Reflections on the metaphysical principles of the Infinitesimal Calculus

Lazare Carnot wrote Réflexions sur la métaphysique du calcul infinitésimal in 1797. It was translated into English by the Rev W R Browell M.A., Fellow of Pembroke College, Oxford, and published in 1832 as Reflexions on the metaphysical principles of the Infinitesimal Calculus. Let us look first at the background to this work.

In 1784 the Berlin Academy announced a prize problem requiring entries giving "a clear and precise theory of what is called the infinite in mathematics." The prize was awarded to Simon Lhuilier in 1786 but Lazare Carnot had also submitted an entry. Carnot revised and extended his entry and published it as Réflexions sur la métaphysique du calcul infinitésimal in 1797. Ivor Grattan-Guinness writes in his book From the Calculus to set Theory 1630-1910 (1980):-
Carnot surveyed various known methods of founding the calculus, including Berkeley's doctrine of the compensation of errors and Euler's view on calculation with zeros. He supported the compensation of errors, but his most valuable discussions concerned the definition and use of infinitesimals, differentials and higher order differentials.
Carnot's work, although interesting, is not considered a major step forward in the development of the differential calculus. It is worth remembering that Carnot's interests were mainly in geometry and applications of mathematics to engineering problems. The work is, however, an excellent survey clearly showing the ideas that were around in the development of the calculus at the end of the 18th century. Jacques Harthong, in 'Lazare Carnot et le calcul infinitésimal', Séminaires de mathématiques. Science, histoire, société (Rennes, 1984), 1-4, takes a much more positive view of Carnot's contribution:-
The main interest of this book is to reveal the fallacy of a very common idea: that the infinitesimal calculus would not have become fully rigorous (that is to say, based on logic alone without any recourse to intuition or flair) only with Weierstrass. It is true that Weierstrass's point of view is the dominant one today; it is also true that it gives a satisfactory approach; but it is no less true that that of Carnot (who, moreover, owes a lot to Lagrange) could have given a just as satisfactory approach, and another explanation must be found for the choice ultimately made by history. Indeed Carnot offers a true axiomatic approach.
Below we give the translator's Preface, Carnot's Preface, and an extract from Carnot's 'General principles', and his 'Conclusions'.

1. Translator's Preface

In offering this Work to the public, the translator is aware that little advantage will be derived from its appearing in English. The French language is now so generally understood, that some apology may be required for producing it in any other form than the original: yet there may still be some, who would rather have an argumentative treatise in their own than in a foreign language: for this reason, whilst multiplying copies of a work which appears singularly calculated to be useful, and is in some degree difficult to be procured, the translation has been attempted.

Although the treatise will be found particularly adapted for the Mathematical student, yet it is brought forward not without a hope that many, who are curious to know the principles of the New calculus, without wishing to enter deeply into the science of abstract quantity, may find in it an answer to their enquiries. But few examples have been given out of the numerous list produced by the Author: they may all be found in the several elementary works on the Differential and Integral Calculus, and would, if introduced here, swell the volume unnecessarily.

2. Reflections on the metaphysical principles of the Infinitesimal Calculus by Lazare Carnot: Preface

It is my object to ascertain in what the true spirit of the Infinitesimal Analysis consists. My reflexions on this subject will be divided into three chapters: in the first, I intend to point out the general principles of this Analysis: in the second, I shall examine how it has been reduced into Algorithm by the invention of the Differential and Integral Calculus: in the third, I shall compare it with the other methods, such as the method of Exhaustion, that of Indivisibles, of Indeterminate quantities, &c. which might be substituted for it.

3. General principles of the Infinitesimal Calculus

There is no discovery which has produced in the science of Mathematics a change so important and so sudden, as that of the Infinitesimal Analysis: no method has furnished means more simple, or more successful, in promoting our knowledge of the laws of nature. By decomposing bodies (as it may be called) into their elements, it appears to have shown their inward structure and organisation: but as all extremes escape the senses and the imagination, none but an imperfect idea can be formed of these elements; a peculiar species of quantity, which at one time bear the character of real quantities, at another must be treated as absolutely nothing, and which seem by their equivocal proper ties to hold a middle rank between magnitude and zero, between existence and non-existence. [I am here speaking in conformity to the vague ideas that are commonly formed of quantities called Infinitesimals, when no pains have been taken to ascertain their nature: but, in reality, nothing is more simple than the exact idea of this sort of quantities.]

