158 SCIENCE IN GRECO-ROMAN ANTIQUITY 



two angles adjacent to this side equal each to each, 

 are equal. To affirm the contrary would be to admit 

 that the two triangles cannot be exactly superposed, 

 and that the angles supposed to be equal are not so in 

 reality, which is not in agreement with the data of the 

 question. 



If we now consider Greek geometry, having no 

 longer regard to its particular methods, but to its 

 spirit, there are other characteristics yet to be noted. 

 The demonstrations are always instinctively based on 

 logical and statical ideas ; they generally avoid making 

 any appeal to considerations which, in spite of their 

 evidence, arise from intuitive perception. It is thus 

 that Euclid demonstrates the following fact which 

 might appear however unquestionably evident : if 

 from a given point a perpendicular and two oblique 

 lines are let fall on a straight line, of those two oblique 

 lines that which diverges most from the perpendicular 

 will be the longer. 



As far as possible Euclid also avoids, if not the dis- 

 placement, the turning over of a figure, although this 

 operation, now considered correct, allows of a more 

 rapid demonstration. For instance, it is enough to 

 turn over an isosceles triangle in order to demonstrate 

 that the angles opposite to the equal sides are them- 

 selves equal. Euclid however prefers to decompose 

 the isosceles triangle into two right-angled triangles, 

 whose equality he then proves. It is the same when 

 he wishes to demonstrate, pair by pair, the equality of 

 the angles formed by a secant which cuts two parallel 

 straight lines. The simplest method would be to 

 displace one of the parallels until it coincides with the 

 other. Euclid here again brings in two right-angled 

 triangles, of which he establishes the equality. In this 

 way the demonstration preserves a static character 

 more in agreement with the exigencies of logic. This is 

 so true that wherever displacement occurs in plane 



