When two points coincide, or are at an infinitely small distance:

Some aspects of the relation between the works of Leibniz, Pascal (and

Desargues)1

Despite the fact that Leibniz’s manuscripts on perspective are not yet completely published, the

question of their importance has already been raised by some studies2. In addition, recent works

by V. Debuiche have presented some aspects of the relation between Desargues’ and Pascal’s

perspective geometry and Leibniz’s geometria situs3. Motivated by these developments, the

current paper discusses the importance for Leibniz of Pascal's works on conic sections,

concentrating on the relational aspects of geometrical elements and the treatment of infinitesimal

distances. I restrict myself to Leibniz’s works in the period between 1674 and 1683.

1. Ambiguity and De la méthode de l’universalité

I shall first consider the text De la méthode de l’universalité, from 16744. In this text, Leibniz

proposes a mathematical notation that makes new generalizations possible, in part due to the

ambiguity of signs and of letters used in the formulas5. Leibniz’s aim is to reduce differences to

“Harmony”, and this will be related to the problem of how to express this mathematical content.

Leibniz’s remarks principally concern algebraic notations for equations, but he also considers the

representation of geometrical curves, and puts forward a general formula for conic sections,

possible thanks to the consideration of cases involving infinite and infinitely small quantities.

In De la méthode de l’universalité, one has the impression that Leibniz produced a synthesis of

two contemporary mathematical traditions, each of which dealt with conic sections in different

ways: the “algebraic” tradition (namely with Descartes) and the “projective” tradition (with the

1 This text was published in Für unser Glück oder das Glück anderer: Vorträge des X. Internationalen Leibniz-

Kongresses, vol. 4, Olms, 2016, pp. 165-178. The present version introduces minor alterations. 2 In particular, Javier Echeverría: „Recherches inconnues de Leibniz sur la géométrie perspective”, in Leibniz et la

Renaissance. Studia Leibnitiana Supplementa 23 (1983), S. 191–201; Echeverría, J., 1994. „Leibniz, interprète de

Desargues”, in: J. Dhombres/J. Sakarovitch (Hrsg.): Desargues en son temps, Paris, 1994, S. 283–293. 3 On this issue, see in particular Valérie Debuiche: „Perspective in Leibniz’s invention of Characteristica

Geometrica: The problem of Desargues’ influence”, in: Historia Mathematica 40 (2013), S. 359-385; Valérie

Debuiche: „L’expression leibnizienne et ses modèles mathématiques”, in: Journal of the History of Philosophy 51/3

(2013), 409–439; Valérie Debuiche: „L’invention d’une géométrie pure au 17e siècle: Leibniz, lecteur de Pascal”

(forthcoming). 4 We know two different versions of this text. The Academy edition (in the preliminary version of the volume VII 7

from 28.10.2015 which I quote here) gives June-August 1674 for De la méthode de l’universalité I, and May-June

1674 for De la méthode de l’universalité II, giving a chronological order inverse to that of Couturat. My remarks

concern De la méthode de l’universalité II. 5 Cf. Emily R. Grosholz: Representation and productive ambiguity in mathematics and the sciences, Oxford 2007,

hier S. 207-213.

works of Desargues and Pascal)6. I do not mean that these two traditions were necessarily

opposed to one another, but simply that they were usually not used together by the same

mathematician.

But while Leibniz undeniably makes use here of algebraic expressions, had projective geometry

played any role in his unification of conics? The first approach to this question is pragmatic:

what did Leibniz know of Desargues and Pascal at that time? As we will see, Leibniz only had

access to Pascal’s works on conics from 1676. But what about Desargues? There are some notes

by Leibniz referring to works on perspective that date from 1673-1676, including notes on

Desargues’ optics and perspective7. But did Leibniz read the Brouillon project d’une atteinte aux

événements des rencontres du cône avec un plan (1639), in which conics are treated

projectively? The only evidence we have is an extract (copied) from the final paragraph of the

Brouillon project, made by Leibniz in January - August 1676, but it is not certain that he had

read the work in its entirety (A VII 7, 111). We can suspect that by 1674 he had just heard of it.

