Differential equation is that which contains differential quantities. The quantities are called of the first order if the differentials are of the first order; the second, if they are of the second, etc.

Differential equations with two variables belong to mechanical curves, which is how they differ from geometric curves. Their construction will be found in the word Curve. However, this construction supposes that the indeterminate would be separated, and this is the object of the integral calculus. See Integral.

In differential equations of the second order, in which d x , for example, is supposed constant, if it is desired that it would no longer be constant, one has only to divide the whole by d x , and then, in place of
, place d
or
, and we have an equation in which nothing will be constant. This rule is explained in several works, and above all in the second partie du traite du calcul integral of Monsieur de Bougainville, which will appear soon. In the meantime, we can have recourse to the oeuvres of Jean Bernoulli, v. IV. page 77, and can note that
, in supposing dx constant, is the same as
, in supposing d x constant. Now
is the same, whether we take d x as constant, or if we make it variable. Because, remaining the same,
does not change, provided that d x is infinitely small. To get a good sense of this, one has only to suppose dy = zdx or
, we shall have d z in place of
in the equation. Now, this d z is the same thing as
, without supposing anything to be constant. And so on, etc.

I must also speak of the differentiation of quantities under the sign
. For example it has been proposed to differentiate
in making only y vary, A being a function of x and of y : this difference is
being the coefficient of d y in the differential of A . This method will be found explained in the mem. de l'acad. de 1740, page 296 , after a memoir of Monsieur Nicolas Bernoulli, and this method will be detailed in the work of Monsieur de Bougainville. I lightly treat these objects, treated elsewhere, coming to the question of the inventor of the differential calculus.

The fact that Leibniz published it first remains constant. It appears that today we generally concede what Newton discovered to support it. It remains to be known if Leibniz has taken from Newton. The pieces of this great process are found in the commercium epistolicum de analysi promota, 1712, Londini . We relate Newton's letter of the 10th of December 1672, which Leibniz is said to have known, and which includes the means of finding curve tangents. But this method, in the cited letter, is only applied to curves of which the equations have no radicals; it does not contain the differential calculus, it is nothing other than a slightly simplified method of Barrow, for tangents. Newton speaks to the truth in this letter, that by his (Barrow's) method are found the tangents of all types of curves, geometric, mechanic, whether they should be radicals, or should they not be in the equation. [1] Still, he is content to say so. [2] Therefore, when Leibniz would have seen this 1672 letter, he would not have taken the differential calculus from Newton, he would have taken all this and more from Barrow, and in this case it would be neither Newton, nor Leibniz, but Barrow who would have discovered the differential calculus. Ultimately, we say in passing, the differential calculus is nothing other than the method of Barrow for tangents, generalized. See this method of Barrow for tangents explained in his lectiones geometricae [3] , and at the end of V. livre des sections coniques de Monsieur de L'Hopital [4], and you shall be convinced of what we advance here. It is only made general by applying it to curves of which the equations are radicals, and for this it suffices to note that mx ^{m –1} dx is the differential of x ^m , not only when m is a completely positive number (as in the case of Barrow), but yet when m is a completely unspecified number, either rounded, positive, or negative. This step was apparently easy, and it was however necessary to do to discover the whole of the differential calculus. So regardless, the invention of the differential calculus only extends and finishes what Barrow had almost done, and the calculus of exponents, discovered by Descartes, made this easy enough to perfect. See Exponent. It is thus often that the most considerable discoveries, prepared by labor of preceding centuries, do not depend on more than a very simple idea. See Discovery.

This generalization of the method of Barrow, which appropriately contains the differential calculus, or (which has the same result) the method of tangents in general, is found within a letter of Leibniz from the 21st of June 1677, related in the same collection, p. 90. It is from this letter that we must date [5], and not from the Leipzig Acta of 1684, when Leibniz had published the first rules of the differential calculus, that he evidently knew seven years before, as we see in the cited letter. Let's go over the other facts against Leibniz.

