246 



ELEMENTARY DEMONSTRATION OF THE 



Let ABCD be a rec- 

 tangle, and AC the dia- 

 gonal passing through 

 A ; draw EAF perpen- 

 dicular to AC, and let 

 fall the perpendiculars 

 BF, BH, DG, DE : then E 



shall ABCD, AFBH and XT' - c 



AEDG be similar rectangles ; and if, in each of these, the 

 equivalent of the pressures represented in direction and quan- 

 tity by the sides, be represented in direction by the diagonal, 

 it shall be represented by the same in quantity also. For if 

 the forces AH and AF be equivalent to m AB ; AE and AG 

 shall be equivalent to m AD ; and AB and AD to m AC ; or 

 m AB and m AD to ra'AC : that is, the forces AH, AF, AE, 

 and AG, will be equivalent to m % AC : But AE and AF are 

 equal and opposite : hence the forces AH and AG are equi- 

 valent to m 1 AC. But AH and AG are equivalent to AC ; 

 therefore m — 1. 



FIG. 2 



A. B 



E 





^**. 





C 



D 



I 



r 



11 



1. Now let ABDC be any square ; and let the sides AB and 

 €D be produced indefinitely towards B and D ; draw the dia- 

 gonal AD ', in CD produced, take DF equal to AD ; join AF; 

 .take FH equal to AF ; join AH, and so on. And complete 

 the rectangles ACFE, ACHG, &c. 



It is obvious, that AD, AF, AH, &c. bisect the angles 

 BAC, BAD, BAF, &c. respectively. Hence the resultant of 



AB 



