Mr. J. Snart on the Qiiadmture of the Circle. 343 



and 6, and whose diiFerences are therefore found by subtracting 

 them from 360 degrees. 



Tlien, as the area of any circle is found by multiplyino- half 

 the circumference (m this case = 1 -570796325) by half the (fiame- 

 te)-{=z-5) we obtain '7853981625 for area of such circle, whose 

 'v/( = -886226925) squared is •785398162594.955625, which 

 product may be called a quadrature of the circle, as being a 

 square, whose area is so nearly equivalent thereto, that it is 

 peifect up to the 1000 millionth place of figures. And as the 

 thmg cannot be done to absolute peifection, discretion must 

 always adjudge the proper maxunum oi approximation thereto, 

 which may be exemplified by the following 



Scholium. 



As it has been demonstrated that the circumference of the 

 circle can only be mensurated by the sum of the sides and 

 the differences of its greatest inscribing and inscribed poly- 

 gons; so, if those of the inscribed hexagon were to be taken, 

 the perimeter would be exactly the same as the triple diame- 

 ter.- because each of the six sides thereof is equal to radius, 

 or half the diameter. Therefore sweeping the outer circle 

 circumscribing the hexagon (fig. 1), any right line from the 

 centre to the circumference may be taken for radius ; then, 

 without going into abstrusities, we may easily demonstrate, by 

 the extraction of the square root only, any augmentation o'f 

 the polygon we please ; all of whose sides being rectilineal are 

 completely mensurable. Thus, 



Let radius of the circle, and of the hexagon (either of whose 

 sides, is =60° at the centre), be 1. The sine of half the arc 

 will then be -5 ( = sine 30°), the square of which ( = -25) sub- 

 tracted (by 47th 1st EucUd) from 1- (the square of 1-) =-75, 

 y -8660254 ( = cosine 30°)— 1-0000000 =-1339746 = versed 

 sme 30° = depth of segment cut off by the chord of any hexa- 

 gon inscribed in a circle whose radius is one. 



But if the segments cut off by so simple a figure as that of 

 the hexagon, be too great to produce any similitude between 

 that figure and the circle, they become lessened by every aug- 

 mentation of the number of the sides, until approximation 

 takes place. Therefore, bisecting the above figure through- 

 out, the dodecagon is produced ; each of whose 12 sides makes 

 an angle with the centre of 30°. Then, as before. If from 

 the square of radius (1-) be subducted the square of the sine 

 of half the arc (or 15°) = -0669873, we obtain -9330127, 

 \/ -9659258 (=cosine 15°) — 1-0000000 =-0340742= versed 

 sine 15°=depth o\' segment cut off by tlie chord of any do- 

 decagon inscribed in a circle whose radius is one. 



Thus, 



