecedent defined, 1. 158. Anticipation of nature, il. 137. Apparent eqaalitf, 1, 319; sequence of events, ii. 13. Approximation, principles o£ ii. 93 ; to exact lam, 79. 81. a logieal division, i. Ango, photometer,!. 335; rotating disc, u. 170, 340. Archimedes, ds sranE nnmero, L ti* ; centre of gravity, 433. Aristotle, £ctum, i- )6 ; overiooked simple Identities, 46, 48 ; imperfM syllogistic conclusion. 71; on time, 3S9. Arithmetic, reuonmg in, 1. 188 > at ap- proximate quaotitiea, ii. 103. Arithmetica] triangle, t 108, 308, 330. Asteroids, discoveiy of, ii. 17, 444. Astronomy, f. 185 ; ii loi. Atmospheric tides, ii. 101. Atoms, iL 10; lice «t,\. 331; wngbta of, S05. Angnstin, on time, ]. 359. Aurora, I. 333. Aversige, 1.417 ; divergence &om, 313 ; derivation ot word, 4) I . Axes of crystals, il 350. Axioms of algebra, i. 185. Babbage, Charies, ii. 455 ; calenlatliig ma(£ine, L 113, 437 ; ^hthouse sig- nals, no; change of Uw, 165; U. 437 ; nataral constants, I. 38I ; hu- man remans, li. 17; gtoraal prin- ciples, 309. Baoon, Francis, biliteral dpher, i. 310 ; on caoses, 154 ; Copemican system, 187; insUncM, 313; the senses. 331; fiillscies, ii. 5; method of, ii. 134, 310; use of hypothesis, 137; expen- mentnm cruds. 149 ; latens proces- sus, 373. Sacon, Boger, on rainbow, ii :59, 347. Buly. Francis, L 316; ii. 171 ; pen- dulum experimente, i. 463 ; experi- ments on denn^ of euth, i. 430, 463; ii. 41, 108. by Google Bain, on powen of mind, i. Bkllot-box, hvpoUiMiB of^ L 169, 175, iSo. 191 ; li. 156, 464. Barometer, i. 455; ii. 315; ituidaril, OAf-Lunsc'i,i-4oi; vBtutionB 0(389. BayDOB, ProfesBor T. S., u. 387, 393. Belief, quantity of, i. 3 16. Beoeke, on Subatitutioii, L 16, Bentham, George, t. iS ; bifiircnte olaS' uficatioD.ii. 373, 387; quantifintio of prediatto, 398. Bontham. Jeretnj, ii. 385. Benienberg, i. 453: ii. 311, BemonilU. Daniel, planetary orbita, L 189; resiBtiag media, ii. 86; vibra- tioTu, 97. BemouilU, Jamea, i. 30] 1 oombina- tiona, 19S 1 aritbineticU triangle, 106 ; theorem of, 138; erroneous solution, >44- Bemoiiilli, nuinbere of, i. 143. Beseel, fbrmuU o^ ii. 1 1 1 ; observation of oomBta, 133 ; sun's parallai, 103 ; fig;ure oF earth, 107; pendulum, 155. BieU'a Comet, ii. 316, 440. Bifurcate claeeiGcatiaD, U. 3^1. Binomial theorem, L 316 ; ducovery ot. Bismuth, ii. 161. Bode'i law, t. 165, igj, Boethiua, i 40, 83 ; on meaos, 418. Boiling point, ii. 54, 315, Boole, George, his logic, i. iS, 37, 18, 39' '74 i "■ 333; CommntaaTeneM, 1.4*! indetermfiiate adjective »ome,49, Jo; mga of addition. Si; exclusive diviiions, 81, 83 ; value of bis logic, 13O; statistical conditians, 1901 nu- toerioallj definite propomtions, 194; probaluUtf, tij, 135 ; on inverse me- thod, 396. Botany, 11.338,350,353.388; nomen. Backle, li. 311, 459. Buffon. i. 137, 346. BoneeD, Bobwt.speetnim. i. 39i : light, 316, 378; ii. 51 ; oJorimeter, i. 397- Calo-spar, IL 335, 36*. CaloreKeiice. iL 333. Calorimeter, i. 405. Canton, John, oompreanon of walv, L 390- . . Capillary attraction, li. 57. Cubon, condnctiluli^ ofi iL 53 ; die- mietiy of, 418. Oarbonie acid, li. 334. Camot, a. 3S7- Catait^ea, ii. 40r. Caachy, ii. BS. Cause, i. 363 : pmbabili^ of; J79. Cavendish,!. 