u154In 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.

PROBLEM II.

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 , 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 otherwise,[Pg 572] a new number V is to be assumed, and a new trajectory to be found; and then the place of the comet in this trajectory to the time of that probatory observation, and the intersection of a right line drawn to the comet with the plane of the trajectory, are to be determined as before; and by comparing the variation of the error with the variation of the other quantities, we may conclude, by the Rule of Three, how far those other quantities ought to be varied or corrected, so as the error may become as small as possible. And by means of these corrections we may have the trajectory exactly, providing the observations upon which the computation was founded were exact, and that we did not err much in the assumption of the quantity V; for if we did, the operation is to be repeated till the trajectory is exactly enough determined. Q.E.F.

END OF THE SYSTEM OF THE WORLD.

[Pg 573]

[Pg 575]

TRANSCRIBER’S NOTES

Simple typographical errors have been silently corrected; unbalanced quotation marks were remedied when the change was obvious, and otherwise left unbalanced.

Punctuation and spelling were made consistent when a predominant preference was found in the original book; otherwise they were not changed.

Inconsistent hyphens left as printed.

Due to lack of clarity, the figures on pages 82 and 144 have been replaced by the corresponding figures from the Latin version.

In the table on page 318, in the fifth row, the third column has been repositioned in the fourth column, according to the text that follows on page 319.