By Newton's letter of the 13th of June, 1676, p. 49 of this collection, we see that this great geometer has imagined one method of series, which led him to the differential and integral calculus. [6] Newton, however, does not explain how this method led him there, he is content to give examples [7], and besides, the inspectors of the Royal Society did not say if Leibniz had seen this letter, or to speak more exactly, did not say that he had seen it: a remarkable and important observation, as we shall now see. It is only mentioned in the congress ( rapport ) of inspectors that the letter of Newton, of 1672, had been seen by Leibniz, and they concluded nothing against him, as we have proved. See p. 121 of this collection, the congress of inspectors named to the Royal Society, art. II. and III. It seems however, by the title of the Newton's 1676 letter, printed on page 49 of this collection, that Leibniz had seen this letter before his own letter of 1677 [8]; but this letter of 1676 principally treats series, and the differential calculus is only found in a very distant sense, less-understood, and presupposed [9]. It is apparently for this reason that the inspectors do not mention it. Since, by the Leibniz's next letter, on page 58 , it appears that he had seen Newton's 1676 letter, and another of October 24th of the same year, which rests upon the same method of series. Nor is much more said, and still less done about whether Leibniz had seen another writing of Newton from 1669, which contains, a little more clearly, but always implicitly, the differential calculus, and which is found at the beginning of the same collection.

For this reason, if the glory of the invention is to be refused Newton, there is no longer sufficient proof for taking it away from Leibniz. If Leibniz had not seen the writings of 1669 and 1676, he is absolutely the inventor. If he had seen them, he could yet pass for the inventor, at least in the tacit view of the inspectors, since these writings do not clearly enough contain the differential calculus, or the inspectors would have reproached him after reading them. We must grant however that these two writings, most of all, that of 1669 - if he had indeed read it - could have given him ideas [10] ( see page 19 of the collection ), but the merit of having had, of having developed, and of having drawn the general method of differentiating all sorts of quantities will remain with him. One might object, in vain, that the metaphysics of the differential calculus of Leibniz was not so good, as we've seen above. That may be; still, this does not establish any fact against him. He could have discovered the calculus that concerns us here in considering differential quantities as actually ( réellement) infinitely small, just as many other geometers have done. He could then, shocked by the objections, have tottered on this metaphysics. Finally, one may object that this method must have been more fertile in his hands, than in the hands of Newton. This objection is, perhaps, one of the strongest for those who know the nature of the true genius of invention. Still, Leibniz, as we know, was a philosopher with tons of projects on all sorts of matters: he sought rather to propose novel views, than to perfect and develop those that he proposed.

It is in the 1684 Leipzig Acta, as we've said above, that Leibniz offered the differential calculus of ordinary quantities. The calculus of exponential quantities, missing from the writing of Leibniz, has since been given by Monsieur Jean Bernoulli, in 1697 in the Leipzig Acta. So this calculus, in truth, belongs to this last author. [11]

Notes

1. Collins, John. Commercium Epistolicum de Analysi Promotoa, etc . London: J. Tonson and J. Watts, 1712. (see also the non-facsimile reprint of the 1712 edition, Paris: J. B. Biot, 1856). After demonstrating in his own words, based upon Barrow's Tangentes Methodum, how to use a small numerical series multiplied by terms of a polynomial equation to find the tangent to the curve described by this equation, Newton remarks on page 105, (Collins, 1712) “Hoc est unum particulare, vel corollarium potius Methodi generalis, quae extendit se, citra molestum ullum calculum, non modo ad ducendum Tangentes ad quasvis Curvas, sive Geometricas, sive Mechanicas, vel quomodocunque rectas lineas aliasve Curvas respicientes; verum etiam ad resolvendum alia abstrusiora Problematum genera de Curvitatibus, Areis, Longitudinibus, Centris Gravitatis Curvarum, etc. Neque (quemadmodum Huddenii methodus de Maximus et Minimis ) ad solas restringitur aequationes illas, quae quantitatibus surdis sunt immunes.” [This is one part or corollary, rather, of a general method that is extended, short of any tiresome calculation, not only through drawing tangents near curves, whether geometric or mechanical, but through drawing other straight lines near to curves receiving them. This remains true for the resolution of other obscure types of problems through curves, areas, lengths, centers of gravity of curves, etc. and not (as in Hudden's method of Maximum and Minimum) according to drawing up only those equations that don't include surd quantities.] “Surd” etymologically means “muted”, but in mathematical discourse refers to imaginary numbers.

