Page:Newton's Principia (1846).djvu/577

 is likewise given both in magnitude and position, together with the distance ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved about the sun in a circle, at the same distance SP, as V to 1; and we shall have a line pPῶ to be determined, of this condition, that the space pῶ, described by the comet in two hours, may be to the space V $$\scriptstyle \times$$ tτ (that is, to the space which the earth describes in the same time multiplied by the number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun; and that the space pP, described by the comet in the first hour, may be to the space Pῶ, described by the comet in the second hour, as the velocity in p to the velocity in P; that is, in the subduplicate ratio of the distance SP to the distance Sp, or in the ratio of 2Sp to SP + Sp; for in this whole work I neglect small fractions that can produce no sensible error.

In the first place, then, as mathematicians, in the resolution of affected equations, are wont, for the first essay, to assume the root by conjecture, so, in this analytical operation, I judge of the sought distance TR as I best can by conjecture. Then, by Lem. II. I draw rρ, first supposing rR equal to Rρ, and again (after the ratio of SP to Sp is discovered) so as rR may be to Rρ as 2SP to SP + Sp, and I find the ratios of the lines pῶ, rρ, and OR, one to the other. Let M be to V $$\scriptstyle \times$$ tτ as OR to pῶ; and because the square of pῶ is to the square of V $$\scriptstyle \times$$ tτ as ST to SP, we shall have, ex æquo, OR² to M² as ST to SP, and therefore the solid OR² $$\scriptstyle \times$$ SP equal to the given solid M² $$\scriptstyle \times$$ ST; whence (supposing the triangles STP, PTR, to be now placed in the same plane) TR, TP, SP, PR, will be given, by Lem. I. All this I do, first by delineation in a rude and hasty way; then by a new delineation with greater care; and, lastly, by an arithmetical computation. Then I proceed to determine the position of the lines rρ, pῶ, with the greatest accuracy, together with the nodes and inclination of the plane Spῶ to the plane of the ecliptic; and in that plane Spῶ I describe the trajectory in which a body let go from the place P in the direction of the given right line pῶ would be carried with a velocity that is to the velocity of the earth as pῶ to V $$\scriptstyle \times$$ tτ. Q.E.F.

To correct the assumed ratio of the velocity and the trajectory thence found.

Take an observation of the comet about the end of its appearance, or any other observation at a very great distance from the observations used before, and find the intersection of a right line drawn to the comet, in that observation with the plane Spῶ, as well as the comet's place in its trajectory to the time of the observation. If that intersection happens in this place, it is a proof that the trajectory was rightly determined; if other-