25 S

PlVlSlON OF CIRCLES AND ELLIPSES. TRAC* 14.

ivonld still hold good, if ab were any other diameter of theellipse, instead of the axis ; describing on the parts of it semi-ellipses which shall be similar to those into which the diame-ter ab divides the given ellipse.

10. And further, if a circle be described about the ellipse,on the diameter ab, and lines be drawn similar to those in thesecond figure; then, by a process the very same as in Art. 4,et seq. substituting only semiellipse for semicircle, it is foundthat the space

pq is equal to the similar ellipse on the diameter be,tqrs is equal to the similar ellipse on the diameter bf,

Rs is equal to the similar ellipse on the diameter ah,or to the difference of the ellipses on bf and be ;also the elliptic spaces - - - pq, pqrs, rs, tv,

are respectively as the lines - bc, bd, dc, ad,the same ratio as the circular spaces. And hence an ellipseis divided into any number of parts, in any assigned ratios,in the same manner as the circle is divided, namely, dividingthe axis, or any diameter in the same manner, and on the parts-of it describing similar semiellipses.

TRACT XV.

An approximate geometrical division of the circle,

The solution, here improved, of the following problem, Ifirst gave in my Miscellanea Mathematica, published in theyear 1775, pa. 311. The problem is as follows.

To find whether there is any such fixedpoint e, in the radius bd produced, bi-secting the semicircle abc, so that anyline efg being drawn from it, this lineshall always cut the perpendicular radiusad and the quadrantal arc ab, propor-tionally in the two points f and g; viz. sothat DF shall be to BG in a constant ratio.