MATHEMATICAL PAPERS. 



33 



begins. Alfo let the given angles A B C, B C D, C D E 



&c. be exchanged for the 

 angles of commutation are 



F B C, G G D, & 



obtained bv methods 



:. Theft 

 eafter de- 



\^ 



fcribed 



In order to perform the operation, the ft 



pofitions of the bafes, and perpendiculars, mufl be difcovered, 

 and deiignated. The perpendiculars on one fide of the bafes, 

 as F B, G G are denoted by A ; and the perpendiculars on the 

 oppofite fide, as H D, I E by B, prefixed to the angle of com- 

 mutation. Likewife the bafes on one fide of the perpendicu- 

 lars, as F G, G D are denoted by A ; and the bafes on the op- 

 pofite fide, as H E, I A by B, fuffixed to the angle of commu- 

 tation. It is evident, that when a fide happens to be at right 

 angles with the prime fide, the hypothenufe and its bafe be- 

 come equal ; and when a fide is~parallel with the prime fide, the 

 hypothenufe and its perpendicular become equal : and in the 

 former cafe the perpendicular, and in the latter the bafe and 

 angle of commutation evanifli, and become equal to O. Yet 



» 



for the fake of difcovering the pofitions of the fequent bafes, 

 and perpendiculars, they mud be retained, and brought into 

 the calculation. For this end they may be confidered as in- 

 definitely fmall quantities, or quantities lefs than any afTign- 



L 



able one ; and will therefore be expreffed by O. Place either 

 A, or B before, and cither A, or B after the evanefccnt angle 



J 



of commutation belonging to the prime fide. Then the an- 



F 



gles 



