GRAPHICAL STATICS. 



fulness of the relations between the force and equilibrium polygon, upon 

 which most of the practical solutions depend. He developed these rela- 

 tions, applied them in the theory of moments by the introduction of the 

 closing line (Schlvss Linie), and, accepting the rule of signs, obtained gen- 

 eral points of view for the discussion of the most diverse figures which 

 could arise in the same problem. In this and in many other respects even 

 geometrical statics can profit from Culmann's work, as, for instance, in the 

 investigation of the projective relations between the force and equilibrium 

 polygon. 



The fundamental importance of the force and equilibrium polygon was 

 also recognized by those who, after Culmann, occupied themselves with 

 the graphical method. Here we may notice two works of special influence 

 upon the development of the graphical statics those of Mohr and Cre- 

 mona. The idea of Mohr, that the elastic line is an equilibrium polygon or 

 curve (" Beitrag zur Theorie der Holz und Eisenconstructionen." Zeitschr,. 

 d. Hannov. Ing. und Arch. Ver., 1868) is of special significance for graphi. 

 cal statics. 



That from it Molir obtained the graphical determination of the moments 

 at the supports of a continuous girder, is an example both useful as well 

 as interesting. Already it has been endeavored to utilize the same idea in 

 other cases (Frarikel : "zur Theorie der Elastischen Bogentrager," Zeitschr. 

 d. Hannov. Ing. u. Arch. Ver., 1869, p. 115), and by it an impulse has been 

 given to similar investigations. 



Cremona has kept more especially in view the geometrical side of graphi- 

 cal statics. Starting from the theory of reciprocal polyhedrons, he gave 

 the reciprocal relations between the force and equilibrium polygon with a 

 generality and elegance to be expected from this distinguished Italian 

 mathematician (Le figure reciproche nelle statica grnfica. Milan, La'nger, 

 1872). By this investigation the theoretical development of the graphical 

 statics is essentially anticipated. 



It was under the most unfavorable circumstances that Culmann intro- 

 duced his graphical statics in the engineering department of the Zurich 

 Polytechnic in the year 1860. It was finally, indeed, admitted as a regular 

 study, but not the geometry of position which he premised. It was not 

 till 1864 that this last was given in a series of lectures by Reye, and then 

 the time at disposition for both courses was insufficient. Meanwhile the 

 method spread, crept into the construction department of the engineering 

 school, and wherever it came, even in the other departments of the Poly- 

 technic, gained friends. Finally, at the present time, it is to be found, to- 

 gether with the modern geometry of position, upon which it was based, in 

 every Polytechnic throughout Germany. 



According to the above given definition of graphical statics, the methods 

 of the graphical calculus, as far as applied in statical investigations, may 

 also be regarded as belonging to graphical statics, and justly so ; for 

 these methods follow directly from geometrical principles, and can be ap- 

 plied by any one acquainted with geometry, without being collected under 

 the special name of the " graphical calculus." Thus, for instance, Bawch- 

 inger, in his " Elemente der graphischen Statik" (Mi'inchen, 1871), disre- 



