Desargues’s concepts of involution and transversal, their origin, and possible sources of inspiration

Abstract

In this paper, we try to understand what considerations and possible sources of inspiration Desargues used to formulate his concepts of involution and transversal, and to state the related theorems that are at the basis of his Brouillon project. To this end, we trace some clues which are found scattered throughout his works, we connect them together in the light of his experience and knowledge in the field of perspective, and we investigate what were his motivations within Mersenne’s academy. As a result of our research, we can safely say that were his great geometrical insight and his projective vision of space which, guided by some classical theorems, led him to these completely new concepts in the panorama of the geometry of that time that were destined to remain misunderstood for about two centuries.

Appendices

Appendix 1

1.1 Desargues’s geometrical propositions and concept of transversal rooted in perspective

One of the basic problem in perspective was to draw correct images of simple plane figures, as triangles, quadrilaterals, and circles; and the theoretical perspective, which developed in the late sixteenth century, aimed at giving mathematical evidence to support the experimental techniques of practitioners. In Giovanni Battista Benedetti’s Diversarum (1585) and Guidobaldo del Monte’s Perspective libri sex (1600), which according to Andersen (2007) greatly influenced theorists of perspective in the seventeenth century, geometrical constructions are found concerning the aforementioned issues, which were also re-proposed some years later in François Aguilon’s treatise (1613), a widespread book in the Flanders and France. We believe that these constructions, and the related figures, were an inspiration for Desargues in conceiving the first two geometrical propositions and the notion of the transversal of an ordinance of lines.

The first one we consider is the one del Monte provided for proposition VI in Book II, concerning the drawing of the perspective image in the plane of picture VBO of the triangle CDE, as seen from the eye A. The related original figures, here reproduced in Fig. 19a, b, make clear that the corresponding sides of the two triangles (or their prolongations) meet along the ground line BO.

Fig. 19

a Reproduces the figure for proposition VI in del Monte (1600, 62), for the construction of the image LMN of the triangle CDE on the plane VBO seen from the eye A. The two triangles in perspective are not emphasized in the original figure. b Reproduces the figure illustrating the “praxis” for the previous construction (del Monte 1600, 63). The dashed lines, not drawn in the original figure, make clear that the two triangles are in perspective with respect to the eye A

The second construction we want draw attention to is related to the perspective image of a (complete) quadrilateral in a plane not orthogonal to the ground plane as described in Benedetti (1585), and illustrated in his figure D, here reproduced in Fig. 20.

Fig. 20

Reproduces the figure “D” in Benedetti (1585), showing two (complete) quadrilaterals in perspective from the eye o, uadq and its image ercm, whose sides, or prolongations, meet along the ground line qx. In this figure, we have added the tiny straight lines not drawn in the original figure, and marked μ, φ, ϕ, ϛ the corresponding points of intersection

This figure, which clearly represents the vertical version of the same construction as performed with the “sportello”—an instruments often used by the practitioners of perspective (Andersen 2007)—shows, or at least suggests, that the corresponding sides (or their prolongations) of two perspective quadrilateral meet along the ground line.

Although we cannot claim that Desargues knew del Monte and Benedetti’s works, these constructions, or similar ones which found place in various manual on perspective after (Barozzi 1583), did not escape to him (Fig. 21).

Fig. 21

This is the “horizontal” version of the previous figure, which show the construction as it was performed by means of the “sportello”. The corresponding lines of the two perspective quadrilaterals meet along the ground line qx

Finally, we want to mention another very inspiring construction which is presented in del Monte (1600, 217), and in Aguilon (1613, 669), regarding the perspective image of a circle, and whose related figure is here reproduced in Fig. 22. Without entering into a detailed description of such a construction, for which we refer the reader to the just-mentioned works, we briefly comment on the figure illustrating it, but having in mind that Desargues was not only well versed in the art of perspective but also an expert geometer accustomed to dealing with points at infinity.

When the circle BCDE is represented in the plane of picture GRVX as seen from the “eye” A, parallel tangents to the circle are projected in converging tangents to its image; to the directions, CG and DF correspond the vanishing points V and X, respectively, and so, the line at infinity of the plane of the circle is projected in the line VX of the plane of picture; the centre H of the circle, “pole” of the line at infinity with respect to the circle, is projected in the point Q, “pole” of the straight line VX with respect to the conic section image of the circle. All this appears very clear if one keeps to mind the figure IV attached to the Brouillon project (Fig. 12).

