34!2 Mr. J. Snart on the Quadrature of the Circle. 



to arrive at perfection, is pretty evident, or else they would 

 scarcely have carried the solution to so useless an extent. 

 Indeed Van-eick, the Dutch mathematician, declared that he 

 absolutely had effected it ; and pertinaciously insisted on the 

 correctness of his construction ; and had it not been for the 

 acumen of his cotemporaries in science, who were not so 

 easily convinced, he, without benefiting the world in the 

 least, might have made himself master of the Imperial douceur, 

 or 100,000 crowns promised by the Emperor Charles the 

 Fifth ! as well as the })remium offered by the States of Hol- 

 land. However, the imperial and princely bounties were 

 both withheld, as unavailing to procure impossibilities, and 

 since that time some mathematician has declared the thing 

 " impracticable, because the proportion is a surd number" But 

 thinking an ipse dixit, without proof, insufficient to put a stop 

 to this Utopian labour, I have attempted in the simplest 

 manner to demonstrate why it is a surd number. How well 

 I have succeeded, must be left to a learned and dispassionate 

 public to determine. However, to make the matter more in- 

 teresting, I beg leave to subjoin a few words on the nature of the 

 circle, and its measure, as derived from the various polj'gons. 



The difference between the perimeter of the hexagon (each 

 of whose six sides is equal to radius) and that of the circle 

 as derived from the sines and tangents of every second of 

 some portion of the quadrant ( = to a polygon of 1.296,000 

 sides ; or, if taken from the sines, which are always equal to 

 half the chords of double the arcs, = to a polygon of 648,000 

 sides) is but a trifle less than a twenty-second part of the 

 whole circumference thereof, beuig 16"225323 degrees = 

 16° 13' 31" 09'" 46"" 04" ' 48""". A diflTerence of nearly one- 

 seventh of the whole diameter, and which difference subtracted 

 from 360° 00' 00" 00"' 00"" 00' " 00""" as measured on the 

 arc, leaves for the triple diameter, or perimeter of the hexagon 

 = 343-774677° = 343° 46' 28" 50'" 13"" 55"" 12""" of the arc. 



Divided by 6, for length of radius on ditto 



57-2957795° = 57° 17' 44" 48'" 22"" 19'"" 12""' of the arc. 



Length of diameter on ditto 



114-591559°= 114° 35' 29" 36'" 44"" 38'"'" 34""" of the arc. 

 The length of the radius ( = to the chord of 60 degrees) as 

 herein measured upon the arc of the circle, is the quotient 

 arising from dividing 180 degrees by 3*141592653, being a 

 competent part of the great series of 1 28 figures, or the cir- 

 cumference of a circle whose diameter is one, as derived from 

 a polygon so augmented in the number of its sides as to vie 

 with the circle itself. 



The other proportions are multiples of that radius by 2 



and 



