260 GEOMETRICAL DIVISION OF THE CIRCLE. TRACT 15 .

of parts, and through the 2d point of division draw efg, sowill ag be one of the equal parts very nearly.

Corol. 3.—The number T752 being equal to \/3 nearly,for V3 = T732; therefore, if De be taken to da as to1, the point e will be found answering the same purpose asbefore, but not quite so near as the former. And here, be-cause da : de :: 1 : \/3, therefore de is the perpendicularof an equilateral triangle described on ac. Hence then, ifwith the centres a, c, and radius ac, two arcs be described,they will intersect in the point e, nearly the same as before.And this is the method in common practice; but it is not sonear the truth as the construction in the 2d Corollary.

Corol, 4.—Hence also a right line is found equal to the arcof a circle nearly : for bg is = y df nearly. And this is thesame as the ratio of 11 to 1, which Archimedes gave for theratio of the semicircumference to the diameter, or 22 to 7 theratio of the whole circumference to the diameter. But theproportion is here rendered general for any arc of the circle,as well as for the whole circumference.

TRACT XVI.

ON PLANE trigonometry without tables.

The cases of trigonometry are usually calculated by meansof tables of sines, tangents or secants, either of their naturalnumbers, or their logarithms. But the calculations may alsobe made without any such tables, to a tolerable degree of ac-curacy, by means of the theorems and rules contained in thefollowing propositions and corollaries.

froposition.

If 2 a denote a side of any triangle, A the number of degreescontained in its opposite angle, and r the radius of the circle