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diurnal motion in |
its own orbit. Its distance from |
the earth in parts |
whereof the radius |
of the orbis magnus |
contains 1000. |
60° |
65 |
70 |
72 |
74 |
76 |
78 |
80 |
82 |
84 |
86 |
88 |
90 Direct. |
2° 18′ |
2 33 |
2 55 |
3 07 |
3 23 |
3 43 |
4 10 |
4 57 |
5 45 |
7 18 |
10 27 |
18 37 |
Infinite Retrog. |
00° 20′ |
00 35 |
00 57 |
01 09 |
01 25 |
01 45 |
02 12 |
02 49 |
03 47 |
05 20 |
08 19 |
16 39 |
Infinite |
1000 |
845 |
684 |
618 |
651 |
484 |
416 |
347 |
278 |
209 |
140 |
70 |
00 |
The ingress of a comet into the sphere of the orbis magnus, or its egress from the same, happens at the time of its elongation from the sun, expressed in col. 1, against its diurnal motion. So in the comet of 1681. Jan. 4, O.S. the apparent diurnal motion in its orbit was about 3° 5′, and the corresponding elongation 71⅔°; and the comet had acquired this elongation from the sun Jan. 4, about six in the evening. Again, in the year 1680, Nov. 11, the diurnal motion of the comet that then appeared was about 4⅔°; and the corresponding elongation 79⅔ happened Nov. 10, a little before midnight. Now at the times named these comets had arrived at an equal distance from the sun with the earth, and the earth was then almost in its perihelion. But the first table is fitted to the earth's mean distance from the sun assumed of 1000 parts; and this distance is greater by such an excess of space as the earth might describe by its annual motion in one day's time, or the comet by its motion in 16 hours. To reduce the comet to this mean distance of 1000 parts, we add those 16 hours to the former time, and subduct them from the latter; and thus the former becomes Jan. 4d. 10h. afternoon; the latter Nov. 10, about six in the morning. But from the tenor and progress of the diurnal motions it appears that both comets were in conjunction with the sun between Dec. 7 and Dec. 8; and from thence to Jan. 4d.10h. afternoon on one side, and to Nov. 10d.6h. of the morning on the other, there are about 28 days. And so many days (by Table 1) the motions in parabolic trajectories do require. |
But though we have hitherto considered those comets as two, yet, from the coincidence of their perihelions and agreement of their velocities, it is probable that in effect they were but one and the same; and if so, the orbit of this comet must have either been a parabola, or at least a conic section very little differing from a parabola, and at its vertex almost in contact with the surface of the sun. For (by Tab. 2) the distance of the comet from the earth, Nov. 10, was about 360 parts, and Jan. 4, about 630. From which distances, together with its longitudes and latitudes, we infer the distance of the places in which the comet was at those times to have been about 280: the half of which, viz., 140, is an ordinate to the comet's orbit, cutting off a portion of its axis nearly equal to the radius of the orbis magnus, that is, to 1000 parts. And, therefore, dividing the square of the ordinate 140 by 1000, the segment of the axis, we find the latus rectum 19,16, or in a round number 20; the fourth part whereof, 5, is the distance of the vertex of the orbit from the sun's centre. But the time corresponding to the distance of 5 parts in Tab. 1 is 27d.16h.7′. In which time, if the comet moved in a parabolic orbit, it would have been carried from its perihelion to the surface of the sphere of the orbis magnus described with the radius 1000, and would have spent the double of that time, viz., 55d.8¼h. in the whole course of its motion within that sphere: and so in fact it did; for from Nov. 10d. 6h. of the morning, the time of the comet's ingress into the sphere of the orbis magnus, to Jan. 4d. 10h. afternoon, the time of its egress from the same, there are 55d. 16h. The small difference of 7¾h. in this rude way of computing is to be neglected, and perhaps may arise from the comet's motion being some small matter slower, as it must have been if the true orbit in which it was carried was an ellipsis. The middle time between its ingress and egress was December 8d.2h. of the morning; and therefore at this time the comet ought to have been in its perihelion. And accordingly that very day, just before sunrising, Dr. Halley (as we said) saw the tail short and broad, but very bright, rising perpendicularly from the horizon. From the position of the tail it is certain that the comet had then crossed over the ecliptic, and got into north latitude, and therefore had passed by its perihelion, which lay on the other side of the ecliptic, though it had not yet come into conjunction with the sun; and the comet [see more of this famous comet, p. 475 to 486] being at this time between its perihelion and its conjunction with the sun, must have been in its perihelion a few hours before; for in so near a distance from the sun it must have been carried with great velocity, and have apparently described almost half a degree every hour. |