The fact is that in De la méthode de l’universalité Leibniz criticizes the limits of Desargues’ and

Pascal’s works on conic sections, it not being possible to find the “harmony” of these curves

through their approach. This limitation, according to Leibniz, is due to the exclusive

consideration of particular properties of these curves, and to the concrete aspect of this approach

(A VII 7, 49). It would be difficult to solve complex problems with this approach, unless we find

the solution “by chance”, by means of a theorem demonstrated elsewhere. Leibniz’s own

method, on the other hand, would spare the mind and the imagination8, in the same way as the

other parts of Analysis do.

Should we thus dismiss the importance of Desargues and Pascal here? Even if we cannot know

exactly what Leibniz had read from both of them by this time, I would like to show the similarity

of a passage in this text to Leibniz’s notes on Pascal. I argue that either there was already an

influence from projective geometry in De la méthode de l’universalité, even if this is not

admitted by Leibniz; or rather that he later found something in Pascal’s works that agreed with

some of his own ideas9. The relevant passage is one in which the coincidence of points is

associated by Leibniz with these points being infinitely close to one another.

6 To categorize Desargues’ and Pascal’s works on conics is not an easy task. A perspective approach is surely

present. Could these works be considered as “projective geometry”, in the same sense as the discipline formalized in

the XIX century? In the sense that they consider relations that are preserved under projection, I will use this term. 7 “Zu Girard Desargues, Maniere universelle pour pratiquer la perspective” (A VIII 1, 210-226). “Note sur l’optique

de Desargues” (A VIII 1, 227). “Zu J. Aleaume. La perspective speculative et pratique” (A VIII 1, 228–232). “Zu J.

Dubreuil. La perspective practique” (A VIII 1, 233–234). 8 “elle espargne l’esprit et l’imagination” (A VII 7, 49). 9 V. Debuiche showed that it is not easy to know whether Leibniz read Desargues’ Brouillon Project, and if he was

influenced by him or if he just considered “Desargues’ works as some particularly interesting illustrations of his

geometrical project” (Valérie Debuiche: „Perspective in Leibniz’s invention of Characteristica Geometrica: The

problem of Desargues’ influence”, S. 366).

In De la méthode de l’universalité, Leibniz distinguishes the ambiguity of signs and of letters.

He states that while the ambiguity of the signs should be made to disappear in their

interpretation, the ambiguity of the letters can be maintained, allowing the formula to express a

universal case. In particular, letters can signify lines, and the lines can be conceived in three

ways: as finite, as infinitely great or as infinitely small.

Leibniz thus talks about infinitely small lines, “vulgarly called indivisibles” (A VII 7, 59). If a

straight line DE intersects a curve at two points B and (B), we can “imagine” that the distance

between B and (B) is “infinitely small”, and in this case we find the tangent.

Figure illustrating a straight line DB(B)E secant to a curve AC (image from A VII 7, 59)

Leibniz emphasizes in this passage that the method of indivisibles is based on the method “of

Infinities”. In particular, the “Geometry of Archimedes” restored by Guldin, Gregoire de S.

Vincent and Cavalieri makes usage of “infinitely small magnitudes”. Leibniz suggests that we

consider this “supposition” to see the usage it could have in a case considered in a previous

passage: the composition of a straight line by the addition and subtraction of its segments.

(Image from A VII 7, 60)

In this figure, points A and B are considered as fixed, whereas C is considered as “moveable”

(“ambulatoire”). The question is to see how the line AC can be determined by lines AB and BC,

and Leibniz considers the equation

(1)

Where the first sign means the equality, and the third is an ambiguous sign for addition or

subtraction.

The case where AC equals AB corresponds to three possibilities: either point 3C is properly

under B, the distance BC being “nothing” (“BC egale à rien”); or the point is before B, and the

distance (3C)B is infinitely small; or the point is beyond B, and the distance ((3C))B is infinitely

small. These three cases can be represented respectively by the equations (with modern

notation):

(2)

(3)

(4)

Where BC is infinitely small.

The coincidence of two points (and the proper equality in an equation) is thus equivalent to an

infinitely small distance existing between these two points (its addition or subtraction being

neglected). Not only this, but the ambiguity of the sign for addition/subtraction in equation 1

makes it possible to express the last two cases with the same equation.

The method of universality, says Leibniz, allows the application of a general formula to a

particular case. Not only this, but Leibniz writes that

“one cannot understand the case of the coincidence of points B and C in the

general equation [equation 1], unless if one supposes that BC is infinitely

small”10.