3161 ii 308. Centre of gravity, percussion. Ac, L 433; of parallel foreaa, ii. 3 1 7. Centrobaric bodies, i. 433; iL 84. BowoD, on infareno*, I. 136; method of least aquiires, 448 ; classification, ii. 346. Boyle, law of, 11. S7, 91, 174; on hypo- theaia, 136; banuaeter, 335. Bradley, i. 314; ii, •.6g; onaben»tion, BrawBter, Kr David, refractive indicea, i. 13 ; iL 159 ; iridescent colours, 16 ; ■pectram, 39 ; optic axes. 59 ; natnral coloura, 148. British Museum, (Stalogne of, ii. 403, '434- 4°S- Brodie. Sir B. C., errors of eiperimi 831 ozone. 331. Brown, on Cauae. i. 358. .lu. Chalmers, o: Cbance, L 135. Character, human, ii. 415. Cbaracteri/ittcs. ii 39 j. Chemical affinity, ii. 167, Chemistry, ii. 160, aoj, 185, 347, 364, 399 ; organic 181, 419. 4*7. Cblorofoim, discovery of; ii. 165. Cirotn, ratio of diameter and drcnmftr- ence, i. 369. Clairaot.ii. 314, 315 ; on gravity, 81. CLisalGcatiOQ, il. 344. Clocks, astronomical, i. 33a. 394 ; mu- tual influence of, ii 70 ; oonneiion of, 106. Cloud^ii. 14. IS. 16. Coincidence*, i. 301. Collective terms, i. 3$. Colloeatlana of matter, ii. 434. Colours, iridescent, ii. 3fi ; natnral, 1 47; apectroaoopic, 1)9. Combinations, i. 300 ; caloulation of, 104. Comets, iL 348, 315, 316, 3J4, 357 ; hy- perbolic, II; nomber of, II. 14; conflict with, 440, 44}. Comma tativeness, i. 85. 180. lOO. Compamtive use of instruments, i. 350. Compass, variation of, i. 317. Comte, Augnate, L 117, 14s ; 11. 171, 380,449. Concrete number. 1. 17S. CouditiaDB, i. 360; removal of naoal, iL 35 ; uDBuspected, 37 ; maintenance of simtlar, 55 ; approiimatioa to natural. 64. Confusion of eubetancea, 1. 173. . Conical refiMtion, ii. 175, 318. Oonjnnction of planets, i. 34) i ii. 331. DijiiiMb, Google ConMrration of energj, ii. 83, 431. IB qnuitity, il. loS. Contrspositlve propoaitioiu, i ConTCTfflon, 1 ss ; oontnpoaitive, 97. Capemioui theory, i. 183 ; iL i jj, 181, ^ 198. 3'° Copulft, I. 19. Corpuscular theoty, U. 150, 173, 304. CorreotloD, method of, L 406. Correlntiaii, ii. 350, 354. Cotea, Soger, meuia, I. 416 ; method of least sqoana, 437 ; weighted ab- oeirationi, 450. Conpls, mechaDical, ii iiS. Creation, L 170 ; U. 416. CrTBtallographj, i. 153; il. 161, 184, 3«i. 319. 359.398, 450- CrjBtali, pseudomorphtc, il. 3)4. CazvtB, nature of, ii. 99 ; diacovecy of, lis- CuTier, L 355 ; IL 31. Cyanite, ii. 161. Cycloid, ii. 391. D. D'Alembert, probability, i. 144, 345 ; granhr, ii. 81. Dalbiii, UwB of, iL 8j, 91, 174, 3*1). Darwin, Charlea, theory, ii. 8, 48', 165, 3tS, 405, 411; orchids, ip, 4091 cU«^Gcation,4lo ; reptwinction, 411. Sa^, Sir H., iiutrntneiitg, L 313 ; heat of friction, i. 397 ; ii. 33 ; electro- lyiia, 19, 38. Decandolte'8 ayBtem, ii. 38 7. DedaclfoD, 1. 13,59; probable, 139. Defiaitioii, L 64; ii 397. De la Rus, ii 67, 109. De MorgRn. sign of equality, L iSj Ariatotle'a logic, i } j relatirea. 3 7 ; Umited unirene, 53 ; eomplei pm- bleou, 90 ; eoDtnporitiTe oonvenion, iff ■, Euclid's Indirect proo^ 98 ; Dgical problems, 1161 error of sys- tem, 135 1 numerically detinite ayUo- gimn. 190: probability, aiGj eiperi' ments on protmbility, 337; prob&ble argament, ^39 ; tri«ectioa of angle, 168; finite eiperience, 3D0 ; arcual unit, 358; peTBounI error, 403 J means, 419; average, 4)1 : weighted obsei^ vations, 450; works on probability, 459 1 appkrentsequanccu. 