Perspective in Leibniz’s Invention of Characteristica Geometrica : The Problem of Desargues’ Influence

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  This article appeared in a journal published by Elsevier. The attachedcopy is furnished to the author for internal non-commercial researchand education use, including for instruction at the authors institutionand sharing with colleagues.Other uses, including reproduction and distribution, or selling orlicensing copies, or posting to personal, institutional or third partywebsites are prohibited.In most cases authors are permitted to post their version of thearticle (e.g. in Word or Tex form) to their personal website orinstitutional repository. Authors requiring further informationregarding Elsevier’s archiving and manuscript policies areencouraged to visit:  Author's personal copy Available online at ScienceDirect  Historia Mathematica 40 (2013) 359–385 Perspective in Leibniz’s invention of   Characteristica Geometrica :The problem of Desargues’ influence Valérie Debuiche SPHERE (UMR7219), Université de Paris 7-Denis Diderot, Bâtiment Condorcet,5 rue Thomas Mann, 75205 Paris Cedex 13, France Available online 1 November 2013 Abstract During his whole life, Leibniz attempted to elaborate a new kind of geometry devoted to relations and not to magnitudes,based on space and situation, independent of shapes and quantities, and endowed with a symbolic calculus. Such a “geometriccharacteristic” shares some elements with the perspective geometry: they both are geometries of situational relations, founded in atransformation preserving some invariants, using infinity, and constituting a general method of knowledge. Hence, the aim of thispaper is to determine the nature of the relation between Leibniz’s new geometry and the works on perspective, namely Desargues’ones. © 2013 Elsevier Inc. All rights reserved. Résumé Durant toute sa vie, Leibniz a cherché à élaborer une nouvelle géométrie, consacrée aux relations et non aux grandeurs : unegéométrie de l’espace et de la situation, indépendante des figures et des quantités, et dotée d’un calcul symbolique. Une telle“caractéristique géométrique” partage certains éléments avec la géométrie perspective. En effet, toutes deux sont des géométriesdes relations situationnelles, fondées dans une transformation qui préserve des invariants, emploie l’infini et constitue une méthodegénérale de connaissance. Dès lors, le but de cet article est de déterminer la nature de la relation qui existe entre cette nouvellegéométrie leibnizienne et les travaux sur la perspective, notamment ceux de Desargues. © 2013 Elsevier Inc. All rights reserved.  MSC:  01A45; 51-03 Keywords:  Leibniz; Desargues; Perspective; Geometric characteristic; 17th century; Invention 1. Introduction At the end of the 1670s, after he came back from Paris, where between 1672 and 1676 he greatlydeveloped his knowledge of mathematics, Leibniz initiated his work on  Characteristica Geometrica . Hismain purpose was to form a geometry of relations, without magnitudes, independent of figures, embedded  E-mail address:$ – see front matter  © 2013 Elsevier Inc. All rights reserved.  Author's personal copy 360  V. Debuiche / Historia Mathematica 40 (2013) 359–385 in a complete axiomatic and endowed with an expressive symbolism. It was supposed to be both a calculusand a new geometry which would surpass the Cartesian one and include the Euclidean one. In order toelaborate such an innovative geometry restricted to qualitative relations, Leibniz developed a geometry of “space” and “situation.”Despite appearing to be a fundamental innovation, the Leibnizian geometry exposed in the first texts of  Characteristica Geometrica  seems to be close to the almost contemporary works of Desargues and Pascalconcerning geometry in general and perspective in particular. Indeed, as a geometry of quality, anchoredin the essentially relational concept of   situs , in the notion of   extensum , and in the notion of   congruentia  asa preserving transformation which, as such, is propitious to the knowledge of unvarying things, geometriccharacteristic resembles the perspective projection of Desargues and the geometry of conics of Pascal. Thisresemblance, together with explicit references to perspective by Leibniz (1675a, 359; 1677, 62), whilehe was forming the idea of his innovative geometry of situation, suggests the possibility of a relationbetween the two approaches to geometry. This possibility has been studied deeply by Javier Echeverría(1982, 1983, 1994), who presented the objective elements of Leibniz’s knowledge of Desargues’ workson perspective and concluded in favour of the real but limited “influence” of Arguesian 1 perspective onLeibniz’s invention of   Characteristica Geometrica .In general, it is problematic to affirm that Leibniz discovered in Desargues (or in Pascal) the germ of hisinvention of   Characteristica Geometrica . Usually Leibniz willingly admits what he owes to the theoreti-cians who influenced him in one way or another. But, in this matter, he never says, to my knowledge, thathis ideas concerning geometric characteristic or parts of it come from Desargues’ works, even though hehas no reason whatsoever to hide his continuity with Desargues because of the obvious srcinality of hisown work. Moreover, Leibniz frequently refers to Viète and Descartes whose algebraic method, accordingto him, needs to be overtaken by a new modern geometry. He also mentions Euclid whose  Elements  offer acriterion to evaluate the efficiency of his geometric characteristic. Within the framework of these