METHODS AND LIMITS OF GRAPHICAL STATICS. xliii 



gards entirely the graphical calculus, and also cuts loose from the modern 

 geometry ; he develops the elementary principles of the subject in a logi- 

 cal and easily comprehended, if not purely geometrical manner, and thus 

 brings the subject within the reach of those persons for whom it seems so 

 especially designed. The work is remarkable for clear presentation, but 

 expressly avoids all special investigations and practical applications, for 

 which it is merely intended to prepare the way. In the present work, also, 

 a similar plan is pursued, but all such applications as are of most value to 

 the engineer or mechanic find likewise a place. Thus, combining the 

 method of presentation of Bauschinger and the practical applications of 

 Culmann, it has been endeavored to make it a practical manual, as well as 

 a text-book of elementary principles to serve the wants of the practical 

 engineer, and also meet the requirements of the engineering student. How 

 far this twofold design has been realized, the judgment of the reader 

 must decide. 



VIL 



THE METHODS AND LIMITS OF THE GRAPHICAL STATICS. 



The most perfect method of the graphical statics is the synthetic or geo- 

 metric, since in geometrical statics the solution must always, when possi- 

 ble, rest upon pure mechanical or geometrical reasoning. Culmann pre- 

 sents his graphical statics to practitioners "as an attempt to solve by the 

 aid of the modern geometry such problems pertaining to engineering prac- 

 tice as are susceptible of geometrical treatment." 



The graphical statics, however, is not in and of itself the product of 

 endeavors to make the modern geometry of service in applied mechanics ; 

 graphical solutions merely were required. How to obtain these, was 

 another question. Thus it is that Poncelet's solutions consist almost en- 

 tirely of graphical representations of analytical relations ; that Oousinery 

 avoided all use of formulas ; that Culmann made use of the new geometry 

 wherever it was possible ; that Bauschinger and others make use only of 

 the ancient geometry ; and that the latest graphical solutions in a certain 

 degree, those of Mohr also entirely in the spirit of Poncelet's, rest again 

 upon analysis. The pure geometric solution is, indeed, desirable, but is not 

 always attainable. 



If now we review all the cases in which direct and exclusively geomet- 

 rical solutions are not possible, we see at once that this occurs when it is 

 required to make use of the physical properties of bodies, as elasticity, co- 

 hesion, etc. Why ? The actual condition of a body after equilibrium is 

 attained, is a consequence of the motion of a variable system of points. 

 The theory of the motion of variable systems has, however, by no means, as 

 yet, been brought to practical efficiency (II.). We are therefore obliged to 

 start from an hypothetical condition or state of the body (in the theory of 

 flexure, for instance, we rest upon the assumption that all plane cross-sec- 

 tions made before the action of the outer forces remain plane after their 