This declaration shows that this is not only a possibility, but a constraint: the only way of

considering the coincidence of points in this equation is to also admit infinitely small distances.

The generality of this formula achieved by Leibniz is thus possible thanks to:

1) Having ambiguous letters

2) Admitting geometrical elements “as if” they had an specific role, point C being

considered as moveable (relational aspect)

3) Associating the case of the coincidence of points and that of an infinitely small distance

10 “Car on ne sçauroit comprendre le Cas de la coincidence des points B et C. dans l’équation générale [equation 1]

qu’en supposant la ligne BC infiniment petite” (A VII 7, 61).

Leibniz declares he is searching for a “harmony”11. This, we can see, is made possible by the

ambiguity of letters and by the identification of a certain continuity between secant lines and

tangent lines, on the one hand, and between the different cases of equation 112.

While Leibniz’s predecessors considered only infinitesimal cases, Leibniz expresses them side

by side with the finite, which the ambiguous letters allow him to do, particularly in the

arrangements of points on a line.

Before considering Leibniz’s notes on Pascal, I shall briefly consider some other passages of De

la méthode de l’universalité regarding the relational aspect of geometrical elements.

Leibniz declares that mathematicians such as Cavalieri, Fermat and Wallis used letters to

represent “infinitely small lines or equal to nothing”13. Leibniz claims to do the same, also using

letters to represent infinitely great magnitudes, such as the asymptotes of the Hyperbola. Not

only this, but he also says that “we can say in a certain way that the Asymptote of the Hyperbola,

or the tangent drawn from the center to the curve, is an infinite line equal to a finite rectangle”14.

We can recognize here something similar to the paradox of Torricelli’s infinitely long solid

(which has in a certain way already been considered by Oresme15), where an infinite surface is

found to be equal to a finite solid. I want to call attention to the usage of the expression “in a

certain way” (“en quelque façon”), that indicates a relational understanding of geometrical

elements by Leibniz.

Finally, one of the greatest achievements of this text is to present a common equation valid to

determine all conic sections16:

(5)

11 In De la méthode de l’universalité Leibniz writes the words « Harmonie » and « Asymptotes » sometimes

beginning with uppercase letters and sometimes with lowercase letters. 12 Emily R. Grosholz: Representation and productive ambiguity in mathematics and the sciences, S. 209, identifies

here an application by Leibniz of the Law of Continuity. However, Leibniz only conceived this law in the 1680’s,

after the writing of this text. One can discuss whether the notion of continuity already appears here, but as for the

terminology, we cannot talk here about the Law of Continuity. 13 “supposent des certaines lettres, ou lignes infiniment petites ou egales a rien” (A VII 7, 51). 14 “les Geometres n’ignorent pas qu’on peut dire en quelque façon que l’Asymptote de l’Hyperbole, ou la touchante

menée du centre à la courbe est une ligne infinie égale à un rectangle fini” (A VII 7, 61). 15 Cf. Paolo Mancosu: Philosophy of mathematics and mathematical practice in the seventeenth century, New York

1996. 16 In fact, Leibniz presents this as a general equation, but he also squares it, in order to have a form of it in which

none “amphibolie” is found. I think we should interpret here “amphibolie” as equivalent to “ambiguïté” and

“équivocité”.

In this context, Leibniz makes reference to a general object: “we will be able to proceed in the

future as if there was a certain particular figure in the world that we called a Conic Section”17.

Still here a relational aspect is considered, and even the possibility of using a mathematical

fiction, a question that will be of fundamental importance in Leibniz’s considerations on the

status of infinitely small elements.

2. Leibniz’s notes on Pascal’s treatise on conic sections

In De la méthode de l’universalité we saw the possibility of considering coincident points as

equivalent to points at an infinitely small distance, which allowed a general treatment of an

arrangement of points on a line.

Regarding generality, we should bear in mind that in Pascal’s lost Traité des Coniques the

“mystical hexagram” alone would, according to Mersenne, make it possible to demonstrate 400

corollaries, a fact which could well impress Leibniz. He indeed wrote in 1675:

“For messieurs des Argues and Pascal have done very well to take generally the

ordinates as convergent or parallel lines, especially as parallel lines can be taken

as kinds of convergent lines, from which the converging point is infinitely

distant”18.