2. Ibid. p. 106. “Memini me ex occasione aliquando narrasse D. Barrovio, edendis Lectionibus suis occupato, instructum me ess hujusmodi methodo Tangentes ducendi: Sed nescio quo diverticulo ab ea ipsi describenda fuerim avocatus.” [I recall the occasion when Barrow told me, when I related the reading then occupying me, that my goal was whatever has lead me to the method of tangents. But I did not understand from his description where I should have gone from there.]

3. Barrow, Isaac. Lectiones Opticae & Geometricae : In Quibus Phaenomenoon Opticorum Genuinae Rationes Investigantur, Ac Exponuntur, Et Generalia Curvarum Linearum Symptomata Declarantur . Londini: typis Guilielmi Godbid [etc.], 1674.

4. L'Hôpital, Guillaume François Antoine de. Traité Analytique Des Sections Coniques Et De Leur Usage Pour La Résolution Des Équations Dans Les Problémes Tant Déterminez Qu' Indéterminez . Paris: Boudet, 1707.

5. Collins, John. Commercium Epistolicum de Analysi Promotoa, etc . London: J. Tonson and J. Watts, 1712. pp. 192-193, 198, 199. That the calculus is fully formed at this point can be shown on pages 192 – 193 of the Commercium (Collins, 1712) when Leibniz clarifies Slusse's method of tangents: "Nam sint duae proximae sibi (id est, differentiam habentes infinite parvam) scilicet A 1B = y; & A 2B = y + dy. Quoniam ponimus dy ² esse differentiam quadratorum ab his duabus rectis, AEquatio erit (193) dy ² = y ² + 2ydy + dydy – y ² . Seu omissis y ² – y ² quae se destruunt, item omisso quadrato quantitatis infinite parvae (ob rationes ex Methodo de Maximis & Minimis notas,) erit dy ² = 2ydy." [For should there be two near each other (when they have an infinitely small difference), then most certainly, A1B = y, and A2B = y + dy. Since the 2dy I write is a quadratic difference of this second rectangle, the equation shall be (p. 193) dy ² = y ² + 2ydy + dydy – y ² . Or if I omit y ² – y ² , which then disappears, and with similar omission I square these infinitely small quantities (note, to the ratios of the method of maximum and minimum) dy ² shall be = 2ydy.] Not only does a calculus as known to contemporary mathematicians stand out, but Leibniz is careful to show his contribution in relation to Slusse and Newton, as on page 198, “Agnosco interim pulcherrima & utilissima ab eo annotari. Elegantissima & minime expectata est via qua seriem meam
&c. deduxit ex sua.” [Still, I recognize, in my annotation, the most beautiful and useful. The way my series of
&c is deduced from his is elegant and less anticipated.], and on page 199, "Cum ait Newtonus , investigationem Curvae, quando Tangens, vel Intervallum Tangentis & Ordinatae in Axe sumptum, est recta constans, non indigere his Methodus: innuit credo se intelligere Methodum Tangentium Inversam generalem in potestate esse per Methodos Serierum appropinquativas..." [With, says Newton, the investigation of Curves, when tangents or even tangential and ordinate intervals on a straight axis, the rectangle remains constant, though he does not indicate this method: I think it beckons to understand the power of the general method of inverse tangents [as in Leibniz] comes from its approaching the method of series [as in Newton] ... (My emphasis)]. And finally, this historical epistle of Leibniz's, intensely focused on where his contributions indeed sit, makes a case for his method through this same contrast between the methods of others and his own, in the conclusion of the letter: "Extrahatur jam & radix ex Binomio altero
, fiet illa
&c. ut facile demonstrari potest ex calculo: ergo * adendo haec duo extracta, destruenter imaginariae quantitates, & siet z = 2l + 2n &c. quae sunt eae Seriei portiones in quibus nulla reperitur imaginaria. Invento ergo valore ipsius 'z quantum fatis est propinquo, quemadmodum Schotenius postulat, reliqua methodo Schoteniana , perinde ac in illis Binomiorum extrahendorum generibus, transigentur." [Were I to now perform the extraction, and the root from the binomial, I would change
, making it
etc., so that one can easily demonstrate from the calculation, I thus add this second extract, destroying imaginary quantities, and making z = 2l + 2n etc., which is the portion of its series in which no imaginaries are found. So I find the value z, sketching in its quantity just as Schoten has postulated, accomplishing his method just as it will have been in those types of extracted binomials.]

