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epithets pure and exact cannot be ap . There is, however, no other branch plied.
of knowledge which does not excluSo far, therefore, as our knowledge sively rest on that induction which ob relates to magnitude, figure, and num servation and experiment supply. The ber-that is, so far as it is conversant laws of motion perhaps approach nearwith mathematics, it appears to us est in simplicity and universality of apthat it consists in resolving the evi- plication to mathematical propositions; dence on which it rests into identical and these will be found, if carefully propositions: the steps by which this examined, to rest entirely and excluis accomplished may be long: the pro- sively on observation and experiment. cess may be extremely involved and A name of great and deserved celes difficult; but the object and end of brity is indeed opposed to this opi. all, is to establish an identical propo- nion : Professor Robison maintains, sition. « Le Geometre avance de sup that the first two laws of motion are not position en supposition. Et retour matters of experience or contingency, nant sa pensee sous mille formes, depending on the properties which it c'est en repetant sans cesse, le meme has pleased that author of nature to est le meme, qu'il opere tous ses pro- bestow upon body; but that they are diges.” This character of mathematical to us necessary truths. The proposievidence cannot be thought to lower its tions announcing them do not so much importance or utility, or the talents and express anything with regard to body, acquirements of those who have dis as they do the operations of our mind - tinguished themselves in its cause : when contemplating body. Hence he the truths to which it conducts us, consistently regards the first and sethough of the simplest form, when dis- cond laws as identical propositions ; covered, rather gain than lose in subli- but, with respect to the third law, he is mity on that account. Unless all ma- unwilling to regard it in that light; thematical evidence is reducible into because, though it is really a law of identical proportions, it appears to us, nature, it is not a law of human indeed, that it cannot amount to de- thought; it is a discovery. The conmonstration ; and that mathematical trary involves no absurdity or contratruths cannot be regarded as absolute- diction. It would indeed be contrary ly necessary, in the strictest sense of to experience ; but things might have the term, unless the reverse of thein been otherwise. If, however, we eximplies a contradiction; and if the de- amine the first and second laws, we nial of any proposition implies a con shall be convinced that they also are tradiction, that proposition must in the results of observation; but of obreality, and when traced to its sim- servation so easy, so universally, and plest form and turns, though it may so inperceptibly made, that we are not in appearance, be identical. not aware of it, and regard the truths
If this view of the nature of mathe- it teaches as innate and self-evident. matical evidence be correct, it follows Both these laws rest on this most gethat this branch of human knowledge neral principle, that every effect must does not necessarily depend either on have a cause ; but this principle is asthe permanency and stability of the or- suredly gathered from what we obder of nature, or on that fundamental
serve and experience. law of the mind from which the associ After, however, the laws of motion ation of ideas springs. It is possible, and and the other laws of matter are estawe can conceive, that the appearances blished from an induction of facts; they and operations of nature were without come within the scope and application order and uniformity, that under ex of mathematics, and consequently so actly the same circumstances, various far lead to certain and necessary conand opposite events might occur ; but clusions. Experiment, for instance, we cannot conceive of any proposition, having established this as an undoubtthe terms of which are contradictory. ed and unvarying fact, that the power If the association of ideas ceased to of gravity is directly as the masses, and take place in the mind, our mathema- inversely as the square of the distance ; tical knowledge, so far as it was the all the possible and actual consequenresult of mere induction and experi- ces of gravity may be calculated with ment, would be annihilated; but its mathematical certainty, provided the peculiar and firmest foundation, that masses and distances are known. S:ill, evidence, which is resolvable into iden- however, that portion of human knowtical propositions, would still remain. ledge, which is included in the term phy
sical philosophy, isinferior, in respect to directly as the masses, and Inversely certainty, to that which we derive from as the square of the distance. His as mathematical investigations ;' or, presumption of these laws was correct; haps, to define the distinction between his calculations were correct; but them more accurately, mathematical his computation did not agree with truths are necessary; they could not the phenomena. This arose from his possibly be otherwise : so long as mag- ignorance of the real magnitude of the nitude and figure exist, or can be con- earth: some years afterwards this was ceived to exist, they must be truths. ascertained by Picard ; and Newton There are no extraneous circumstances « had the inexpressible satisfaction of which can alter or modify them; they finding that his calculation agreed exare in fact an enumeration of the pro actly with what it ought to be, if the perties that belong to magnitude and opinion he had formed was correct. figure. In the circle, for example, we He therefore concluded, that his conbegin with the radius as the most jecture was correct, and that the moon simple, and deduce all the other pro- was really kept in her orbit by the force perties of it ; but we might begin with of gravity," acting exactly on the same any other, and thence deduce the equa- laws as near the surface of the earth. lity of the radii. In the most simple This is an instance of an error in truths of physical science, we depend physical researches arising from a misentirely on observation and experi- take with regard to a fact. Newton's ment; in the most sublime and asto- law of gravity was true in both its parnishing application of these truths, ticulars ; his observations on the effect entirely on observation; but unless we of gravity at the moon were also corobserve accurately, and observe all rect; but this effect did not agree with that can modify the result, the law, or what his calculations, grounded on a general fact we deduce, must be erro- mistaken notion of the earth's magnineous; and the application of that law, tude, led him to expect. even when assisted by the most pro In the history of astronomy we have found and accurate mathematical rea- also an instance of error proceeding soning, leads to error.
