114 THE SQUARING OF TFIE CIRCLE. 



matical point of view than the work of Ludolf. luliis book Cyclometri- 

 cus Snell took the j^osition that the method of couiparison of polygons, 

 which originated with Archimedes and was employed by Ludolf, need 

 by no means be the best method of attaining the end sought; and he 

 succeeded, by the employment of j^ropositions which state that certain 

 arcs of a circle are greater or smaller than certain straight lines con- 

 nected with the circle, in obtaining methods that make it possible to 

 reach results like the Ludolflan with much less labor of calculation. 

 The beautiful theorems of Snell were proved a second time, and better 

 j)roved, by the celebrated Dutch promoter of the science of optics, 

 Huygeus {Opera Varia, pp. 3G5 et seq. ; Theoremata I)e Circuli et Hy- 

 perbolae Quadratura^lG^l), as well as perfected in many ways. Snell 

 and Huygens were fully aware that they had advanced only the prob- 

 lem of numerical quadrature, and not that of the constructive quadra- 

 ture. This, in nuygeus's case, plainly appeared from the vehement 

 dispute he conducted with the English mathematician, James Gregory. 

 This controversy has some significance for the history of our problem, 

 from the fact that Gregory made the first attempt to prove that the 

 squaring of the circle with ruler and compasses must be impossible. 



The controversy between Huygens and Gregory. — The result of the con- 

 troversy, to which we owe many valuable treatises, was that Huygeus 

 finally demonstrated in an incontrovertible manner the incorrectness of 

 Gregory's proof of impossibility, adding that he also was of opinion 

 that the solution of the problem with ruler and compasses was impossi- 

 ble, but nevertheless was not himself able to demonstrate this fact. 

 And Newton, later, expressed himself to a similar effect. As a matter 

 of fact it took till the most recent period, that is over 200 years, until 

 higher mathematics was far enough advanced, to furnish a rigid dem- 

 onstration of impossibility. 



Before we proceed to consider the promotive influence which the in- 

 vention of the differential and the integral calculus had upon our prob- 

 lem, we shall enumerate a few at least of that never-ending line of 

 mistaken quadrators who have delighted the world by the fruits of their 

 ingenuity from the time of Newton to the present period ; and out of a 

 pious and sincere consideration for the contemporary world, we shall 

 entirely omit in this to speak of the circle-squarers of our own time. 



Hobbes^s quadrature. — First to be mentioned is the celebrated English 

 philosopher Hobbes. In his book, De Problematis Physicis, in which he 

 chiefly proposes to explain the phenomena of gravity and of ocean 

 tides, he also takes up the quadrature of the circle and gives a very 

 trivial construction that in his opinion definitively solved the problem, 

 making 7r=:3i. In view of Hobbes's importance as a philosopher, two 

 mathematicians, Huygens and Wallis, thought it proper to refute 



