epithets pure and exact cannot be applied. So far, therefore, as our knowledge relates to magnitude, figure, and number-that is, so far as it is conversant with mathematics, it appears to us that it consists in resolving the evidence on which it rests into identical propositions: the steps by which this is accomplished may be long: the process may be extremely involved and difficult; but the object and end of all, is to establish an identical proposition. "Le Geometre avance de supposition en supposition. Et retournant sa pensee sous mille formes, c'est en repetant sans cesse, le meme est le meme, qu'il opere tous ses prodiges." This character of mathematical evidence cannot be thought to lower its importance or utility, or the talents and acquirements of those who have distinguished themselves in its cause: the truths to which it conducts us, though of the simplest form, when discovered, rather gain than lose in sublimity on that account. Unless all mathematical evidence is reducible into identical proportions, it appears to us, indeed, that it cannot amount to demonstration; and that mathematical truths cannot be regarded as absolutely necessary, in the strictest sense of the term, unless the reverse of them implies a contradiction; and if the denial of any proposition implies a contradiction, that proposition must in reality, and when traced to its simplest form and turns, though it may not in appearance, be identical. If this view of the nature of mathematical evidence be correct, it follows that this branch of human knowledge does not necessarily depend either on the permanency and stability of the order of nature, or on that fundamental law of the mind from which the association of ideas springs. It is possible, and we can conceive, that the appearances and operations of nature, were without order and uniformity, that under exactly the same circumstances, various and opposite events might occur; but we cannot conceive of any proposition, the terms of which are contradictory. If the association of ideas ceased to take place in the mind, our mathematical knowledge, so far as it was the result of mere induction and experiment, would be annihilated; but its peculiar and firmest foundation, that evidence, which is resolvable into identical propositions, would still remain. There is, however, no other branch of knowledge which does not exclusively rest on that induction which observation and experiment supply. The laws of motion perhaps approach nearest in simplicity and universality of application to mathematical propositions; and these will be found, if carefully examined, to rest entirely and exclusively on observation and experiment. A name of great and deserved celebrity is indeed opposed to this opinion Professor Robison maintains, that the first two laws of motion are not matters of experience or contingency, depending on the properties which it has pleased that author of nature to bestow upon body; but that they are to us necessary truths. The propositions announcing them do not so much express anything with regard to body, as they do the operations of our mind when contemplating body. Hence he consistently regards the first and second laws as identical propositions; but, with respect to the third law, he is unwilling to regard it in that light, because, though it is really a law of nature, it is not a law of human thought; it is a discovery. The contrary involves no absurdity or contradiction. It would indeed be contrary to experience; but things might have been otherwise. If, however, we examine the first and second laws, we shall be convinced that they also are the results of observation; but of observation so easy, so universally, and so imperceptibly made, that we are not aware of it, and regard the truths it teaches as innate and self-evident. Both these laws rest on this most general principle, that every effect must have a cause; but this principle is assuredly gathered from what we ob serve and experience. After, however, the laws of motion and the other laws of matter are established from an induction of facts; they come within the scope and application of mathematics, and consequently so far lead to certain and necessary conclusions. Experiment, for instance, having established this as an undoubted and unvarying fact, that the power of gravity is directly as the masses, and inversely as the square of the distance; all the possible and actual consequences of gravity may be calculated with mathematical certainty, provided the masses and distances are known. Still, however, that portion of human knowledge, which is included in the term phy directly as the masses, and inversely as the square of the distance. His as sical philosophy, is inferior, in respect to certainty, to that which we derive from mathematical investigations; or, presumption of these laws was correct; haps, to define the distinction between them more accurately, mathematical truths are necessary; they could not possibly be otherwise: so long as magnitude and figure exist, or can be conceived to exist, they must be truths. There are no extraneous circumstances which can alter or modify them; they are in fact an enumeration of the properties that belong to magnitude and figure. In the circle, for example, we begin with the radius as the most simple, and deduce all the other properties of it; but we might begin with any other, and thence deduce the equality of the radii.In the most simple truths of physical science, we depend entirely on observation and experiment; in the most sublime and astonishing application of these truths, entirely on observation; but unless we