Jean Le Rond D'Alembert, (1717--1783)

This was an era of intellectual ferment in France, known as the Enlightenment. One manifestation was the publication of the Encyclopédie edited by Diderot in 28 Volumes, beginning in 1750. D'Alembert wrote nearly all the articles on mathematics and its applications.

As we have seen, there were extensive developments in the techniques of calculus in the 18th Century, but little progress on the Foundations. Compared to Euclid's Elements which was an ideal model of Mathematics, there was no logical framework for the central concepts. Even the greatest mathematicians such as Newton, Leibniz and Euler made statements that left them open to criticism. Maclaurin's great efforts were not appreciated, probably because most mathematicians wanted to break new ground, not retreat to the methods of the Ancients.

The text: Article on Differentials from the Encyclopédie (1750)
From Struik 'A Sourcebook in Mathematics, 1200--1800'

Leibniz, was embarrassed by the objections he felt to exist against infinitely small quantities, as they appear in the differential calculus; thus he preferred to reduce infinitely small to merely incomparable quantities. This, however, would ruin the geometric exactness of the calculations; is it possible, said Fontenelle, that the authority of the inventor would outweigh the invention itself? Others, like Nieuwentijt, admitted only differentials of the first order and rejected all others of higher order. This is impossible; indeed, considering an infinitely small chord of first order in a circle, the corresponding abcissa or versed sine is infinitely small of second order; and if the chord is of the second.order, the abscissa mentioned will be of the fourth order, etc. This is proved easily by elementary geometry, since the diameter of a circle (taken as a finite quantity) is always to the chord as the chord to the corresponding abscissas. Thus, if one admits the infinitely small of the first order, one must admit all the others, though in the end one can rather easily dispense with all this metaphysics of the infinite in the differential calculus, as we shall see below.

Newton started out from another principle; and one can say that the metaphysics of this great mathematician on the calculus of fluxions is very exact and illuminating, even though he allowed us only an imperfect glimpse of his thoughts.

He never considered the differential calculus as the study of infinitely small quantities, but as the method of first and ultimate ratios, that is to say, the method of finding the limits of ratios. Thus this famous author has never differentiated quantities but only equations; in fact, every equation involves a relation between two variables and the differentiation of equations consists merely in finding the limit of the ratio of the finite differences of the two quantities contained in the equation. Let us illustrate this by an example which will yield the clearest idea as well as the most exact description of the method of the differential calculus.

Let AM (Fig. 1) be an ordinary parabola, the equation of which is yy = ax; here we assume that AP = x and PM = y, and a is a parameter. Let us draw

the tangent MQ to this parabola at the point M. Let us suppose that the problem is solved and let us take an ordinate pm at any finite distance from PM; furthermore, let us draw the line mMR through the points M, m. It is evident, first, that the ratio MP/PQ of the ordinate to the subtangent is greater than the ratio MP/PR or mO/MO, which is equal to it because of the similarity of the triangles MOm, MPR; second, that the closer the point m is to the point M, the closer will be the point R to the point Q, consequently the closer will be the ratio MP/PR or mO/MO to the ratio MP/PQ; finally, that the first of these ratios approaches the second one as closely as we please, since PR may differ as little as we please from PQ. Therefore, the ratio MP/PQ is the limit of the ratio of mO to OM. Thus, if we are able to represent the ratio mO/OM in algebraic form, then we shall have the algebraic expression of the ratio of MP to PQ and consequently the algebraic representation of the ratio of the ordinate to the subtangent, which will enable us to find this subtangent. Let now MO = u, Om = z; we shall have ax = yy, and ax + au = yy + 2yz + zz. Then in view of ax = yy it follows that au = 2yx + zz and z/u = a/(2y + z).

This value a/(2y + z) is, therefore, in general the ratio of mO to OM, wherever one may choose the point m. This ratio is always smaller than a/2y; but the smaller z is, the greater the ratio will be and, since one may choose z as small as one pleases, the ratio a/(2y + z) can be brought as close to the ratio a/2y as we like. Consequently a/2y is the limit of the ratio a/(2y + z), that is to say, of the ratio mO/OM. Hence a/2y is equal to the ratio MP/PQ, which we have found to be also the limit of the ratio of mO to Om since two quantities'that are the limits of the same quantity are necessarily equal to each other. To prove this, let X and Z be the limits of the same quantity Y. Then I say that X = Z; indeed, if they were to have the difference V, let X = Z + V: by hypothesis the quantity Y may approach X as closely as one may wish; that is to say, the difference between Y and X may be as small as one may wish. But, since Z differs from X by the quantity V, it follows that Y cannot approach Z closer than the quantity V and consequently Z would not be the limit of Y, which is contrary to the hypothesis.

