THE MATHEMATICAL SCIENCES 117 



rectilinear axis. 1 At the same time conic sections took 

 their rightful place in geometry, since they could be 

 obtained, as we have seen, by the intersection of a cone 

 and a plane suitably placed. Such curves as the 

 quadratrix of Hippias, the conchoid of Nicomedes and 

 the cissoid of Diodes remained rather on the margin 

 of the pure and officially recognized science ; they were 

 considered too mechanical because instruments other 

 than the rule and compass were needed for their con- 

 struction. 



Descartes rightly points out how arbitrary such a 

 distinction is. I cannot understand, he says in effect, 

 why the Ancients called these curves mechanical rather 

 than geometrical. " For if we say that it might have 

 been because it is necessary to use some instrument to 

 describe them, it would be necessary to reject for the 

 same reasons circles and straight lines, since these can 

 only be described on paper by means of compass and 

 rule, which may also be called instruments." 2 The 

 argument of Descartes appears to be unanswerable. 

 But then, whence came the self-imposed limitation of 

 the Greek geometer ? According to P. Boutroux there 

 was no other reason for this but the desire to obtain 

 a science which was simple and well arranged and 

 consequently beautiful and harmonious. This reason 

 does not seem absolutely decisive. Certainly the 

 tracing of a straight line or a circumference is done by 

 means of a very simple process ; besides, the straight 

 line and the circumference represent perfect and 

 homogeneous mathematical facts, for two arcs of the 

 same circumference can be superposed just as two 

 sections of the straight line, but there the simplicity 

 ends. As soon as the relation of the radius to the 

 circumference is sought, the problem becomes obscure. 

 Hence the fruitless attempt to effect the quadrature of 



1 4 Boutroux, Ideal, p. 40. 



2 Geometry, Bk. II ; edit. Adam and P. Tannery, VI, p. 388. 



