Differential calculus, the manner of differentiating quantities, that is to say, of finding the infinitely small difference from a variable finite quantity.

It is one of the most beautiful and fecund methods in all Mathematics. Monsieur Leibniz, who was the first to publish it, calls it differential calculus , considering infinitely small magnitudes as differences between finite quantities. For this reason, he expresses them by the letter d, which he puts before the differentiated quantity. So the differential of x is expressed by dx , that of y by dy , etc.

Monsieur Newton calls the differential calculus, fluxion method , because he utilizes, we say, infinitely small quantities for fluxions, or momentary growths. He considers, for example, a line engendered by a fluxion of a point, a surface by the fluxion of a line, a solid by the fluxion of a surface. And in place of the letter d , he marks fluxions by a point placed above the differentiated magnitude. For example, for the fluxion of x , he writes
; for that of y ,
, etc., providing the sole difference between the differential calculus and the fluxion method. [1] See Fluxion.

All the rules of the differential calculus may be reduced to the following:

1. The difference of the sum of several quantities is equal to the sum of their differences. So d ( x + y + z )= dx + dy + dz .
2. The difference xy is ydx + xdy .
3. The difference x ^m , where m is positive and whole, is mx ^{m –1} dx .

There is no quantity that cannot be differentiated by these three rules. One writes, for example,
. See Exponent. The difference is therefore (rule 2) y ^–1 Xdx + xXd ( y ^–1 ) = ( rule 3. )
. The differential of
is
. Since
would be = x , we have z = x ^q  [2]and dz = qx ^{( q –1)} dx and
 [3] . Even
. So the difference is
, and so on.

The three rules above are demonstrated in a very simple manner in an infinity of works ( ouvrages ), and most of all in the first analysis section of Infiniment petits of Monsieur de l'Hopital [4], to which we presently refer. This section lacks a differential calculus of logarithmic and exponential quantities, which are seen in I. Volume des oeuvres of Jean Bernoulli, and in I. partie du traité du calcul intégral of Monsieur de Bougainville the younger [5]. We may consult these works ( ouvrages ), now in the hands of the entire world. See Exponential. The metaphysics of the differential calculus is the most important to treat here.

This metaphysics, of which so much has been written, is even more important, and perhaps as difficult to develop as these same rules of the calculus. Several geometers, among them, Monsieur Rolle, unable to admit the supposition that there are infinitely small magnitudes, rejected it completely, and claimed that this principle was faulty and capable of inducing error [6]. Yet when we notice that all truths discovered through the assistance of ordinary Geometry are discovered similarly and much more easily with the differential calculus [7], it is impossible to not conclude that this calculus furnishes certain( sûres) , simple and exact methods, with principles that must also be simple and certain ( certains ).

Monsieur Leibniz complicated the objections that he felt might be made against infinitely small quantities, as considered by the differential calculus, having preferred to reduce his infinitely small to the not-comparable, which ruined the geometric exactitude of the calculus. And what weight, according to Monsieur Fontenelle, must not the authority of the inventor have against the invention? Others, namely Monsieur Nieuwentyt [8], admit differentials of the first order alone, and reject all those of higher order – without foundation. Since imagining in a circle an infinitely small cord of the first order, an abscissa or sinus, and the corresponding line of the second, infinitely small; if the cord of the second is infinitely small, the abscissa of the fourth is infinitely small, etc. This is easily demonstrated by elementary Geometry, since the diameter of a finite circle is always in relation to the chord as the chord is to the corresponding abscissa. So if we admit, one time, the infinitely small of the first order, all others become necessary. Our remarks here are only to show that, in admitting the infinitely small of the first order, we must admit all others to infinity, because one can after all, easily pass from the metaphysics of infinity into the differential calculus, as we demonstrate further below [9].

Monsieur Newton subscribes to another principle, and we can say that the metaphysics of this great geometer on the calculus of fluxions is very exact and very enlightening ( lumineuse ), though he'd be satisfied to give us just a glimpse.

