Book 3 — PROPOSITIONS. (continued 3)

COR. 9. The mean distance of the centres of the earth and moon in the syzygies of the moon is 60 mean semi-diameters of the earth, and a 10th part of one semi-diameter; and in the moon's quadratures the mean distance of the same centres is 61 mean semi-diameters of the earth, subducting one 30th part of one semi-diameter.

COR. 10. In the moon's syzygies its mean horizontal parallax in the latitudes of 0, 30, 38, 45, 52, 60, 90 degrees is 57' 20'', 57' 16'', 57' 14'', 57' 12'', 57' 10'', 57' 8'', 57' 4'', respectively.

In these computations I do not consider the magnetic attraction of the earth, whose quantity is very small and unknown: if this quantity should ever be found out, and the measures of degrees upon the meridian, the lengths of isochronous pendulums in different parallels, the laws of the motions of the sea, and the moon's parallax, with the apparent diameters of the sun and moon, should be more exactly determined from phænomena: we should then be enabled to bring this calculation to a greater accuracy.

PROPOSITION XXXVIII. PROBLEM XIX.

To find the figure of the moon's body.

If the moon's body were fluid like our sea, the force of the earth to raise that fluid in the nearest and remotest parts would be to the force of the moon by which our sea is raised in the places under and opposite to the moon as the accelerative gravity of the moon towards the earth to the accelerative gravity of the earth towards the moon, and the diameter of the moon to the diameter of the earth conjunctly; that is, as 39,788 to 1, and 100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, by the force of the moon, is raised to feet, the lunar fluid would be raised by the force of the earth to 93 feet; and upon this account the figure of the moon would be a spheroid, whose greatest diameter produced would pass through the centre of the earth, and exceed the diameters perpendicular thereto by 186 feet. Such a figure, therefore, the moon affects, and must have put on from the beginning. Q.E.I.

COR. Hence it is that the same face of the moon always respects the earth; nor can the body of the moon possibly rest in any other position, but would return always by a libratory motion to this situation; but those librations, however, must be exceedingly slow, because of the weakness of the forces which excite them; so that the face of the moon, which should be always obverted to the earth, may, for the reason assigned in Prop. XVII. be turned towards the other focus of the moon's orbit, without being immediately drawn back, and converted again towards the earth.

LEMMA I.

If APEp represent the earth uniformly dense, marked with the centre C, the poles P, p, and the equator AE; and if about the centre C, with the radius CP, we suppose the sphere Pape to be described, and QR to denote the plane on which a right line, drawn from the centre of the sun to the centre of the earth, insists at right angles; and further suppose that the several particles of the whole exterior earth PapAPepE, without the height of the said sphere, endeavour to recede towards this side and that side from the plane QR, every particle by a force proportional to its distance from that plane; I say, in the first place, that the whole force and efficacy of all the particles that are situate in AE, the circle of the equator, and disposed uniformly without the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from the plane QR, to wheel the earth about its centre with a like circular motion, as 1 to 2. And that circular motion will be performed about an axis lying in the common section of the equator and the plane QR.

For let there be described from the centre K, with the diameter IL, the semi-circle INL. Suppose the semi-circumference INL to be divided into innumerable equal parts, and from the several parts N to the diameter IL let fall the sines NM. Then the sums of the squares of all the sines NM will be equal to the sums of the squares of the sines KM, and both sums together will be equal to the sums of the squares of as many semi-diameters KN; and therefore the sum of the squares of all the sines NM will be but half so great as the sum of the squares of as many semi-diameters KN.

Suppose now the circumference of the circle AE to be divided into the like number of little equal parts, and from every such part F a perpendicular FG to be let fall upon the plane QR, as well as the perpendicular AH from the point A. Then the force by which the particle F recedes from the plane QR will (by supposition) be as that perpendicular FG; and this force multiplied by the distance CG will represent the power of the particle F to turn the earth round its centre. And, therefore, the power of a particle in the place F will be to the power of a particle in the place A as FG × GC to AH × HC; that is, as FC² to AC²: and therefore the whole power of all the particles F, in their proper places F, will be to the power of the like number of particles in the place A as the sum of all the FC² to the sum of all the AC², that is (by what we have demonstrated before) as 1 to 2. Q.E.D.

And because the action of those particles is exerted in the direction of lines perpendicularly receding from the plane QR, and that equally from each side of this plane, they will wheel about the circumference of the circle of the equator, together with the adherent body of the earth, round an axis which lies as well in the plane QR as in that of the equator.

