METHODS AND LIMITS OF GRAPHICAL STATICS. 



action). To deduce now from this general condition the special relations 

 necessary for solution, demands an essentially analytical process (I). 

 Hence the dependence of the graphical solutions in such caws upon ana- 

 lytical relations relations which, when the body is assumed to be rigid, 

 us in the arch, in frame work, or the simple girder, no longer exist. 



The sphere of action of an independent graphical statics is, then, con- 

 fined to those problems which, under the assumption of inflexibility, are 

 determined by a sufficient number of conditions. Beyond this point we 

 have chiefly graphical interpretations only. 



It has been already noticed that graphical statics, without the application 

 of algebraic operations, can furnish no general laws (IV.). From relatively 

 simple figures, indeed, here and there, general formulae of metrical relations 

 have been derived, as is, in fact, not theoretically impossible (I.), but such 

 formulas were always previously known. Such a result holds, in general, 

 immediately good only for that form of figure which has been discussed, 

 or, according to the terminology of Carnot, only for the existing " primi- 

 tive figure," and must be proved or transformed for all " correlative 

 figures " which can occur in accordance with the conditions of the prob- 

 lem. When the graphical investigation is guided by analytical opera- 

 tions, it is these last which render possible the deduction of general metri- 

 cal relations. 



Thus, in the theory of structures, there remains subject to pure graphical 

 treatment only the general relations of form and position. Here we have 

 the elegant deductions upon unfavorable loading, and here the graphical 

 method often attains its end in a more elegant manner than the analytical. 

 A complete exploration and development of such form and place relations, 

 without a geometry of position, would evidently be impossible (IX.). The 

 scientific future of the graphical statics, therefore, rests essentially upon 

 the influence of the modern geometry. To endeavor to separate the higher 

 geometry from the graphical method would be as unwise and fruitless as 

 the attempt to exclude the higher analysis from analytical investigations. 

 As, however, for certain purposes an elementary presentation of analytical 

 theories relating to engineering practice will ever be acceptable, so also an 

 elementary development of graphical methods is not without justification, 

 the more so as long as the modern geometry itself is not sufficiently well 

 known. 



Culmann says of the graphical statics : " It includes, thus far, only the 

 general part which we need in the investigation of problems in construc- 

 tion, but it must and will extend, as graphical methods find ever wider 

 acceptance. Then, however, it will escape the hands of the practitioner, 

 and must be built up by the geometer and mechanic to a symmetrical 

 whole, which shall bear the same relation to the new geometry that analyti- 

 cal mechanics does to the higher analysis." Such an estimation does not 

 appear to be entirely correct. It is geometrical statics (or mechanics) for 

 which the above relation may subsist, and to this, indeed, Culmaun's valu- 

 able work has itself greatly contributed. It was, moreover, developed 

 quite independently of and much earlier than graphical statics (III.). In 

 this respect, therefore, the spread of graphical methods is of less impor- 



