THE GOLDEN AGE OF GREECE 71 



Such axioms as "Equals subtracted from equals leave equals" 

 date from this period. The analytical method is developed, con- 

 necting that which is to be proved with that which is already 

 known. Another principle carefully observed is to isolate the 

 problem by removing all non-essential elements, and a third con- 

 sists in proving that assumptions inconsistent with that which is 

 to be proved are impossible. 



THE ANALYTIC METHOD. The analytic method, proceed- 

 ing from the unknown to the known, depends for its validity on 

 the reversibility of the steps ; the synthetic method on the contrary 

 proceeds from the known to the unknown, with unimpeachable 

 validity. It was characteristic of the Greek geometers to aim 

 at this form for their demonstrations, even if the results had been 

 first obtained analytically. The two methods are well illustrated 

 by the following : 



A circle is given and two external points A and B. It is required 

 to draw straight lines AC and BC meeting the circle in C, D, and E 

 so that DE shall be parallel to AB. It is 

 shown that if the construction can be made, 

 the tangent to the circle at D will meet AB 

 (produced if necessary) in a point F which 

 will lie on a new circle passing through A, C, 

 and D. This analysis of consequences is the 

 desired clue on which the following synthesis F "" /A B 



of the construction is then based. Starting 



again with A, B and the circle, we locate F so that BA X BF 

 BC X BD = square of the tangent BG from B. Then drawing a 

 tangent from F to the circle, D is determined and with it the re- 

 quired line DE. 



A solution of the " duplication of the cube " problem is also 

 attributed to Plato, though the mechanical process employed is so 

 much at variance with his usual teachings that the correctness of 

 the attribution is seriously questioned. 



SPQR is a frame in which SPQ and PQR are always right angles, 

 while PQ may be varied, and SQ and PR can be revolved about Q 



