136 SCIENCE IN GRECO-ROMAN ANTIQUITY 



as Cavalieri, Fermat and specially Pascal did later, 

 the nature of the progressions which represent the 

 decomposition of the geometrical figure. It would 

 have been necessary to establish, once for all, the con- 

 ditions which these progressions must satisfy in order 

 to be used in the solution of any problem of quadrature. 

 By following this path, the Greek geometers would 

 perhaps have discovered some device similar to that 

 used by Newton and Leibnitz, and they would have 

 brought into their method a generalization of which 

 they possessed the essential elements. But, being 

 desirous above all to avoid the direct use of infinity, 

 they were so intent on ensuring the rigour of the 

 method of exhaustion in each particular case " that 

 it left them no room to develop, beyond the need of 

 the moment, the methods they employed to prove 

 their results, or to create new methods." x Already 

 necessitating lengthy demonstrations for relatively 

 simple cases, the method of exhaustion became most 

 complicated when used for the integration of surfaces 

 and volumes of which the elements are connected by 

 complex relations. So it is not astonishing that the 

 successors of Archimedes, adhering to this method, were 

 not able to carry on the brilliant work of their master, 

 notwithstanding the time and knowledge at their 

 disposal. 



4. GEOMETRICAL ALGEBRA 



Although the way opened up by Archimedes was 

 but little followed, the comparative study of lines, 

 surfaces and volumes nevertheless made real progress 

 by means of what may be called geometrical algebra. 



The Pythagoreans had already employed geometry 

 in the study of the numerical properties of magnitudes 

 regarded as commensurables, and thereby, as we have 

 seen, they were restricted in spatial arithmetic. 

 1 29 Zeuthen, Histoire des mathematiques, p. 142. 



