Page:Newton's Principia (1846).djvu/277

 has in H. If that lineola IN is of a finite magnitude, it will be expressed by the third term, together with those that follow in infinitum. But if that lineola be diminished in infinitum, the terms following become in finitely less than the third term, and therefore may be neglected. The fourth term determines the variation of the curvature; the fifth, the variation of the variation; and so on. Whence, by the way, appears no contemptible use of these series in the solution of problems that depend upon tangents, and the curvature of curves.

Now compare the series $$\scriptstyle e-\frac{ao}{e}-\frac{nnoo}{2e^{3}}-\frac{anno^{3}}{2e^{5}}-$$ &c., with the series P - Qo - Roo - So³ - &c., and for P, Q, R and S, put e, $$\scriptstyle \frac{a}{e}$$, $$\scriptstyle \frac{nn}{2e^{3}}$$ and $$\scriptstyle \frac{ann}{2e^{5}}$$, and for $$\scriptstyle \sqrt{1+QQ}$$ put $$\scriptstyle \sqrt{1+\frac{aa}{ee}}$$ or $$\scriptstyle \frac{n}{e}$$: and the density of the medium will come out as $$\scriptstyle \frac{a}{ne}$$; that is (because n is given), as $$\scriptstyle \frac{a}{e}$$ or $$\scriptstyle \frac{AC}{CH}$$, that is, as that length of the tangent HT, which is terminated at the semi-diameter AF standing perpendicularly on PQ: and the resistance will be to the gravity as 3a to 2n, that is, as 3AC to the diameter PQ of the circle; and the velocity will be as $$\scriptstyle \sqrt{CH}$$. Therefore if the body goes from the place F, with a due velocity, in the direction of a line parallel to PQ, and the density of the medium in each of the places H is as the length of the tangent HT, and the resistance also in any place H is to the force of gravity as 3AC to PQ, that body will describe the quadrant FHQ of a circle. Q.E.I.

But if the same body should go from the place P, in the direction of a line perpendicular to PQ, and should begin to move in an arc of the semi circle PFQ, we must take AC or a on the contrary side of the centre A; and therefore its sign must be changed, and we must put -a for +a. Then the density of the medium would come out as -$$\scriptstyle \frac{a}{e}$$. But nature does not admit of a negative density, that is, a density which accelerates the motion of bodies; and therefore it cannot naturally come to pass that a body by ascending from P should describe the quadrant PF of a circle. To produce such an effect, a body ought to be accelerated by an impelling medium, and not impeded by a resisting one.

2. Let the line PFQ be a parabola, having its axis AF perpendicular