By like computations I find that the comet of 1618 entered the sphere of the orbis magnus December 7, towards sun-setting; but its conjunction with the sun was Nov. 9, or 10, about 28 days intervening, as in the preceding comet; for from the size of the tail of this, in which it was equal to the preceding, it is probable that this comet likewise did come almost into a contact with the sun. Four comets were seen that year of which this was the last. The second, which made its first appearance October 31, in the neighbourhood of the rising sun, and was soon after hid under the sun's rays, I suspect to have been the same with the fourth, which emerged out of the sun's rays about Nov. 9. To these we may add the comet of 1607, which entered the sphere of the orbis magnus Sept. 14, O.S. and arrived at its perihelion distance from the sun about October 19, 35 days intervening. Its perihelion distance subtended an apparent angle at the earth of about 23 degrees, and was therefore of 390 parts. And to this number of parts about 34 days correspond in Tab. 1. Farther; the comet of 1665 entered the sphere of the orbis magnus about March 17, and came to its perihelion about April 16, 30 days intervening. Its perihelion distance subtended an angle at the earth of about seven degrees, and therefore was of 122 parts: and corresponding to this number of parts, in Tab. 1, we find 30 days. Again; the comet of 1682 entered the sphere of the orbis magnus about Aug. 11, and arrived at its perihelion about Sep. 16, being then distant from the sun by about 350 parts, to which, in Tab. I, belong 33½ days. Lastly; that memorable comet of Regiomontanus, which in 1472 was carried through the circum-polar parts of our northern hemisphere with such rapidity as to describe 40 degrees in one day, entered the sphere of the orbis magnus Jan 21, about the time that it was passing by the pole, and, hastening from thence towards the sun, was hid under the sun's rays about the end of Feb.; whence it is probable that 30 days, or a few more, were spent between its ingress into the sphere of the orbis magnus and its perihelion. Nor did this comet truly move with more velocity than other comets, but owed the greatness of its apparent velocity to its passing by the earth at a near distance. |
It appears, then, that the velocity of comets (p. 471), so far as it can be determined by these rude ways of computing, is that very velocity with which parabolas, or ellipses near to parabolas, ought to be described; and therefore the distance between a comet and the sun being given, the velocity of the comet is nearly given. And hence arises this problem. |
PROBLEM. |
The relation betwixt the velocity of a comet and its distance from the sun's centre being given, the comet's trajectory is required. |
If this problem was resolved, we should thence have a method of determining the trajectories of comets to the greatest accuracy; for if that relation be twice assumed, and from thence the trajectory be twice computed, and the error of each trajectory be found from observations, the assumption may be corrected by the Rule of False, and a third trajectory may thence be found that will exactly agree with the observations. And by determining the trajectories of comets after this method, we may come, at last, to a more exact knowledge of the parts through which those bodies travel, of the velocities with which they are carried, what sort of trajectories they describe, and what are the true magnitudes and forms of their tails according to the various distances of their heads from the sun; whether, after certain intervals of time, the same comets do return again, and in what periods they complete their several revolutions. But the problem may be resolved by determining, first, the hourly motion of a comet to a given time from three or more observations, and then deriving the trajectory from this motion. And thus the invention of the trajectory, depending on one observation, and its hourly motion at the time of this observation, will either confirm or disprove itself; for the conclusion that is drawn from the motion only of an hour or two and a false hypothesis, will never agree with the motions of the comets from beginning to end. The method of the whole computation is this. |
LEMMA I. |
