u152And hence it follows that the comets, during the whole time of their appearance unto us, being within the sphere of activity of the circum-solar force, and therefore agitated by the impulse of that force, will (by Cor. 1, Prop. XII, Book I, for the same reason as the planets) be made to[Pg 562] move in conic sections that have one focus in the centre of the sun, and by radii drawn to the sun, to describe areas proportional to the times; for that force is propagated to an immense distance, and will govern the motions of bodies far beyond the orbit of Saturn.

There are three hypotheses about comets (p. 466); for some will have it that they are generated and perish as often as they appear and vanish; others, that they come from the regions of the fixed stars, and are seen by us in their passage through the system of our planets; and, lastly, others, that they are bodies perpetually revolving about the sun in very eccentric orbits. In the first case, the comets, according to their different velocities, will move in conic sections of all sorts; in the second, they will describe hyperbolas, and in either of the two will frequent indifferently all quarters of the heavens, as well those about the poles as those towards the ecliptic; in the third, their motions will be performed in ellipses very eccentric, and very nearly approaching to parabolas. But (if the law of the planets is observed) their orbits will not much decline from the plane of the ecliptic; and, so far as I could hitherto observe, the third case obtains; for the comets do, indeed, chiefly frequent the zodiac, and scarcely ever attain to a heliocentric latitude of 40°. And that they move in orbits very nearly parabolical, I infer from their velocity; for the velocity with which a parabola is described is every where to the velocity with which a comet or planet may be revolved about the sun in a circle at the same distance in the subduplicate ratio of 2 to 1 (by Cor. VII, Prop. XVI); and, by my computation, the velocity of comets is found to be much about the same. I examined the thing by inferring nearly the velocities from the distances, and the distances both from the parallaxes and the phænomena of the tails, and never found the errors of excess or defect in the velocities greater than what might have arose from the errors in the distances collected after that manner. But I likewise made use of the reasoning that follows.

[Pg 563]

Supposing the radius of the orbis magnus to be divided into 1000 parts: let the numbers in the first column of the following table represent the distance of the vertex of the parabola from the sun's centre, expressed by those parts: and a comet in the times expressed in col. 2, will pass from its perihelion to the surface of the sphere which is described about the sun as a centre with the radius of the orbis magnus; and in the times expressed in col. 3, 4, and 5, it will double, triple, and quadruple, that its distance from the sun.

TABLE I.

[This table, here corrected, is made on the supposition that the earth's diurnal motion is just 59', and the measure of one minute loosely 0,2909, in respect of the radius 1000. If those measures are taken true, the true numbers of the table will all come out less. But the difference, even when greatest, and to the quadruple of the earth’s distance from the sun, amounts only to 16h. 55'.]

The time of a comet's ingress into the sphere of the orbis magnus, or of its egress from the same, may be inferred nearly from its parallax, but with more expedition by the following

TABLE II.

[Pg 564]

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[Pg 565] 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[Pg 566] 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.