Mr. J. Siiart on the Quadrature of the Circle. 339 



(which could never terminate) until the inscribed and in- 

 scribing polygons have been augmented to half a million of 

 sides to lessen their differences. And by the tangent of 30 

 degrees, or by constantly bisecting the arc, the decimals have 

 been wrought out to 128 places of figures ! (a number far too 

 great, either to be used or appreciated), without coming to a 

 conclusion. For still the Utopian phantom, like the " Cube's 

 Duplicature" has always eluded theii" most sanguine gi'asp, 

 although the imperial largess of Charles the Fifth tempted their 

 efforts with a bait of a hundred thousand crowns ! And the 

 States of Holland, at a respectfiil distance followed the Em- 

 peror's example. 



The last of these elaborate operations was that of De 

 Lagny, a late mathematician of France, and is one of the 

 many proofs in Nature, that great learning is not always 

 competent, nor needful^ to explain simple matters; for had he 

 been less erudite, he might probably by dint of reason alone, 

 have hit upon, at least the negation of, that plain matter of 

 fact at once, by seeing its impracticability. Instead of which, 

 his very scientific process only tends to seduce himself and 

 others into the unfathomable abyss of iiifinitij. 



And yet reflection must needs tell all who use it, that by 

 continually bisecting and comparing the inscribing and in- 

 scribed polygons, we only increase the approximation, and 

 with it the difficulties : because by bringing them nearer to 

 equality, we do but remove the decimal-differences further 

 off from the separatrix. And then, as every finite number 

 must bear some proportion to every other finite number, it is 

 but a sophistication of science to expect a perfect conclusion 

 to a process which so obviously leads deeper and more deep 

 at every step, until thought itself is lost and bewildered in the 

 impalpable mazes of iiifinitely approximating decimals, without 

 even the forlorn relief of a circulate, or the delusive ground of 

 hope, that a nonary instead of a decary scale might effect our 

 object and enable us to mensurate the circle! 



Descending therefore from these sublime heights, let us 

 seek the truth of the matter in the more humble paths of 

 arithmetic. Analysing first the functions of those })owers 

 with which we would make the comparison. 



There are one or two properties of figures, in viensuration, 

 which though not latent, are notwithstanding pretty much 

 overlooked. That in the^rs^ power is this: ^\\<ifrst, second 

 and third powers of numbers are not only lineal, supcyfcial 

 and solid, but they are rectilinear also, in their operations ! 

 And unless the integrity of this (and another) essential feature 

 be preserved inviolate, they cannot produce their plenary 

 U u 2 effect. 