13: uuall CTTorv, lOl ; subequatity, 101 : gene- nditatioD, 349 ; catalogaee, 403. Density, nnity of^ i. 371 ; of etuih, ii 307 ; oegKtire, 304. Beptb of oceans, i. 347. Descartes, ii. 135, 390. Development, logical, i. 104. DingnoBiB, ii. 394, Di^iond, ii. t£9, 361. DiatomaceEB, ii. 9, 410. Difference, law of, i. 6, 87, 95 ; .sign of, 30,54; of numbers, 110: oaloaltuof, ii. 131 ; logical, 377. Difierential calculus, iL 99 ; thermo- meter, i. 400. Difirrtctioa of light, ii. 17. Discontinuity, ii. 374. DiscovfHea, accidental, ii. 163 ; pre- dicted, 171. Disjunctive tonus, i. 79 ; conjunction. So; proposition, 89, 117; syllogism, 93. Donkin, i. 336, 337, 143, 34S. Double refraction, iL 174, 331, 361. Dove's law, iL 168. Draper's law, ii. 357. Dnalitr, law of, L 87, 95. Duration, i. 360. Eclipses, i 343 i ii 333. Electric acid, ii. 38. Electricity, ii. 163, 187, 337, 164; unit of, i. 379 ; theory o^ iL 154. Electrolyms, ii. 37, 163. Btectro-magnstism, ii. 164. Electrometer, i. 350. Glementa, claaification of, Ii. 347, 349, 364. 374- Ellicott, on Clocks, ii. 70. Ellipms, logiosl, i. 69. Elliptic variation, ii. 94. Ellis. A. J,, L 37, 100. 194. Ellis, W., effect of full moon, iL 14. Emanation, law of; ii. 81. Empirical knowledge, ii. 133, 157, 158; measuremente. 190. Encke, Comet, i. 363 ; ii 313; law of error, i. ^45 ; mean, 449 ; reusting medium, li. 1511. Energy, unit of, i 376 ; conservation of. ii 83, 431- Equality, i. 56, 183 ; sign of, 18 ; meaoings of, ii. to3. Equations, L 141, 180; ii. 51. Equilibrium, unstable, i. 330 ; ii. 319- Equivsleuce, logicsL i '3>. 134 1 ^^ markable ciwe of, 163 ; iL 333. Eratosthenes, sieve, i. 96, 141, 160; on latitude, 315, 341. Error, function, i. 383 ; avoidance of; 393 ; personsj, 403 ; rules Ibr elimi- nating, 409 ; taw of, 434 i formula ot by Google 44); probsble, 451 i 460; Etfa«r, mction of, ii. 954. Eaclid, kiioDU, L 183 ; ftbl«g, 31Q; right tragle, 35S ; uu- logj, U. 189. Enlar, on hnowledge, i. 173; gisyity, 374: iL Si; medium of Ught, 143; coipoBcalar theoiy. 15a. Exceptions, in indnctioB, !. i>i) ; cImbob i. - i» aa pu. J30. Btdaded middle, l«w o( L 7. Eiclonre iUt«ini*tiT«, i. 134. Eiperiment, u, t,ia ; riiDplifioition of, 30; hflnre in, 33; bEad ot tert, i. 40« ; ii. 43 : negative remits of, 45 ; limit! of, 4S ; collectiTe, 57 ; disoora- anoe Ot. 198. Eiperimeutum cnicia, ii. 135, 337. Eipliui&tion, a. ij7, 166. Extent oC logte*! tenns, !. 31 j'pTopo- ritiona, 57, Eiti^iolAtian, Ei. 1 10. Skotorikls, i. Joa- Fallaciea, i. 74.117. FbdmIsj, Miduwl, od gold-Ie*i', L 346 ; grxn^i 396 ; ii 136 ; magnetism of gueg, i. 407; Ijcopodium, Ii. a; ; electroljw*, ig, 31, S3 ; dectrio polo, 19 ; eloctro-magnetiKni. 31, :B4, >74 ; polarized light. 31, 318 ; preotutioua Id experiment, 40 1 lines of Force. 