Fortunately this difficulty has not impeded the progress of the discovery: there are certain primitive ideas which always leave some obscurity in the mind, but which, when their first deductions have once been made, open a field both extensive, and free from obstacles. Such we have seen to be the idea of infinity; and many geometricians, who never perhaps had investigated it, have made the most fortunate application of that idea: philosophers, however, could not content themselves with a notion so vague; they have been desirous of returning to principles, but they have found themselves divided in their opinions, or rather in their manner of regarding the objects of their speculation. My intention in this work is, to concentrate these different points of view, to show their mutual relations, and to propose some afresh.

The difficulty which is often experienced in expressing exactly the different conditions of a problem by equations, and in solving these equations, might have given rise to the first idea of the Infinitesimal Calculus: indeed, when it is impossible to find the exact solution of a question, it is natural to endeavour to approach to it, as nearly as possible, by neglecting quantities which embarrass the combinations, if it be foreseen that these quantities which have been put aside cannot, by reason of their small value, produce more than a trifling error in the result of the calculation. For example, as the properties of curves are with difficulty discovered, it is necessary to consider them as polygons of a great number of sides. By way of illustration, if a regular polygon be supposed to be inscribed in a circle, it is evident, that these two figures, although always different, and incapable of becoming identical, are nevertheless more and more alike, according as the number of the sides of the polygon increases: it is evident, that their perimeters, their areas, the solids formed by their revolving round a given axis, the analogous lines drawn within or without these figures, the angles formed by these lines, &c. are, if not respectively equal, at any rate so much the nearer approaching to equality as the number of sides becomes increased; whence it follows, that, by supposing the number of these sides really very great, it will be possible, without any perceptible error, to assign to the circumscribed circle the properties that shall have been found belonging to the in scribed polygon. Besides, each of the sides of this polygon evidently diminishes in size, according as the number of the sides increases; and consequently, if it be supposed that the polygon be really composed of a great number of sides, we may assert also, that each of them is really very small.

This being established, if by chance in the course of a calculation a particular circumstance were discovered, by means of which the operations might be greatly simplified, by neglecting (for example) one of these little sides in comparison with a given line, such as the radius; i.e. by employing in the calculation this given line instead of a quantity equal to the sum of the given line, and the small side spoken of above; it clearly might be done without impropriety; for the amount of error which would result from it could only be extremely small, and would not repay any pains bestowed on learning its value.

4. Carnot's Conclusions

The different methods which we have described in this work are, properly speaking, one and the same method, presented under different points of view: it is always the method of exhaustion of the ancients more or less simplified, more or less happily applied to the wants of the calculus, and at last reduced to a regular notation. But this notation is of great importance, it is to the method of exhaustion what common algebra is to synthesis; the ancients knew only of synthesis and the method of exhaustion, which is itself but a branch of synthesis: the moderns by discovering common algebra and the Infinitesimal Analysis have gained immense advantages. It is a means by which they abridge and facilitate the operations of the mind, by reducing them, if we may so express ourselves, to mechanical trouble. The symbols are not only what writing is to the thoughts a means of describing and of fixing them; they react upon the thoughts themselves, they direct them up to a certain point, and it is sufficient to arrange them on paper according to certain very simple rules, in order to arrive infallibly at new truths.

The algebraic symbols do more, they introduce into combinations purely imaginary forms, fictitious beings, which can neither exist nor even be conceived, and which notwithstanding are of use. We employ them as auxiliaries or terms of comparison, to facilitate the approximation to real quantities which are unknown, and then we eliminate them by transformations which themselves are only manual operations. This wonderful instrument of the pure sciences could not be produced but by reiterated attempts of the greatest geniuses, and perhaps by some fortunate chances. But we must not forget, that it is but an instrument formed to aid the imagination, and not to weaken the springs of it; it is always an indirect means of obviating the weakness of our minds: we must employ it only with reluctance, and with a view of overcoming difficulties or of generalising questions; and to have recourse to it unnecessarily, as in the first elements of science, where it causes more embarrassment than real assistance, is clearly an abuse of it.

Far from using the analysis to establish elementary truths, we ought to disengage from them every thing which hinders us from perceiving them as distinctly as possible, and from recurring to the path which leads us to them. As for those who succeed in shewing us almost intuitively such results as we had not arrived at before them, but by the aid of a complicated analysis, do they not occasion as much pleasure to us as surprise, provided always that this is done in a simple manner, and without increasing the difficulties?

