THE MATHEMATICAL SCIENCES 133 



and (C, D) be two pairs of magnitudes, the proportions 

 A/B and C/D will be equal, if, whatever may be the 

 whole numbers m and p, we always have 



wA wC 

 ~pB ~" pD 



In this way the ratios of magnitudes become geo- 

 metrical, and no longer simply arithmetical, as they 

 had been to the Pythagoreans. 



Having established this point, Eudoxus laid the 

 foundations of an infinitesimal method by which it 

 would be possible to pass gradually from a regular 

 figure to the figure which circumscribes it. This 

 method, called the method of exhaustion, is based on 

 the following principles which are derived from the 

 lemmas formulated for geometrical proportions. 1 



1. If two magnitudes a and b be unequal, the 

 lesser repeated a sufficient number of times (n) will 

 end by equalling or exceeding the greater. In other 

 terms if a < b, na > b. 



2. If from a magnitude there be taken more than its 

 half, then from the remainder a part greater than 

 half of this remainder, and so on indefinitely, there will 

 be finally obtained a remainder less than any given 

 magnitude. 



It was by taking these principles as a basis that 

 Eudoxus demonstrated, amongst other things, that 

 circles have areas proportional to the squares of their 

 diameter. The proposition is true for regular figures 

 of 4, 8, 16, 32, etc., sides which are successively in- 

 scribed in the circles. Now, at each operation, the 

 difference between the area of the circles and that of 

 the new polygons inscribed is diminished by more than 

 half. It tends to become zero, so that the properties 

 established for polygons hold good for circles. 



The method of exhaustion was taken up and given 



1 26 Tannery, Geo. grecque, p. 96. 



