THE MATHEMATICAL SCIENCES 135 



This same exactitude prevented the Greek geometers 

 from looking in another direction for the solution of 

 the problem of areas and curvilinear volumes. By a 

 stroke of genius Archimedes invented a method of 

 integration based on the comparative study of the 

 static moments of two figures, and which necessitates 

 for this study the use of an infinite number of lines 

 or parallel planes ; the comparison of suitably selected 

 sections then gives the equation of equilibrium between 

 the known surface or volume of one of the figures and 

 the unknown surface or volume of the other. Thus 

 to have equilibrium with a sphere, it is necessary to 

 have four cones having as base the great circle and 

 as height the radius of the sphere. The sphere has 

 therefore a volume four times greater than that of the 

 cone constructed with its radius. Archimedes, how- 

 ever, would not acknowledge any power of demon- 

 stration in this mechanical method, whose results, to 

 be valid in his eyes, had to be confirmed by exhaustive 

 reasoning. In fact, the Greek geometers considered 

 that it was only by this reasoning that the dialectic 

 of Zeno could be successfully refuted. On the one 

 hand, the condition imposed on the difference (line 

 or surface) of always diminishing by more than its 

 half ensures that this difference can become less than 

 any given quantity, after a finite number of operations. 

 On the other hand, the method of construction em- 

 ployed in each problem ensures that the law of diminu- 

 tion is really obeyed by the decreasing magnitudes ; 

 hence the terms which form the numerical representa- 

 tion of these constitute a series the convergence of 

 which is evident and has no need of proof. In every 

 way the direct use of infinity, which results from 

 dichotomy, and which Zeno had criticized, is avoided. 

 However, the method of exhaustion thus understood 

 remains difficult to manipulate. To make its applica- 

 tion general, it would have been necessary to examine, 



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