PROPOSITION XLI. PROBLEM XXI.
From three observations given to determine the orbit of a comet moving in a parabola.
This being a Problem of very great difficulty, I tried many methods of resolving it; and several of these Problems, the composition whereof I have given in the first Book, tended to this purpose. But afterwards I contrived the following solution, which is something more simple.
Select three observations distant one from another by intervals of time nearly equal; but let that interval of time in which the comet moves more slowly be somewhat greater than the other; so, to wit, that the difference of the times may be to the sum of the times as the sum of the
times to about 600 days; or that the point E may fall upon M nearly, and may err therefrom rather towards I than towards A. If such direct observations are not at hand, a new place of the comet must be found, by Lem. VI.
Let S represent the sun; T, t, τ, three places of the earth in the orbis magnus; TA, tB, τC, three observed longitudes of the comet; V the time between the first observation and the second; W the time between the second and the third; X the length which in the whole time V + W
the comet might describe with that velocity which it hath in the mean distance of the earth from the sun, which length is to be found by Cor. 3,
