THEOREM X11. 1ZULE 5, is a very near approximation to the quadrature of the segment of a circle. it is much ea.ier than any other rule yet shown for the same purpose. It was invented, and first published in the article MENSURATION, of The principles of Architecture, by Nicholson ; sinee that time, it has been copied into the new edition of Ilawney's Mensuration. The rule was first given without a demonstra tion; but it is now supplied with the following investigation.
be the value of the segment, the diameter of which is d, and
its versed sine v; then by comparing these two series they
will be found to be nearly equal, the former being the greater; Now, let ABCDEF
the suns of all these areas will be the area of the whole c + 4 d + e curve ;
.3" m+
X m e + 4 f +g4b+4d+4f+2c+2e+a+g X 7n, &e.
3 3 4 x (6+ d + f) 2 x (c + + a + g X m = x m 3 equal to the area of the curve.
This not only holds true in superficies, but also in solids.
Then, if the intermediate ordinates be called sections, and be numbered 1, 2, 3, &c., the first being called odd, the next even, and so on alternately • the last section will always be odd ; then to four times the sum of the odd sections add twice the sum of the even sections, and the two ends ; then one-third of the suns being multiplied by the common distance gives the area, which is a near approximation for any curvi linear figure whatever.