302

*Circular bodies and their Squares.*

[Oct.

*vice versa*, knowing the square, the circle can be obtained and the difference 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*^{[1]} 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 25², is equal to AE² + EB²; that is 625=AE²+EB²; but as AE and EB are radii of the same circle and equal, their squares are equal also, and 625 = 2EB²; or 6252 = EB²; or V6252 = EB; or again V312.5 = EB (Extract root of 312—5).

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312.5 (17,6776 the root, 1

27 212

189

346 2350

2076

3527 27400

24689

35347 271100

247429

353546 2367100

2121276

That is 17.6776 is equal to EB; but EB, or radius of the circle, being half the diameter DB, then the diani. DB, is equal to 35,355.

Again (as before) unit is to 3.1416 as diani. DB, or 35,355, is to the circumference of the circle ABCD; or thus, 1 : 3.1416 : : 35.355 (by common rule of three), to 111.072. QED.

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.^{[2]}

A sphere or round body *cubed,* would of course suffer double the loss, or give a difference of 22 per cent.

Cochin—Malabar Coast, 16*th July* 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. - ↑ The decimal .072, or 721000
*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 × 11 = 15 + ₄1 = 155; or as 14oo, is to 1555: and a half or double, &c, in proportion can also be used if necessary.