ing is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries cannot have occasioned other than infinitely small errors in all the equations; and when the relations of finite quantities are reached, these relations must be rigorously exact, since the only errors then possible must be finite ones, which cannot have entered: and thus the final equations become perfect. Carnot's theory is doubtless more subtle than solid; but it has no other radical logical vice than that of the infinitesimal method itself, of which it is, as it seems to me, the natural development and general explanation; so that it must be adopted as long as that method is directly employed. The philosophical character of the transcendental analysis has now been sufficiently exhibited to allow of my giving only the principal idea of the other two methods. NEWTON'S Newton offered his conception under several different forms in succession. That which is now most commonly adopted, at least on the continent, was called by himself, sometimes the Method of prime and ultimate Ratios, sometimes the Method of Limits, by which last term it is now usually known. Method of limits. Under this Method, the auxiliaries introduced are the limits of the ratios of the simultaneous increments of the primitive quantities; or, in other words, the final ratios of these increments; limits or final ratios which we can easily show to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is afterwards employed, to rise from the equations between these limits to the corresponding equations between the primitive quantities themselves. The power of easy expression of the mathematical laws of phenomena given by this analysis arises from the calculus applying, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments; and from our being therefore able always to substitute for each increment any other magnitude more easy to treat, provided their final ratio is the ratio of equality; or, in other words, that the limit of their ratio is unity. It NEWTON'S METHOD OF LIMITS. 6.9 is clear, in fact, that the calculus of limits can be in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offered by the analysis of Leibnitz, which are merely considered from another point of view. Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uniform motions of continually nearer approximation, etc., etc. Such is, in substance, Newton's conception; or rather, that which Maclaurin and d'Alembert have offered as the most rational basis of the transcendental analysis, in the endeavour to fix and arrange Newton's ideas on the subject. Newton had another view, however, which ought to be presented here, because it is still the special form of the calculus of indirect functions commonly adopted by English geometers; and also, on account of its ingenious clearness in some cases, and of its having furnished the notation best adapted to this manner of regarding the transcendental analysis. I mean the Calculus of fluxions and of fluents, founded on the general notion of velocities. Fluxions and fluents. To facilitate the conception of the fundamental idea, let us conceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the curve, the abscissa, the ordinate, the arc, the area, etc., will be regarded as simultaneously produced by successive degrees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluent. Henceforth, the transcendental analysis will, according to this conception, consist in forming directly the equations between the fluxions of the proposed quantities, to deduce from them afterwards, by a special Calculus, the equations between the fluents themselves. What has just been stated respecting curves may evidently be transferred to any magnitudes whatever, regarded, by the help of a suitable image, as some being produced by the motion of others. This method is evidently the same with that of limits complicated with. the foreign idea of motion. It is, in fact, only a way of representing, by a comparison derived from mechanics, the method of prime and ultimate ratios, which alone is reducible to a calculus. It therefore necessarily admits of the same general advantages in the various principal applications of the transcendental analysis, without its being requisite for us to offer special proofs of this. LAGRANGE'S Lagrange's conception consists, in its admirable simplicity, in considering the transcendental analysis to be a great algebraic artifice, by which, to facilitate the establishment of equations, we must introduce, in the place of or with the primitive functions, their derived functions; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The Calculus of indirect functions, properly so called, is destined here, as well as in the conceptions of Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equations between the primitive magnitudes. The transcendental analysis is then only a simple, but very considerable extension of ordinary analysis. It has long been a common practice with geometers to introduce, in analytical investigations, in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simplify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater extent, and commanding, in consequence, much more important resources for this common object. But, though we may easily conceive, à priori, that the auxiliary use of these derivatives may facilitate the study of equations, it is not easy to explain why it must be so under this method of derivation, rather than any other transformation. This is the weak side of Lagrange's great idea. We have not yet become able to lay hold of its precise advantages, in an abstract manner, and without recurrence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate consideration of each principal question; and this verification becomes laborious, in the treatment of a complex problem. COMPARISON OF THE THREE METHODS. 71 Other theories have been proposed, such as Euler's Calculus of vanishing quantities: but they are merely modifications of the three just exhibited. We must next compare and estimate these methods; and in the first place observe their perfect and necessary conformity. Considering the three methods in regard to their destination, independently of pre- three methods. Identity of the liminary ideas, it is clear that they all con sist in the same general logical artifice; that is, the introduction of a certain system of auxiliary magnitudes uniformly correlative with those under investigation; the auxiliaries being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated by the help of a special calculus. It was this which determined me to define the transcendental analysis as the Calculus of indirect functions, in order to mark its true philosophical character, while excluding all discussion about the best manner of conceiving and applying it. Whatever may be the method employed, the general effect of this analysis is to bring every mathematical question more speedily into the domain of the calculus, and thus to lessen considerably the grand difficulty of the passage from the concrete to the abstract. We cannot hope that the Calculus will ever lay hold of all questions of natural philosophy— geometrical, mechanical, thermological, etc.-from their birth. That would be a contradiction. In every problem there must be a certain preliminary operation before the calculus can be of any use, and one which could not by its nature be subjected to abstract and invariable rules :-it is that which has for its object the establishment of equations, which are the indispensable point of departure for all analytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcendental analysis, which has thus hastened the moment at which general and abstract processes may be uniformly and exactly applied to the solution, by reducing the operation to finding the equations between auxiliary magnitudes, whence the Calculus leads to equations directly relating to the proposed magnitudes, which had formerly to be established directly. Whether these indirect equations are differential equations, according to Leibnitz, or equations of limits, according to Newton, or derived equations, according to Lagrange, the general procedure is evidently always the same. The coincidence is not only in the result but in the process; for the auxiliaries introduced are really identical, being only regarded from different points of view. The conceptions of Leibnitz and of Newton consist in making known in any case two general necessary properties of the derived function of Lagrange. The transcendental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted; and the processes of the Calculus of indirect functions are necessarily identical in these different methods, which must therefore, under any application whatever, lead to rigorously uniform results. tive value. If we endeavour to estimate their comparaTheir compara- tive value, we shall find in each of the three conceptions advantages and inconveniences which are peculiar to it, and which prevent geometers from adhering to any one of them, as exclusive and final. The method of Leibnitz has eminently the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitudes. We owe to its use the high perfection attained by all the general theories of geometry and mechanics. Whatever may be the speculative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed the analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his "Analytical Mechanics." Such a fact needs no comment. Yet are we obliged to admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical relations. He himself declared the notion of infinitely small quantities to be a false idea: and it is in fact impossible to conceive of them clearly, though we may sometimes fancy that we do. This false idea bears, to my mind, the characteristic impress of the metaphysical age of its birth and tendencies of its originator. By the ingenious principle of the compensation of errors, we may, as we have already seen, explain the necessary exactness of |