THE GRAPHICAL CALCULUS. XXXIX 



upon a point of our discussion ; for the introduction of the modern 

 geometry in the graphical method by Culmann, is still, thirty years after 

 Cousinery, a chief hindrance to its rapid spread.* 



After Cousinery, no one occupied himself with the graphical calculus 

 till Culmann gave it a place in his Graphischs Statik. The presenta- 

 tion is here far better, and especially shorter. The rule of signs, which 

 was unknown to Cousinery, is at once brought out. Instead of such 

 long and tedious applications as the graphical interpolation, a few 

 examples from engineering practice are given, among which we may 

 especially notice earth-work "calculations. In the extensive earth works of 

 roads, canals, and railways, the method shows not only most plainly the 

 extent and best arrangement of transport, but also allows, with the aid of 

 the plani metre, the cost of transport to be determined. 



As to the rest, it would appear as if the graphical calculus should play 

 an important pail in engineering practice. This circumstance, as well as 

 the interesting problems which present themselves in connection, haa 

 gained for the Ariihmography many friends. Several publications 

 have since sought to win for it a wider recognition without furnishing 

 anything essentially new. [H. Eggers : " Grundzuge einer graphischen 

 Arithmetic," Schaffhausen, 1865. J. Schlesinger : " Ueber Potenzcurven," 

 Zeitschr. d. osterr. Arch. u. Ing. Ver., 1866. E. Jdger : " Das graphischen 

 Rechnen," Speier, 1867. K. von Ott : " Grundziige des graphischen Rech- 

 nens und der graphischen Statik," Prag, 1871.] 



Recently the method of the graphical calculus has been applied to Dif- 

 ferentiation and Integration. A treatise by Solin shows the first exact, so 

 far as possible in a construction, the last approximate only (" Ueber graph. 

 Integr. ein Beitrag z. Arithmographie, Abhand. d. konigl. bohm. Gesellsch. 

 d. Wissenbach." VI. Folge, 5 Bd. Separate reprint by Rivnac, Prag, 

 1871). It is to be remarked also that examples of double integration and 

 differentiation were given by Mohr in 1868. The graphical construction 

 of the elastic line, and the determination of the moments at the supports 

 of a continuous girder, are essentially examples in point (MoJir : "Bei- 

 trag zur Theorie der Holz und Eisenconstructionen," Zeitschr. d. Hannov. 

 Ing. und Arch. Ver., 1869 ; or W. Bitter : " Die elastische Linie," Zurich, 

 1871.) 



As to the importance of the graphical calculus as an independent study 

 or discipline, it is, as we believe, often exaggerated. The theoretical value is 

 but little, for graphical constructions, as given by the graphical calculus, 

 offer in no respect anything new. That which pertains to practical applica- 

 tions may be easily based directly upon geometry, and is nowhere found 

 as a consequence of the method itself. If it is considered advisable to call 

 special attention to a few general points before making such applications, 

 all that can be desired can be easily presented in ten or a dozen pages 

 octavo. 



* See Preface ; also Chaps. VII. and VIII. of this Introduction, 

 c 



