138 Mb WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULA, 



AAA 

 cx±bx= cb. 



AAA 

 px±qx=pq. 



A A A 



yx + bx = Tr—pb, 



A A A 



cx + qx = ir—cq. 



A . A . A 



sin ex sin erp . sin cq , 



~ A" ~~ A \ A" ^^^• 



sin bx sin aq . sin bp 



A A . A 



sin wo: sin «c.sin bp ,^, 



7A = . A . A (2)- 



sin qx sin ab . sin cq 



A . A . A 



sin px sin ac . sin pa 



-^ = .A .A ^^)- 



sin ix sin be .sin ag^ 



A A A 



sin ex sin i c . sin «» , , . 



A = . A . A W- 



sin qx sin a& . sin pq 



When the angles are known, the line x may be found from any 



AAA 



one of the four angles bx, ex, j)^^ Q-^' by two proportions which when 

 united give these four values of x, viz. 



A A 



sin bp.sin ac 



A _ A ' 



sin^j.f.sin be 



A .A 



sin eq . sin ab 



— ^~7~^' 



sin qx .sin be 



A A 



, _ sin bp.sm aq 



~ A a" 



sin bx . sm pq 



A A 



sin eq . sin ap 



'' A a"' 



sin ex . sin pq 



36. Application to a problem in Algebra. 



The Angular Calculus was applied with advantage to the reso- 

 lution of Quadratic Equations, first, I believe, by Dr Halley, in 

 Lectures given at Oxford in 1704. From this it might be inferred 

 that it may be applied to the solution of every Algebraic problem 

 which produces a Quadratic Equation, without a previous reduction to 

 that form, although I do not know that this application has been 

 expressly treated of, and examples given. The formulae which have 

 been investigated in this paper apply with peculiar advantage to the 

 solution of a known problem in Algebra, which appears at first sight 

 to be by no means easy. It is this: 