58; Arago'g experiment, 170; velocltj of electricity, I So i hii reaaanthes, J13T renerrstion of jndgiiiBnt, 140; beaiy glasa, 36 1 ; electricity, 164 ; radiant matter, 304 ; hydrogen, 364. Fatali^, i. 305. Figurate numben, i. ill, 114. Figure of earth, ii. 76, 107. Fiieau, Newton's rings, i. 347 ; U. 117 ; quartz, i. 367 ; revolving minor, 349. Flunsteed, i. .114 ; use of wells, 341 ; ■taodard stare, 3jl ; paralUi of pole star, 391 ; choice of observatioDl, 4:5 ; instruments, 456 ; wlar eclipses. Fluorescence, iL 331. Forbes, J. D., i. 186 ; It. S9, 454. Force, unit of, I. 375. FoesQa, ii. 3*7. Foncault, revolving mirmr, i. 349; pendulum, 396 ; ii. 41 ; relodty of light, S3, JO». Fourier, theory of beat, ii. 89, 438. Fowler, Profeoor, Indaotive infemiMe, i. i6r ; method of Tariatiomt iL 51. Freeiiiig mixtuies, Ii. 183. Fresnel.in9eiionoflight,iL 17; donhla refraction, 59 ; imdnlatoTy tliawy, 173- Friction, detenntnaticn ot, L 401 ; heat of, ii. »7. 187. Functions, definition ot, U. 113 : dis- Galileo, &lling bodies, i. 333; diffo^ ential meUiod, 399 ; [oojectilea, iL S5 ; gravity, 154 ; eonUnui^, 170. Galton, Francis, L 114, 373; ii. 3>l. Galvanometer, i. 407. Gas, il 90, 150, )66, 310, 334. GrauBS, use of mirror, 1. 334 ; pandalnm experiments, 370: ii. St; latv of error, i. 436 ; constant airon, 461. Gay-Luisac, barometer, L 401 ; law of, iL 174 ; boiling point, 315. . Genealogical tree, iL 407. General names, twofold meaning of, reaaoning 14S ; hasty. General prlndples, ii. 309 . by, )43. Generalisation, ii. 141, 389 ; ings 0^ 146; valne 0^ Genius, iu 119, 311. Genus, ii. 376 ; generalisdmall), 379, 3S1 ; natural, 414. Geolo^, iL 3»7, 335, 337 ; negatiTe argu mania in, 18. Geometric mean, i. 418. Geometry, reasoning in. L 183,168,309. Gilbert, Copemican system, L 1S7 ; magnetism, ii. 41 ; on experiment, 55. Oold-aasny process, ii. 45. Gold-leaf, i. 346. Gradation of character, U. 410. Graham, Profeasor, iL 167, 366. Grammar, rules of, i. 37; ii. 3*8; equivalents in, i. 13S. Granite, classification of, ii. 411. Graphical method, ii. 1 1 6. Gravity, (i. 19, 75, 95, 06, 141, I44. 154, 313; measure of density, L 371, 375; nnifbrmitjof, ii.36, 56; Hookah experiments, 46 ; law ot; So ; Pant- day's experiments, 136. Great Britain steamship, voyages ot, i. 453- Grove, iL i6j, 26S ; magnetism, L 397 ; medium of light, ii. I43. b, Google iind librations, ii. 8 I. 136 ; frM will, 357; qnantifioation of predic&to, ii. 387. Hamilton, Sir W. Bswmi, ii. 175. Haoghton, Rot. S., on mnscniar exer- Heat, iL 197 ; adt of, i. 378 ; meoBiiie- ment of, 403 ; mechuiical eqnivalsnt of. ii. III. UekTT slasd, ii. 335> 160. Helmliolti, 99. Horeditory deicent. ii, 407. Herschel, Sir J. F. W., plagihedral quarts crystala, i. 14S, 1S3 ; liensiW □f eartb, 110; ptioCometrj', 316, 351 ; numsrical predsion, 317 ; anit of length, 367 ; actiaometer, 3S9 ; use of mean, 411; metfaod of leaat ■qoarw, 437 ; double stars, 457: ii, Ufi ; active and pasaiTe obaerratjon. a; fluor«gceiice, »; fuU-moon, 15; cometa, iG; spectrum, 30 ; collective instances, Ji) ; principle of forced vibrations, 65, 311 ; meteorological variations, III; oirect action, lig; nae of thBoriea. 136 ; etliereal mediuiD, 145 ; eipsrimeatnm cmciB. 149; iDteifsrenoii of light, 175; interfereDce of sonnd, 176: residual phenomena, iti ; discovery bj aoa- loCT, 186. Hindenborg, combinat{ emanation, ii. Bi ; axiom of siinplicitj, 3S1. Kepler, on star ijigcB, i. 454 ; comets, ii. IJ; laws of, 7*; refraction, laS; philoaophic method of, iji, i8a. Kind, derivatton of word, ii. 406. L. geometry, igi. Laoeuage, ii. 184, 305. Laplace, probabilitj, i. 137, 147; of inverae method, 1 79 ; plauetar}' moTementa, 18S, iSg ; solutioii of inTeraa problem, 196, 311 i long beqmJity of Jupiter and Saturn, 341 ; atmoapberic tidoe, 416 1 obaer- vatioD of tities, 433 ; law of error, 438; works on probability, 460; dark Btare, ii. 7 ; hyperbolic oomete, 11 i knowledge, 13; velocity of 8™*''}'. 