PRACTICAL SIGNIFICANCE OF GRAPHICAL STATICS. xlix 



"how many superfluous definitions and demonstrations could net be 

 spared, if they were already completely comprehended and recognized by 

 the scholar as parts of a higher whole" (Culmann "Die Qraphischo 

 Statik"). At no very distant day it will no' longer be possible to read a 

 scientific work upon applied mathematics without familiarity with- the 

 principles of the modern geometry.* Permitting pure graphical applica- 

 tions, without the aid of analytic symbols, it forms the common point o 

 view for descriptive geometry, practical geometry, and graphical statics. 



Descriptive geometry existed before the modern, and this last has sprung 

 from it. Now, reversely, the geometry of position comes to the aid of 

 descriptive geometry, and offers in return its most fruitful principles and 

 efficient aid. Thus in descriptive geometry we may refer to the works of 

 Pohlke, Schlesinger, and Fiedler. The effect of the geometry of position in 

 this direction to simplify and condense may be seen from the work of 

 Staudigl ("Ueber die Identitat von Constructionen in perspective, schiei'er 

 und orthogonaler Projection"), where it is proved that " all problems of 

 the descriptive geometry, in which neither linear nor angular measure are 

 considered therefore all problems which belong to the geometry of posi- 

 tion can in similar manner and by precisely similar constructions be solved 

 as well in perspective as in oblique and orthagonal projection." In shades 

 and shadows and in geometrical drawing, Burmeister and Paulus owe to 

 the modern geometry the simplicity of their constructions. 



In the department of practical geometry also, in geodesy, perspective, 

 surveying, we mark the influence of the modern geometry in the works of 

 Mutter and Spangeriberg, of Franke and Baur. 



In mechanics and physics, we see it again in the works of Lindemann, 

 Burmeister and Zech. 



X. 



PRACTICAL SIGNIFICANCE OF THE GRAPHICAL SVATICS. 



We have already remarked (VII.) that the importance of graphical 

 statics is in great part dependent upon its advantages as compared with 

 the analytical method, and have reserved for this place a comparison from 

 a practical point of view. 



Here, first of all, we have to notice the independence of the graphical 

 construction of the regularity or irregularity of the given relations. 

 Whether the forces are equal or not, whether they act at equal or varying 

 distances, even their relative position, are matters of indifference. Centre 

 of gravity, central ellipse, kernel for all, even the most irregular figures, 

 are found in similar manner, with equal ease, even when exact analytical 

 solutions are hardly conceivable. Thus a process, a routine almost 

 mechanical is rendered possible in many investigations of stability, with- 

 out losing sight of interior relations ; for in the repeated and independent 

 compositions of the forces we always perceive the origin, connection and 



* Well illustrated in Gillespie's Land Surveying. New York, 1870. 



