u127Upon AC, bisected in I, erect the perpendicular Ii. Through B draw the obscure line Bi parallel to AC. Join the obscure line Si, cutting AC in , and complete the parallelogram iI . Take equal to ; and through the sun S draw the obscure line equal to . Then, cancelling the letters A, E, C, I, from the point B towards the point , draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity . And through the point E draw again the right line AEC by the same rule as before; that is, so as its parts AE and EC may be one to the other as the times V and W between the observations. Thus A and C will be the places of the comet more accurately.

Upon AC, bisected in I, erect the perpendiculars AM, CN, IO, of which AM and CN may be the tangents of the latitudes in the first and third observations, to the radii TA and . Join MN, cutting IO in O. Draw the rectangular parallelogram , as before. In IA produced take ID equal to . Then in MN, towards N, take MP, which may be to the above found length X in the subduplicate proportion of the mean distance of the earth from the sun (or of the semi-diameter of the orbis magnus) to the distance OD. If the point P fall upon the point N; A, B, and C, will be three places of the comet, through which its orbit is to be described in the plane of the ecliptic. But if the point P falls not upon the point N, in the right line AC take CG equal to NP, so as the points G and P may lie on the same side of the line NC.

By the same method as the points E, A, C, G, were found from the assumed point B, from other points b and assumed at pleasure, find out the new points e, a, c, g; and , , , . Then through G, g, and , draw the circumference of a circle , cutting the right line in Z: and Z will be one place of the comet in the plane of the ecliptic. And in AC, ac, , making AF, af, , equal respectively to CG, cg, ; through the points F, f, and , draw the circumference of a circle , cutting the right line AT in X; and the point X will be another place of the comet in the plane of[Pg 473] the ecliptic. And at the points X and Z, erecting the tangents of the latitudes of the comet to the radii TX and , two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book I) to the focus S a parabola is described passing through those two places, this parabola will be the orbit of the comet. Q.E.I.

The demonstration of this construction follows from the preceding Lemmas, because the right line AC is cut in E in the proportion of the times, by Lem. VII, as it ought to be, by Lem. VIII.; and BE, by Lem. XI., is a portion of the right line BS or 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 first 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, , not at random, but nearly true. If the angle AQt, at which the projection of the orbit in the plane of the ecliptic cuts the right line tB, is rudely known, at that angle with Bt draw the obscure line AC, which may be to in the subduplicate proportion of SQ to St; and, drawing the right line SEB so as its part EB may be equal to the length Vt, 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, moreover, finding the length MP, in tB take the point b, by this rule, that, if TA and intersect each other in Y, the distance Yb 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 , 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, f, and G, g, are found, draw the right lines Ff and Gg, and they will cut TA and in the points required, X and Z.

EXAMPLE.

[Pg 474]

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 observations.

To these you may add some observations of mine.

These observations were made by a telescope of 7 feet, with a micrometer and threads placed in the focus of the telescope; by which instruments we determined the positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ), C a star of the sixth magnitude (Bayer's ) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, , , , , other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places of the comet in the observations above set down; and, reckoning the distance AB of parts, AC was of those parts; BC, ; AD, ; BD, ; CD, ; AE, ; CE, ; DE, ; AI, ; BI, ; CI, ; DI, ; AK, ; BK, 43; CK, ; FK, 29; FB, 23; FC, ; AH, ; DH, ; BN, ; CN, ; BL, ; NL, . HO was to HI as 7 to 6, and, produced, did pass between the stars D and E, so as the distance of the star D from this right line was . LM was to LN as 2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.

[Pg 475]

Mr. Pound has since observed a second time the positions of those fixed stars amongst themselves, and collected their longitudes and latitudes according to the following table.

[Pg 476]

The positions of the comet to these fixed stars were observed to be as follow:

Friday, February 25, O.S. at P.M. the distance of the comet in p from the star E was less than , and greater than , and therefore nearly equal to ; and the angle ApE was a little obtuse, but almost right. For from A, letting fall a perpendicular on pE, the distance of the comet from that perpendicular was .

The same night, at , the distance of the comet in P from the star E was greater than , and less than , and therefore nearly equal to of AE, or . But the distance of the comet from the perpendicular let fall from the star A upon the right line PE was .

Sunday, February 27, P. M. the distance of the comet in Q, from the star O was equal to the distance of the stars O and H; and the right line QO produced passed between the stars K and B. I could not, by reason of intervening clouds, determine the position of the star to greater accuracy.

Tuesday, March 1, 11h. P. M. the comet in R lay exactly in a line between the stars K and C, so as the part CR of the right line CRK was a little greater than , and a little less than , and therefore = , or .

Wednesday, March 2, 8h. P. M. the distance of the comet in S from the star C was nearly ; the distance of the star F from the right line CS produced was ; and the distance of the star B from the same right line was five times greater than the distance of the star F; and the right line NS produced passed between the stars H and I five or six times nearer to the star H than to the star I.