VI.—On the relative proportion between Circular bodies and their Squares.—By Captain Cortlandt Taylor, of the Madras Artillery.
Being on a mission to the Malabar Coast connected with timber for the Ordnance Department at Madras, I found the native dealers in that article at Calicut disposed of timber cut to the square, varying about thirty per cent. from their measurement in the round log. I was induced in consequence to fall back upon school recollections, to ascertain how far this method was correct or borne out by theory, and which the subjoined mathematical process enabled me to do.[1] As the results, for comparative data, may be of practical utility, perhaps its dissemination will not be unattended with benefit.
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Let ABCD be a circle, and the diameters AC, and BD, perpendicular to each other; then are the triangles AEB, BEC, CED, and DEA (formed by the radii of the circle and the points ABC and D being joined), right angle triangles and similar; having the common centre E a right angle in each; the legs AE, BE, CE, and DE (or circle's radii), equal; and the lines or hypothenuses AB, BC, CD, and AD, also equal, and together forming an inscribed square to the circle ABCD.
Then, as the circumferences of all circles are to their diameters, as 3.1416 is to unit; so 3.1416 is to 1, as the circumference of any given circle ABCD, is to its diameter, DB: but DB the diameter is the double of EB the radius, and EB is one leg or side of the right angle triangle AEB, of which AE is the other leg, and AB, is the third side, or hypothenuse.
Then, as in right angle triangles, the square of the hypothenuse is equal to the sum of the squares of the other two sides; in the right angle triangle AEB,-BE2+ AE2=AB2 but as BE, and AE are equal, their squares are equal also, and 2, EB2=AB2 or V2, EB2=AB, the one-fourth of the pevimeter of the required square.
Thus knowing the circumference of any circle, the pevimeter, or measurement of the inscribed square, can be easily ascertained; and
- ↑ The salo of the slabs pays the sawing expenses of squaring.