observe accurately, and observe all that can modify the result, the law, or general fact we deduce, must be erroneous; and the application of that law, even when assisted by the most profound and accurate mathematical reasoning, leads to error. To attain physical truth, therefore, two things are indispensably requisite; that our knowledge of facts be accurate, and that our mathematical reasoning be without mistake. To confine ourselves to the law of gravity: In the history of this branch of physical science, there are two facts strikingly illustrative of the remarks we have just made. Newton might have been in error regarding the laws of gravity, or, they being well founded, he might have been in error with respect to facts, when he wished to apply them; or, these facts also being correct, he might commit mistakes in the process of his mathematical reasoning. He was naturally very anxious to ascertain whether the laws of gravity extended to the heavenly bodies, in the hope that thus he might account for their motions, and perhaps because gravity, as displayed by their mutual actions, would necessarily be free from these extra neous circumstances which interfered with its operation near the surface of the earth. Accordingly he endeavoured to compute the force of gravity at the moon, of course proceeding on the supposition that it operated by the same laws there as near the earth-that is, his calculations were correct; but his computation did not agree with the phenomena. This arose from his ignorance of the real magnitude of the earth: some years afterwards this was ascertained by Picard; and Newton had the inexpressible satisfaction of finding that his calculation agreed exactly with what it ought to be, if the opinion he had formed was correct. He therefore concluded, that his conjecture was correct, and that the moon was really kept in her orbit by the force of gravity," acting exactly on the same laws as near the surface of the earth. This is an instance of an error in physical researches arising from a mistake with regard to a fact. Newton's law of gravity was true in both its particulars; his observations on the effect of gravity at the moon were also correct; but this effect did not agree with what his calculations, grounded on a mistaken notion of the earth's magnitude, led him to expect. In the history of astronomy we have also an instance of error proceeding from the other cause to which we alluded. Euler, D'Alembert, and Clairault, resolved the celebrated problem of the three bodies, in order to investigate all the lunar inequalities to which gravity could give rise: the result was, that they agreed in finding, by the theory of gravitation, the motion of the lunar perigee only half as great as it appears to be from observation; it seemed, therefore, that gravity did not diminish in the inverse ratio of the square of the distance. And Clairault concluded, "that the law of attraction was not quite so simple as had been imagined; he supposed it to consist of two parts, one varying inversely as the square of the distance, and sensible only at the great distance of the planets from the sun; and the other increasing in a greater ratio, sensible at the distance of the moon from the earth." Clairault first detected the error which he and the other two mathematicians had committed, in having neglected some small quantities in the approximation of the series which represented the motion of the apogee-rectified it, reconciled observation and the theory of gravity, and thus added a new proof to the universality of this law of nature. Perhaps in no branch of science have systematic theory, aided by mathematical investigations and observations, mutually illustrated and confirmed each other so much as in astronomy. Sometimes the former has pointed out the fact long before observation and experiment have detected it; but more frequently what has long been observed, but unaccounted for, has been proved to be the legitimate and necessary result of the laws of nature, by mathematical investigations. Of the former case, the conclusion to which Newton was led by theory and calculation alone, regarding the figure of the earth, is a striking and most happy instance: at the time," 1686, when he computed the ratio of the polar and equatorial diameters, no evidence from actual admeasurement existed; but he lived till it was ascertained by observation, that the ratio of the polar and equatorial diameters of Jupiter was nearly such as his theory gave on the hypothesis of an uniform density. He also lived till the results of actual admeasurement, made in France, appeared entirely inconsistent with the form which he had assigned. Subsequent measurements, made soon after Newton's death, fully established that the equatorial exceeded the polar diameter." (Brinkley's Astronomy, p. 251.) The periodical inequalities of the moon had long puzzled astronomers: these were all reconciled to the theory of gravity by the labours of La Place, &c. But in no instance have the investigations of this celebrated philosopher been more successful, or tended more to illustrate the application of profound mathematical knowledge to account for embarrassing facts, and reconcile them to the laws of nature, than in his labours regarding the secular equation of the moon. "What exquisite delight," observes Mr Stewart, must La Place have felt, when, by deducing from the theory of gravitation, the cause of the acceleration of the moon's mean motion-an acceleration which proceeds at the rate of little more than 11" in a century, he accounted, with such mathematical precision, for all the recorded