THE SQUARING OF THE CIRCLE. 135 



employing propositions which state that certain arcs of a circle are 

 greater or smaller than certain straight lines connected with the 

 circle, in obtaining methods that make it possible to reach results 

 like the Ludolfian with much less labor of calculation. The beau- 

 tiful theorems of Snell were proved a second time, and better 

 proved, by the celebrated Dutch promoter of the science of optics, 

 Huygens {Opera Varta, p. 365 et seq. ; Theoremata De Circuli et 

 Hyperbolae Quadratura, 1651), as well as perfected in many ways 

 by him. Snell and Huygens were fully aware that they had ad- 

 vanced the problem of numerical quadrature only, and not that of 

 the constructive quadrature. This plainly appeared in Huygens's 

 case from the vehement dispute which he conducted with the Eng- 

 lish mathematician James Gregory. This controversy is significant 

 for the history of our problem, from the fact that Gregory made 

 the first attempt to prove that the squaring of the circle with straight 

 edge and compasses was impossible. The result of the contro- 

 versy, to which we owe many valuable tracts, was, that Huygens 

 finally demonstrated in an incontrovertible manner the incorrect- 

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

 opinion that the solution of the problem with straight edge and 

 compasses was impossible, but nevertheless was not himself able 

 to demonstrate this fact. And Newton later expressed himself to 

 the same effect. As a matter of fact a period of over 200 years 

 elapsed before higher mathematics was far enough advanced to 

 furnish a rigorous demonstration of impossibility. 



v. 



FROM NEWTON TO THE PRESENT. 



Before we proceed to consider the promotive influence which 

 the invention of the differential and the integral calculus exercised 

 upon our problem, we shall enumerate a few at least of that never- 

 ending succession of erring quadrators who delighted the world 

 with the products of their ingenuity from the time of Newton to 

 the present ; and out of a pious and sincere regard for the contem- 

 porary world, we shall omit entirely to speak of the circle -squarera 

 of our own time. 