But what had Leibniz read from these authors by the end of 1675? What we know is that Leibniz

had access to Pascal’s treatise on conics in 1676 thanks to the Périer brothers, his notes being

from January 1676 to August of the same year19 – that is, after he had written De la méthode de

l’universalité. The only part we known from Pascal’s lost treatise is entitled Generatio

conisectionum, which came to us from a copy made by Leibniz in his own hand20.

Leibniz seems to be impressed by the generality achieved by Pascal’s approach, in which “two

parallel [straight] lines are understood as concurrent”21 – indeed, the acceptance of points at an

17 “nous pourrons procéder à l'avenir, comme s'il y avoit une certaine figure particuliere dans le monde, qu'on

appellat Section Conique” (A VII 7, 77). 18 “Car messieurs des Argues et Pascal ont fort bien fait de prendre les ordonnées generalement par des lignes

convergentes ou paralleles, d’autant plus que les paralleles peuvent estre prises pour une espece de convergentes,

dont le point de concours est éloigné infiniment” (Leibniz to Gallois, Ende 1675. A III 1, 359). 19 Cf. Mesnard in Pascal, Oeuvres Complètes, Bd 2, S. 1102. 20 Blaise Pascal: Oeuvres Complètes, hrsg. v. Jean Mesnard, Bd 2, Paris, 1970, S. 1108-1131. Both of them will

appear in A VII 7. 21 “Duae linae parallelae concurrere intelliguntur” (in: Pierre Costabel: „Traduction française de notes de Leibniz sur

les “Coniques” de Pascal”, in: Revue d’histoire des sciences et de leurs applications 15/3-4 (1962), S. 253-268, hier

S. 260).

infinite distance is an essential characteristic in Desargues’ and Pascal’s work on conic

sections22, making possible the generalization of some theorems for all conic sections.

But for Leibniz the generality of this approach seems to go beyond the unification of the

treatment of conic sections:

“In geometry, any method of invention by means of situation, and then without

calculation, consists in simultaneously encompassing several [objects] in the same

situation; this sometimes happens by means of a figure which contains some

others, where the use of solids can be found, sometimes by means of motion or

mutation. Besides, amongst motions and mutations, it seems very useful to resort

to the mutation of appearance, i.e. the optical transformation of figures; it remains

to be seen whether by these means we might surpass the earlier [treatment of] the

Cone and rise to higher [considerations]”. (translation by Valérie Debuiche)23

We cannot be sure what Leibniz is thinking of here, but we see well that the mutation of

appearance is very important and leads him to think of “higher considerations” – perhaps even of

a more general geometry, such as the geometry of situation.

The generality achieved in Pascal’s work should be regarded in connection with his relational

consideration of geometrical elements. In the Generatio conisectionum, Pascal addresses the

points of the circle that have a projective image on the hyperbola only at an infinite distance. He

calls these points “puncta non apparentia circuli, et respectu hyperbolae, puncta deficientiae”24.

We can thus see that Pascal gives two distinct names to these points: they are called the “points

without a projected image” (“puncta non apparentia”) of the circle, but “regarding the

hyperbola” (“respectu hyperbolae”) they are called “missing points” (“puncta deficientiae”).

While the first of these two names is based on the circle’s projective relation, the second is a sort

of relational denomination.

The relational aspect also appears on a scholium of the same treatise, in respect to projected

lines: “Est ergo in parabola recta deficiens, quae quidem vice fungitur tangentis, cum tangentis

sit apparentia”25. The verb “fungor” can be translated as “to perform a function”, “play a role”.

In this case, the “missing straight line” plays the role of the tangent, because (“cum”) it is the

22 I discussed this problem in João F. N. Cortese: „Infinity between mathematics and apologetics: Pascal’s notion of

infinite distance”, in: Synthese 192/8 (2015), S. 2379-2393. 23 “Omnis in Geometricis ope situs inveniendi ratio adeoque sine calculo in eo consistet, ut plura simul eodem situ

complectamur, quod fit tum ope figurae cujusdam plures includentis, ubi usus solidorum patet, tum ope motus sive

mutationis. Porro ex motibus et mutationibus utilissime videtur adhiberi mutatio apparentiae, seu optica figurarum

transformatio, nam et videndum an ejus ope possimus ultra Conum ad altiora quoque assurgere”. (A VII 7, 109) 24 Pascal: Oeuvres Complètes, Bd 2, S. 1114. 25 Pascal: Oeuvres Complètes, Bd 2, S. 1117.

projected image (“apparentia”) of the tangent. Projection thus makes it possible to conserve the

“role” played by a curve, a relational structure being preserved.