from the other cause to which we alTo attain physical truth, therefore, luded. Euler, D'Alembert, and Clairtwo things are indispensably requisite; ault, resolved the celebrated problem that our knowledge of facts be accu- of the three bodies, in order to investirate, and that our mathematical rea gate all the lunar inequalities to which soning be without mistake. To con- gravity could give rise : the result was, fine ourselves to the law of gravity : In that they agreed in finding, by the the history of this branch of physical theory of gravitation, the motion of the science, there are two facts strikingly lunar perigee only half as great as it illustrative of the remarks we have just appears to be from observation; it made. Newton might have been in seemed, therefore, that gravity did not error regarding the laws of gravity, or, diminish in the inverse ratio of the they being well founded, he might square of the distance. And Clairhave been in error with respect 'to facts, ault concluded, “ that the law of atwhen he wished to apply them; or, these traction was not quite so simple as had facts also being correct, he might com- been imagined ; he supposed it to conmit mistakes in the process of his ma- sist of two parts, one varying inverse thematical reasoning. He was natural- ly as the square of the distance, and ly very anxious to ascertain whether sensible only at the great distance of the laws of gravity
extended to the the planets from the sun ; and the heavenly bodies, in the hope that thus other increasing in a greater ratio, senhe might account for their motions, sible at the distance of the moon from and perhaps because gravity, as dis- the earth.” Clairault first detected the played by their mutual actions, would error which he and the other two manecessarily be free from these extra- thematicians had committed, in haneous circumstances which interfered ving neglected some small quantities with its operation near the surface of in the approximation of the series the earth.
which represented the motion of the Accordingly he endeavoured to apogee-rectified it, reconciled obsercompute the force of gravity at the vation and the theory of gravity, and inoon, of course proceeding on the sap- thus added a new proof to the univerposition that it operated by the same sality of this law of nature. laws there as near the earth--that is, Perhaps in no branch of science have
systematic theory, aided by mathema- ing process, and from the powerful tical investigations and observations, effect produced on the imagination, by mutually illustrated and confirmed a calculus which brings into immediate each other so much as in astronomy. contrast with the immensity of time, Sometimes the former has pointed out such evanescent elements as the fracthe fact long before observation and tional parts of a second, that the coexperiment have detected it; but more incidence between the computation frequently what has long been obser- and the event appears in this inved, but unaccounted for, has been stance so peculiarly striking.” proved to be the legitimate and ne When we reflect that the perfection cessary result of the laws of nature, to which astronomical instruments are by mathematical investigations. Of now brought—the effect of which is, the former case, the conclusion to in reality, to render our observations which Newton was led by theory and more accurate, and to extend them to calculation alone, regarding the figure objects and motions that they could of the earth, is a striking and most not reach before and that the applihappy instance: at the time, “ 1686, cation of mathematical investigations when he computed the ratio of the po to such observations so made, have lar and equatorial diameters, no evi- enabled the moderns to compute the dence from actual admeasurement ex- weights and densities of most of the isted; but he lived till it was ascertain- planets—to ascertain their respective ed by observation, that the ratio of the sizes and distances from the sun, and polar and equatorial diameters of Ju- their mutual actions, and the result piter was nearly such as his theory of these actions on their orbits and gave on the hypothesis of an uniform motions ;—that no motion is now density. He also lived till the results known to exist in the system that canof actual admeasurement, made in not be demonstrated to be conformable France, appeared entirely inconsistent to the laws of universal gravitation, with the form which he had assigned. and the result of it ;-that the mean Subsequent measurements, made soon motions and the mean distances of all after Newton's death, fully established the planets are to be considered invathat the equatorial exceeded the polar riable, and the effects of their mutual diameter.” (Brinkley's Astronomy, p. actions are all periodical ;-that the 251.)
celebrated dispute between Leibnitz The periodical inequalities of the and Newton, regarding the permamoon had long puzzled astronomers : nency of the system of the universe, these were all reconciled to the theory is thus settled;*
-we shall not hesiof gravity by the labours of La Place, tate to acknowledge that this branch &c. But in no instance have the in- of physical science, resting on the obvestigations of this celebrated philoso- servation and experience of those propher been more successful, or tended perties of matter, which are the most more to illustrate the application of simple and universal, and which are profound mathematical knowledge to the least liable to be counteracted or account for embarrassing facts, and re- suspended by extraneous and inappreconcile them to the laws of nature, ciable circumstances, and on the apthan in his labours regarding the se plication of mathematical investigacular equation of the moon. “ What tions to these laws, is, next to pure exquisite delight," observes Mr Stew. mathematics, the most certain kind of art, “ must La Place have felt, when, human knowledge. by deducing from the theory of gravi. After this full explanation of the tation, the cause of the acceleration of nature of the evidence on which our the moon's mean motion—an accelera- acquaintance with this most sublime, tion which proceeds at the rate of little interesting, and important division of more than 11" in a century, he account- mechanical philosophy depends, the ed, with such mathematical precision, manner in which this evidence is obfor all the recorded observations of her tained, and the most comprehensive place from the infancy of astronomical views of the universe to which, by its science ! It is from the length and union of observation and mathematiabstruseness, however, of the reason cal investigations, it has already con
In the Edinburgh Review, vol. XIV. p. 80, see some excellent remarks on the opposite opinions of these two great men.
ducted us, it is unnecessary to go into perties. Alchemy, afterwards, the detail with respect to the other divi- Offspring of ignorance, avarice, and sions of mechanical philosophy. Op- superstition, conducted its votaries to ties, Acoustics, Hydronymics, &c. are some of the first experimental truths all similar to Astronomy in the nature of Chemistry. Then its own wonders, of their evidence, and in the certainty acting on the mind of the philosopher, of the doctrines and facts about which and the advantages it held out to those they are conversant. They all relate arts of life that are connected with our to the sensible motions of matter, health, comforts, and luxuries, tendwhich can be measured ; consequent- ed to enlarge the boundaries of this ly, so far as these motions are accu- science, till it arrived at its present rately ascertained, and in proportion state. It is, however, entirely a science as they are least liable to be counter- of observation and experiment, almost acted or modified by accidental and entirely of experiment- except so far extraneous circumstances, so will the as the recent doctrine of equivalents particular conclusions and general prin- and the atomic theory may place it on ciples to which mathematical investi- the basis of mathematics. Astronomy gations applied to them conduct us, is a science of observation; the other be conformable to fact, and our sure branches of mechanical philosophy, of guides in predicting what will occur, observation and experiment; but Cheand in guiding our operations. As we mistry allows experiment a much wider have already remarked, so far as ma- range than any of these. thematical investigations are concern To it alone are analysis and synthesis ed, we tread on sure ground ;--but if applicable; and hence, by their means, our data are inaccurate, or, though though it is conversant with the inte accurate in themselves, we do not allow grant parts of bodies, and with the for particular circumstances, our ma most minute and rapid operations of thematical investigations, proceeding nature, and, from these causes, liable on wrong principles, must lead to to frequent sources of mistake and error; or, even when proceeding on a error, that cannot, without much difsound general principle, must equally ficulty and care, be either detected or lead to error, when the particular cir- accounted for,-yet the great and pecumstances which take the case out of culiar advantage it derives from anathe range of this principle are not spe- lytical, as well as synthetical expericially noticed and allowed for. ments, bestows on it a degree of cer.
