ON GRAPHIC METHODS IN MECHANICAL SCIENCE, 591 



Take OA,, OA2 (fig. 12) along the line A, and along any other line take 



BC=K, 

 CD=K, 



Take any point P at a distance of unity (P H) from BCD taken 

 parallel to O A, . . . and join P B, P C, P D, . . . 



Draw parallels to these, as shown in the figure, to meet other parallels 

 to B C D E . . . through Aj Ao A3 . . . Then the segment cut off by 

 the first and last ray on the parallel through O gives the sum of the 

 products. 



The reason for this construction is obvious, since it is merely a device 

 for obtaining the necessary similar triangles required, in such a position 

 as to enable the pi-oducts to form one continuous segment. These similar 

 triangles are shown in dotted lines, and it is obvious that since O' A/ L 

 and P B C are similar triangles • 



0'A/_OAi . O'V^OAj^ 

 BC PH °^' K, 1 ' 



that is 



0'A',_OA, 



or 



0'Ai'=OA, .K, 



Ai'A2' = OA2.K2 



Helice 0'A,/=20A.K. 



The figure is usually taken, so that OA] A2 . . . is horizontal, and 

 B C D . . . is vertical, as in this case there are many direct applications ; 

 the general nature of the proposition in which the lines are taken at any 

 angle is, however, apparent. 



This important construction appears to have been first given by 

 Culmann. In the first edition of his work it is stated in paragraph 26, 

 under the title ' Funicular Polygon or Line of Pressure and Force 

 Polygon,' but in the second edition it is placed in the preliminary chapter 

 entitled ' Operations upon Lines,* under the heading (paragraph 4) 

 ' Summation Polygon, or Funicular Polygon,' and it forms the basis of 

 much of graphic statics. It is there given in a very general form, which, 

 notwithstanding the number of yeai's that have elapsed, has not appeared 

 in any English work ; and as the style and method of Culmann are, to 

 say the least, original, the following free translation of that article is 

 given : — 



Summation or Puniculae Polygon. 



After stating that in statics we often have to multiply, not only 

 separate lines, but whole rows of lines by different ratios, and to add up 

 these ratios as well as the products, it is stated that if we execute the 

 summation of these products in such a way that the resulting triangles 