4S i Btabilitry of planetary Byatem, 61, 441 ; phases of Venus, 63; corpuscular theory, 151; figure of Joprter. 196; figure of e^rUi, 107; yelodty of sound. Jit: chemical affinity, )68 ; laws of force, 391 ; on UiuTeree. 431. Latent heat of rtsam, ii. 80. Lavotaier, oijgsn, i. J74; Ii. 33S; decomposition of water, 3 j dmplifi. cation of eiperiments. 31 ; defitutjon of element, 36. Law, of identity, difference, duality, excluded middle, of thought, Ac, i. 6. 7, 87, 88, 95 ; of things. 8 ; disjunctive relation, 85 ; commuta- tiveness, 85, iSo, loO; Bode's, 165. 197 ; homogeneity, 1 79 ; error, 434 ; eiact, ii. 79 ; discovery of, 90 ; Daiton'p, 81, 91, J74, 399; empiricaj matbematic^, no; empirical quan- titative, 115: of emanation, 83; of nature. 143; Dove's, i68; Draper's »ij; Carnot'a, 957; of continuity, itrtJ, 419; of motion, 370; reign of, 418; nataral. 41Q. Least squBrcH, method of, i. 137, 458. Legendr- ' ■ -._ . . squB s. 43;, tinuity, 171. Lenlie, i. 400 ; ±33. 104. Leiell's Comet, ii. jig. Light, unit of, i. 377 ; raediuni of, ii, "4', 145; predictions in, 173; wa»e- len^hs, 19S ; velocity of, 103 ; mac- nctM influence on, 334 ; colours, 433 Lindsay. T. M., i. 36, LinnKOH, iL 399, 415, 416. liquids, ii. 151, 367, 310, 334. Locke, on number, i. 1 76 ; prolnI»]itr. 346; power. 154. Logarithms, errors in tables ot I. 378; calculation nf, ii. 6. Logical Abaoedarium, i. 107, log, 314, =3i; "- .167. 380, 387; abacna, L 119; slate, no; machine, II3. Lucretius, atoms, i. 156 ; ii. 435 ; in- deatrucUbilityofmAtter,377; S^^^ij, M. Machine, logical, i. 1J3 ; Smee's, 134. Maclcay's ajatom, ii. 407. Magnetism, and light, ii. 334 ; aUnc- tion of. 356; univerwJity ttt. 374; animal, 343, Mallet, on earthquakes, i. 368. Malua, polarixatioD of light, ii. 163. Mammalia, - tive iadactive arguments in, iL to ; incompleteness oC 451. MaiweU. Clerk, on BalanoBs, L 355 ; speed of electricity, ii, 54 ; on Fai«- day. 334. Mean, method of, i. 414; derivatlcm of word, 418 ; Gctitions, 413 ; ptwrise, 434; probable, 447. Measurement. e:tact, i, 313 ; oonditiooa of accurate, 338 ; instruments o^ jjo ; by natural coincidence, 341 ; moda of indirect, 34s ; syatematic perjbrm- anoe of, 351; attainable accuracy, 354! units and standards ot, 357: expliuned reatdts of, ii. 193; ae arice of, iol"; beat mode of, agreement of distinet n Melvill, Thomas, ii. 38. Metals, i. 398 ; ii, 365, Meteorology, interpolation in, i results in, 191. Meteors, obsenratioD o£ 1. 433 , number ot 1 1. Method, inverse, i. 179 ; of n ment, 331 ; repetition, 335 ; measurement. 345 ; of avoidance ot 'V™''' 393 ; differential, 398 ; corrcc- tion, 400; compensation, 406; ro- v6rBal.4io; meBna,4i6; least Bqnar«. 