OF ARTS AND SCIENCES. 137 



" Prop. 1. To express any side of a triangle in terms of the other 

 sides and their included angle. 



" Let AB C he any triangle ; and suppose its sides B C, A C, A B 

 severally contain some assumed length (considered as the unit of 



m 



length) a, b, c times, then will the sides be expressed by a, b, c ; 

 where it may be observed that a, b, c are positive, and that they 

 may be integral or fractional, rational or surd, according to the nature 

 of the case ; we shall denote the angle B AC hy A, and shall sup- 

 pose C-4 to be produced (in the direction C A) to B\ so that AB' — 

 B A, then (Simson's Euclid, Book I., prop. 13) the angle BAB' is the 

 supplement of A, or the sum of A and B A B is equal to two right 

 angles. 



" By Sim., B. I., p. 20, we have the inequalities a -\-b^c,a-^c^b, 

 or (which is equivalent to them), we have a'^^(c — Z»), (1); in 

 which we must use thq, upper sign when c is greater than b, and the 

 lower sign must be taken when c is less than b ; and it is manifest 

 that (1) exists even when c = &. In order to remove the ambiguous 

 sign, we may (by taking the second power of a, and + (c — b) put 

 (1) under the form a^y {c — bf, ox a^ — {c — b)^yO, (2). If 

 the angle ^=0, AB falls on AC, and (2) evidently becomes 

 a^ — (c — b)- = 0, which is its least value; and, Sim., B. I., p. 24, 

 if we suppose b and c each invariable, and the angle A to be in- 

 creased, then A will be increased, and the greatest value that a can 

 have will be when the angle A equals two right angles, or when 

 AB coincides with AB', and a == b -\- c, so that (2) becomes 

 {b-\-c)'^ — (b — c)2 = 4Z»c, which is its greatest value; hence and 

 by (2) if we put °^~^^''~^^' = p, (3), p cannot be less than (or 

 cannot be negative), nor greater than 2, or p has for its lesser, and 

 2 for its greater limit. From (3) we get a-= {c—b)^ -\-2p b c = 

 b^ + c^ — 2 ( 1 — p) & c, or if we put 1 — p = n, (4), then a~ = 

 b- -f c- — 2nbc,{5); where, since p never passes the limits and 2, 



18 