The principles of common algebra are much less clear and less firmly established than those of the Infinitesimal Analysis, in that which distinguishes the latter from the former. The metaphysical principle of the rule for signs is much more difficult to discover than that of infinitely small quantities. This rule has never been demonstrated in a satisfactory manner: it does not appear that it can be, and nevertheless it serves as the foundation of all algebra. What then would be gained by substituting this latter for the Infinitesimal Analysis, (since the operations of the former are much more complicated than those of the second,) in order to attain those objects which. naturally spring from the latter? The analytic expression of an object can never be so clear as the immediate perception of the object itself: it is looking at a thing reflected by a mirror, which we might see directly. This analytic expression may be embarrassed by imaginary forms, or point out operations to be performed of which we are incapable. To define an object of sense by similar expressions is not only an useless employment of an indirect method, but it is representing a thing clear of itself by a symbol which is much less so. The following is a very remarkable passage from Euler on this subject, who was the first analyst of the last century. (Memoire de l'Academie de Berlin, année 1754.)
There are persons," he says, "who maintain, that geometry and analysis do not require much reasoning. They imagine, that the rules which these sciences prescribe to us already comprise the knowledge necessary for arriving at the solution, and that we have only to perform these operations in conformity with these rules, without troubling one's self with the reasoning on which these rules are founded. This opinion, if it were well founded, would be contrary to the almost universal sentiment; namely, that geometry and analysis are the fittest means of employing the mind, and of exercising the reasoning faculties. Although these people have a smattering of mathematics, they can have applied themselves but very little to the solution of difficult problems; for they would have perceived, that the mere application of prescribed rules is a very weak auxiliary in solving this sort of problems; and that it is necessary, in the first place, to examine attentively all the circumstances of the problem, and previously to employing these rules, to draw many arguments, because these contain the rest of the reasoning which we with difficulty observe whilst pursuing the course of the calculation. It is this necessary preparative, before having recourse to the calculus, which very often requires a longer train of arguments than perhaps any other science ever did. And in this we are possessed of the great advantage of being satisfied of their propriety, whilst in other sciences we are frequently obliged to stop short at unconvincing arguments. But the calculus itself, although analysis prescribes to it its rules, ought to be sustained by solid reasoning, in the absence of which we run the risk of committing errors every moment. The geometrician then finds on all sides opportunities of exercising his mind by reasoning, which ought to lead to the solution of all the difficult problems which he undertakes; and unless he be on his guard, there is a chance of his coming to false conclusions.
The rule, as it appears to us, ought to be always to take the simplest method; and, when the difficulties are equal, to take the most perspicuous: no method should be used exclusively. Thus, to return to our subject, amongst the methods spoken of above, it is necessary, with a view to the habitual employment of it, to choose that which effects its end in general by the shortest and easiest way, but not so as to reject any of the others, since they are fine speculations for the mind; and besides, there is not one amongst them, perhaps, which may not lead to some truth hitherto unknown, or procure in certain cases an unobserved result, or a solution more elegant than any other. But amongst all these methods deriving their origin in common from the method of exhaustion of the ancients, which is that which offers most advantages for habitual employment? It seems generally agreed, that it is the Analysis of Leibniz. The works of Descartes, of Pascal, Fermat, Huygens, Barrow, Roberval, Wallis, and particularly those of Newton, prove, that for a long time they had touched on this great discovery when it was announced by Leibniz; and it appears that there is no one amongst these illustrious geometricians who would not have done it, if he had suspected that he could make a great discovery in this quarter; which is as much as to say, that there is not one amongst them who would not have found means of reducing the methods of exhaustion to notation, if the idea had struck them of doing so, and they had foreseen all the fertility with which a similar method would one day be invested. Perhaps too, amongst the several notations invented by so many original geniuses, there would have been found some which would have obtained the preference over that of Leibniz, which habit has endeared to us not less than the precious and costly labours, which in the present day are dressed up in the forms of this notation.

"We may," says Lagrange, "consider Fermat as the first inventor of these new calculations." Barrow conceived the substitution of real but infinitely small quantities for quantities which, according to Fermat, ought to be taken at zero; and in 1674 he produced his method of tangents, where he considers the curve as a polygon of an infinite Dumber of sides: but this calculus was but sketched out; for it applied to rational expressions only. It remained still to find out a simple and general notation applicable to all sorts of expressions, by which one might pass immediately, and without any reduction, from algebraic formulas to their differentials: this Leibniz effected ten years subsequently. It appeared that Newton had arrived at the same method of shortening the calculus for differentiation during the same time, or a little before. But it is in the formation of differential equations, and in their integration, that the great merit and principal force of the new calculations consist; and on this point it seems that the glory of the discovery is due to Leibniz almost entirely, and for the rest to the two Bernoullis.