6. Ibid. pp. 132 -144

7. Ibid. pp. 132-134, 135-141, 142-144.

8. Ibid. p. 131. This title in the Commercium reads: “No. XLVIII. Epistola prior D. Isaaci Newton, Matheseos Professoris in Celeberrima Academia Cantabrigiensi; ad D. Henricum Oldenburg, Regalis Societatis Londini Secretarium; 13 Junii 1676, cum Illustrissimo Viro D. Godfredo Guilielmo Leibnitio ( eo mediante) communicanda. Literis Oldenburgi, (26 Junii) ad Leibnitium missa. [No. XLVIII. Previous letter of Isaac Newton, Professor of Mathematics of the celebrated Cantabridge Academy , to Henrick Oldenberg, Secretary of the Royal Society of London. June 13, 1676, shared by him with the illustrious Mr. Gottfried William Leibniz. Sent in a letter of Oldenburg to Leibniz June 26th.]

9. Ibid. pp. 131-132. While the inspectors decided that the calculus could only be seen in a distant sense in the letter of June 13, 1676, there are several things in this letter which look very much like the beginnings of Leibnizian if not contemporary calculus. Newton introduces a theorem to abbreviate extractions of radicals: “Sed Extractiones Radicum multum abbreviantur per hoc * Theorema ,
+ etc. ... Ubi P + PQ significat Quantitatem cujus Radix, vel etiam Dimensio quaevis, vel Radix Dimensionis, investiganda est. P, Primum Terminum quantitatis ejus; Q, reliquos terminos divisos per primum. Et m/n, numeralem Indicem dimensionis ipsius P + PQ: sive dimensio illa Integra sit, sive (ut ita loquar) Fracta; sive Affirmativa, sive Negativa.” [Yet many extractions of radicals may be abbreviated by this * Theorem,
+ etc.... Where P + PQ signifies the quantities of that root, which can be any dimension, or root dimensions to be investigated. P represents the principal terms of these quantities, Q represents the next term divided by the first. And m/n is a number indicating the dimension of P + PQ, either the dimension it is integrated with, or, (so to speak) a fraction, either negative or positive. ] Newton remarks that in the same way terms such as “aa, aaa, etc.” may be written as a ² a ³ , and the terms
as
,
, as well as the terms
,
,
etc. may be written as a ^–1 , a ^–2 , a ^–3 , “Nam, sicut Analystae, pro aa, aaa, etc. scribere solent a ² , a ³ , etc. sic ego, pro

,
, etc. scribo
,

; et pro
,
,
, scribo a ^–1 , a ^–2 , a ^–3 ....”(p. 103). Not only through the latter condition does it appear that Newton has co-opted an exponential notation for writing radicals and fractions similar to that used in the calculus proper, but this theorem curiously approaches both the third rule of the calculus (that the difference of x ^m is mx ^{m –1} dx ) and points to an inchoate concept of limit with a slope-like dy/dx, derived analogously to performing rule three on a polynomial equation under the abstraction of P as the limit of a flexible, arbitrary, and tiny difference, PQ, which reduces to P ^m/n from
.