Pascal calls the ellipse antobola, because it “comes back to itself”. In his notes on Pascal’s

treatise on conics, Leibniz writes that the parabola shall be “conceived as if” (“concipit velut”) it

was an ellipse which comes back to itself at infinity26. This idea was indeed already present in

Pascal’s Generatio conisectionum: one can find a correspondence between the parabola and the

ellipse if one conceives the first together with a point at infinite distance, by the means of which

the parabola can be conceived as a “closed” curve. But what matters here is to discuss the

manner by which this correspondence, and the acceptance of this point, is made. For Leibniz,

this seems to be possible if we conceive something “as if” it were something else. This kind of

conception is surely relational; would it also be a way of considering mathematical objects as

fictions, as the verb “fungor” suggests?

I shall now address a passage in Leibniz’s notes about the relation between the coincidence of

points and the infinitely small distance, the same question we saw in De la méthode de

l’universalité.

For Leibniz, the parabola is considered as an intermediate curve between the ellipse and the

hyperbola, as it was for Pascal. The plane of the parabola is parallel to one of the “verticales”

(that is, the straight lines that generate the cone27), whereas the plan of the ellipse is not parallel

to any of the “verticales”, and the plan of the hyperbola is parallel to two of them.

Leibniz considers the circle BDC at an infinite distance, with center E, and the conic surface

generated by the movement of the straight line AB around this circle28. The straight lines AB, AD,

AT and AS are “verticales”. The plane RD is parallel to two of them, AT and AS, and thus

generates a hyperbola29.

26 “Ellipsin antobolam vocat quia in se recurrit: parabolam concipit velut Ellipsin infinite ab hinc in se recurrentem.

Hyperbola re vera non una linea curva in se rediens, sed duae” (in Pierre Costabel: „Traduction française de notes

de Leibniz sur les “Coniques” de Pascal”, S. 258. Leibniz read Pascal’s treatise together with Tschirnhaus, but most

part of the notes are from the hand of Leibniz (cf. Mesnard, in Pascal: Oeuvres Complètes, Bd 2, S. 1106). 27 Leibniz follows Pascal, calling this line “verticalis”, since it is a straight line that passes by the “vertex” of the

cone. 28 Differently from Pascal, Leibniz considers the circle at an infinite distance in order to generate an infinite cone.

Pascal himself also considered the cone as infinite, but generating it by the circumvolution of an infinite straight line

(Pascal: Oeuvres Complètes, Bd 2, S. 1108-1109). Leibniz makes reference to an “axem A”, and not to a point A,

which is confirmed by the drawing. I have no hypothesis about why he does this. 29 Leibniz’s representation is of half of the cone, making it easy to see how the plane RD intersects with it generating

one of the branches of the hyperbola. But one should bear in mind that the hyperbola is constituted by two branches,

both of them generated by the intersection of the same plane and the cone. Pascal’s works make an advance in

considering the hyperbola as one curve, and not two, but in his notes Leibniz seems to have difficulty in accepting

this, as we see in some of his notes on Pascal: “Hyperbola revera non una linea curva in se rediens, sed duae” (A VII

7, 108).

Figure from Conica pascaliana (image from A VII 7, 107)

Leibniz then says that it is not important whether the plane RD passes through TS or not – if this

is the case, we have a conic section constituted by one triangle in each half of the cone, which

Pascal called a “rectilinear angle”.

But Leibniz notices an exception:

“Excipe hunc unum casum, quo duo verticales AT, AS, infinite parvam habent

distantiam seu coincidunt in unum extremum, seu ipsam generatricem ut AC. et

tunc Hyperbola degenerat in parabolam (...)”. (A VII 7, 108)

The fact that in this same phrase Leibniz uses both the words “verticalis” and “generatrix” in

regard to straight lines that we would consider as having equivalent roles is puzzling30. But on

the whole the passage is a difficult one31.

