1 24 THE SQUARING OF THE CIRCLE. 



later Greek mathematicians, and by the Romans called "quadrat 

 rix." Regarding the nature of this curve we have exact knowledge 

 from Pappus. But it will be sufficient here to state that the quad- 

 ratrix is not a circle nor a portion of a circle, so that its construc- 

 tion is not possible by means of the postulates enumerated in the 

 preceding section. And therefore the solution of the quadrature 

 of the circle founded on the construction of the quadratrix is not 

 an elementary solution in the sense discussed in the last section. 

 We can, it is true, conceive a mechanism that will draw this curve 

 as well as compasses draw a circle ; and with the assistance of a 

 mechanism of this description the squaring of the circle is solvable 

 with exactitude. But if it be allowed to employ in a solution an 

 apparatus especially adapted thereto, every problem may be said to 

 be solvable. Strictly taken, the invention of the curve of Hippias 

 substitutes for one insuperable difficulty another equally insuper- 

 able. Some time afterwards, about the year 350 B. C., the mathe- 

 matician Dinostratus showed that the quadratrix could also be used 

 to solve the problem of rectification, and from that time on this 

 problem plays almost the same role in Grecian mathematics as the 

 related problem of quadrature. 



As these problems gradually became known to the non-math- 

 ematicians of Greece, attempts at solution at once sprang up 

 that are worthy of a place by the side of the solutions of modern 

 amateur circle-squarers. The Sophists especially believed them- 

 selves competent by seductive dialectic to take the Stronghold that 

 had defied the intellectual onslaughts of the greatest mathemati- 

 cians. With verbal nicety, amounting to puerility, it was said that 

 the squaring of the circle depended upon the finding of a number 

 which represented in itself both a square and a circle ; a square by 

 being a square number, a circle in that it ended with the same 

 number as the root number from which, by multiplication with it- 

 self, it was produced. The number 36, accordingly, was, as they 

 thought, the one that embodied the solution of the famoup prob- 

 lem. 



Contrasted with this twisting of words the speculations of Bry- 

 son and Antiphon, both contemporaries of Socrates, though inex- 