He has never looked at the differential calculus as the calculus of infinitely small quantities, but as the method of first and second reasons [10], that is to say, the method of finding the limits of relations. So, this illustrious author has never differentiated using quantities, only equations, because all equations contain a relation between two variables, and the differentiation of equations only consists in finding the limits of relation between finite differences of two variables that the equation contains [11]. This is what we must clarify with an example that will simultaneously give us the clearest idea and the most exact demonstration of the method of the differential calculus.

Let A M ( figure 3. analys. ) be an ordinary parabola, of which the equation, in labeling A P, x , P M, y, and a, the parameter [12] , is yy = ax [13] . Let us propose to draw a tangent M Q from this parabola at point M. Suppose that the problem is solved, and imagine an ordinate p m at any finite distance of P M; and for the points M, m, we draw the line m M R . It is evident that: 1. the relation
, from the ordinate to the sub tangent, is greater than the relation
or
, which is equal to it due to similar triangles, M O m, M P R [14]: 2. the closer the point m is to point M , the closer the point R will be to point Q , and consequently, the relation
or
will approach the relation
, and that the first of the equations could approach the second, as closely as desired, since P R can differ as little from PQ as desired. Therefore, the relation
is the limit of the relation of m O to O M [15] . Thus, if the limit of the relation between m O to O M is found, expressed algebraically, we obtain the algebraic expression of the relation of MP to PQ and, consequently, the algebraic expression of the ordinate relation at the subtangent, where this sub tangent will be found [16]. Letting therefore MO = u , Om = z , gives us ax = yy , and ax + au = yy + 2yz + zz [17] . So, because ax = yy , it yields au = 2yz + zz and
 [18] .

figure 3. analys.

So
is in general the relation of m O to O M, at any part for which we take point m . This relation is always smaller than
, but the smaller z , the more this relation will be augmented. And, as we can extend z as little as is desired, we can approach the relation
as close as we want to the relation
. Thus,
is the limit of the relation of
, which is to say of the relation
. So,
is equal to
, which we have found also to be the limit of the relation of m O to OM , because two magnitudes which make the limit of the same magnitude, are necessarily equal between them [19]. To prove this, given Z and X , limits of the same quantity Y, I say that X = Z, because if there is some difference between them, such as V, it would be X = Z ± V. By hypothesis, the quantity Y can approach Z as closely as we desire. That is to say that the difference between Y and X can be as small as wished. Therefore, since Z differs from X by the quantity V , it follows that Y cannot approach Z any closer than the quantity V, and consequently, that of Z is not the limit of Y , which is contrary to the hypothesis [20]. See. Limit , Exhaustion.

Thus, the result is that
is equal to
. Therefore,
. Now, following the method of the differential calculus , the relation of M P to P Q is equal to that of dy to dx , and the equation ax = yy gives adx = 2ydy and
. [21] And so
is the limit of the relation of z to u , and this limit is found in making z = 0 in the fraction
. But should we say it is necessary that z = 0 and u = 0 also, in the fraction
, and then that we shall have
? What does that mean? [22] I say 1. There is no absurdity in this, because
can be equal to anything desired: it therefore can be
. [23] I say, 2. because although the limit of the relation of z to u is found when z = 0 and u = 0, this limit is not properly the relation of z = 0 to u = 0 , because that does not present a clear idea; it is no longer known that it is a relation of two empty terms. This limit is the quantity of which the relation
keeps approaching in supposing z and u both real and decreasing, and of which this relation approaches almost anything desired. [24] Nothing is clearer than this idea; it can be applied to an infinity of other cases. See Limit, Series, Progression, etc.

Following the differentiation method, which begins the tract on the quadrature of curves by Monsieur Newton, this great geometer, in place of the equation ax + au = yy + 2yz + zz , would have written ax + a 0′ = yy + 2y0 + 00, considering, some way, z and u as zeros, which would give him
. [25] One must feel, from all this, the impression of what we said above of the advantages and disadvantages of this denomination: the advantage of z being = 0 disappears without any other supposition of the relation
; it is a disadvantage in that these two terms are supposed to be zeros, which does not, at first glance, present a very clear idea [26].