How can we interpret it? The case of the “degeneration” of the hyperbola into the parabola

corresponds to the case when the two verticales are at an “infinitely small distance” from one

another. But is this case equivalent to them “coinciding at one extremity”, or to them coinciding

at a “generatrix such as AC”? The double usage of “seu” makes this phrase extremely ambiguous

– in principle, we could have one, two or three cases here32. But what about the geometrical

interpretation of this passage? I can only make sense of it if the three conditions are equivalent.

We should then say that the case of an infinitely small distance is said to be equivalent to the

case where there is a coincidence – just as we had in the De la méthode de l’universalité.

30 As the expression “vestigium generatricis AB” was crossed out some lines earlier, Costabel thinks that there was a

discussion between Leibniz and Tschirnhaus on the usage of this vocabulary, “verticalis” being more general than

“generatrix” (Pierre Costabel: „Traduction française de notes de Leibniz sur les “Coniques” de Pascal”, S. 262). 31 Costabel translates: “Le seul cas d’exception est celui où les deux génératrices AT, AS ont une distance infiniment

petite ou coïncident en un extrême ou en une même génératrice comme AC: et alors l’hyperbole dégenère en

parabole”. 32 In the continuation of the notes, Leibniz writes: “RD Hyperbola. AS verticalis sectioni Hyperbolae parallela

superior et inferior quae in parabola in unum coincidit” (A VII 7, 108).

This identification of the case of two verticales/generatrices coinciding and the case in which

they are at an infinitesimal distance to one another is apparently original from Leibniz in relation

to Pascal (even if Leibniz follows Pascal’s approach in several aspects, including the relational

presentation of geometrical elements)33. Apparently, from the point at an infinite distance, as

well as the correspondence between the infinitely great and the infinitely small (essential

according to Pascal)34, Leibniz arrives at an infinitely small distance that, perhaps for the sake of

the generalization, is said to be equivalent to the coincidence of the lines.

Desargues, rather than Pascal, had indicated something close to this in his own terminology. In

some passages of the Brouillon Project, Desargues, considering pairs of points in relation to a

reference point (the “stump”), writes that if one them is at an infinite distance, the other is

“joined” or “united” to the stump:

“And when the inner knot of a pair of extreme knots is joined or united to the

stump of the tree, the outer knot of the same pair is at an infinite distance on the

trunk: And contrariwise”35.

Not only this, but in two well-known paragraphs at the beginning and at the end to the Brouillon

Project Desargues wrote about “quantities so small that their opposing extremities are united”36.

Thus for Desargues, the reciprocal of an infinite distance is found when two points are “united” –

do we have here a clue for understanding Leibniz’s association of coincidence and infinitely

small distances? Even if we do not know exactly what Leibniz had read of the Brouillon Project,

this could be an interesting aspect of similarity between them.

3. Elementa nova matheseos universalis

33 Cf. Valérie Debuiche: „L’expression leibnizienne et ses modèles mathématiques”, S. 420. 34 Cf. the text De l’esprit géométrique, in Pascal, Oeuvres Complètes, Bd. 3, S. 390-412. 35 Translation from Judith Field and Jeremy Gray: The Geometrical Work of Girard Desargues, New York 1987, S.

81, with modifications – I write “is joined or united” rather than “is coincident or identified”, preserving the

literality of the text . 36 A detail here is of the major importance, as for this passage at the beginning of the text “Desargues originally

wrote ' ... [quantities] which decrease so as to reduce their two opposing extremities to one' (' ... qui s'apetissent

jusques it reduire leurs deux extremitez opposees en une seule .. .'). We have adopted his revised version, given in

the Notice appended to the work, which reads ' ... si petites que leurs deux extremitez opposees sont unies entre

elles'. It will be noted that this second version removes the reference to change” (Judith Field and Jeremy Gray: The

Geometrical Work of Girard Desargues, S. 204, note). This means that Desargues hesitated before choosing the

vocabulary of “to unite” (“unir”). Field and Gray choose to translate this verb by “coincide”, whereas I preserve

“united”.

I shall now consider the text Idea libri cui titulus erit Elementa Nova Matheseos Universalis (A

VI 4, 513-524)37. This text was written in the summer of 1683, and thus not very long after

Leibniz’s Parisian period where he was in contact with the works of Desargues and Pascal.

