122 Treatment of Geometry as a branch of Analysis. [No. 134. 



knowledge inform us at once, that if a be increased p times and b, q 

 times, the area is increased pq times, hence jo^A = <j> (pa y qb). 



pq 

 Hence $ (p«> qb) must be divisible by pq with a quotient inde- 

 pendent of pq. Symmetrically therefore it must also be divisible by ab 

 with a quotient independent of a and b ; let the quotient of both divi- 

 sions be k. Then 



<j> (j)a, qb) = kabpq. 

 .*. jo^A = kabpq or A = #a£>. 

 Assuming now as is usual, that the superficial unit is the square on the 

 linear unit, we find k by making a = b = 1 (the linear unit) .*. A 

 = 1 (the superficial unit). Hence k = 1 and therefore 

 A = ab. 



17. From this well known theorem, the various properties of rec- 

 tangular areas flow with the utmost facility. The first ten of Euclid's 

 second book are reduced to the results of algebraic multiplication and 

 division, remembering that area of square on a equals a x « = a 2 . 



Recurring to equations y in (12), if a perpendicular be dropped on 

 a from A, the segment between it and B is c cos B ; call it s, 



... & = a 2 + c 2 H- 2 as 

 the double sign depending on the species of B. If it be obtuse 2 as 

 is additive (II. 12) ; if acute, 2 as is subtractive (II. 13) ; if it be right 

 s = or b 2 = av + c 2 , (I. 47). Similarly if b* = a 2 + c\ 



cos B = and .-. B = \ ir (I. 48.) 



18. A triangle is easily shewn to be half a rectangle on the same 

 base, and with the same altitude, hence a triangle = 1 altitude x 

 base. The following consequences immediately result. Triangles or 

 parallelograms on equal bases vary as their altitudes and vice versa 

 (Young VI. 12). Triangles and parallelograms having equal bases 

 and equal altitudes are equal, and the contrary (I. 35, 36, 37, 38 

 39, 40). If a be the base of a triangle A, the altitude or perp. from 

 A = c sin B 



.*. A = i ac sin B. 

 .-. A : A' = ac sin B : a'c' sin B' 

 If then the triangles (A, A') are equal and an angle in each (B, B ; ) 

 equal, ac = a'c' or the sides are reciprocally proportional (VI. 1 5). 



