Publisher's Synopsis
This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1801 edition. Excerpt: ...by flowing uniformly acquires the augment Dtt, a to the space R as DG is to Dd (by theor. 2), or as I'J is to Eo, and therefore is equal tathearea.o.f a circle whose radius is a mean proportional betwixt do and 3Ea; but the the area of this circle is greater than the surface described by the right line Eo, by lemma 8, and the conical surface aEe increases with an accelerated motion when the axis increases uniformly, by art. 217. Therefore a greater space is generated in the same time by a motion continued uniformly, than when the same motion is continually accelerated, against the first axiom. In the same manner it appears, from the second axiom, that the space R is not less than the area of a circle the radius of which is a mean proportional betwixt DE and PROP. XIX. 229. Let DE and GH (fg. 74) perpendicular to the axis meet the curve in E and H, and let GH meet the tangent at E in t; then, the fluxion of the axis being represented byDG, the fluxion of the surface described by the curve FE shall be accurately measured by the area of a circle the radius of rchich is a mean proportional betwixt DE and 2Ef. Let the ordinate PM always meet the curve in M and the tangent aE in N; and the motions with which the surfaces F/inM, an flow shall be equal at the term when M and N come together to E, or the fluxion of the surface TfeE shall be equal to the fluxion of the surface aEe. For, the construction being the same as in the 226th and 228th articles, first let the arch CEH be convex towards the axis BG, and the surface F/wiM shall flow with an accelerated motion while P and M describe BG and CH, by art. 226. Suppose, first, BD to be so small, according to the 225th article, that the perpendicular troui T upon the tangent at C may meet DE in D, ..