437 i >■■ 1 16; variations. iLfOigiapfaj. oi 304: ?d by Google Metre, error in. L ^ Uiohdl, on nrotnbilities, i. 141 ; aUi^ Byrtenw, aSs ; Btar-digCB, 455 ; tonion UUooe, il. 108 ; Fleiftdea, 199. MHkj Way, ii, iga. Hill, J. S., on exoluHiTe <arDativea, i. 83; prolNibiiity, 117, 145; cause, 154; indnotlva infraence, 161; ii. 34} ; dedaotive method, i. 307 ; Ii. 136; errDQeoos reTo&rks od jsean, i. 446 ; joint method of ■greement, &a., il. 34 ; method of concomitADt vuia' tionB, 106; coUooatioua, 434. Hinenlogy, clawificstion in, ii 349, ****■ , . HomentDm, unit 01^ 1. 375. Hood, foUac; oonceming, ii. 14; atmo- ■pheK of, 45 ; periods of, 63 ; motions Huaoular, nuoirui, i. 348 ; exertion, ii. N. NegaUre ^5 = 17, SB ; pretni*ei u. 16, ajG; reauh }f eipmment, 45. Newton, Sir IwukO, binomial theorem, i. 166 ; pluidtarj moTements, 38S { interraii of ocUve, 303 ; valocdtjr of eound, 344; ii. S7, 114; meaaure- ment of light wavea, i. 346 ; tides, 347; pendnlum siperimenti, 354: ii' ££• '54 i >l>Bolute time, i. 360 ; Impact, 403 ; eiperimenta on speo- tmm, ii. 15, iB, 31 i Newton's ringi, *7.£9.6o,89; inflexion of light, 37; gravity, 19; achromatio lenses, 41; refdsting ether, 46 ; abflorptioD of light, £8 ; theory of pluietuy motiou, 73, 84, 86; redatiug media, 86; diSereatial calculus, 99 ; alchemy, 1331 knowledge of Bacon'a norks, 134 ! on hypothMee. 144 ; natural Dolonn, 147 ; vortices, 147 ; corpus- cular theory, 151;' fits of easy re- flection, &c., 154; oombaitible nib- ■tances, 159; discoveij ofgnritation, 194 ; ruled of philosophiiing, 158, 180: andulatoiy theory, 195; nega- tive density, 304. Newtonlui method, ii. ]i6. Koble'a chronoacope, 1. 360; 11. 17a. Nomenclature, ii. 41 B. Numbers, prime, 141 ; of BemouiUi, 143; natare of, 175; conoete and aUtntct, 178; (riangnlar, 1091 Sgu- Nnmerioallj definite r. ■■ '90- instnimental and ■ennui condiljons □f, 7; external oonditions ofi 10; weighted abwrvationi, i. 449. Odours, ii- 4M- Oented,ii. 164, 169. 1S4. Order, of terms, i. 40 ; of premises, 131. Oscillation, centre of, i. 413. Osteudva instances, ii. 159. Ozone, ii 331. P. PaiKbola, ii. 74 ; orders of, 95 ; approu- ParaliaT of sun, ii. 303. Panllel fbnwc, i 431 ; ii 317. Paralogiim, i. 75. 1 1 8. Purity of reasoning, i 310. Partial identities, 1.471 inference &om, 64, 66, 70, 71 ! induction of, 149. Particular reasoning, ii. I43. Pascal, orithmeticat macblne, i. 133! arithmetical triimgle, 106, 31 1 ; prob- ability, 344,346; barometer, ii. 149. Passive state of steel, ii. 316. FecnhiiT propertj, ii. 377. Puree, 1. 17. Pendulum, 1. 339, 351, 3C9, 413; li. 79. »54- . Perfect induction, i. 164. Perigon, i. 358. Permutations, of versea, 1. 197 ; alpha- bet, 303 ; cards, 377. Perpetual motion, i. 356 ; Ii. 177. PeiBonal error, i. 40}. Physical astronomy, ii, 76. FUgihedtal crystals, 11. 387. Flaneta. conjunctions o^ i. 305, 113; ii. 311 ; coinddeDcee concerning, i. 304 ; ii- 356- PUteau's eiperimenta, ii. 36. Plattes. Qabriel, on divining rod, ii. 45 ; gradual effects, 49. PlumbUne, divergence of, i, 4I9. Poisson, on probability, i. i3o ; sidereal day, 361; works oC, 460; Newton's rings, ii 89 ; inflexion of light, 1 74 ; crystals, 180. Polariied light, 11 163, 134, 187, 196, 31B. Pole, of magnet, i. 444 ; of battery, il. 19. ) of. i of Pole-star, ... observation ol, 446 ; singularity of, ii- 317- Porphyry,I»agoge,ii. 376; tree of. 3S1. Port Boyal Logic, i. 16. Pouillet's Pyrheliometer, i 390. Powell, Baden, ii 378, 317. Prediciblea, 11-375. PredicUon, ii. 157, 