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For if we join EO, cutting the parabolic arc ABC in Y, and draw μX touching the same arc in the vertex μ, and meeting EO in X, the curvilinear area AEXμA will be to the curvilinear area ACYμA as AE to AC; and, therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area ASEXμA will be to the whole area ASCYμA as |
AE to AC. But, because ξO is to SO as 3 to 1, and EO to XO in the same proportion, SX will be parallel to EB; and, therefore, joining BX, the triangle SEB will be equal to the triangle XEB. Wherefore if to the area ASEXμA we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBXμA, equal to the area ASEXμA, and therefore in proportion to the area ASCYμA as AE to AC. But the area ASBYμA is nearly equal to the area ASBXμA; and this area ASBYμA is to the area ASCYμA as the time of description of the arc AB to the time of description of the whole arc AC; and, therefore, AE is to AC nearly in the proportion of the times. Q.E.D. Cor. When the point B falls upon the vertex μ of the parabola, AE is to AC accurately in the proportion of the times. |
SCHOLIUM. |
If we join μξ cutting AC in δ, and in it take ξn in proportion to μB as 27MI to 16Mμ, and draw Bn, this Bn will cut the chord AC, in the proportion of the times, more accurately than before; but the point n is to be taken beyond or on this side the point ξ, according as the point B is more or less distant from the principal vertex of the parabola than the point μ. |
LEMMA IX. |
The right lines Iμ and μM, and the length , are equal among themselves. |
For 4Sμ is the latus rectum of the parabola belonging to the vertex μ. |
LEMMA X. |
Produce Sμ to N and P, so as μN may be one third of μI, and SP may be to SN as SN to Sμ; and in the time that a comet would describe the arc AμC, if it was supposed to move always forwards with the velocity which it hath in a height equal to SP, it would describe a length equal to the chord AC. |
For if the comet with the velocity which it hath in μ was in the said time supposed to move uniformly forward in the right line which touches the parabola in μ, the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ASCμA; and therefore the space contained under the length described in the tangent and the length Sμ would be to the space contained under the lengths AC and SM as the area ASCμA to the triangle ASC, that is, as SN to SM. Wherefore AC is to the length described in the tangent as Sμ to SN. But since the velocity of the comet in the height SP (by Cor. 6, Prop. XVI., Book I) is to the velocity of the same in the height Sμ in the reciprocal subduplicate proportion of SP to Sμ, that is, in the proportion of Sμ to SN, the length described with this velocity will be to the length in the same time described in the tangent as Sμ to SN. Wherefore since AC, and the length described with this new velocity, are in the same proportion to the length described in the tangent, they mast be equal betwixt themselves. Q.E.D. |
Cor. Therefore a comet, with that velocity which it hath in the height Sμ + ⅔Iμ, would in the same time describe the chord AC nearly. |
LEMMA XI. |
If a comet void of all motion was let fall from, the height SN, or Sμ + ⅓Iμ, towards the sun, and was still impelled to the sun by the same force uniformly continued by which it was impelled at first, the same, in one half of that time in which it might describe the arc AC in its own orbit, would in descending describe a space equal to the length Iμ. |
For in the same time that the comet would require to describe the parabolic arc AC, it would (by the last Lemma), with that velocity which it hath in the height SP, describe the chord AC; and, therefore (by Cor. 7, Prop. XVI, Book 1), if it was in the same time supposed to revolve by the force of its own gravity in a circle whose semi-diameter was SP, it would describe an arc of that circle, the length of which would be to the chord of the parabolic arc AC in the subduplicate proportion of 1 to 2. Wherefore if with that weight, which in the height SP it hath towards the sun, it should fall from that height towards the sun, it would (by Cor. 9, Prop. XVI, Book 1) in half the said time describe a space equal to the square of half the said chord applied to quadruple the height SP, that is, it would describe the space . But since the weight of the comet towards the sun in the height SN is to the weight of the same towards the sun in the height SP as SP to Sμ, the comet, by the weight which it hath in the height SN, in falling from that height towards the sun, would in the same time describe the space ; that is, a space equal to the length Iμ or μM . Q.E.D. |
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, Prop. XL, Book III; and tV a perpendicular upon the chord Tτ. In the mean observed longitude tB take at pleasure the point B, for the place of the comet in the plane of the ecliptic; and from thence, towards the sun S, draw the line BE, which may be to the perpendicular tV as the content under SB and St² to the cube of the hypothenuse of the right angled triangle, whose sides are SB, and the tangent of the latitude of the comet in the second observation to the radius tB. And through the point E (by Lemma VII) draw the right line AEC, whose parts AE and EC, terminating in the right lines TA and τC, may be one to the other as the times V and W: then A and C will be nearly the places of the comet in the plane of the ecliptic in the first and third observations, if B was its place rightly assumed in the second. |
Upon 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 λ, and complete the parallelogram iI λμ. Take Iσ equal to 3Iλ; and through the sun S draw the obscure line σξ equal to 3Sσ + 3iλ. Then, cancelling the letters A, E, C, I, from the point B towards the point ξ, draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity Sμ + ⅓iλ. 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 τC. Join MN, cutting IO in O. Draw the rectangular parallelogram iIλμ, as before. In IA produced take ID equal to Sμ + ⅔iλ. 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 β assumed at pleasure, find out the new points e, a, c, g; and ε, α, κ, γ. Then through G, g, and γ, draw the circumference of a circle Ggγ, cutting the right line τC in Z: and Z will he one place of the comet in the plane of the ecliptic. And in AC, ac, ακ, taking AF, af, αϕ, equal respectively to CG, cg, κγ; through the points F, f, and ϕ, draw the circumference of a circle Ffϕ, cutting the right line AT in X; and the point X will be another place of the comet in the plane of the ecliptic. And at the points X and Z, erecting the tangents of the latitudes of the comet to the radii TX and τZ, 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.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 Bξ 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, β, 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 4⁄3Tτ 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 τC 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 β, 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 τC 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 observations. |
1680, Dec. 12 |
21 |
24 |
26 |
29 |
30 |
1681, Jan. 5 |
9 |
10 |
13 |
25 |
30 |
Feb. 2 |
5 Time Sun's |
Longitude Comet's |
Appar. True. Longitude. Lat. N. |
h. ″ |
4.46 |
6.32½ |
6.12 |
5.14 |
7.55 |
8.02 |
5.51 |
6.49 |
5.54 |
6.56 |
7.44 |
8.07 |
6.20 |
6.50 h. ′ ″ |
4.46.0 |
6.36.59 |
6.17.52 |
5.20.44 |
8.03.02 |
8.10.26 |
6.01.38 |
7.00.53 |
6.06.10 |
7.08.55 |
7.58.42 |
8.21.53 |
6.34.51 |
7.04.41 ° ′ ″ |
♑ 1.51.23 |
11.06.44 |
14.09.26 |
16.09.22 |
19.19.43 |
20.21.09 |
26.22.18 |
♒ 0.29.02 |
1.27.43 |
4.33.20 |
16.45.36 |
21.49.58 |
24.46.59 |
27.49.51 ° ′ ″ |
♑ 6.32.30 |
♒ 5.08.12 |
18.49.23 |
28.24.13 |
♓ 13.10.41 |
17.38.20 |
♈ 8.48.53 |
18.44.04 |
20.40.50 |
25.59.48 |
♉ 9.35.0 |
13.19.51 |
15.13.53 |