From this it follows that MP/PQ is equal to a/2y. Hence PQ = 2yy/a = 2x. Now, according to the methocl of the differential calculus, the ratio of MP to PQ is equal to that of dy to dx; and the equation ax = yy yields a dx = 2y dy and dy/dx = a/2y. So dy/dx is the limit of the ratio of z to u, and this limit is found by making z = 0 in the fraction a/(2y + z).

But, one may say, is it not necessary also to make z = 0 and u = 0 in the fraction z/u = a/(2y + z), which would yield 0/0 = a/2y? What does this mean? My answer is as follows. First, there is no absurdity involved; indeed 0/0 may be equal to any quantity one may wish: thus it may be = a/2y. Secondly, although the limit of the ratio of z to u has been found when z = 0 and u = 0, this limit is in fact not the ratio of z = 0 to u = 0, because the latter one is not clearly defined; one does not know what is the ratio of two quantities that are both zero. This limit is the quantity to which the ratio z/u approaches more and more closely if we suppose z and u to be real and decreasing. Nothing is clearer than this; one may apply this idea to an infinity of other cases.

Following the method of differentiation (which opens the treatise on the quadrature of curves by the great mathematician Newton), instead of the equation ax + au = yy + 2yz + zz we might write ax + aO = yy + 2yO + 00, thus, so to speak, considering z and u equal to zero; this would have yielded 0/0 = a/2y. What we have said above indicates both the advantage and the inconveniences of this notation: the advantage is that z, being equal to 0, disappears without any other assumption from the ratio a/(2y + 0); the inconvenience is that the two terms of the ratio are supposed to be equal to zero, which at first glance does not present a very clear idea.

From all that has been said we see that the method of the differential calculus offers us exactly the same ratio that has been given by the preceding calculation. It will be the same with other more complicated examples. This should be sufficient to give beginners an understanding of the true metaphysics of the differential calculus. Once this is well understood, one will feel that the assumption made concerning infinitely small quantities serves only to abbreviate and simplify the reasoning; but that the differential calculus does not necessarily suppose the existence of those quantities; and that moreover this calculus merely con.sists in algebraically determining the limit of a ratio, for which we already have the expression in terms of lines, and in equating those two expressions. This will provide. us with one of the lines we are looking for. This is perhaps the most precise and neatest possible definition of the differential calculus; but it can be understood only when one is well acquainted with this calculus, because often the true nature of a science can be understood only by those who have studied this science.

In the preceding example the known geometric limit of the ratio of z to u is the ratio of the ordinate to the subtangent; in the differential calculus we look for the algebraic limit of the ratio z to u and we find a/2y. Then, calling s the subtangent, one has y/s = a/2y; hence s = 2yy/a = 2x. This example is sufficient to understand the others. It will, therefore, be sufficient to make oneself familiar with the previous example concerning the tangents of the parabola, and, since the whole differential calculus can be reduced to the problem of the tangents, it follows that one could always apply the preceding principles to various problems of this calculus, for instance to find maxima and minima, points of inflection, cusps, etc....

What does it mean, in fact, to find a maximum or a minimum? It consists, it is said, in setting the difference dy equal to zero or to infinity; but it is more precise to say that it means to look for the quantity dy/dx which expresses the limit of the ratio of finite dy to finite dx, and to make this quantity zero or infinite. In this way all the mystery is explained; it is not dy that one makes equal to infinity: that would be absurd, since dy is taken as infinitely small and hence cannot be infinite; it is dy/dx: that is to say, one looks for the value of x that renders the limit of the ratio of finite dy to finite dz infinite.

In Limits he said

One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude however small, although the first magnitude may never exceed the magnitude it approaches.

Note that this definition is geometric rather than algebraic and was not followed up by others in the eighteenth century, who continued to work with fluxions and infinitesimals.

Some important features of the extract are: