Justification of the Differential Calculus
This section presents several texts written by Leibniz in the context of the famous “Querelle des infiniment petits”, that agitated the French Académie royale des Sciences, after the attacks launched by Michel Rolle and Thomas Gouye in 1700. Leibniz first sent a letter which was published almost in extenso in the Mémoire de Trevoux [Leibniz 1701b]. However, this text raised a lot of perplexity, even amongst Leibniz’ supporters, since it contained a comparison between the various orders of differentials and fixed and finite entities such as a grain of sand and the sun (GM V 96). Being asked about these comparisons by Varignon, Leibniz replied in the famous letter of 2 February 1702, that it was a coarse way of speaking and that infinitesimals should not be seen as fixed entities (published in the Journal des Sçavans [Leibniz 1702]). The Justification was conceived as a follow up to this letter aimed specifically at critique coming from Rolle. It was sent again to Pinsson for Varignon for publication in the Journal des Sçavans, but the project did not succeed. The Cum Prodiisset, written at the same period, provides a different strategy for the foundation of the differential calculus (based not only on the law of continuity, but also on the use of varying ratios of finite quantities). The beginning of The Defense du calcul summarizes these various strategies and testifies to Leibniz’ pluralistic approach to the justification of his calculus.
Quaestio de jure negligendi quantitates infiniti parvas
Overview of Quaestio de jure negligendi quantitates infiniti parvas (> 1702)
This manuscript has been already transcribed by Pasini (App., [Pasini 1985, 40-47).
One of the main interests of Quaestio de Jure is that it contains most of the arguments put forward by Leibniz to legitimize his calculus between 1701 and 1702. Moreover, Leibniz made an allusion to “ingeniosissimos viros” just before developing examples, which are based on series and which illustrate the computational intractability of zero and infinity when considered as absolutes. These developments are extremely close to those exchanged in October 1702 with Philip Joseph Jenisch [LH 35, 7, 17, 2ro-5ro] and Philippe Naudé [LH 35, 7, 17, 7ro] (transcribed both in App. [Pasini 1985, 19-31] and [AIII 9, 216-226]. This provides a terminus a quo for the dating of our text.
The main criticism of Leibniz’ calculus is the question of the elimination of infinitely small quantities of any order during the computing process. This criticism is crucial because it is intimately linked to the status of differentials and in particular to the existence of higher-order differentials. This subject, already raised during the confrontation with the Dutchman Bernard Nieuwentijt, was taken up forcefully by opponents of calculus at the Académie royale des sciences (Michel Rolle, Jean Gallois and Thomas Gouye). It is also mentioned at the beginning of the Defense du calcul des differences (available here) as one of the main criticisms addressed to Leibnizian calculus.
Quaestio is divided into two parts. In the first, Leibniz clarifies the status of differentials and their operations. In the second he shows, through an application to numerical series, the impossibility of dealing with absolute zero or absolute infinity, and then uses this last explanation as an additional argument to argue in favor of the fruitful use of infinitesimals.
Before starting these two developments, Leibniz points out that mathematics should not depend on metaphysics. It does not matter whether the infinitesimals are real or not: in practice it is enough – and it is even necessary – that they are considered and manipulated as are the imaginary roots in solving algebraic equations. This compendium of thought is legitimized by the possibility of reducing it to a demonstration ad absurdum, by substituting the idea of infinitely small quantity by that of quantity determined but less than a given assignable quantity [commutando indefinite parvam, in parvam definite, sive assignata minus].
The safeguard constituted by this compendium allows flexibility in the way of considering the infinitesimal as incomparable, so that Leibniz allows himself two analogies with finite quantities (the diameter of a grain of sand compared to the diameter of the Earth or the latter in relation to the distance from the fixed stars). Leibniz knows that these analogies are not without danger if they are introduced without precaution. Such was the case during his intervention in the Journal de Trévoux in November-December 1701 [Leibniz 1701b] which shocked both opponents and defenders of differential calculus (See also here)
The force of his argument is that it makes it possible to consider the infinitely small or infinite compared to finite quantities but also infinitely infinitely small. The Lemmas on Incomparables [Leibniz 1689] evolved over time from a simple practical prescription to a key concept: by forging a new kind of law of homogeneity, Leibniz endowed transcendental analysis with a more general rule than the one introduced by François Viète for the dimensions of finite quantities.
Leibniz evoked another utility of differentials: they make it possible to state universal propositions. In fact, this argument is not new and Leibniz here revives an idea he had developed in 1674 in “Méthode de l’universalité” [A VII, 7, 102-105]: it is possible to consider the equation of a parabola, a triangle or even a point as special cases of the general equation of a conic by introducing infinite and infinitely small quantities. But while he described this general equation as “the key of all the harmonies and differences between things” [clef de toutes les harmonies et différences de choses], Leibniz now insists on linking this kind of situation as falling under the Law of Continuity.
The perfect adequacy of infinite quantities to express continuous phenomena shows that, despite their imaginary character, they are quam maxime reales.
It is not the same for zero and absolute infinity because their analytical processing leads to difficulties, even to impossibilities. In a certain sense, the notions of absolute zero and infinity would therefore be completely useless in mathematics and would instead fall under metaphysics.
Qvelques remarques sur les Memoires de Trevoux de l'an 1701
Overview of Qvelques remarques sur les Memoires de Trevoux de l'an 1701 (1702)
The Mémoires pour l’histoire des sciences et des beaux-arts, better known as the Journal de Trévoux or Mémoires de Trévoux, is a journal founded by the Jesuits in 1701 in Trévoux. This bimonthly publication was composed of articles on various literary, scientific and religious topics.
As soon as the first issues of the Journal appeared, Leibniz wished to be kept informed of articles published in it and he obtained them through his friend François Pinsson. His interest was aroused by certain articles and he did not hesitate to send his point of view to the Journal on many occasions.
The present text is an extract from a manuscript of notes and remarks that Leibniz wrote about articles published in the issues of September-October and November-December 1701, in which he also contributed [Leibniz 1701a], [Leibniz 1701b].
The article entitled “Mémoire de Mr Leibnitz touchant son sentiment sur le Calcul différentiel” [Leibniz 1701b] is an answer to an article from the months of May-June 1701 [Gouye 1701]. In it, Father Gouye criticized the Analyse des infinis in which, according to the Preface to the book by L’Hospital [L’Hospital 1696], “things got taken further” [a porté les choses plus loin], “not only embracing the infinite, but the infinity of the infinite, or an infinity of infinities” [n’embrassant pas seulement l’infini, mais l’infini de l’infini ou une infinité d’infinis]. Nevertheless, despite “his admirable fecundity”, Gouye judged that the new analysis lacked “in its demonstrations the evidence that one expects from them” [dans ses démonstrations cette évidence que l’on attend] and that it was therefore preferable to favour other safer methods than to “to embark on the new roads of the infinity of the infinite” [s’engager dans les nouvelles routes de l’infini de l’infini] “where one can easily get lost, without noticing” [où l’on peut aisément s’égarer, sans qu’on s’en apperçoive] [Gouye 1701, 430]. Leibniz’s well-known answer [Leibniz 1701b] consists in explaining that mathematics does not depend on metaphysical considerations, and that in the practice of calculus it is not necessary to consider rigorous infinities (but quantities as small or large as desired to be less than a given one). By doing “so that one only differs from the manner of Archimedes only in the “expressions”.
Leibniz compares the consideration of the infinity of infinities with what happens when one considers the ratio between a ball and the diameter of the Earth or the ratio between the Earth and the distance of the fixed stars – or when one considers that the ratio between the ball and the distance from the fixed stars is infinitely infinitely small. Its development, however, based on a comparison with finite quantities, caused confusion amongst the members of the Académie des sciences, who understood, to their great surprise, that Leibniz was claiming that a differential is a fixed and determined finite. Leibniz’ text is followed by a comment which is almost a request for a reply
“Some geometers, who have examined with great care the Analyse des infiniment petits by Mr. Le Marquis de l’Hôpital, & who declare following his method, say that it is necessary to take infinity à la rigueur, & not as Mr. Leibnitz explains here”.
The text we publish, Qvelqves remarques sur les Memoires de Trevoux de l’an 1701, has never been published. In the marginalia, Leibniz wrote that he had lost the piece of paper and thus the remarks written on it have not been communicated. Leibniz would finally respond to this last comment in March 1702 [Leibniz 1702] and we find in our short extract some ideas which are developed at greater length in the March response. Differentials are not rigorous infinites; they lack reality as much as the imaginary roots of algebra. The calculus is well founded because “the thing can be reduced to the incomparable”.
Cum prodiisset
Overview of Cum prodiisset (>1702)
The Cum Prodiisset manuscript was first published by Gerhardt, with no indication of its date and a few transcription errors [Leibniz,1846]. Child proposed an English translation in 1920 [Child, 1920, pp. 144-158]. He dated the manuscript from the time of the controversy with Bernard Nieuwentijt (1694-1695). This dating is not satisfactory since at the beginning of the text, Leibniz refers to the article by Thomas Gouye published in the Journal des Sçavans in May 1701 [Gouye 1701] and then to Hermann's answer to Nieuwentijt [Hermann, 1700]. Additional evidence suggests that Cum Prodiisset is, in fact; intimately related to the twin texts presented in this section : Justification du Calcul des infinitésimales par celuy de l’Algèbre ordinaire and Defense du calcul des différences. Its date would thus be after 1702.
In Defense du calcul des différences, Leibniz announces that it is possible to offer two additional ways to justify his calculus, besides that grounded on the Lemmas on Incomparables and arguments in the manner of the Ancients. The first is a justification "by the ordinary calculus of Algebra" that is based on the principle of continuity. Leibniz expressly chooses a geometric configuration that appears intractable by using ordinary algebra (involving the quotient of two zeros). This example and the reasoning behind it are the same as those developed in Justification. The second way consists in giving an “interpretation” (interprétation is Leibniz's word) of the computational process in which "very small magnitudes" (très petites grandeurs) are used instead of infinitesimals. Leibniz claims that this interpretation sheds more light on the reason why he spoke of “incomparables” (incomparables) in other places. However, the latter point is not developed here as the text ends with an uncompleted sentence: "En voicy un essay.” This second approach, involving “very small magnitudes”, reminds one of the strategy developed in the Cum Prodiisset. Indeed, in marginalia of this last text (neither reproduced nor translated by Child), Leibniz wrote
"all this must be edited very carefully so it can be published, omitting what is harsher in contradicting others. It is to be joined by my method for showing the law of continuity by the move of lines, in conformity with the tract [schediasma] I sent the Parisians in order to show that in the common example the ratio between nothings is feigned to be something"
It is possible to conceive of the Defense as one such attempt in which Leibniz intended to put together both the Justification, the only known schediasma sent to his Parisians in which the ratio of two hypothetic zeros is considered, and the Cum Prodiisset, the only known text where he implements a method based on “very small magnitudes.” [Rabouin & Arthur 2020].
Leibniz often argued that the rules of his calculus may be demonstrated by Archimedes-style arguments and the use of The Lemmas on Incomparables [AIII 7, 235, 576, 618, 857 ; AIII 8, 91], but by his own admission he never provided such a proof. In Cum Prodiisset, he does not do so either, but he presents two ways of justifying his calculus. One is based explicitly on the principle of continuity, while the other involves "finite witnesses" and avoids the use of infinitely small or vanishing quantities.
Leibniz prefaces his first method with a commentary on the principle of continuity. He had already formulated this principle (which he calls a postulatum) in other writings. The formulation in our text is the following: “in any proposed continuous transition ending in a terminus, one may institute a common reasoning in which the ultimate terminus is included”.
Reasoning based on this principle, he argues, is not new. He himself established results in geometry and other fields by drawing on it. In physics, for example, this principle allowed him to highlight errors that derived from applying Descartes’ and Malebranche’s rules for motion. He also points out that establishing results through arguments based on the principle of continuity is in fact common practice among other authors. Descartes recognized such a practice among the Ancients, which he referred to as "metaphysics of geometry". Leibniz goes further and interprets the way some of his contemporaries (he cites Descartes, Christiaan Huygens and Philippe de La Hire) establish results as also falling within this type of reasoning. There are many well-known examples of the application of this principle — what can be said of a polygon can be said of a polygon with an infinitely-sided polygon, that is to say a curve; what applies to an ellipse applies to an ellipse whose focus is at infinity, i.e. a parabola, etc. It must be stressed, however, that Leibniz was the first to explicitly formulate this principle.
How does he use this principle to present his first method?
He presents his method by applying it directly to determining the tangents of the common parabola and the cubic of equation [latex]x ^ 3= aay[/latex]. His method relies on a geometric configuration, usual at the time, consisting of two similar triangles, one formed by the subsecant [latex]T1X[/latex] and the ordinate [latex]T1Y[/latex] (Leibniz indicates indices by a number before the letter) and the other formed by the differences [latex]dy[/latex] and [latex]dx [/latex]. His purpose is to establish a general expression for [latex]\frac{dy}{dx}[/latex]. Leibniz calls this expression a "general rule" [regula generalis], and then appeals to the principle of continuity: the reasoning is valid for the extreme case where the ordinate [latex]2X2Y[/latex] moves until it is very close to the fixed ordinate [latex]1X1Y[/latex]. In this case, the differences can be considered to be infinitely small and the secant [latex]T1Y[/latex] is the tangent. He therefore allows himself to use that [latex] dx = 0 [/latex] to obtain the expression of [latex]\frac{dy}{dx}[/latex] corresponding to the tangent. Leibniz takes up the idea developed in the example of Justification, also taken up in Defense, in which a ratio between two vanishing quantities is always interpretable even as they vanish because it is equal to a proportion between two finite quantities.
This last reasoning involves infinitesimals which make it possible to imagine the case where the differences are zero as if they were not. However, Leibniz, probably anticipating criticisms, points out that it is quite possible to obtain the same computational results by considering that the differences are not evanescent but arbitrarily small finite quantities. The key to his method lies in the introduction of a segment which is finite and constant, denoted [latex](d)x[/latex]. In Nova Methodus, Leibniz had also introduced a finite segment to present his calculus, probably with a view to excluding other presentations involving infinitely small quantities [Hess 1984]. However, in Cum Prodiisset he took fuller advantage of this idea. He considers a segment [latex] (d) y [/latex] such that [latex] (d) y: (d) x = dy: dx = 1XT: 1X1Y [/latex]. The segment [latex] (d) y [/latex] is finite, its trivial construction depends on [latex] dx [/latex]. For any other curve [latex] VV [/latex] with the same axis as [latex] YY [/latex] with [latex] 1X1V = v [/latex] and [latex] 1X1V = v + dv [/latex], he also considers [latex](d) v [/latex] such that [latex] (d) v: (d) x = dv: dx = 1XU: 1X1U [/latex]. The triangles with sides [latex] dx [/latex] and [latex] dy [/latex] or [latex] dx [/latex] and [latex] dv [/latex],… are each similar to a finite "witness" triangle.
Leibniz demonstrates the differentiation rules for finite quantities [latex] (d) x, (d) y, (d) v [/latex] which are valid even in the case where the triangles (of sides [latex] dx, dy, dv[/latex], …) vanish since the similarity is supposed to remain. He points out that this reasoning could be imagined [fingendo] by using unassignable quantities since their ratio would be the same as that of finite and real quantities. As pointed out by Rabouin and Arthur, Leibniz’s procedure facilitates Archimedian-style demonstration. Here, obtaining this type of equality would be conducive to an Archimedean demonstration of the rules of the calculus [Rabouin & Arthur 2020].
Leibniz similarly demonstrates the rules of second-order differentiation even if, as Bos showed in a seminal article which did much for the fame of our text [Bos 1974], his reasoning is insufficient. Indeed, Leibniz assumes without making it explicit that the differential of one variable is constant [Bos 1974, 59-62].
Justification du Calcul des infinitésimales par celuy de l’Algèbre ordinaire
Overview of Justification du Calcul des infinitésimales par celuy de l’Algèbre ordinaire (1702)
There are three versions of the text we present: the manuscripts LBr. 951, Bl. 14, LBr. 951, Bl. 12-13, LBr. 951 Bl. 15-16. The last one has already been edited by Gerhardt [GM IV, 104-106]. We will offer soon a new transcription integrating the variants which seem significant to us.
This text is to be situated among a series of responses that Leibniz formulated particularly for the attention of the members of the Académie des sciences during la querelle des infiniment petits. Indeed, barely one year earlier, in a short article in the Journal de Trévoux [Leibniz 1701b], Leibniz had explained that the practice of his calculation did not require taking the infinitely small in a rigorous way, but that it was sufficient to consider quantities as small as needed. This way of proceeding differs from the manner of the Ancients, he argued, only in the “expressions”. Using the examples illustrated by his “Lemma of Incomparables” [Leibniz 1689], he maintained that it is thus possible to consider that the ratio between a ball and the diameter of the Earth or that of the Earth and the distance from the fixed stars as infinitely small, or that the ratio between the ball and the distance from the fixed stars as infinitely infinitely small. Its development, however, based on a comparison with finite quantities, caused confusion amongst the members of the Académie des sciences, whether be they opponents and defenders of differential calculus. In particular, Varignon asked Leibniz for clarification. On 2 February 1702, Leibniz hastily wrote a response to his friend, who made it public in the Journal des Sçavans in March [Leibniz 1702]. In this response, Leibniz first tried to clarify what he meant by “to explain the infinite by the incomparable” [expliquer l’infini par l’incomparable] and what is meant by “incomparable”. This intervention, according to Varignon, calmed those attacking differential calculus, in particular Father Gouye. But Leibniz wanted to be sure that he had been understood and it was probably partly for this reason that he adopted a new strategy for his calculus and sent it on 21 April to Varignon, wishing it to be published (but this was not the case). In this later text, Leibniz intended to show that infinitesimals were not the privilege of differential calculus but had been used since algebra had been applied to geometry, so that “we found in the calculus of ordinary Algebra traces of the transcendental calculus of differences” [on trouve dans le calcul de l’Algebre ordinaire les traces du calcul transcendant des differences]. To justify this point Leibniz essentially relied on a geometric configuration, which he chose for its simplicity and in which a triangle vanishes while keeping the same shape. This example can be found almost identically (except for the notation of points) in another manuscript, written at the same period and entitled Defense du Calcul des différences (available here) (Note that Leibniz had chosen to entitle the first version V1 “Défense des infinitésimales par le calcul de l’Algèbre ordinaire suivant son usage conforme à la Loy de la continuité”).
To understand this geometrical situation using calculus in all its generality, that is to say until the triangle vanishes completely, it must be accepted that the lengths of the sides never become absolutely zero, but that ‘”at the last moment” they are infinitesimal. Thus, thanks to the Law of Continuity, they keep among themselves the essential property of magnitudes which is to have a ratio. It is the existence of this “last ratio” that allows us to understand the geometrical situation in all its generality. Conversely, Leibniz shows by a reasoning ad absurdum – which, in passing, is not conclusive – that such a conclusion would not hold considering that the sides become absolutely zero.
The interest of this text among the other texts of our section Justification of differential calculus is that Leibniz adopts a strategy of justification, different from the the call to the argument of the incomparable, and that it is based, for this purpose, on the principle of continuity applied to algebraic calculus.
Defense du calcul des Differences
Overview of Defense du calcul des Differences (1702)
Defense du calcul des Differences is clearly to be situated at the same time and in the same context as Justification du Calcul des infinitésimales par celuy de l’Algèbre ordinaire which we also present in this section (available here). While the notation is different, their developments are based on identical geometric configuration.
It is possible that Leibniz wrote Justification first and then used it later to write a longer text in which he kept the same example. Indeed, although very similar, Defense du calcul des Differences, unlike Justification, contains a preamble in which Leibniz announces his intention to show that in his calculus the elisions of infinitely small quantities are not made in an arbitrary manner as argued by "des personnes habiles qui s’opposent au Calcul des Différences" but according to rules that can be justified in the manner of Archimedes. These rules are based on his famous Lemmas on Incomparables [Leibniz, 1689] whose application makes it possible "to know what can be neglected with impunity without it being able to cause any error".
By invoking the "des personnes habiles qui s’opposent au Calcul", Leibniz was addressing academicians like Rolle and Galloys, who refused to recognize the exactness of differential calculus.
Leibniz had already addressed a justification of his calculus to the French savants. A year earlier, in a short article in the Journal de Trévoux [Leibniz 1701b], he explained that the practice of his calculus does not require taking the infinitely small in a "rigorous" manner but as quantities as small as needed. Taking the same comparison with finite magnitudes as he did in his inaugural article presenting the Lemmas on Incomparables [Leibniz 1689] (the diameter of a grain of sand, the diameter of the Earth and the distance from the fixed stars), he caused confusion amongst the members of the Académie des sciences, both opponents and defenders of differential calculus alike. This intervention was followed by an article in the Journal des Sçavans [Leibniz 1702] in which he clarified what he meant by "to explain the infinite by the incomparable" [expliquer l’infini par lincomparable] and how differentials could be considered as “ideas suitable for abbreviating reasoning, and grounded in reality” [des idées propres à abréger les raisonnemens, & fondées en realitez].
In Defense du calcul des différences, Leibniz announces that it is possible to propose two other ways of justifying his calculus (in addition of the one that based on the Lemmas on Incomparables and reasoning in the manner of the Ancients). The first is a methodological justification "by the ordinary calculus of Algebra" and which is based on the principle of continuity. His reasoning is the same as that developed in Justification. The second is an interpretation of the computational process: considering "very small quantities" instead of infinitesimals. Leibniz claims this would shed more light on what made him speak of incomparables in other times. However, this last point is not developed here since its text ends leaving "En voicy un essay" pending. This last strategy brings to mind that developed in the Cum Prodiisset [Leibniz 1702b]. In marginalia of this last text, Leibniz wrote “all this must be edited very carefully so it could be published, omitting what is harsher in contradicting others. It is to be joined by my method for the law of continuity shown by the tracing of lines, and also by the tract [schediasma] I sent the Parisians in order to show that in the common example the ratio between nothings is feigned to be something” (translate in [Rabouin & Arthur 2020]). It is therefore possible that the Defense is an attempt in which Leibniz intended to bring together both the Justification (the only known schediasma sent to Parisians) and the Cum Prodiisset (the only known text where he implements a method based on “finite witnesses”) [Rabouin & Arthur 2020].
By “justifying our way of calculating by that of ordinary Algebra”, Leibniz wants to make it clear that those who practice the ordinary calculus of algebra, already use infinitely small quantities even if some of them claim that these quantities are in fact zero. Presenting his calculus, criticized in such a vivid manner by the algebraists, as based on the very same foundations as those they need for their own practice, Leibniz adopts a new strategy of justification, different from the one he had developed a few months earlier.
This second justification, in support of geometrical configuration, is practically the same as that of Justification. To understand this geometrical situation using calculus in all its generality, that is to say until the triangle vanishes completely, it must be accepted that the lengths of the sides never become absolutely zero, but that '"at the last moment" they are infinitesimal. Thus, thanks to the Law of Continuity, they keep among themselves the essential property of magnitudes which is to have a ratio. It is the existence of this “last ratio” that allows us to understand the geometrical situation in all its generality.