references,we may wonder whether it is justified to claim the existence of a ‘positive’ inspiration of Desargues, ratherthan the existence of a ‘negative’ influence of Viète and Descartes, or of Euclid’s methodological limita-tions. Nonetheless, the question of Desargues’ influence is still interesting, not only in order to determinethe effective degree of such an influence on Leibniz’s invention of   Characteristica Geometrica , but also soas not to miss a possible connection between perspective and geometric characteristic. My purpose is thento read the first texts of geometric characteristic in the light of Desargues’ methods, above all to be ableto delineate the specific srcinality of Leibniz and the very nature of his geometry. Indeed, the perspectivemethod is certainly at least a model for Leibniz, his doctrine of science and knowledge, and his conceptionof space as related to  situs . But, to find a theory to be a model is not to use it as a foundation nor, even, doesit imply influence. As a matter of fact, while Leibniz was aware of the power resulting from the innovationsof perspective – such as, for instance, the identification of the points at infinity and the points at a finitedistance – and had the intellectual capacity to extrapolate them, including in his own new geometry, yet hedid not. Hence, the question becomes: what kind of clues does such a lack give us for our understanding of Leibniz’s ingenuity in geometry?A substantial secondary literature about the relation between Leibniz and Desargues exists, namely inthe French language. A first period (covering the 1960s and 1970s), depending on the edition of Desargues’texts, includes the fundamental works of Jean Mesnard (1978) and René Taton (1951, 1978). The impor- tant essay by Joseph E. Hofmann,  Leibniz in Paris. 1672–1676   (Hofmann, 1974) should be considered aspart of this set of French commentaries for the precision and the completeness of its analysis. 2 A second 1 In analogy with the term ‘Cartesian,’ I have chosen to use ‘Arguesian’ to refer to the ideas or elements coming from Desargues’posterity. 2 I may also refer to the papers by Hans Freudenthal (1972), Herbert H. Knecht (1974) and Hans Peter Münzenmayer (1979) which offer different studies about the Leibnizian  Analysis situs  itself.  Author's personal copy V. Debuiche / Historia Mathematica 40 (2013) 359–385  361 period of secondary literature (at the end of the 1980s and 1990s) is devoted to the invention of geometriccharacteristic.Javier Echeverría produced acrucial piece oftranscription workandedition ofthefirstessayson  Characteristica Geometrica  (1995). Graham Salomon’s dissertation (Salomon, 1989) and the papers by G.G. Wallwitz (1991) or E. Giusti (1992) deal with some aspects of its genesis or its posterity. 3 Exceptfor the previously cited papers by J. Echeverría about the relation between Leibniz and Desargues, 4 noneof the previous contributions presents a detailed analysis of the relation between Leibniz and Desargues.A third period is currently beginning with the important work of Kirsti Andersen (2007) about perspectiveand an srcinal reading of the later Leibnizian  Analysis Situs  by Vincenzo De Risi (2007). But, once again,the link between perspective and  Geometria Situs  is not the main topic.Thus, my goal is to show that, while inventing his  Char acteristica Geometrica , even if Leibniz some-times presents the perspective method as a model for his own project of geometric characteristic, or assome anticipatory applications of it, his invention of a new geometry is relative to prior requirements con-cerning general characteristic rather than to the geometrical innovations and specificities of perspective,although perspective contains some great inventions for the emergence of a qualitative geometry of situa-tion and space. In order to establish the undeniable methodological and conceptual srcinality of Leibniz’swork, not only despite the similarities between his and Desargues’ conceptions, but also  thanks to the ever-meaningful incompleteness ofthesesimilarities , Iwillpresent, inSection 2,the  Characteristica Geometrica as it is exposed in the first formulations that Leibniz produced in 1677–1680. I will insist on the doublenature of geometric characteristic as ‘characteristic’ and as ‘geometry.’ Then, in Section 3, I will elabo-rate on the nature of the continuity which might be asserted from Desargues to Leibniz. This will requirespecifying some philological difficulties and, depending on them, drawing as precisely as possible the paththat goes from Arguesian perspective to Leibnizian invention. In Section 4, I will establish that this pathis not sufficient to unconditionally link Leibniz to Desargues, since their shared ideas are not specificallyrelated to Leibniz’s knowledge of previous Arguesian works. They also depend on his own conceptions of geometrical methods and symbolic requirements in his earlier thoughts. Finally, in Section 5, I will drawconclusions about the possibility of others sources for Leibniz’s invention of   Characteristica Geometrica . 2.  Characteristica Geometrica  from 1677 to 1680 2.1. The project of Characteristica Geometrica In January 1677, in Hannover, Leibniz wrote a text entitled  Characteristica Geometrica  (Leibniz, 1677),which begins with a clear explanation for the Leibnizian project of a new geometry: Analysis Geometrica nondum habetur absoluta. [ ... ] Cogitavi dudum mederi huic imperfectioni, et efficere,ut in calculo tota figurae ratio situsque appareat, quod alioqui fieri non solet. Contenti enim sunt Analyticimagnitudines ad calculum revocare situs vero in figura supponere, unde figuris et linerarum ductu atqueimaginationis opera carere non possunt. 