171 i in Bcienoe of light, 173 ; in theory of Dndolations, 176; in other sciences, 17S; bj ' Luss aude^t. iBl. by Google Prime numbeta, i. i^i, 365. PrincipU, ii. 317. Priiiciple, of probability, L 3a8 ; of invarse method, 179 ; of fortied Tibia- tioua, ii. 65, 33) i of coexiatenoe of ■nuill vibrations, 97 ; of Bapsr-poaitian of fliualleffteta, 9S. PiobBbili^, meaniiig of, i 194 ; priod- plea of, uB; rulea of, 331 j cotn- pariBOQ with aiperience, 138 ; diffi- cultiea of theory, 143 ; indactive applicHtioii uf, 17G; inveiw method of, a 79 ; ii. 944 ; Hpplictttion to utronomy, i. 385 ; inrerae problem, l8g; general eolutioD, 195; rulea of inverae method, 197 ; works on, 459. Probable error, i. 451 ; ii. 194. Produa, conunentariea of, i. 167. Proctor, R. A., on star drifts, i. 187. Prqectilea, theory of, ii, 85. PropertJee, generality of, ii. 149; uni- fbrm, 354 ; variable. 15S ; extreme, 359 ; correlation of, 353. Property, logical, ii. 377; peculiM, 377- ProposttioiiB, i. 43 ; negative, $1 ; oon- vetaion of, 55; twofold iatlnpretiitioD of, 57: diajuiictive, 89; eqaivalenoy of, I J J. Protean veraea, i. 197. Protoplasm, ii. 155. Prout, law of. i. 304 ; ii. St. Pythagoras, on duality, i. tloj on number seven, ii, 379. Qnadric varlaUone, H. gj. QuantiBcBtion of predicate, i. 41 387. Quaahty, continaous, i. 318; ii. of revolution, i. 358. Quartz oystals, i. 367 ; ii. 361, 3i Quaternions, i. 180; U. 457. Queteiet, on avera^ i. 114; e: ments on probaUlity, 137; u meao, 411 ; law of error, 441 311; errors of observatioD, i. letters on probability, 459. Banldne. ii. 197 ; recoooentration of energy. 447- BatiomU formuln, ii. 1 1 3. BeductioD, indireot, i. loo; ad abeur> Reflection, total, ii, 314. Kefraction, atmuspheric, i. 413; ii. 1161 double, ii. 341 law oC, tsS. Regnault, dilatation of : __^, _ ■ 3^6; moaaurement of hent, "40$; dilatation of gases. 461 ; iL 87 ; latent heat of steam, 1 1 1 ,- gnphical method, 118 ; apedGoheat of air, 19J. Regular system of ciTstals, ii. 360. Beign of uw, ii. 43S. SejecCionaf obsOTvations, i. 45G. Btdation, sign of, L zo ; lo^c of. aj, licpetition, method o^ L 335, 336, 35J. Representative bypothesea, n. 156. Residual phenomena, il. 199, 101. Resisting medium, i. 3G3; ii. 155, sij. Retrograde motion, ii. 317. Reversal, method of^ i. 410. Revolution, quantity of; i. 358. Robiaon, electrio curves, ii. 58. Bock-salt, ii. 40, 361. Roemer, divided circle, i. 41 1 1 veloci^ of light, ii. 1C9. Boacos, chemical action of light, i, 316; eipeiimenia on solubility, 3151 standard unit of light, 37S ; ii. 53 ; KSesfcbes on vanadium, i. 457 ; iL 313; abaorptioD of gasea, 135. Rotation of plane of polariud ligh^ ii. 318. Rouaseau on ffeomeliy, i. Rules, for oalculatiOD o( i. 304 ; of probabilitieB, 13 1 ; of in- verse method, 197 i for eilmioatioti of error, 409, Romford, L 397, 405 ; ii. 86. Ruminant anunala, ii. 356, 391. , perigon. L 358; • arithmetic. iL 104. Saturn, eatcJlite 7, 171 i lelooity of, i. 41a; iL 113; interfeninc« of, 176; dus- fic&tdon ot, 413. Species, iL 376; iufimk, jSo ; natural, 414- Specific haat of air, ii. 197. Spectrum, L 319, 348; ii. 9, 97, 1^)9, 151 ; Thonuw Melvill on, 38. Spenoer, Herbert, laws of tbought, i, 9; sign of equality, iS ; rhTthjnical motioa. Ii- 61 ; abttraction uid ggae- ralizatioD, 3901 thooriea of, 405; quantification of predicate, 3S7. Spontaneoui generation, ii. 43. Standards, i, 365. Stan, motion <^, 1. 