124 



A SHORT HISTORY OF SCIENCE 



With it two poles, bearing disks, were used, exactly as by modern 

 surveyors. A cyclometer for a carriage is also described, with a 

 series of cog-wheels and an index. 



In optics he shows that under the law of equal angles of incidence 

 and reflection, the path described by the ray is a minimum. 



HERO'S TRIANGLE FORMULA. - - His Geodesy, also the Dioptra 

 contains the well-known formula for the area of a triangle 



K = 



a + b + c , x a +6 - c fe + c a e + a 6 



which, since it involves the multiplication of four lengths together, 

 is heterodox from the Euclidean standpoint. 



ABC is the given triangle of sides a, b, c, touching its inscribed circle 

 f.t D, E, and F. Taking BJ = AD, we have CJ = %(a + b + c) and 



area ABC = twice area CJM. 



Draw perpendiculars to CM 

 at M and to CJ at B, meet- 

 ing in H. A semicircle on the 

 diameter CH will pass through 

 both M and B. The sum 

 of the angles CHB and CMS 

 is 180; the triangles BCH 

 and MAD arc therefore simi- 

 lar, 



so that BC : BH = AD : MD, or BC : B J = BH : ME. 

 Also the triangles BGH and EGM are similar, 



so that BH : ME = BG : EG and BC : BJ = EG : EG, 

 whence BC + BJ : BJ = BG +EG : EG, that is, 



CJ:BJ = BE: EG and CJ 2 : B J X CJ = CE X BE : CE X EG, 



that is, CJ 2 : B J X CJ = BE X CE : EM 2 , which is equivalent to 



BE X CE : C J X EM = CJ X EM : BJ X CJ. 



But CJxEM^-, CE = (a + b - c), etc., 



whence 4 K" = (a + b + c) (a + b - c) (6 + c - a) (c + a - 6). 