5 [Leibniz, 1677, 50–52] 3 Around the same time (in the 90s), a new interest appeared in Desargues’ work and the geometrical aspects of perspective,thanks to the English translation of the  Brouillon Project   (Field and Gray, 1987), the collective book  Desargues en son temps (Dhombres and Sakarovitch, 1994), and some papers about the projective method (Bkouche, 1991; Le Goff, 1994). But, in none of  these references, was the relation between the invention of   Characteristica Geometrica  and the perspective or projective methodexamined. 4 Some details about that subject can also be found in an article on Desargues’ posterity (Lanier and Le Goff, 1991). 5 “AperfectGeometricalAnalysishasnotsofarbeenachieved[ ... ]Irecentlythoughttoamendthisimperfection,andaccomplishwhat is besides not habitual, so that all element concerning figure and situation appears in calculus. Indeed, analysts are satisfiedwith introducing magnitudes in calculus, while subsuming situation to figures, so that they cannot abstain from the drawing figuresand lines, and from the labor of imagination.”  Author's personal copy 362  V. Debuiche / Historia Mathematica 40 (2013) 359–385 In fact, Leibniz envisioned the possibility of a new geometrical method without the two main flaws inCartesian geometry: the lack of completed analysis and the use of figures. He then strove to elaboratea geometry of   new  elements out of which a complete analysis could be elaborated. As such, this newgeometry had to become characteristic, that is to say analytically founded, combinatorial, and symbolic.Thus, Leibniz’s project appears doubly innovative: it is to include both a new “geometry” and an srcinal“characteristic,” as will be presented now. 2.1.1. A characteristic project as a heuristic method  The 1677 text clearly reveals the characteristic dimension of the  Characteristica Geometrica  project.Right from the beginning, the author suggests some rules for the use of characters: a point is designated bya letter  A ,  B , a line is expressed by a formula  1 B 2 B 3 B , and a straight line by an equation  1 B 2 B + 2 B 3 B = 1 B 3 B  (Leibniz, 1677, 52). Leibniz also gives several equations for other  loci : a circle, an angle, and a rightangle. Then, characters must “express” (“ exprimere ”) the nature of objects, that is to say, in Leibnizianterms, the composition of the notional content of an object from more elementary notions. For example,the notion of ‘line’ is broken into “ multorum punctorum locus ” (Leibniz, 1677, 52), so that the notion of ‘point’ appears more primary than that of ‘line.’ Thus, in order to choose characters which are the most ableto express the nature of geometrical objects, it is required to form the alphabet of the fundamental notions,which enables Leibniz to elaborate the set of corresponding characters. Nonetheless the relation betweenthe characteristic expression and the geometrical object is not only a matter of translation from geometricalconcepts into symbolic expressions. It is also and above all a method of knowledge by transposing the relations  between geometrical objects into  relations  between signs in order to discover, through a simplecalculus, something new about signs which corresponds to something intrinsic to objects themselves. Forinstance, to prove the similarity between two surfaces or sectors, it is sufficient to remark or establishthe similarity between their expressions (Leibniz, 1677, 56–58). Indeed, similar objects are supposed toreceive similar expressions, if these expressions are well-established and rightly represent the geometricalcomposition of objects. Reciprocally, any proof of the similarity of expressions is sufficient to prove thesimilarity of objects.The heuristic power of characteristic implies a double invariance: the preservation of conceptual com-position, i.e. of internal conceptual relations, from the objects into their expressions, and the preservationof external relations between geometrical objects in their corresponding formula. In any case, it suggeststhe preservation of relations. The question of preservation actually is the core of the characteristic method,since without such an invariance, nothing could be known, but without the incompleteness of preservation,expression and the thing expressed could not be distinguished. Thus, the issue is to delineate what hasto be preserved from the object into its expression. That implies geometrically defining objects, in theirinternal but also relational nature, with regard to the necessity of invariance; establishing the modality of their external relations according to some invariant elements; and setting the rules of calculation, i.e. of the passage from an expression into another, without losing the invariant elements in and between objects.All these elements concern the geometrical part of   Characteristica Geometrica , as will be developed in thefollowing section. 2.1.2. A new, tentative geometry As far as the “geometric” aspect is concerned, the 1677 essay is not really completed, or even deeplydeveloped. Nonetheless, it shows some hesitancy which reveals what Leibniz considers to be important. 6 Indeed, he defines the line as the  locus  of many points (Leibniz, 1677, 52), and he later corrects himself,when he mentions the straight line but without any specific consideration of the property of straightness: 6 The consequences of this point will be developed in Section 4.2.1.
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