u126Let HI, IK, KL, LM (in the preceding Fig.), represent the times between the observations; HA, IB, KC, LD, ME, five observed longitudes of the comet; and HS the given time between the first observation and the longitude required. Then if a regular curve ABCDE is supposed to be drawn through the points A, B, C, D, E, and the ordinate RS is found out by the preceding lemma, RS will be the longitude required.

[Pg 468]

After the same method, from five observed latitudes, we may find the latitude to a given time.

If the differences of the observed longitudes are small, suppose of 4 or 5 degrees, three or four observations will be sufficient to find a new longitude and latitude; but if the differences are greater, as of 10 or 20 degrees, five observations ought to be used.

LEMMA VII.

Through a given point P to draw a right line BC, whose parts PB, PC, cut off by two right lines AB, AC, given in position, may be one to the other in a given proportion.

From the given point P suppose any right line PD to be drawn to either of the right lines given, as AB; and produce the same towards AC, the other given right line, as far as E, so as PE may be to PD in the given proportion. Let EC be parallel to AD. Draw CPB, and PC will be to PB as PE to PD. Q.E.F.

LEMMA VIII.

Let ABC be a parabola, having its focus in S. By the chord AC bisected in I cut off the segment ABCI, whose diameter is and vertex . In produced take equal to one half of . Join OS, and produce it to as may be equal to 2SO. Now, supposing a comet to revolve in the arc CBA, draw , cutting AC in E; I say, the point E will cut off from the chord AC the segment AE, nearly proportional to the time.

For if we join EO, cutting the parabolic arc ABC in Y, and draw touching the same arc in the vertex , and meeting EO in X, the curvilinear area will be to the curvilinear area as AE to AC; and, therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area will be to the whole area as[Pg 469] AE to AC. But, because 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 we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area , equal to the area ; and therefore in proportion to the area as AE to AC. But the area is nearly equal to the area ; and this area is to the area 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 of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

If we join cutting AC in ; and in it take in proportion to as 27MI to , 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 , according as the point B is more or less distant from the principal vertex of the parabola than the point .

LEMMA IX.

The right lines and , and the length , are equal among themselves.

For is the latus rectum of the parabola belonging to the vertex .

LEMMA X.

Produce to N and P, so as may be one third of , and SP may be to SN as SN to ; and in the time that a comet would describe the arc , 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 was in the said time supposed to move uniformly forward in the right line which touches the parabola in , the area which it would describe by a radius drawn to the point S would be equal to the parabolic area ; and therefore the space contained under the length described in the tangent and the length would be to the space contained under the lengths AC and SM as the[Pg 470] area to the triangle ASC, that is, as SN to SM. Wherefore AC is to the length described in the tangent as 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 in the reciprocal subduplicate proportion of SP to , that is, in the proportion of to SN, the length described with this velocity will be to the length in the same time described in the tangent as 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 must be equal betwixt themselves. Q.E.D.

COR. Therefore a comet, with that velocity which it hath in the height , 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 , 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 .

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 I), 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 I) 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 , 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 or . Q.E.D.

[Pg 471]

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, , three places of the earth in the orbis magnus; TA, tB, , 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,[Pg 472] Prop. XL, Book III; and tV a perpendicular upon the chord . 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 , 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.