One gets the impression from all this, that we intend to say that the method of the differential calculus gives us exactly the same relation that comes from that given by the preceding calculus. Other examples become even more complicated. The latter appears to us to suffice to make the true metaphysics of the differential calculus initially understood. When understood well once, the supposition that one has made of infinitely small quantities will be felt to be only for abridging and simplifying reasoning. However, as the foundation of the differential calculus does not necessarily presume the existence of these quantities, as the calculus only consists in algebraically determining the limit of a relation that has already been expressed in lines, and in equaling these two limits, which allows us to find one of the lines for which we search [27]. This definition is, perhaps, the most precise and clearest that can be given for the differential calculus. Still, it cannot be very well understood when the calculus has been made familiar, because often the true definition of a science can only be very sensible to those who have studied science. See Preliminary Discourse, p. xxxvii. [28]

In the preceding example, the known geometric limit of the relation of z to u is the relation of the ordinate to the subtangent. With the differential calculus is sought the algebraic limit of the relation of z to u , and
is found. Thus naming s the subtangent, we have
. So,
. [29] This example is sufficient to understand the others. It thus will suffice to make more familiar in the example above, the tangents of the parabola, and as the entire differential calculus can be reduced to a problem of tangents, it follows that the preceding principles may always be applied to different problems that are resolved by the calculus, as the invention of maxima and minima , points of inflection and folding ( rebroussement), Etc. See these words .

How, in effect, can one find the maximum or minimum ? It is, evidently, to give the difference of dy equal to zero or to infinity, but to speak more exactly, it is to search for the quantity
, which expresses the limit of the relation of a finite dy to a finite dx , and then makes this quantity either nothing or infinite. And there, the mystery is completely explained. It's not dy that is made = to infinity; that would be absurd, because dy being taken for infinitely small, cannot be infinite. It's
: that's to say that we look for the value of x which makes the limit of the relation of finite dy to finite dx infinite. [30]

We've seen above that there is no clean point of infinitely small quantities of the first order within the differential calculus, that the quantities that one therefore names are supposed to be divided by other quantities which are supposed to be infinitely small, and that in this state, they mark, not infinitely small quantities, nor even fractions, of which the numerator and the denominator are infinitely small, but the limit of a relation between two finite quantities. There are even second differences, and others of a more elevated order. In geometry, there is no true point d d y , but when d d y is encountered in an equation, it is supposed to be divided by a quantity, dx ² , or another of the same order. In this state, what is
? It is the limit of the relation,
, divided by d x , or what will be still clearer, it is, in making
a finite quantity, the limit
.

The differentio-differential calculus is the method of differentiating differential magnitudes, and the differentio-differential quantity is called the differential of a differential .

As the letter d denotes a differential , that of the differential of dx is ddx , and the differential of ddx is dddx , or d ² x, d ³ x , Etc., or
.. , or
..., Etc. in place of ddy , d ³ x , etc.

The differential of a finite ordinary quantity is called a differential of the first degree or of the first order, like dx .

A differential of the second degree or the second order, that one also calls, as we shall come to see, differentio-differential quantity, is the infinitely small part of a differential quantity of the first degree, like ddx , dxdx , or dx ² , dxdy , etc.

A differential of the third degree is an infinitely small part of a differential quantity of the second degree, like dddx , dx ³ , dxdydz , and so on.

Differentials of the first order are nevertheless called, first differences ; those of the second, second differences ; those of the third, third differences .

The second power of dx ² of a differential of the first order is an infinitely small quantity of the second order. This is because dx ² : dx :: dx .1; so dx ² is supposed to be infinitely small in relation to dx. [31] It will even be found that dx ³ or dx ² dy , is an infinitely small of the first order, Etc. We speak here of infinitely small quantities, and we've spoken earlier in this article to conform to ordinary language; because by what we've already said of metaphysics of the differential calculus, and by what we have still to say, it will be seen that this way of speaking is only an expression abridged and obscure in appearance, corresponding to something very clear and very simple.