We come next to another great di- tainty, which, without the union of vision of human knowledge, quite dis- these modes of proof, it could not postinct in the nature of the evidence on sibly have attained. which it rests, as well as in the nature We are well aware that some of the of the truths about which it is con- truths of Chemistry rest only on analyversant, from mechanical philosophy: tical proof, and that in some cases anawe mean Chemistry. The motions lysis, as where it is applied to mineral that take place in nature, which are waters and vegetable and animal subthe objects of Astronomy, are sensible, stances, it teaches us only the integrant can be measured, and do not affect the parts of the compound, and can give us properties of bodies, or occur in their little certainty with respect to the parintegrant and constituent parts. Che- ticular combinations of them in these mistry, on the other hand, is that bodies; it bringsoutoxygen, hydrogen, science, “the object of which is to carbon, azoti, &c.; it enables us to asdiscover and explain the changes of certain their respective quantities, but composition that occur among the in- it not unfrequently fails to shew us tegrant and constituent parts of dif- how and in what proportions they ferent bodies."
were combined in the body subjected Probably, long before it was either to analysis. But we are here regardascertained or suspected that bodies, ing Chemistry generally, and therefore which to all appearance are simple and our remarks on the nature of the evi. uncompounded, were in reality con- dence on which it rests are sufficiently stituted of various elements, it had applicable and correct. been found that the union of two or We are also aware that the terms more bodies, as they exist in nature, analysis and synthesis are used to dein some cases did not merely increase note modes of proof, of which other their bulk, but also altered their pro- sciences are susceptible. That they
cannot be applied, with any propriety, cal truth are well founded, the whole to metaphysical or moral investiga difference between these two modes of tions, though sometimes loosely se proof will amount to this : That in done, so very little reflection on the the case of analysis we assume the nature of the process which the terms more complicated property, and thence respectively imply, will convince any deduce the more simple ; whereas, in one, who will employ it, that we deem synthesis, we deduce the more comit unnecessary to prove their total inap- plicated from the more simple. Thus, plicability to those branches of know, from the equality of the radii .of a ledge.
circle, we may deduce all the other Nor, in our opinion, can synthesis properties of it, which are not so apand analysis be deemed processes by parent and simple; or taking one of which we attain any kind of mathe- these latter complicated properties for matical truth, either as respects their granted, we may prove that it must strict and etymological meaning, or as be such as the proposition lays down, they are employed in explaining those by its involving and necessarily supfacts that relate to the composition and , posing the equality of the radii. The decomposition of bodies. In Chemis: evidence, by whatever steps it protry, bodies formed of different ele, ceeds, ultimately resolves itself into ments are the subject of our observas the perception of identity. In the case tion and experiment; our object is todes of analysis, as it is called, the steps compound them if we can, or, in other lead us from what is more to what is words, to analyse them so as to ascer less complicated, till we reach the tain the elements of which they are most simple ; in synthesis, as it is formed ; and, in order to put the ac. called, the steps lead us from the most curacy of our analysis to the test, we simple truths, gradually to the more take the elements which it exhibits, complicated ; but the result is the and by synthesis, or putting them to same--the perception of identity. We gether, reproduce a compound ; if, are apt to be led astray from the real when this is done, the same compound nature of mathematical evidence, by is formed, we conclude that our ana denominating one proposition the con lysis has been accurate, and conducted sequence of another; whereas, as all us, not only to the simple elements, the truths in pure mathematics are but also to the proportions in which co-existent in point of time, this can they existed in the compound. Both justly be predicated of them, only these modes of proof are not applicable with a reference to our established to all chemical researches ; and in the arrangements, by which we proceed same manner, as agents must be used from the more simple to the more in our analysis, so agents must be used complex properties of figure and magto re-unite, by synthesis, the elements nitude. into the same compound. But our re The algebraical analysis may also marks are sufficiently accurate and ac- be shewn to be essentially different cordant with chemical investigations, from that employed in Chemistry to illustrate the nature of analysis not to be consonant to the spirit and and synthesis, when employed in this etymological meaning of the term, and science.
in reality to conduct us only to an The geometrical analysis is very dif- identical proposition. To take a plain ferent from this. Assuming the truth and simple case, which, however, will of the proposition, its object is to explain the real nature of algebraical prove, that it leads either to another analysis in its most complex form. problem previously known to be true, The resolution of an equation amounts or to a theorem previously demonstra- to this, the proof of the identity of the ted, or to one which involves an ope two sides of it: Before it is resolved, ration known to be impracticable, or one side contains a known quantity : a theorem which involves a contradic the other side two or more quantities, tion, or is known to be false. Synthe all of which except one is known; and tical demonstration reverses this, by these, when certain operations are persetting out from the more simple pro formed upon them, of addition, subblem or 'theorem, anil by means of traction, &c. are held, by the propothem arriving at the proof of the more sition, to be equal to the quantity on complicated proposition. But if our the other side of the equation. It will remarks on the nature of mathemati- be admitted that 6=6 is an identical