The System of the World
Table II
The apparent
elongation of
a comet from
the sun. Its apparent diurnal
motion in its
own orbit. Its distance from
the earth in parts
whereof the radius
of the orbis magnus
contains 1000.
Direct. Retrog.
60° 2° 18’ 00° 20’ 1000
65 2 33 00 35 845
70 2 55 00 57 684
72 3 07 01 09 618
74 3 23 01 25 551
76 3 43 01 45 484
78 4 10 02 12 416
80 4 57 02 49 347
82 5 45 03 47 278
84 7 18 05 20 209
86 10 27 08 19 140
88 18 37 16 39 70
90 Infinite Infinite 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 °; 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 °; and the corresponding elongation 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. 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 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. 1, belong 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.