Differential powers, like dx ² , are differentiated in the same way as the powers of ordinary quantities. And as composite differentials are each multiplied or divided, or are powers of first-degree differentials , these differentials are differentiated even as ordinary magnitudes. Thus, the difference of dx ^m is m ( dx ) ^{m –1} ddx , and so on. For this reason, the differentio-differential calculus is basically the same as the differential calculus.

A well-known author of our day remarks in the preface of a work on Geometrie de l'Infini , that there will be found no geometer who could have precisely explained what the difference of d y is , having become equal to infinity within certain points of inflection. Nothing, however, is simpler: at the point of inflection, the quantity
is a maximum or a minimum. So the difference divided by dx is = 0 or = to infinity. Thus, in taking dx as constant, we have the quantity
= to zero or to infinity; this quantity is not an infinitely small quantity, it is a quantity which is necessarily either finite, or infinite, or zero, because the numerator ddy which is infinitely small, of the second order, is divided by dx ² , which is also of the second order. To abbreviate, we say that d d y is = to infinity, but ddy is supposed to be multiplied by the quantity
, which makes all mystery disappear. In general ddy = to infinity signifies no other thing than
= to infinity, or in this equation where no differential enters. For example, let
. We shall have
and
= to infinity something other than
= to infinity. That is to say
= to infinity, which occurs when x = a. We see that the differential does not enter into the quantity 20 / (a - x) ⁶ , which represents ddy / dx ² or the limit of the limit of
.We eliminate the dx ² to abbreviate, but it is not really supposed to exist. It is then that we are often ready, in the Sciences, for ways of abbreviated speaking, which can induce an error when we do not understand the true sense. [32] See Elements.

Given all that we have said, 1., that in the differential calculus, neglected quantities are neglected, not as we ordinarily say, because they are infinitely small in relation to those left remaining, which only produces an infinitely small or empty error, but because they must be neglected for rigorous exactitude. It is seen above, in effect that
is the true and exact value of
. So, in differentiating ax = yy , it is 2y dy , and not 2ydy + dy ² that we must take for the differential of y ² , [33] finally, yielding, as it ought,
. 2. It is not a matter, as we say ordinarily, of infinitely small quantities in the differential calculus, but, uniquely, a matter of the limits of finite quantities. And so the metaphysics of infinity and infinitely small quantities each larger or smaller, is totally useless to the differential calculus [34]. The term infinitely small only makes us ready to abbreviate its expression. We therefore say, with most geometers, that one quantity is infinitely small, not before it vanishes, not after it vanishes, but at the very instant it vanishes. Since, who wants to give a definition so false, and a hundred times more obscure than what we wish to define? We say that there are no infinitely small quantities in the differential calculus. Nevertheless, we speak at length of the metaphysics of quantities in the article Infinity. Those who will read with attention what we shall say, and who shall join the use of calculus and reflections, will have no more difficulty in any case, and will easily find responses to the objections of Rolle and other adversaries of the differential calculus, if any remain. We must grant that, if the calculus has had enemies at its birth, it is the fault of geometers, these partisans, some of whom have understood it poorly; others who've not explained it enough. But its inventors sought to put as much mystery as they could into their discoveries, and in general, men do not hate obscurity, provided that it results in something marvelous. What charlatanism all this is! Truth is simple, and can always be carried to everyone [35], when the care is taken.

