Mr. J. Snart oti the Qttaihature of the Circle. 345 



The proportion of the area of the circle, to that of a tetra- 

 gon, or square whose side or root is equal to that of the cir- 

 cle's diameter, though not so simple, may notwithstanding 

 be worth the notice of those to whom it is addressed. Its 

 area being J|)^^ of that of the square, that is, as 3927 is to 

 5000, or nearly j^j ths. So that when any diameter is squared 

 and multiplied by eleven^ the quotient arising from dividing 

 such product by l* will be but a trifle 7nore than the true 

 area of the circle as found by the quadrating factor '7851?. 



The proportion of the sphere's solidity/ in comparison witli 

 that of a cube, whose side or root is equal to the diameter of 

 such sphere, and whose decimal expression of solidity is '5236 

 or xVmjD' ™^y ^^ ^'^^^ ^® expressed, for common use, by the 

 equivalent vulgar fraction ^f-^^, or nearly so, by the very low 

 fraction of Ji; or as 11 is to 21 : : the solidity of the sphere 

 to that of the cube. 



Therefore, when any diameter is cubed, or twice involved 

 by its own root, and that product multiplied by 11, the Cjuo- 

 tient arising from dividing the latter product by 21 will be 

 but a trifle more than that found by the decimal mode. Thus 

 2x2x2xll---21 = 4.-190 instead of 4-1888. That is, by 

 either mode, the solidity of the sphere is but little more than 

 half that of the cube of the same depth or thickness. 



The supetjicies of any sphere is always equal to four times 

 the area of one great circle thereof, and is therefore found by 

 multiplying those areas by 4, or by squaring the diameter and 

 multiplying the product thereof by 3—^^^. 



Thus, 8x8x3^/^ = 201x1-3^ ( = 201-06195+ by the deci- 

 mal process) = the superficies of a sphere whose diameter 

 is 8. 



And 12x12 x3xVt = *52xV? ( = 4'52-38938+ by the de- 

 cimal process) = the superficies of a sphere whose diameter is 

 12. N. B. The numbers indicate the same kind of measure 

 as those in which the diameters were first taken, whether leet, 

 inches, or whatever. 



How far these little matters may be called discoveries, in 

 tliis enlightened age, when it is difficult to produce any thing 

 wesx'j or show any thing that has not been done by others, is a 

 hard matter even for tlie author himself to determine, unless 

 he were acquainted with all that others have done. I shall 

 therefore claim no merit beyond that of an unpirated attempt 

 to inform those who most need it, and therefore delivered in 

 terms best suited to their capacities. 



I remain yours, &c. 

 21oTooley-street,Oct. 13, 182.3. JoHN SnarT. 



Vol. G2. No. 307. AW. 1823. Xx LXIX. .4/)- 