LEMMA II.

The same things still supposed, I say, in the second place, that the total force or power of all the particles situated every where about the sphere to turn the earth about the said axis is to the whole force of the like number of particles, uniformly disposed round the whole circumference of the equator AE in the fashion of a ring, to turn the whole earth about with the like circular motion, as 2 to 5.

For let IK be any lesser circle parallel to the equator AE, and let Ll be any two equal particles in this circle, situated without the sphere Pape; and if upon the plane QR, which is at right angles with a radius drawn to the sun, we let fall the perpendiculars LM, lm, the total forces by which these particles recede from the plane QR will be proportional to the perpendiculars LM, lm. Let the right line Ll be drawn parallel to the plane Pape, and bisect the same in X; and through the point X draw Nn parallel to the plane QR, and meeting the perpendiculars LM, lm, in N and n; and upon the plane QR let fall the perpendicular XY. And the contrary forces of the particles L and l to wheel about the earth contrariwise are as LM × MC, and lm × mC; that is, as LN × MC + NM × MC, and ln × mC - nm × mC; or LN × MC + NM × MC, and LN × mC - NM × mC, and , the difference of the two, is the force of both taken together to turn the earth round. The affirmative part of this difference LN × Mm, or 2LN × NX, is to 2AH × HC, the force of two particles of the same size situated in A, as LX² to AC²; and the negative part , or 2XY × CY, is to 2AH × HC, the force of the same two particles situated in A, as CX² to AC². And therefore the difference of the parts, that is, the force of the two particles L and l, taken together, to wheel the earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX² - CX² to AC². But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX² will be to the like number of IX² as 1 to 2 (by Lem. 1); and to the same number of AC² as IX² to 2AC²; and the same number of CX² to as many AC² as 2CX² to 2AC². Wherefore the united forces of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX² - 2CX² to 2AC²; and therefore (by Lem. 1) to the united forces of as many particles in the circumference of the circle AE as IX² - 2CX² to AC².

Now if Pp, the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to insist, the matter in the circumference of every circle IK will be as IX²; and therefore the force of that matter to turn the earth about will be as IX² into IX² - 2CX²; and the force of the same matter, if it was situated in the circumference of the circle AE, would be as IX² into AC². And therefore the force of all the particles of the whole matter situated without the sphere in the circumferences of all the circles is to the force of the like number of particles situated in the circumference of the greatest circle AE as all the IX² into IX² - 2CX² to as many IX² into AC²; that is, as all the AC² - CX² into AC² - 3CX² to as many AC² - CX² into AC²; that is, as all the AC⁴ - 4AC² × CX² + 3CX⁴ to as many AC⁴ - AC² × CX²; that is, as the whole fluent quantity, whose fluxion is AC⁴ - 4AC² × CX² + 3CX4, to the whole fluent quantity, whose fluxion is AC⁴ - AC² × CX²; and, therefore, by the method of fluxions, as to ; that is, if for CX we write the whole Cp, or AC, as to ; that is, as 2 to 5. Q.E.D.

LEMMA III.

The same things still supposed, I say, in the third place, that the motion of the whole earth about the axis above-named arising from the motions of all the particles, will be to the motion of the aforesaid ring about the same axis in a proportion compounded of the proportion of the matter in the earth to the matter in the ring; and the proportion of three squares of the quadrantal arc of any circle to two squares of its diameter, that is, in the proportion of the matter to the matter, and of the number 925275 to the number 1000000.

For the motion of a cylinder revolved about its quiescent axis is to the motion of the inscribed sphere revolved together with it as any four equal squares to three circles inscribed in three of those squares; and the motion of this cylinder is to the motion of an exceedingly thin ring surrounding both sphere and cylinder in their common contact as double the matter in the cylinder to triple the matter in the ring; and this motion of the ring, uniformly continued about the axis of the cylinder, is to the uniform motion of the same about its own diameter performed in the same periodic time as the circumference of a circle to double its diameter.

HYPOTHESIS II.

If the other parts of the earth were taken away, and the remaining ring was carried alone about the sun in the orbit of the earth by the annual motion, while by the diurnal motion it was in the mean time revolved about its own axis inclined to the plane of the ecliptic by an angle of degrees, the motion of the equinoctial points would be the same, whether the ring were fluid, or whether it consisted of a hard and rigid matter.