To cut two right lines OR, TP, given in position, by a third right line RP, so as TRP may be a right angle; and, if another right line SP is drawn to any given point S, the solid contained under this line SP, and the square of the right line OR terminated at a given point O, may be of a given magnitude. |
It is done by linear description thus. Let the given magnitude of the solid be M² N: from any point r of the right line OR erect the perpendicular |
rp meeting TP in p. Then through the point Sp draw the line Sq equal to . In like manner draw three or more right lines S2q, S3q, &c., and a regular line q2q3q, drawn through all the points q2q3q, &c., will cut the right line TP in the point P, from which the perpendicular PR is to be let fall. Q.E.F. |
By trigonometry thus. Assuming the right line TP as found by the preceding method, the perpendiculars TR, SB, in the triangles TPR, TPS, will be thence given; and the side SP in the triangle SBP, as well as the error . Let this error, suppose D, be to a new error, suppose E, as the error 2p2q ± 3p3q to the error 2p3p; or as the error 2p2q ± D to the error 2pP; and this new error added to or subducted from the length TP, will give the correct length TP ± E. The inspection of the figure will shew whether we are to add to or subtract; and if at any time there should be use for a farther correction, the operation may be repeated. |
By arithmetic thus. Let us suppose the thing done, and let TP + e be the correct length of the right line TP as found out by delineation: and thence the correct lengths of the lines OR, BP, and SP, will be OR - e, BP + e, and . Whence, by the method of converging series, we have , &c., , &c. For the given co-efficients , , , putting F, , , and carefully observing the signs, we find , and . Whence, neglecting the very small term , e comes out equal to - G. If the error is not despicable, take . |
And it is to be observed that here a general method is hinted at for solving the more intricate sort of problems, as well by trigonometry as by arithmetic, without those perplexed computations and resolutions of affected equations which hitherto have been in use. |
LEMMA II. |
To cut three right lines given in position, by a fourth right line that shall pass through a point assigned in any of the three, and so as its intercepted parts shall be in a given ratio one to the other. |
Let AB, AC, BC, be the right lines given in position, and suppose D to be the given point in the line AC. Parallel to AB draw DG meeting BC |
in G; and, taking GF to BG in the given ratio, draw FDE; and FD will be to DE as FG to BG. Q.E.F. |
By trigonometry thus. In the triangle CGD all the angles and the side CD are given, and from thence its remaining sides are found; and from the given ratios the lines GF and BE are also given. |
LEMMA III. |
To find and represent by a linear description the hourly motion of a comet to any given time. |
From observations of the best credit, let three longitudes of the comet be given, and, supposing ATR, RTB, to be their differences, let the hourly motion be required to the time of the middle observation TR. By Lem II, draw the right line ARB, so as its intercepted parts AR, RB, may be |
as the times between the observations; and if we suppose a body in the whole time to describe the whole line AB with an equal motion, and to be in the mean time viewed from the place T, the apparent motion of that body about the point R will be nearly the same with that of the comet at the time of the observation TR. |
The same more accurately. |
Let Ta, Tb, be two longitudes given at a greater distance on one side and on the other; and by Lem. II draw the right line aRb so as its intercepted parts aR, Rb may be as the times between the observations aTR, RTb. Suppose this to cut the lines TA, TB, in D and E; and because the error of the inclination TRa increases nearly in the duplicate ratio of the time between the observations, draw FRG, so as either the angle DRF may be to the angle ARF, or the line DF to the line AF, in the duplicate ratio of the whole time between the observations aTB to the whole time between the observations ATB, and use the line thus found FG in place of the line AB found above. |
It will be convenient that the angles ATR, RTB, aTA, BTb, be no less than of ten or fifteen degrees, the times corresponding no greater than of eight or twelve days, and the longitudes taken when the comet moves with the greatest velocity; for thus the errors of the observations will bear a less proportion to the differences of the longitudes. |
LEMMA IV. |
To find the longitudes of a comet to any given times. |
It is done by taking in the line FG the distances Rr, Rρ, proportional to the times, and drawing the lines Tr, Tρ. The way of working by trigonometry is manifest. |
