XXXV111 THE GRAPHICAL CALCULUS. 



transformation plays in the graphical calculus.) Such was the problem 

 which Cousinery proposed, and whose solution he attempted in his " Calcul 

 par le trait " (Ses Elements et aes applications. Paris, 1839). 



Cousinery applied the graphical calculus to powers, roots, proportion 

 and progression; to the measure of lines, surfaces, cubes, graphic inter- 

 polation, and the stability of retaining walls. The presentation is nat- 

 urally by no means complete, and labors also under a prolixity and 

 minuteness of detail to which the results obtained are by no means com- 

 mensurate. It sounds somewhat comic when Cousiuery, in his " Calcul par 

 le trait" claims the then already-existing graphical solutions of Poncelet 

 (" Memoire sur la stdbilite den revetements, in Memorial de Voff du genie' 1 ' 1 ) 

 as an elegant example of the application of his graphical calculus. 



While Cousinery thus sought to apply geometry in a direction where 

 until then analysis had held sway, he acted in entire accordance with the 

 spirit of his age, though without making use of those means for aid which 

 lay at his disposal. " Without effect upon him," says Culmann, " were 

 the researches of Steiner, already published in 1832, as well as those of 

 his predecessor ; and instead of simply premising the elementary prin- 

 ciples of the modern geometry, he laboriously sought to deduce them in- 

 dependently by the aid of perspective." The works, at least, of the French 

 predecessors of Steiner were, at any rate, well known to Cousinery. In his 

 preface we read : " Peut-6tre me'me nos efforts eussent-ils fite" complete- 

 ment infructueux, sans les ressources que nous ont procurers et les annales 

 de M. Gergonne et les travaux de M. Brianchon, et ceux plus re"cents de 

 M. Poncelet. Nous avons envers M. Chasles une obligation encore plus 

 droite, car outre les prgcieux documents que renfernie son ' Histoire des 

 methodes en geometric? nous avous fi lui faire agrer un tgrnoignage par- 

 ticulier de reconnaissance pour la maniere dont il a bien voulu mentionner 

 nos premiers essais sur le systeme de projection polaire." 



Why Cousinery made use of perspective and not of the modern geome- 

 try, is easily understood. The development of geometry at that time, as to- 

 day, proceeded in various almost independent directions, and Cousinery 

 himself had the pleasure of seeing his " Geometric perspective " (Paris, 

 1828) designated by the reporters for the Academy, Fresnel and Matthieu, 

 as new and ingenious, as well as favorably noticed by Chasleit.* He 

 sought, therefore, naturally to develop and render fruitful his own method, 

 so much the more as the true significance and value of the various growing 

 branches of geometry could not then, as now, be correctly estimated. Ac- 

 cordingly, the Inge"nieur-en-chef. B. E. Cousinery, wrote avowedly for his 

 colleagues, and did not feel justified in directly premising a knowledge of 

 the newest investigations, more especially of his own. 



We have noticed the above somewhat in detail, because it bears directly 



* Its newness, at least, is not without doubt. According to Fiedler, the 

 principles are completely given in Lambert's celebrated work, " Die freie 

 Perspective" (Ziirich, 1759). Poncelet also takes issue with the estimation of 

 the " Geometric perspective" by Chasles (" Traite des propr. prey.," IL, 6d. 

 1865, p. 412> 