We shall address here the subject of differential quantities of the second order, and others of higher orders, a remark which will be very useful to beginners. In the mem. de l'acad. des Sciences de 1711 , and in the 1st tome de oeuvres de Monsieur Jean Bernoulli, are found a memoir in which it is remarked, with reason, that Newton is deceived, when he believes the second difference of z ⁿ , in supposing dz constant, is
in place of n .( n –1) z ^{n –2} dz ² , as it does result from the rules enunciated above, and conforms to ordinary principles of the differential calculus. Against this, we must take guard, and this gives us yet the occasion to insist upon the differences between polygonal curves, and rigorous curves, of which we have already spoken in the articles Central and Curve . Let there be, for example, y = x ² , the equation of a parabola. Suppose d x constant, that is to say, all d x 's equal. It will be found that x + dx gives for the corresponding exact ordinate that I call y ′, x ² + 2xdx + dx ² , and that x + 2dx gives the corresponding ordinate that I call y ″, exactly equal to x ² + 4xdx + 4dx ² . Thus 2xdx + dx ² is the excess of the second ordinate upon the first, and 2xdx ² + 3dx ² is the excess of the third on the second. The difference of these two excesses is 2dx ² ,and it is the ddy , such as the differential calculus gives. Now if by the extremity of the second ordinate, the tangent which comes to slice through the third ordinate is taken, we find that this tangent would divide the ddy in two equal parts, of which each would be, consequently dx ² or
. [36] It is this half of true ddy that Monsieur Newton has taken for a completely true ddy , and here is what occasions this mistake: the veritable ddy is found by the means of the tangent considered as the secant in a rigorous curve, because, in making dx constant, and seeing the curve as a polygon, the ddy will be given by prolonging one side of the curve, up until this side encounters an infinitely close ordinate, just as prolonged. Now the rigorous tangent in the rigorous curve being similarly prolonged, gives this half of ddy , and Monsieur Newton has believed that this half of ddy would express the true ddy , because it was formed by the subtangent. Therefore, he has confounded the polygonal curve with a rigorous one. A very simple shape will be easily understood, to those who engage in few exercises of the geometry of curves and the differential calculus. See Polygonal curve in the word Curve, l'Histoire de l'académie des Sciences de 1722 [37], and my Traité de Dynamique [38], I. partie, in the article on central forces .

Notes

1. Two ideological strands come together in d'Alembert's commencement of “Différentiel”. First, as a mathematician, reducing the calculus to three rules and reducing the distinction between Newtonian and Leibnizian methods to one mark (dot or “d”) he foregrounds the calculus as a mathematical artifact deductively prior to the history of its development. He not only compresses the whole trajectory of the development of the calculus to a metaphorical point that may easily be re-appropriated into science, but prepares the reader to move from this focus of compression back to its conditions of genesis and historical development as his article proceeds foreword. Thus secondly, in this presentation of a “bottom line” as a punch line of 18th Century info-speak, d'Alembert demonstrates a possible aspect of the encyclopedic program, which is to provide information in a scientific manner and to plant subversive material in the inductive nooks and crannies of an ostentatiously, deductively presented argument. In this same moment of inventing specialist info-speak of 18th Century science, d'Alembert makes a major contribution to 18th century collective notions of the symbolic, announcing one direction while not neglecting to propose many only seemingly subordinate others.

2.
is the qth root of z. So if the qth root of z = x, then z is in turn equal to x ^q

3. If one divides both sides of dz = qx ^{q –1} dx by qx ^{q –1} , one gets
. Leaving off q as a coefficient under dz and converting
to a whole number and exponent by multiplying the exponent by -1, we get x ^{– q +1} , which then can be submitted to the calculus:

4. Analyse Des Infiniment Petits, Pour L'intelligence Des Lignes Courbes . [by G. F. A. De L'hôpital, Marquis De Sainte Mesme.]. Paris: Imprimerie Royale, 1696.

5. Bougainville, Louis-Antoine de. Traité Du Calcul Intégral : Pour Servir De Suite À L'analyse Des Infiniment-Petits De M. Le Marquis De L'hôpital. Paris: Desaint & Sulliant, 1754.