As the ordinates of a general ordinance are divided in equal parts by the orthogonal diameter WHW′ of butt Σ_∞, and vice versa, the corresponding points on the image, as w, Q, w′, Σ form a harmonic set. This is graphically illustrated in Fig. 23, which also shows that since the fourth harmonic point of the centre of a circle (which divides in two equal parts any diameter) with respect to any two diametrical points lies on the line at infinity, the fourth harmonic of their perspective images lie on the horizon line XV.

Fig. 22

This diagram is an advanced version of the figure in del Monte (1600, 217), giving the construction of the perspective image of a circle from the eye A in the plane of picture YXVR. We see that the centre H of the circle, pole of the line at infinity of the ground plane, in which lie the butt Y_∞ of the straight lines GC, YL, ID, etc. and the butt Z_∞ of the straight lines FD, ZC, etc., is projected onto the point Q “pole” of the straight line XV, into which the line at infinity is transformed. To the polars ID and CZ, of the points Z_∞ and Y_∞, correspond the polars IV and XZ, of the points X and V, respectively

Fig. 23

This diagram illustrates the graphical construction in the plane of picture of the fourth harmonic Σ of Q with respect to w and w′

We cannot fail to say that Fig. 22 brings to mind the phrase, here reported in Sect. 1, with which Desargues ended his first essay on perspective (1636).

Often in geometry, a drawing can say more that many words.

Therefore, we believe that also the theory of the transversal of an ordinance of straight lines (that is of “polar”) that Desargues expounded in the Brouillon project found its origin in the practice of perspective (Fig. 23).

Appendix 2

2.1 Extracts from Desargues’s letter to Mersenne

In this second appendix, we record and briefly comment upon some extracts from the letter that Desargues addressed to Mersenne on the 4th of April 1638. As already said, Desargues wrote this letter on the occasion of returning to Mersenne a certain manuscript of Descartes, that the Minim had given him to read. The main object of the letter concerned the dispute between Descartes and Fermat about their different methods for constructing the tangents to a conic section from a given point outside the curve, or at a point on it, and also the motivations that Roberval and Étienne Pascal had made known to him in support of Fermat’s thesis. Much of the question regarded the use of specific properties of the curves in question, and Fermat’s methods for treating the cases of the ellipse and of the parabola separately. We do not enter here in technical details about this subject, referring for them to Grégoire (1992), but we notice that Desargues took the opportunity to express his own opinion on the controversial matter, and in doing that to stress his view about the concept of “general method” for the problems concerning conic sections, and, above all, to anticipate some of his achievements in this field.

Desargues did not share Roberval and Pascal’s opinion, and giving more credit to Descartes he specified:

Selon ma maniere de proceder universelle j’auray raisonné selon cette façon, tant au sujet de la parabole que de l’autres coupes de cone, comme estant une chose commune a toute les coupes [dont je scay bien que ils n’ont pas accoustumé de faire mention comme d’une proprieté generalement commune a toutes coupes, mais ils en font deux especes de proprietez, une particuliere a la parabole et l’autre particuliere aux autres coupes où je voy qu’il n’ont pas…]⁶⁵

thus, according to Desargues, as regard the question of tangents, one should look at the ellipse and the parabola as the same kind of curve, because tangency had to be treated in the same way for all conic sections. In fact, few lines below he added:

car si la methode est generale les mesmes motz exprimants une mesme proprieté doivent convenier et servir a chacune espece de coupe.

Then, Desargues continued by saying that he had told Mydorge how he was surprised in seeing that such well-versed mathematicians had a veil in front their eyes:

qui leur face constituer un genre particulier de ligne des seules touchantes aux coupes de cone, different en toutes choses d’avec celles qui traversent les mesme coupe de cone quand ces lignes (que j’entens droites) viennent d’un mesme poinct.