LEMMA V. |
To find the latitudes. |
On TF, TR, TG, as radiuses, at right angles erect Ff, RP, Gg, tangents of the observed latitudes; and parallel to fg draw PH. The perpendiculars rp, ρῶ, meeting PH, will be the tangents of the sought latitudes to Tr and Tρ as radiuses. |
PROBLEM I. |
From the assumed ratio of the velocity to determine the trajectory of a comet. |
Let S represent the sun; t, T, τ, three places of the earth in its orbit at equal distances; p, P, ῶ, as many corresponding places of the comet in |
its trajectory, so as the distances interposed betwixt place and place may answer to the motion of one hour; pr, PR, ῶρ, perpendiculars let fall on the plane of the ecliptic, and rRp the vestige of the trajectory in this plane. Join Sp, SP, Sῶ, SR, ST, tr, TR, τρ, TP, and let tr, τρ, meet in O, TR will nearly converge to the same point O, or the error will be in considerable. By the premised lemmas the angles rOR, ROρ, are given, as well as the ratios pr to tr, PR to TR, and ῶρ to τρ. The figure tTτO is likewise given both in magnitude and position, together with the distance ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved about the sun in a circle, at the same distance SP, as V to 1; and we shall have a line pPῶ to be determined, of this condition, that the space pῶ, described by the comet in two hours, may be to the space V tτ (that is, to the space which the earth describes in the same time multiplied by the number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun; and that the space pP, described by the comet in the first hour, may be to the space Pῶ, described by the comet in the second hour, as the velocity in p to the velocity in P; that is, in the subduplicate ratio of the distance SP to the distance Sp, or in the ratio of 2Sp to SP + Sp; for in this whole work I neglect small fractions that can produce no sensible error. |
In the first place, then, as mathematicians, in the resolution of affected equations, are wont, for the first essay, to assume the root by conjecture, so, in this analytical operation, I judge of the sought distance TR as I best can by conjecture. Then, by Lem. II. I draw rρ, first supposing rR equal to Rρ, and again (after the ratio of SP to Sp is discovered) so as rR may be to Rρ as 2SP to SP + Sp, and I find the ratios of the lines pῶ, rρ, and OR, one to the other. Let M be to V tτ as OR to pῶ; and because the square of pῶ is to the square of V tτ as ST to SP, we shall have, ex æquo, OR² to M² as ST to SP, and therefore the solid OR² SP equal to the given solid M² ST; whence (supposing the triangles STP, PTR, to be now placed in the same plane) TR, TP, SP, PR, will be given, by Lem. I. All this I do, first by delineation in a rude and hasty way; then by a new delineation with greater care; and, lastly, by an arithmetical computation. Then I proceed to determine the position of the lines rρ, pῶ, with the greatest accuracy, together with the nodes and inclination of the plane Spῶ to the plane of the ecliptic; and in that plane Spῶ I describe the trajectory in which a body let go from the place P in the direction of the given right line pῶ would be carried with a velocity that is to the velocity of the earth as pῶ to V tτ. Q.E.F. |
PROBLEM II. |
To correct the assumed ratio of the velocity and the trajectory thence found. |
Take an observation of the comet about the end of its appearance, or any other observation at a very great distance from the observations used before, and find the intersection of a right line drawn to the comet, in that observation with the plane Spῶ, as well as the comet's place in its trajectory to the time of the observation. If that intersection happens in this place, it is a proof that the trajectory was rightly determined; if otherwise, a new number V is to be assumed, and a new trajectory to be found; and then the place of the comet in this trajectory to the time of that probatory observation, and the intersection of a right line drawn to the comet with the plane of the trajectory, are to be determined as before; and by comparing the variation of the error with the variation of the other quantities, we may conclude, by the Rule of Three, how far those other quantities ought to be varied or corrected, so as the error may become as small as possible. And by means of these corrections we may have the trajectory exactly, providing the observations upon which the computation was founded were exact, and that we did not err much in the assumption of the quantity V; for if we did, the operation is to be repeated till the trajectory is exactly enough determined. Q.E.F. |
END OF THE SYSTEM OF THE WORLD. |
CONTENTS |