PROPOSITION XXXIX. PROBLEM XX.

To find the precession of the equinoxes.

The middle horary motion of the moon's nodes in a circular orbit, when the nodes are in the quadratures, was 16'' 35''' 16^iv. 36^v.; the half of which, 8'' 17''' 38^iv. 18^v. (for the reasons above explained) is the mean horary motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20° 11' 46''. Because, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11' 46'' in antecedentia; and, if there were more moons, the motion of the nodes of every one (by Cor. 16, Prop. LXVI, Book I) would be as its periodic time; if upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11' 46'' as 23^h. 56', the sidereal day, to 27^d. 7^h. 43', the periodic time of our moon, that is, as 1436 to 39343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other, or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible.

Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth PapAPepE, which lies without the sphere Pape (see fig. Lem. II); and because this sphere is to that exterior earth as aC² to AC² - aC², that is (seeing PC or aC the least semi-diameter of the earth is to AC the greatest semi-diameter of the same as 229 to 230), as 52441 to 459; if this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of the ring (by Lem. III) would be to the motion of the inner sphere as 459 to 52441 and 1000000 to 925275 conjunctly, that is, as 4590 to 485223; and therefore the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 to 489813. Wherefore if the ring adheres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 to 489813; upon which account the motion of the equinoctial points will be diminished in the same proportion. Wherefore the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° 11' 46'' as 1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369. But the forces by which the nodes of a number of moons (as we explained above), and therefore by which the equinoctial points of the ring recede (that is, the forces 3IT, in fig. Prop. XXX), are in the several particles as the distances of those particles from the plane QR; and by these forces the particles recede from that plane: and therefore (by Lem. II) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PapAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5. Wherefore the annual regress of the equinoxes now would be to 20° 11' 46'' as 10 to 73092; that is, would be 9'' 56''' 50^iv.

But because the plane of the equator is inclined to that of the ecliptic, this motion is to be diminished in the proportion of the sine 91706 (which is the co-sine of deg.) to the radius 100000; and the remaining motion will now be 9'' 7''' 20^iv. which is the annual precession of the equinoxes arising from the force of the sun.

But the force of the moon to move the sea was to the force of the sun nearly as 4,4815 to 1; and the force of the moon to move the equinoxes is to that of the sun in the same proportion. Whence the annual precession of the equinoxes proceeding from the force of the moon comes out 40'' 52''' 52^iv. and the total annual precession arising from the united forces of both will be 50'' 00''' 12^iv. the quantity of which motion agrees with the phænomena; for the precession of the equinoxes, by astronomical observations, is about 50'' yearly.

If the height of the earth at the equator exceeds its height at the poles by more than miles, the matter thereof will be more rare near the surface than at the centre; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity.

And now we have described the system of the sun, the earth, moon, and planets, it remains that we add something about the comets.

LEMMA IV.

That the comets are higher than the moon, and in the regions of the planets.

As the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the planets; for all the comets which move in a direct course according to the order of the signs, about the end of their appearance become more than ordinarily slow or retrograde, if the earth is between them and the sun; and more than ordinarily swift, if the earth is approaching to a heliocentric opposition with them; whereas, on the other hand, those which move against the order of the signs, towards the end of their appearance appear swifter than they ought to be, if the earth is between them and the sun; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. And these appearances proceed chiefly from the diverse situations which the earth acquires in the course of its motion, after the same manner as it happens to the planets, which appear sometimes retrograde, sometimes more slowly, and sometimes more swiftly, progressive, according as the motion of the earth falls in with that of the planet, or is directed the contrary way. If the earth move the same way with the comet, but, by an angular motion about the sun, so much swifter that right lines drawn from the earth to the comet converge towards the parts beyond the comet, the comet seen from the earth, because of its slower motion, will appear retrograde; and even if the earth is slower than the comet, the motion of the earth being subducted, the motion of the comet will at least appear retarded; but if the earth tends the contrary way to that of the comet, the motion of the comet will from thence appear accelerated; and from this apparent acceleration, or retardation, or regressive motion, the distance of the comet may be inferred in this manner. Let , , , be three observed longitudes of the comet about the time of its first appearing, and its last observed longitude before its disappearing. Draw the right line ABC, whose parts AB, BC, intercepted between the right lines QA and QB, QB and QC, may be one to the other as the two times between the three first observations. Produce AC to G, so as AG may be to AB as the time between the first and last observation to the time between the first and second; and join QG. Now if the comet did move uniformly in a right line, and the earth either stood still, or was likewise carried forwards in a right line by an uniform motion, the angle would be the longitude of the comet at the time of the last observation. The angle, therefore, FQG, which is the difference of the longitude, proceeds from the inequality of the motions of the comet and the earth; and this angle, if the earth and comet move contrary ways, is added to the angle , and accelerates the apparent motion of the comet; but if the comet move; the same way with the earth, it is subtracted, and either retards the motion of the comet, or perhaps renders it retrograde, as we have but now explained. This angle, therefore, proceeding chiefly from the motion of the earth, is justly to be esteemed the parallax of the comet; neglecting, to wit, some little increment or decrement that may arise from the unequal motion of the comet in its orbit; and from this parallax we thus deduce the distance of the comet.