6. Rolle, Michel. Remarques de M. Rolle de l'Academie Royale des Sciences Touchant le problesme general des tangents . Paris: Jean Boudot, 1703. On page 24, Rolle claims that the first derivative of an equation set equal to 0 makes no tentative plot points known in terms of the analysis of Infiniment petits , art. 49, p. 43, and the second derivative gives only imaginary point values for y. At this juncture, Rolle argues any subsequent derivative will cause the calculus to lose its elegance, since not all substitutions produce the same solutions to their equations. Rolle is particularly uncomfortable with the shared identity of zero and infinity in the theory of the infinitely small, for he writes on page 25: “Ainsi le même dy ou Rm seroit egal à 0 & à l'Infini, toutes choses d'ailleurs étant les mêmes. Car le changement des expressions dans l'égalité proposée ne change rien dans l'état de la Question, ni dans la valeur de Rm . Ce qui marqueroit la plus énorme de toutes les contradictions...” [So the same dy or Rm would be equal to 0 and to infinity, all things the same - because changing the expressions in equality can offer no change in the state of the Question, nor in the value of Rm. It is this which marks the most enormous of all contradictions...].

7. Mechanical inventions attempt to ease repetitive tasks, but the label “easy” applied to mechanical arts is problematic for Denis Diderot (co-founder with d'Alembert of the Encyclopédie), if not for all the encyclopedists. For Diderot, mechanical arts compete on the level of intellectual invention, and the voluminous engravings of the Encyclopédie depicting trades, crafts, and machines attest to this valuation. Thus the differential calculus, represented here as “easier” pushes high geometry to the level of mechanical trade, while at the same time it prioritizes its mechanics for this same higher mathematics, although in greater degrees.

8. Nieuwentyt, Bernard. Bernhardi Nieuwentiit Considerationes Circa Analyseos Ad Quantitates Infinitè Parvas Applicatæ Principia & Calculi Differentialis Usum in Resolvendis Problematibus Geometricis . Amstelædami: J. Wolters, 1694.

9. “So if we admit, one time, the infinitely small of the first order...”. While the abstractions of mathematics are said by Diderot in Pensées sur l'interpr é tation de la nature ( DPV, IX. Paris: Hermann, 1987) to be far from experimental sense data, with a metaphysics of infinity, the realization that the space between the chord of a circle and a segment of its attached arc can be divided infinitely, returns one to corpuscular philosophy and to the modern debate over whether matter is infinitely divisible. This metaphysics arising out of the infinitesimal being matched to the physical world prepares the non-Euclidian features of Leibnizian topology and its recursive folds to become a tool of philosophy. Here the symbols of mathematics are made concrete and therefore become usable by the empiricist.

10. Contrast “reasons” and “differences”, the former seemingly tied to mechanical operations with parts of equations that establish relations as if they were levers and pulleys, and the latter finding a home even today in contemporary notions of informational signs. It is fitting that d'Alembert should commend Newton and Reason, as his character in Diderot's Rêve de d'Alembert ( DPV, XVII. Paris: Hermann, 1987) embodies a mathematical rationalism from which the unconscious secret of sensibilité escapes during the night. From the point of view of Diderot, d'Alembert's dream makes a case for Western science to “swerve” away from systems building and Newton, but to d'Alembert, Newtonian rationalism guides us on the way to living and thinking within the “clear and distinct” - even if d'Alembert co-founded the encyclopédie, a project for which contemporary distinctions between “reasons” and “differences” were less pronounced.

