BOOK III.] OF NATURAL PHILOSOPHY. 4~3 the ecliptic. And at the points X and Z, erecting the tangents of the latitudes of the comet to the radii TX and rZ, two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book 1) to the focus S a parabola is described passing through those two places, this parabola will be the orbit of the comet. Q.E.L The demonstration of this construction follows from the preceding Lem mas, because the right line AC is cut in E in the proportion of the times, by Lem. VI L, as it ought to be, by Lem. VIII. ; and BE, by Lem. XL, is a portion of the right line BS or B£ in the plane of the ecliptic, intercepted between the arc ABC and the chord AEC ; and MP (by Cor. Lem. X.) is the length of the chord of that arc, which the comet should describe in its proper orbit between the firs': and third observation, and therefore is equal to MN, providing B is a true place of the comet in the plane of the ecliptic. But it will be convenient to assume the points B, b, (3, not at random, but nearly true. If the angle AQ/, at which the projection of the orbit in the plane of the ecliptic cuts the right line £B, is rudely known, at that angle with Bt draw the obscure line AC, which may be to -f TT in the subduplicate proportion of SQ, to S/ ; and, drawing the right line SEB so as its part EB may be equal to the length \t, the point B will be determined, which we are to use for the first time. Then, cancelling the right line AC, and drawing anew AC according to the preceding construction, and, aioreover, finding the length MP, in tB take the point b, by this rule, that, if TA and rC intersect each other in Y, the distance Y6 may be to the distance YB in a proportion compounded of the proportion of MP to MN, and the subduplicate proportion of SB to Sb. And by the same method you may find the third point 18, if you please to repeat the operation the third time ; but if this method is followed, two operations generally will be sufficient ; for if the distance Bb happens to be very small, after the points F,/, and G, §•, are found, draw the right lines F/and G^-, and they will cut TA and rC in the points required, X and Z. EXAMPLE. Let the comet of the year 1680 be proposed. The following table shews the motion thereof, as observed by Flamsted, and calculated afterwards by him from his observations, and corrected by Dr. Halley from the same ob servations.