Sentiment de Monsieur Leibnitz
Overview of Sentiment de Monsieur Leibnitz (1705)
In July 1700, at the Académie royale des sciences, the algebraist Michel Rolle launched the first criticisms against differential calculus, based on its foundations and on its use which he thought fraudulent. This intervention marks the beginning of the well-known querelle des infiniment petits. The Sentiment de Monsieur Leibnitz manuscript bears witness to one of the last episodes of this quarrel before its appeasement in 1706.
In 1705 the debate was at its peak. In his essay Remarques de M. Rolle de l’Académie des Sciences touchant le problesme général des tangentes [Rolle, 1703], Rolle completely discredited both the foundations and the exactness of differential calculus Two years later, on 23 April 1705, Saurin published a reply [Saurin 1705] which he ended by imploring the Académie to pass judgment on the discrepancies between him and Rolle, which he listed in seven points. This extremely tense climate led Varignon to write to Leibniz on 10 May 1705 [AIII 9, 549]. Varignon wanted Leibniz to intervene with members of the Academy. He also asked him to solicit an endorsement on the points formulated in Saurin’s article and those of a memoire he attached to this letter, from members of the République des Lettres, who understood his calculus.
Leibniz reacted immediately. He wrote two letters – one to Gallois and the other to Bignon – and an endorsement in defense of his friends. Two versions exist: a Latin version [Gotha FB A 448-449, 41-42] and an abridged version in French, written by a copyist [LH 35 VII 9, 1-2]. Only the Latin version of this text was published under the title “Sentiment de Monsieur Leibnitz” [Leibniz 1706]. This publication is attached to a text by Saurin (which follows his article of 23 April 1705) and two other endorsements: that of Jacob Hermann and those of the Bernoulli brothers. All the publications were, however, confiscated by Jean-Paul Bignon [Bella 2021].
L’Analyse des infiniment petits pour l’intelligence des lignes courbes contains an article (§ 163) [L’Hospital 1696] which explains that when a fractional expression has both the numerator and denominator equal to zero at a certain point, its value can be obtained by differentiating the numerator and the denominator. This result is very useful for the determination of tangents at a crunode. Indeed, the formula of the tangent applied to this type of point leads to the expression [latex]\frac{0}{0}[/latex].
In an article in the Journal des Sçavans published on 3rd August 1702, Joseph Saurin applied article 163 to determine the values of the two tangents at the crunode point [latex]x=2 [/latex] and [latex]y=2 [/latex]) for the quartic equation
[latex]y^4 – 8y^3-16y^2-12xy^2+48xy+4x^2-64x=0[/latex]
However, this ingenious use was harshly criticized by Rolle. He accused Saurin of adding “supplements” to the rules, whenever it pleased him, and also of abusing the rules of algebra. The Sentiment de Monsieur Leibnitz manuscript was the response to a request from Pierre Varignon who had asked Leibniz to support differential calculus within the République des lettres, in particular to members of the Académie royale des sciences. The manuscript has two very distinct parts. In the first, Leibniz elucidates some technical aspects of Rolle’s critique. The second part is what makes the text more interesting. By clarifying article 163, Leibniz shows that differential calculus is the only one which provides an interpretation of the problematic expression [latex]\frac{0}{0}[/latex] (which appeared here in the geometric configuration of the crunode point, although we can also find it in the characteristic triangle).
The association of the ratio sign and zero “0”, which constitutes [latex]\frac{0}{0}[/latex], refers in algebra to an impossibility and indicates a limitation. Using vanishing quantities, Leibniz takes up the challenge of making sense of the expression [latex]\frac{0}{0}[/latex] and goes beyond this limitation. This is an additional argument toward the habiles de la spécieuse ordinaire.
There is a French version of this text by Leibniz himself (available here).
Puisque des personnes que j’estime beaucoup
Overview of Puisque des personnes que j’estime beaucoup (1705)
This manuscript is a shortened French version of Sentiment de Monsieur Leibnitz (available here). It was probably intended for the Parisian savants (the other, in Latin, being reserved for publication in the Acta Eruditorum).