In this text, Leibniz distinguishes between quantities that are “impossible by accident” and those

that are “absolutely impossible” because they imply contradiction. Imaginary quantities are

impossible by accident because it is not possible to exhibit them, lacking the possibility to

“produce an intersection”. In this sense, says Leibniz, they can be compared to infinite and

infinitely small (“parvis”) quantities.

The example given is the following: a straight line AC is perpendicular to an “undefined” straight

line AB – that is, point A is given but point B on this line is undefined (we recall here the

“moveable” points from De la méthode de l’universalité).

(Image from A VI 4, 521)

If a secant is drawn from C to any point B on AB, such as 1B, 2B, or 3B, the more (“prout”) the

angle between CB and AB approaches a right angle, the shorter AB becomes, “so far as in a right

angle, in which 1B falls in A, that is, A1B is infinitely small or null”38. We find here again the

case in which we are interested: an infinitesimal distance is said to be equivalent to a coincidence

of points (when the distance is “null”).

The opposite can also happen: the more “the angle or the inclination” between the lines CB and

AC “approaches the Parallel”, the greater AB will be, so that A4B is bigger than A3B39. “And

37 In discussing this text, I use the French translation to be published by the Centre d’Études Leibniziennes -

Mathesis. Discussion with this group, especially with David Rabouin and Valérie Debuiche, was of great value in

shaping the ideas of the present article. 38 “Patet prout recta CB ad rectam AB angulum facit propiorem recto eo minorem esse AB, adeo ut in casu anguli

recti 1B incidat in A, seu A1B sit infinite parva sive nulla” (A VI 4, 521). 39 “Contra quo angulus vel inclinatio rectae CB ad rectam AC magis accedit ad Parallelam eo major erit AB, ita A4B

major quam A3B” (A VI 4, 521).

when the straight line CB is completely parallel to AB, then the common point B is imaginary or

null, that is at infinite distance, and the straight line AB is infinite”40.

In what follows, Leibniz writes that imaginary quantities of this kind are very useful for

universal constructions, including Conics. Leibniz is interested here in the fact that in one case

the solution comes from the angle, whereas in the other it comes from the parallelism between

the lines – Leibniz writing in the margins: “because parallelism is not truly an angle”41. But the

“truth of the calculation”, says Leibniz, leads us to the necessity of such quantities, which could

seem as an absurdity for some. Nevertheless, this “apparent impossibility” can be surpassed if

one understands that we have to draw parallel lines in this case, and not an angle between the

two lines, and “so the parallelism is the angle or quasi-angle that is searched for”42.

The fact of taking parallelism as if it were an angle (we should notice the usage of the expression

“quasi-angle”) shows that in this text Leibniz also sees geometrical elements as relational, also

regarding the consideration of coincident points as equivalent to an infinitesimal distance.

4. Conclusions

We have seen that in three texts from the period 1674 to 1683 Leibniz associates the coincidence

of points and infinitesimal distances. While it is difficult to determine a direct influence from the

works of Pascal and Desargues, we did see that Leibniz’s approach, which allows the

generalization of the geometrical expressions (arrangements of points on a line, equation for the

conic sections) is at least compatible to the projective works of his predecessors. We can also see

a similarity in the relational aspect of geometrical objects and in the correlation between the

infinitely great and the infinitely small. Even if all these features were not suggested to Leibniz

by projective geometry, he could at least recognize similarities with his works on a geometric

characteristic that would serve a more general geometry.

Would the fact of considering the coincidence of points and infinitesimal distances as equivalent

have influenced Leibniz to consider the infinitesimals as fictions? I won’t discuss this chapter of

the history here. What I claim is that in the texts analyzed we do find a “relational reading” of

geometrical elements, including the case of infinitely small distances, which allows a broader

generality in geometrical expressions to be achieved.

40 “Et quando recta CB fit omnino parallela ipsi AB, tunc punctum commune B, est imaginarium seu nullum, infinite

scilicet hinc distans, et recta AB, est infinita” (A VI 4, 521). 41 “parallelismus enim revera non est angulus” (A VI 4, 521, Am Rande). 42 “hunc parallelismum esse angulum illum seu quasi angulum quaesitum.” (A VI 4, 521).