It would appear very difficult now to quit the route which has been opened to us by these illustrious geometricians, to take, up suddenly a new way of looking at it, a new notation and new expressions. Lagrange himself acknowledged, as we have already seen, that the method of Infinitesimals, such as we employ it in the present day, is a certain means of abbreviation and simplification; and he thought himself bound to use it in his new edition of his Analytic Mechanics, in preference to those which he had just proposed in his Theorie des fonctions Analytiques. We are hence at liberty to presume, that this latter work was but as it were a means of assembling under one point of view many analytical contrivances, 'which in the course of his labours had presented themselves, and with a view to exhibit systematically the great resources of his genius in calculation.

It has been remarked many times by this profound thinker, that the real secret of analysis consists in the art of making one's self acquainted with the different degrees of indetermination, of which the quantity is capable; an idea which always struck us as forcible, and which occasioned our regarding the method of indeterminate quantities of Descartes as the most important corollary to the method of exhaustion.

In all branches of analysis taken generally, we observe that these operations are always founded on the different degrees of indetermination in the quantities compared by it. An abstract number is less determinate than a concrete, since the latter determines not only the quantity but the quality of the object submitted to the calculus. Algebraic quantities are more indeterminate than abstract numbers, because they do not specify the quantity. Amongst these last, variables are more indeterminate than constants, since the latter are considered fixed for a longer period in the calculation. Infinitesimal quantities are more indeterminate than simple variables, since they still continue susceptible of change, even when it has already been agreed on to consider the others as fixed. Lastly, the variations are more indeterminate than the simple differentials, since the latter are restricted to varying according to a given law, instead of which the law according to which the others change is arbitrary. This gradation in the different degrees of indetermination is endless, and it is in this union of quantities more or less defined, more or less arbitrary, that is founded the fertile principle of the general method of indeterminate quantities, of which the Infinitesimal Calculus is in truth but a fortunate application.

These quantities, on one hand united to the conditions of the question, whilst on the other we are free to attribute to them greater or less values, these quantities, which are in some sort arbitrary, make us observe the necessity of drawing that distinction which we have made between "assigned" and "unassigned" quantities; a distinction which is not the same as that between constants and variables; for "assigned" quantities comprise under their term the unknown constants and variables, and their functions; that is to say, all those which can enter into the results of the Calculus, whilst "unassigned" quantities are necessarily excluded from them. The latter can enter then merely as auxiliaries, serving only to facilitate the expression of the conditions of the problem : after which all the energies of the calculator must be employed with a view to their elimination, which is indispensable in every case, and always shows, when it is completed, that this calculation loses from that time forth its characteristic of the Infinitesimal Calculus, and returns to the province of ordinary algebra.

The method of limits, or of prime and ultimate ratios, does not dispense with the distinction of these "assigned" and "unassigned" quantities; for the limit of a quantity is but the term to which this other quantity is supposed to approach continually, until it differs from it as little as we please. This limit is then considered as fixed, and consequently as an "assigned" quantity, whilst the other, being capable of approaching to this limit as much as we please, remains always arbitrary or "unassigned." and cannot enter into the result of the calculation. Hence we may observe, that the expression of limit is neither more nor less difficult to define exactly than that of an infinitely small quantity; and that consequently it is erroneous to think that the method of limits is more rigorous than that of the Infinitesimal Analysis: for, in order to proceed strictly by the method of limits, it is necessary first to define what is a limit. Now the difference between any quantity and its limit is exactly that which we should call an infinitely small quantity: the one is not therefore more difficult to understand than the other; and if the method of limits is exact, as we cannot doubt, there is no reason to doubt whether the Infinitesimal Analysis is so. But the latter has great advantages over the former; because in the method of limits we do not allow ourselves to introduce these semi-arbitrary quantities separately into the calculation; we do not admit even their ratios, but merely the limits of their ratios, which are "unassigned" quantities. Consequently we are deprived of those means of combination and transformation, which procure for the Infinitesimal Analysis the property (which it claims to itself, and that justly) of acting on these auxiliary quantities separately, a property which constitutes one of the principal advantages of its notation. The Infinitesimal Analysis is therefore an improvement on the method of limits; it is a more general application of the latter, and such as nevertheless is neither less exact nor less clear.