Desargues, alluding to his theory of the transversal of an ordinance of lines (polarity) as will be clear shortly, meant that tangents should not be distinguished from secants when all these straight lines belong to the same ordinance; in fact, he continued by saying:

Et moy que vous scavez qui n’ay de connoissance de ces matieres que pour mes propres et particulieres contemplations, je m’enhardy lors de dire à M. Mydorge, contre son attente et son opinion, que par mes contemplations capricieuse du cone rencontré par divers plans en toute façons, et des lignes et des figures qui s’engendrent en cette rencontre,⁶⁶ j’ay trouvé que par un seul et mesme enonciation, constrution et preparation ou pour dire mieux par un seul et mesme discours et sous de mesmes paroles, on declare un moyen de construire ou bien on declare les moyens de faire une construction d’un autre ordre par lequelle on voit egalement une pareille generation en toutes especes de plates coupes de cone, de toutes especes de lignes droites qui ont et reçoivent des ordonnés, comme diametres et autre, et l’on voit semblablement une pareille generation en chaque espece de plate coupe de cone, de toute les especes d’ordonnées qu’il y a pour chaque espece de lignes qui reçoivent des droites ordonnées. Et l’on voit une pareille generation à mesme temps de toutes leurs touchantes, chacune de ces touchantes estant membre d’un des corps de ces diverses especes d’ordonnèes.

As Taton remarked (1951a, 84, footnote 1), Desargues was referring to his notion of ordinates as the straight line passing through the “pole” of a straight line given in the plane of the conic section, which generalizes the notion of ordinates with respect to a diameter of the curve, all parallel to the direction of the conjugate diameter. He went on writing:

Et semblablement par un autre seul et mesme discours et constrution on voit une pareille generation en chaque espece de coupe de cone, des pointz qu’on nomme foyers, et en suite leur situation et quelquez proprietez commune entre eux en chaque espece de coupe de cone. Le tout sans faire bande a part pour la parabole et sans en exclure le cercle, non plus pour les foyers que pour les diverses especes des droites qui reçoivent des ordonnées, ny pour les diverses especes d’ordonnées.

Therefore, in his study of conic sections, which aimed at expounding their common properties, Desargues did not distinguish them in kinds,⁶⁷ and did not consider the parabola and the circle as separate cases, neither for the foci nor for the various species of straight lines which generalize the diameters and their ordinates.

We also understand that Desargues, like Kepler before him, admitted the second focus of the parabola, which though placed at infinity enjoyed the same properties as the ordinary ones.

Remarking the novelty of his method for treating conic sections, Desargues wanted to point out:

Et aussi sans employer pour cela aucun des triangles par l’essieu ny faire distinction d’un principal diametre d’avec les autres entre lesquels on distingue nettement les essieux en chaque figure. Je scay bien qu’ils n’ont faict mention que d’une seule espece de lignes qui reçoivent des ordonnées assavoir les diametres seulement en chaque figure, et d’une seule especes d’ordonnées en chaque figure, de quoy je m’estonne car je trouve que dans un mesme genre il y a deux especes de chaqune de ces sortes de ligne.

It is evident that some of mean Desargues’s geometric conceptions were already outlined at the beginning of 1638, as the fruit of his own particular and capricious contemplations of the sections of cone by different planes in whatever position, and of the lines and curves which these sections generate. We cannot but think to his Leçons de ténèbres, and we wonder whether his “contemplations” referred also to optical experiments he may have conducted.

Afterward, Desargues returned to discuss the objections Fermat had raised to Descartes’s method for the determination of the tangents to a curve as expounded in the Géométrie, and the opinion that Mydorge expressed in this regard, to conclude that his thinking was closer to that of Descartes than Fermat. At the end of the letter, Desargues wrote:

Quand à sa Geométrie [Descartes’s] j’en entens quelque chose, mais si j’osoy l’en importuner ou vous, je seroy bien aise d’en avoir un peu de plus familière explication pour mon esprit grossier, et puisque l’auteur est vivant, estre délivré du travail necessaire à son deffault pour m’ajuster asseurement à sa pensée notamment dès l’entrèe de la matiere. Et quoy que disent nos Mess.rs de Beaugrand et autres, j’ay suject de supçonner qu’ils ne entendent pas a fonds, je veus dire qu’ils ne possedent pas bien plainement toutes les intuitions de Monsieur des Cartes au suject de sa Geometrie.

These final words suggest that, although not all of Descartes’s Géométrie was clear to him, Desargues perceived its spirit and importance, and saw, with some regret, the profound incursion of algebra into geometry; but also that he was eager to go more deeply into the matter.

From the letter as a whole, it seems that Desargues’s desired to demonstrate publicly that his general methods allowed one not only to get where the application of algebra had conducted geometry, but also beyond.