observations of her place from the infancy of astronomical science! It is from the length and abstruseness, however, of the reason 66 ing process, and from the powerful effect produced on the imagination, by a calculus which brings into immediate contrast with the immensity of time, such evanescent elements as the fractional parts of a second, that the coincidence between the computation and the event appears in this instance so peculiarly striking." When we reflect that the perfection to which astronomical instruments are now brought-the effect of which is, in reality, to render our observations more accurate, and to extend them to objects and motions that they could not reach before-and that the application of mathematical investigations to such observations so made, have enabled the moderns to compute the weights and densities of most of the planets-to ascertain their respective sizes and distances from the sun, and their mutual actions, and the result of these actions on their orbits and motions;-that no motion is now known to exist in the system that cannot be demonstrated to be conformable to the laws of universal gravitation, and the result of it ;-that the mean motions and the mean distances of all the planets are to be considered invariable, and the effects of their mutual actions are all periodical;-that the celebrated dispute between Leibnitz and Newton, regarding the permanency of the system of the universe, is thus settled ;* -we shall not hesitate to acknowledge that this branch of physical science, resting on the observation and experience of those properties of matter, which are the most simple and universal, and which are the least liable to be counteracted or suspended by extraneous and inappreciable circumstances, and on the application of mathematical investigations to these laws, is, next to pure mathematics, the most certain kind of human knowledge. After this full explanation of the nature of the evidence on which our acquaintance with this most sublime, interesting, and important division of mechanical philosophy depends, the manner in which this evidence is obtained, and the most comprehensive views of the universe to which, by its union of observation and mathematical 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 detail with respect to the other divisions of mechanical philosophy. Optics, Acoustics, Hydronymics, &c. are all similar to Astronomy in the nature of their evidence, and in the certainty of the doctrines and facts about which they are conversant. They all relate to the sensible motions of matter, which can be measured; consequently, so far as these motions are accurately ascertained, and in proportion as they are least liable to be counteracted or modified by accidental and extraneous circumstances, so will the particular conclusions and general principles to which mathematical investigations applied to them conduct us, be conformable to fact, and our sure guides in predicting what will occur, and in guiding our operations. As we have already remarked, so far as mathematical investigations are concerned, we tread on sure ground;-but if our data are inaccurate, or, though accurate in themselves, we do not allow for particular circumstances, our mathematical investigations, proceeding on wrong principles, must lead to error; or, even when proceeding on a sound general principle, must equally lead to error, when the particular circumstances which take the case out of the range of this principle are not specially noticed and allowed for. We come next to another great division of human knowledge, quite distinct in the nature of the evidence on which it rests, as well as in the nature of the truths about which it is conversant, from mechanical philosophy: we mean Chemistry. The motions that take place in nature, which are the objects of Astronomy, are sensible, can be measured, and do not affect the properties of bodies, or occur in their integrant and constituent parts. Chemistry, on the other hand, is that science," the object of which is to discover and explain the changes of composition that occur among the integrant and constituent parts of different bodies." Probably, long before it was either ascertained or suspected that bodies, which to all appearance are simple and uncompounded, were in reality constituted of various elements, it had been found that the union of two or more bodies, as they exist in nature, in some cases did not merely increase their bulk, but also altered their pro perties. Alchemy, afterwards, the offspring of ignorance, avarice, and superstition, conducted its votaries to some of the first experimental truths of Chemistry. Then its own wonders, acting on the mind of the philosopher, and the advantages it held out to those arts of life that are connected with our health, comforts, and luxuries, tended to enlarge the boundaries of this science, till it arrived at its present state. It is, however, entirely a science of observation and experiment, almost entirely of experiment-except so far as the recent doctrine of equivalents and the atomic theory may place it on the basis of mathematics. Astronomy is a science of observation; the other branches of mechanical philosophy, of observation and experiment; but Chemistry allows experiment a much wider range than any of these. To it alone are analysis and synthesis applicable; and hence, by their means, though it is conversant with the inte grant parts of bodies, and with the most minute and rapid operations of nature, and, from these causes, liable to frequent sources of mistake and error, that cannot, without much difficulty and care, be either