11. As d'Alembert has signed the Extrait des registres de l’académie royale des sciences in the front of M. de Bougainville's Traité Du Calcul Intégral : Pour Servir De Suite À L'analyse Des Infiniment-Petits De M. Le Marquis De L'hôpital . ( Paris : Desaint & Sulliant, 1754 ) , and as the current Encyclopédie entry precedes this text of Bougainville by three years, it is curious that Bougainville's preface should contain the following passage in regard to Newton on page viij: “Le Calcul infinitésimal de Newton est indépendant de la réalité des quantités infiniment petites; réalité que l'Auteur n'admet nulle part comme un principe nécessaire. La supposition qu'il en fait, n'est qu'une hypothèse momentanée pour abréger le procédé & le rendre plus simple. Il ne fait autre chose qu’appliquer le calcul à la méthode d'exhaustion des Anciens, c'est-à-dire, à la méthode de trouver les limites des rapports. Aussi ce grand Philosopher ne différentie-t-il jamais des quantités, mais des équations; parce que toute équation exprime un rapport entre deux indéterminées; & qu'ainsi différentier un équation, c'est trouver les limites du rapport entre les différences finies des deux indéterminées renfermées dans l'equation. [The infinitesimal calculus of Newton is independent of the reality of infinitely small quantities, a reality for which the author admits no part as a necessary principle. The supposition he has made, is only that of a momentary hypothesis for abbreviating the process and rendering it simpler. He has done no other thing than to apply the calculus to the method of exhaustion of the ancients, that is to say, to the method of finding the limits of relations. Also, this great Philosopher, he never differentiates quantities, but equations, since every equation is expressed as a relation between two indeterminates, and so to differentiate an equation is to find the limits of the relation between the finite differences of the two indeterminates contained in the equation.]

12. “Parametre” is used here to denote a fixed limit or line that can be added to an equation of a curve to determine its shape. It is particular to similar equations for the geometrical figure D'Alembert discusses (figure 3, analys.), though its meaning has a more general mathematical or technical sense of a measurable value that establishes conditions in a system and interacts with other parameters in that system.

13. The area (y times y) is equal to the area (a [an arc bridging two sides of the triangle] times x [the shortest side of the triangle]).

14. This is because MP is less than both PQ and PR, and so if we substitute 1 for MP and set it as a numerator over a denominator of either PQ or PR, the value of the fraction with PR as the denominator will be less than that of PQ.

15. A way to conceive of this relation is as two angles having an equal slope in their current position. It holds that if m gets closer to M, then logically, R will similarly get closer to Q, changing its slope but keeping it equally to a similarly changed slope of mM.

16. So if the slope of
is known, then the relation between Om as an ordinate and mM will hold for
.

17. That is y ² + 2yz + z ² , in contemporary notation.

18. Because 2yz +zz (or, that is, 2yz + z ² ) can be factored to z(2y + z). Then both sides of the equation can be divided by 2y + z, equaling
. Then dividing both sides by u equals
.

19. Since the slope relation is equal to ax + au, which in turn is equal to y ² + 2yz + z ² , , then the latter, under the form of
, is equal to both instances of the slope relation, or
, that is, both
and
.

20. If X=Z and X also equals Z ± V , then substituting X for Z in X = Z ± V equals X = X ± V . But then there is a seeming contradiction because Y has the same limit (X and Z), and X = X ± V is not equal to Z = X ± V , that is, unless Z and V are the same distance from Y. This proposition is only possible if, despite a different actual value, V is infinitely small and thus Z = V because V= the infinitely small.

21. The differential of yy is 2y (or 2ydy). Solving for dy,
, so subsequently dividing each side of the equation by dx, we have
.

22. Qualitatively, perhaps something like .01 X 1 ^-∞ , or 1 ^-∞ .

23. The distinction computer science makes between a variable and a literal may be useful here, as
is a container for the literal value of
.

24. The relation of
is infinitely small, thus supposedly virtually equal to 0. This is not to say that 0 cannot be a generic limit with an infinitely small “deviation”. In this sense, a theoretically large number is the “zero” of any one of its infinitely small differences.

25. Newton's equation is found by replacing every u and z in the first equation with 0.

26. Not clear in an abstract mathematical sense but clear if one treats 0 as signifying linguistically, the infinitely small. Paradoxically, on page 205 of Newton's Account of the Commercium Epistolicum , Newton's use of the letter o is in this sense of linguistic signification of infinity: "And whereas it has been represented that the use of the letter o is vulgar... one infinitely little Quantity represented by a symbol, the symbol O... Account of the Commercium Epistolicum , as an appendix of, Hall, A. Rupert. Philosophers at War: The Quarrel Between Newton and Leibniz . Cambridge: Cambridge University Press, 1980.