10. Ibid. pp. 75, 84, 85, 88. D'Alembert may rightfully claim that Leibniz, having knowledge of the 1669 writing, De Analysi per Aequationes Numero Terminorum Infinitas, would more likely have obtained much about the calculus from Newton, given any number of propositions in De Analysi. For instance, after Newton demonstrates his rules for representing curves through polynomial series, he indicates, on page 88 ( Collins, 1712) of de Analysi, “ Et haec de Curvis Geometricis dicta sufficient. Quinetiam Curva etiamsi Mechanica sit, methodum tamen nostram nequaquam respuit. ” [ " And so that of geometrical curves stands sufficient. Moreover such method of ours by no means rejects any curve, even though it be mechanical"], which seems to indicate the wider applicability of a generalized calculus. In addition Newton moves his reader towards an insight about the calculus in 20th century societies through demonstrating the importance of finite variation in the form of ratios that he renders as decimal coefficients substituted into algebraic symbols, as on page 75 (Collins): “Tum pono 2 + p = y , & substituo hunc ipsi valorem in Aequationem, & inde nova prodit p ³ + 6p ² + 10p – 1 = 0, cujus Radix p exquirenda est, ut quotienti addatur... itaque scribo 0,1 in quotiente, & suppono 0,1 + q = p, & hunc ejus valorem, ut prius substituo, unde prodit q ³ + 6,3q ² + 11,23 q + 0,061 = 0.” [Then I put 2 + p = y, and I substitute this very value into the equation, and now I therefore produce p ³ + 6p ² + 10p – 1 = 0, any root p which shall be sought is to be added to the quotient....So, I write 0.1 in the quotient, and place 0.1 + q = p underneath, and so this value, when substituted first, then produces q ³ + 6.3q ² + 11.23q + 0.061 = 0.]. From this importance placed on minute values describing the non-discrete, the continuous, the notion of fluxions leap out of de Analysi, for instance, when Newton describes a set of perpendicular lines that construct any of its tangents through their action as ordinates in motion, or the Newtonian and Leibnizian "moments", (Collins, 84): “Sit ABD curva quaevis, & AHKB rectangulum cujus latus AH vel BK est unitas. Et cogita *rectam DBK uniformiter ab AH motam, areas ABD & AK describere... Jam qua ratione Superficies ABD ex momento suo perpetim dato, per praecedentes regulas elicitur, eadem quaelibet alia quantitas ex momento suo sic dato elicietur...” [Suppose we have any curve ABD, and a rectangle AHKB in which sides AH or BK is one. And the known straight line DBK moves uniformly towards AH, tracing out the areas ABD and AK... Now by reason, the surface ABD is continuously given from its moment, eliciting and similarly drawing out, by the previously mentioned rules, whatever other quantities are so given from its moment."] Similarly on page 89 of (Collins): “Vel brevius sic: Cum recta AK tangenti TD parallela sit, erit AB ad BK sicut momentum lineae AB ad momentum lineae BD, hoc est
, etc.” [ ..."Or briefly, with straight line AK and tangent TD parallel, we have AB to BK as it were the moment of line AB to the moment of line BD, which is
, etc."] . And on page 85 of (Collins), fluents of the system of fluxions are more precisely defined: “Non secus ponendo CB esse x, & radium CA esse 1, invenies Arcum LD esse
, etc.... Sed notandum est quod unitas ista, quae pro momento ponitur, est superficies cum de solidis, & linea cum de superficiebus, et punctum cum de lineis (ut in hoc exemplo) agitur.” ["No differently, I make CB x and root CA 1, the found arc LD is
, etc.... What shall be observed is thus unity: surfaces with solids, lines with surfaces, and points with lines are driven for any designated moment (as in this example)..."]. And finally, the importance of finite dx over finite dy in the calculus proper is shown when Newton connects the fraction as a system of representation which may denote each of its parts without reducing them to smaller factors, to a mechanics of the calculus describing complex geometry as a reduction to bounding boxes of variable x's and y's into which curves are imagined by the geometer, page 85: “Nec vereor loqui de unitate in punctis, sive lineis infinite parvis, si quidem proportiones ibi jam contemplantur Geometrae, dum utuntur methodis Indivisibilium.” [“Fearing neither unity being located in a point, or infinitely small lines, if now most certainly there are proportions for the geometer to contemplate, then indivisible methods must be used...”]. “Indivisible” denotes this fractional construct, materially discrete, but locating mathematical constructs elsewhere, in the continuity of Nature.

11. D'Alembert fascinatingly closes his look at the differential calculus, situated between two dually entertained notions of scientific or mathematical invention, very much locating the 18th Century encyclopedia in an environment of politically motivated democratization of knowledge. D'Alembert both acknowledges the genius of Newton and Leibniz, but at the same time he suggests conditions under which there shall always be some greater genius (for instance, Bernoulli) who improves upon the traditional work of genius of any given time period. Additionally d'Alembert paints invention as collective, arising almost deterministically, independent of individual will, as philosophers, mathematicians, and natural scientists commune together within a sea of surplus knowledge, with its layers folded upon itself, with cross-references autonomously building knowledge webs both erasing and representing the Western subject.