302 Circular bodies and their Squares. [Oct. 



vice versa, knowing the square, the circle can he ohtained and the dif- 

 ference or loss on the latter being squared is known. 



Practical Illustration.— To illustrate this demonstration by 

 numbers. In the above figure, assume the measurement of the 

 inscribed square* ABCD at one hundred (say inches), of which the 

 side AB is one-fourth or twenty-five. Then in the right angle triangle 

 ABE (as before), the square of the hypothenuse (or side AB), or 

 25% is equal to AE* -f EB 2 j that is 625=AE 2 -f-EB 2 ; but as AE 

 and EB are radii of the same circle and equal, their squares are equal 

 also, and 625 =2EB* ; or m = EB*; or V^=EB; or again 



2 7 2 ' O 



V312. 5 = EB (Extract root of 312-5). 

 312.5 (17,6776 the root, 

 1 



27 I 212 



I 189 That is 17.6776 is equal to EB ; but EB, 



or radius of the circle, being half the dia- 



346 ! 2350 meter DB, then the diani. DB, is equal to 



I 20/6 35,355. 



Again (as before) unit is to 3.1416 as 



352/ I 27400 diani. DB, or 35,355, is to the circumference 



] 24689 of the circle ABCD; or thus, 



1 : 3.1416 : .' 35.355 (by common rule of 



27H00 three), to 111.072. QED. 

 247429 



35347 



353546 | 2367100 

 2121276 



So, as an average for any general purposes of the department to 

 which I belong, or for any merchant or artisan in his vocation, 11 per 

 cent may be used as the difference between any cylinder and its 

 square ; or, by the common Rule of Three, as 111 is to 100, so is the 

 measurement of any known circle to the square of that circle. f 



A sphere or round body cubed, would of course suffer double the 

 loss, or give a difference of 22 per cent. 



Cochin — Malabar Coast, \6thJuly 1837. 



* This is working the reverse way, but as it enabled me to come, by a single process, 

 to the required information, it was adopted. 



t The decimal .072, or t ^-q per cent may, for any practical purposes, be omitted as 

 insignificant, but should a greater degree of accuracy be required, it may be obtained, I 

 may say, to fractional nicety, by the addition of unit to the sum of per centage of every 

 1400,— which would so be, 14 x 11 = 15 + 4 1 = 155 ; or as 1400, is to 1555 : and a half or 

 double, &c, in proportion can also be used if necessary. 



