144 SCIENCE IN GRECO-ROMAN ANTIQUITY 



movement is continued until the other mark comes 

 on the second given line." x 



There was very probably a time when intercalation 

 was admitted as a means of construction, together 

 with the rule and compass, but it was soon rejected 

 for reasons we have indicated (p. 119). It became 

 then necessary to have recourse to conic sections, when 

 the rule and compass were obviously insufficient. The 

 consequence of the study of these sections was the 

 development of the fruitful conception of "geometrical 

 loci/' for a conic section may be considered as the 



a. 



. , * * 





JB 



Fig. 25. 



locus in which a cone and a plane meet. Hence there 

 arose the expression of " solid loci," since the cone 

 is a volume. 



However, even in its most developed form, the theory 

 of conic sections is closely connected with the first 

 works on geometrical algebra. This is strikingly 

 shown by the works of Apollonius. 2 



In these, the study of magnitudes and their ratios is 

 always done by geometrical operations, only the field 

 is enlarged thanks to the theory of proportions and 

 similitude. This allows of the construction of surfaces 



1 29 Zeuthen, Histoire des mathematiques, p. 66. 



2 3 Boutroux, Analyse, p. 491 et seq. 