315, 34S ; ii 93, 95.115: variable, i. 316 ; ii.64.35S) diacB, i. 464 ; coloured, ii. jjB ; heat of. i. 430 ; conflict with, 443. StcTinos, ii. 176. Stewart, Balfour, guo-cpot^ ii. 67 ; ethereal friction, 113. Stifela, i. 106. Stokea, on resiatance, ii. 96; fluor^ escence, 331. Stone, B. J., radiant heat of atara, t. 430 ; temperature periods, iL 67 ; tnuuit of Venua, 104. Stnitt, J. W., gn4>Iucal method, ii. 119. Substantial temu, i. 34. Substitution of limilare, i. 11, 15; ii. 345 ; of weights, u 13. 399- Sui generia, ii. 1S6, 418. Solpliur, iL34l. Summum genua, i. 108 ; ii. 379. Sou, distance, Ii. 104 v Swan, W., sodium light, ii. 39. Syllogiam, Barbara, i. 66, 103, 111 ; Celarent, 67; Darii, Feiio, 67, 68; ~ ■ "1, Cesare, <)9 ; : diajnncttve, 9]. Symbola, logical, L 39. S7DtheBi8,oftennB, L 36; of laws, 140. Syren, i. 1 1. 348 ; iL )8. ■ Table-tnming, ii. 34a. Tastes, iL 433' TautologouB propodtione, i. 138. Teeth, as critena of olassification, ii. 395- Tempsrature, u. 68. Terms, i. 19; lubstantlal, 14; collec- tive, 35 ; ayntbesia of, 36 ; abstract, 38; logical and numerical, l8a. Test eiperimsnts, L 401. TetraotT^L tio. Thales,ii. 171. Theory, reaulle of. ii. 168 ; ^ts known hy, 185 ; quantitative, 189, 19) ; dis' oordance of, 198. Thennometer. diCTorential, L 400 ; read- ing of, 404: change of lero, 4$j. Thermopile, i. 348. Thomson, Junes, ii. 178. Thomson, ^r W., use of atoms, L 311 ; tidea, ii. 64; thermal phenomena, 180, 197 ; capillary attraction, 167 ; magnetism, 333: heat-history of uni- ThomwD and lWt,chronoinetry,i. 364; Eroblem of bars, ii. 771 polarised ght, 318. Tide-gnsge, i. 4»7. Tidea, ii. 64, 66, 9B, 178, 195; atmo- spheric, i. 415, 496; ii. 191. Hme, meaBurement of, L 359! equal Todhnnter, on Hicliell a speculations, i. 1S6 ; bia Hiatory, 460. Torricelli, ii. 336 ; theorem of, 67, 156. Tunion-balance. i. 354; ii. 78. Tranait, initrument, i. 4II; ofVenus, i- 343. 399 ; "■ "°^- Tree of Porphyry, ij. 3S1. IViang^e, anthmeticaC i- '08, 439. 444 ; Triangular numbov, i. 309. obliquity of earth's cumpolar stars, 415 ; Sirius, 454. I^ndall, natural constants, i 380; precautions in experiment, ii. 41 ; singing flames. 109 ; use of hypo- theaia. 134: magnetisiu, 16S ; scope for disaovery, 499. 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Weighted obaervationB, i. 449. Wells, on den-, ii. 34. Wenzel. on nontral salts, i, 344. Wheatstone, i. 143! i>. 531 galvano- meter, i. 333 ; rotating mirror, 349; electric ipariiB. 360 ; kaleidophane, ii.s8- ' Wbewell, on tides, i. 431; ii. 178; method of leatrt aqoarea, i. 44S. Wbitwortfa, Kr. J., bj of lengtb, 356 ; n. Boole's method, i. 135. WilbrahMD, WiUiunaoD. i. 374 ; prediction in org>mc iatry, ii. 181. Wollaaton, goniometer, L 334 ; liglit, 3S) : spectrum, U. 38. X, the sabstance. ii. 155. Yard, standard, i. 463. Young, on Diodorua Siculns, oonnexion of lauguagea, 185 ; of aqueoua vapour, ii. I16 : hypothesis, 136 ; ethereal MS- OXFORD: BT B. S. O&BDNKR, ■. PICKAAD HALL, AKD t. H. B FBIMTEBS TO THR DHIVZBBITI. by Google L'ljirz^byGpO^le b, Google b, Google b, Google