The System of the World
Problem I
From the assumed ratio of the velocity to determine the trajectory of a comet.
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Let S represent the sun; t, T, , three places of the earth in its orbit at equal distances; p, P, , as many corresponding places of the comet in its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour; pr, PR, , perpendiculars let fall on the plane of the ecliptic, and the vestige of the trajectory in this plane. Join Sp, SP, , SR, ST, tr, TR, , TP, and let tr, , meet in O, TR will nearly converge to the same point O, or the error will be inconsiderable. By the premised lemmas the angles rOR, , are given, as well as the ratios pr to tr, PR to TR, and to. The figure 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 to be determined, of this condition, that the space, described by the comet in two hours, may be to the space (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, 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, first supposing rR equal to, and again (after the ratio of SP to Sp is discovered) so as rR may be to as 2SP to SP + Sp, and I find the ratios of the lines, , and OR, one to the other. Let M be to as OR to; and because the square of is to the square of as ST to SP, we shall have, ex æquo, OR2 to M2 as ST to SP, and therefore the solid OR2 × SP equal to the given solid M2 × 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, , with the greatest accuracy, together with the nodes and inclination of the plane to the plane of the ecliptic; and in that plane I describe the trajectory in which a body let go from the place P in the direction of the given right line would be carried with a velocity that is to the velocity of the earth as to. Q.E.F.