27. The calculus produces a ratio, like a slope that transparently produces a line in Cartesian space. Appearing distinct from the general project of a “reasoned dictionary”, this definition highlights d'Alembert's situation within a type of proto-positivism from which all things slightly messy (i.e. metaphysics) are discarded.

28. While there seems to be no place for metaphoric knowledge with this statement, and moreover, because any truth which lies beyond metaphor can thus only be found by natural philosophers/scientists who must be forerunners to contemporary scientists who work within disciplines that are highly specialized and incommensurable, d'Alembert's statement partially seems to flow from a sense of loss stemming from an embedded desire that this definition would be simple enough to be understood by anyone (see also note 35.)

29. The distance of y divided by the distance of s when s is positioned as a subtangent is equivalent to the distance of a divided by y doubled.

30. Such a moment might inaugurate a movement of western science away from modeling only what is discrete to modeling what is continuous and from an attempt at absolute measurement to relativized or relational measurement. Infinity is found not in the farthest reaches of the universe, but in a minute local relation between two very finite numbers, and the fact that any integer can have such a relation to another defines a tension between the continuous and discrete.

31. A limit times a limit results in an even a smaller value (given that “limits” are multiplied together in the same way as fractions even if they are not fractions). So since dy has to be fractional/limited, since it cannot be greater than 1 and still be infinitely small, its multiplication by dx in turn multiplied by dx is like dx times dx times dx, since both dy and dx are congruent under a metaphysics of infinity.

32. D'Alembert devalues (an unspoken) sense of metaphor because it leads to error and seems glued to the philosophical sense of the so-called “bad” effects of abbreviations. Contrast this statement to what follows in the next paragraph in which the data of the differential must be redacted to get the “correct” results.

33. In cultural terms, difference (or différance) must be omitted in recourse to some higher principal of neatness, clarity, or order, in this claim of d'Alembert for “rigorous exactitude”. Infinity as order is rather the same as disorder, although too, making all difference representable by zero/infinity seems to be porous enough to give Rolle difficulty indeed.

34. For there are no infinitely small quantities (what would be either the numerator or denominator in the dx/dy relation), but only infinitely small relations, that is dx/dy as a whole.

35. Contrast this statement to the earlier claim about the truth of the calculus being lost when it has become familiar. Does this mean that the techniques of science are then simple and the common human may then learn to do science, or does d'Alembert's claim now seem to argue that truth relies on metaphor because truth is found in simplicity?

36. “Thus 2xdx + dx ² is the excess of the second ordinate upon the first, and 2xdx ² + 3dx ² is the excess of the third on the second... the tangent which comes to slice through the third ordinate is taken, we find that this tangent would be divided by the d d y in two equal parts....” D'Alembert's language interestingly gives an image of contemporary avant-garde musical composition akin to the American composer John Cage, in which not only are lines on a graph the incisors of parabolae and other curves, but these lines may be carried to the two-dimensional plane of a page of the mathematician's notebook, acting as sword-like operators chopping off “excess” terms of an equation.

37. Académie des Sciences. Histoire De L'académie Royale Des Sciences : Avec Les Mémoires De Mathématique Et De Physique ... Tirez Des Registres De Cette Académie. 1699-1704. 2.Éd., Rev. Et Augm. 1718-1722 . Paris, 1722.

38. Alembert, Jean le Rond d, and aîné David l. Traité De Dynamique Dans Lequel Les Loix De L'equilibre & Du Mouvement Des Corps Sont Réduites Au Plus Petit Nombre Possible, & Démontrées D'une Maniére Nouvelle, & Où L'on Donne Un Principe Général Pour Trouver Le Mouvement De Plusieurs Corps Qui Agissent Les Uns Sur Les Autres, D'une Maniére Quelconque . A Paris, chez David l'aîné, Libraire, rue Saint Jacques, à la Plume d'or. MDCCXLIII. Avec approbation et privilege du Roi, 1743.