In conclusion, it is not in the explanation of the principles that one can show the advantage of the Infinitesimal Analysis over all others. All of them are for the most part equally clear in their principles: but it is not equally easy to apply them to particular questions. The principal difficulty is to put problems into equations, which, on the other hand, in the Infinitesimal Analysis is very easy: the other methods refuse to admit into their Calculus the species of quantities, which we have called infinitely small, and their means of comparison are limited: they are obliged to go round in order to arrive at the same end; and this difficulty is much less observable in algebraic phrases and in operations of any kind whatever, than in propositions or arguments which precede them, or are substituted for these operations. One example will be sufficient to prove the superiority of the Infinitesimal Analysis in this respect. Let us take the enunciation of the famous principle of virtual velocities. The following is given by Lagrange in his "Mecanique Analytique."
If any system whatever of as many bodies or points as we please, acted on by any forces whatever, be in equilibrio, and if we give to this system any small motion whatsoever, in consequence of which each point describes an infinitely small space, which will express its virtual velocity; the sum of the forces, multiplied each by the space which the point to which it is applied, describes in the direction of this same force, will equal zero: taking as positive the small spaces described in the path of the forces, and as negative the spaces described in an opposite direction.
Now how will those who reject the expressions admitted into the Infinitesimal Analysis enunciate this proposition, as clearly as we have just done after the example of the celebrated author of the Mecanique Analytique? But all Mathematics are, properly speaking, merely a course of similar expressions: to abandon them would be to plunge into tedious and intricate difficulties: we should still have to dread least some errors might remain in the result: now everyone allows the method to be infallible in its results. It must be observed, that in mathematical researches we naturally fix our minds on quantities called infinitely small, and not on the limits of their ratios. If the volume of a body having a curved surface be required, we really imagine this body divided into a great number of slices, or even of particles: these slices or these particles are what we in fact consider, and not the different ratios which they may have to one another, and still less the limits of these ratios: the imagination looks for an object of sense: purely algebraic forms offer to it merely what is vague. The division of volume into layers or particles presents a picture to the mind, clears and directs its operations, and facilitates the solution. One of these particles is regarded as the element of the whole quantity, which is in fact supposed to be the sum of all these elements; the differential expression is then sought out, which is to represent this particle, neglecting that which the rules of the Calculus justify omitting. The known formulas of the integral Calculus are in the next place applied to this differential expression, and thus, we are enabled without much trouble to solve a problem, which perhaps would have resisted every effort of the method of exhaustion, or of any other which does not allow of using those means of abbreviation and simplification which are supplied by the Infinitesimal Analysis. It is allowable to consider infinitely small quantities either as real quantities, or as absolutely nothing: but the imagination accommodates itself more readily to the method which looks upon objects as efficient, than to that which supposes them reduced to zero. The law of continuity itself, which alone can determine the value of the fractions 0/0 in each case, whereas without it they would remain indeterminate, oblige us to compare them before they vanish altogether: besides, all these quantities may be eliminated, and that with out assigning to them any determinate value; it is therefore at any rate superfluous to suppose them equal to zero: it is particularising the question when we might dispense with doing so, and consequently it is to solve it in a less elegant way.

The essential, the sublime merit, we might almost say, of the Infinitesimal Analysis, is the union of the facility in common processes of approximation with the exactness of results in common analysis. This immense advantage would be lost, or at any rate diminished, if, instead of this pure and simple method such as Leibniz supplied, we were to substitute for it other less natural means under the appearance of maintaining greater rigour throughout the course of the calculations; means which would be less convenient, less in conformity with the probable progress of the discoverers. If this method is exact in its results, as no one of the present day doubts, if we must always have recourse to it in difficult questions, as has likewise been allowed, why should we return to indirect and complicated methods to supply its place? Why be contented with relying on its deductions and the conformity of its results with those which the other methods furnish, when we can demonstrate it directly and generally, perhaps more readily, than by any of these methods? The objections brought against it all go on this false supposition, that the errors made in the course of the calculation by neglecting infinitely small quantities, remain in the result of the Calculus, however small we may suppose them. Now this is not the case; elimination removes them entirely; and it is singular, that in this indispensable condition of elimination the real character of Infinitesimal Quantities, and a decisive answer to all objections, should not hitherto have been discovered.

Last Updated November 2020