detected or accounted for,-yet the great and peculiar advantage it derives from analytical, as well as synthetical experiments, bestows on it a degree of certainty, which, without the union of these modes of proof, it could not pos sibly have attained. We are well aware that some of the truths of Chemistry rest only on analytical proof, and that in some cases analysis, as where it is applied to mineral waters and vegetable and animal substances, it teaches us only the integrant parts of the compound, and can give us little certainty with respect to the particular combinations of them in these bodies; it brings out oxygen, hydrogen, carbon, azoti, &c. ; it enables us to ascertain their respective quantities, but it not unfrequently fails to shew us how and in what proportions they were combined in the body subjected to analysis. But we are here regarding Chemistry generally, and therefore our remarks on the nature of the evidence on which it rests are sufficiently applicable and correct. We are also aware that the terms analysis and synthesis are used to denote modes of proof, of which other sciences are susceptible. That they cannot be applied, with any propriety, to metaphysical or moral investiga tions, though sometimes loosely so done, so very little reflection on the nature of the process which the terms respectively imply, will convince any one, who will employ it, that we deem it unnecessary to prove their total inap plicability to those branches of know ledge. Nor, in our opinion, can synthesis and analysis be deemed processes by which we attain any kind of mathematical truth, either as respects their strict and etymological meaning, or as they are employed in explaining those facts that relate to the composition and decomposition of bodies. In Chemis try, bodies formed of different elements are the subject of our observation and experiment; our object is to de compound them if we can, or, in other words, to analyse them so as to ascertain the elements of which they are formed; and, in order to put the accuracy of our analysis to the test, we take the elements which it exhibits, and by synthesis, or putting them to gether, reproduce a compound; if, when this is done, the same compound is formed, we conclude that our analysis has been accurate, and conducted us, not only to the simple elements, but also to the proportions in which they existed in the compound. Both these modes of proof are not applicable to all chemical researches ; and in the same manner, as agents must be used in our analysis, so agents must be used to re-unite, by synthesis, the elements into the same compound. But our remarks are sufficiently accurate and accordant with chemical investigations, to illustrate the nature of analysis and synthesis, when employed in this science. The geometrical analysis is very different from this. Assuming the truth of the proposition, its object is to prove, that it leads either to another problem previously known to be true, or to a theorem previously demonstra ted, or to one which involves an ope ration known to be impracticable, or a theorem which involves a contradiction, or is known to be false. Synthetical demonstration reverses this, by setting out from the more simple problem or theorem, and by means of them arriving at the proof of the more complicated proposition. But if our remarks on the nature of mathemati cal truth are well founded, the whole difference between these two modes of proof will amount to this: That in the case of analysis we assume the more complicated property, and thence deduce the more simple; whereas, in synthesis, we deduce the more complicated from the more simple. Thus, from the equality of the radii of a circle, we may deduce all the other properties of it, which are not so apparent and simple; or taking one of these latter complicated properties for granted, we may prove that it must be such as the proposition lays down, by its involving and necessarily sup posing the equality of the radii. The evidence, by whatever steps it proceeds, ultimately resolves itself into the perception of identity. In the case of analysis, as it is called, the steps lead us from what is more to what is less complicated, till we reach the most simple; in synthesis, as it is called, the steps lead us from the most simple truths, gradually to the more complicated; but the result is the same the perception of identity. We are apt to be led astray from the real nature of mathematical evidence, by denominating one proposition the consequence of another; whereas, as all the truths in pure mathematics are co-existent in point of time, this can justly be predicated of them, only with a reference to our established arrangements, by which we proceed from the more simple to the more complex properties of figure and magnitude. The algebraical analysis may also be shewn to be essentially different from that employed in Chemistrynot to be consonant to the spirit and etymological meaning of the term, and in reality to conduct us only to an identical proposition. To take a plain and simple case, which, however, will explain the real nature of algebraical analysis in its most complex form. The resolution of an equation amounts to this, the proof of the identity of the two sides of it: Before it is resolved, one side contains a known quantity: the other side two or more quantities, all of which except one is known ; and these, when certain operations are performed upon them, of addition, subtraction, &c. are held, by the propo sition, to be equal to the quantity on the other side of the equation. It will be admitted that 6=6 is an identical |