Let S represent the sun, acT the orbis magnus, a the earth's place in the first observation, c the place of the earth in the third observation, T the place of the earth in the last observation, and a right line drawn to the beginning of Aries. Set off the angle equal to the angle , that is, equal to the longitude of the comet at the time when the earth is in T; join ac, and produce it to g, so as ag may be to ac as AG to AC; and g will be the place at which the earth would have arrived in the time of the last observation, if it had continued to move uniformly in the right line ac. Wherefore, if we draw parallel to , and make the angle equal to the angle , this angle will be equal to the longitude of the comet seen from the place g, and the angle TVg will be the parallax which arises from the earth's being transferred from the place g into the place T; and therefore V will be the place of the comet in the plane of the ecliptic. And this place V is commonly lower than the orb of Jupiter.

The same thing may be deduced from the incurvation of the way of the comets; for these bodies move almost in great circles, while their velocity is great; but about the end of their course, when that part of their apparent motion which arises from the parallax bears a greater proportion to their whole apparent motion, they commonly deviate from those circles, and when the earth goes to one side, they deviate to the other; and this deflexion, because of its corresponding with the motion of the earth, must arise chiefly from the parallax; and the quantity thereof is so considerable, as, by my computation, to place the disappearing comets a good deal lower than Jupiter. Whence it follows that when they approach nearer to us in their perigees and perihelions they often descend below the orbs of Mars and the inferior planets.

The near approach of the comets is farther confirmed from the light of their heads; for the light of a celestial body, illuminated by the sun, and receding to remote parts, is diminished in the quadruplicate proportion of the distance; to wit, in one duplicate proportion, on account of the increase of the distance from the sun, and in another duplicate proportion, on account of the decrease of the apparent diameter. Wherefore if both the quantity of light and the apparent diameter of a comet are given, its distance will be also given, by taking the distance of the comet to the distance of a planet in the direct proportion of their diameters and the reciprocal subduplicate proportion of their lights. Thus, in the comet of the year 1682, Mr. Flamsted observed with a telescope of 16 feet, and measured with a micrometer, the least diameter of its head, 2' 00; but the nucleus or star in the middle of the head scarcely amounted to the tenth part of this measure; and therefore its diameter was only 11'' or 12''; but in the light and splendor of its head it surpassed that of the comet in the year 1680, and might be compared with the stars of the first or second magnitude. Let us suppose that Saturn with its ring was about four times more lucid; and because the light of the ring was almost equal to the light of the globe within, and the apparent diameter of the globe is about 21'', and therefore the united light of both globe and ring would be equal to the light of a globe whose diameter is 30'', it follows that the distance of the comet was to the distance of Saturn as 1 to inversely, and 12'' to 30'' directly; that is, as 24 to 30, or 4 to 5. Again; the comet in the month of April 1665, as Hevelius informs us, excelled almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid colour; for this comet was more lucid than that other which had appeared about the end of the preceding year, and had been compared to the stars of the first magnitude. The diameter of its head was about 6'; but the nucleus, compared with the planets by means of a telescope, was plainly less than Jupiter; and sometimes judged less, sometimes judged equal, to the globe of Saturn within the ring. Since, then, the diameters of the heads of the comets seldom exceed 8' or 12', and the diameter of the nucleus or central star is but about a tenth or perhaps fifteenth part of the diameter of the head, it appears that these stars are generally of about the same apparent magnitude with the planets. But in regard that their light may be often compared with the light of Saturn, yea, and sometimes exceeds it, it is evident that all comets in their perihelions must either be placed below or not far above Saturn; and they are much mistaken who remove them almost as far as the fixed stars; for if it was so, the comets could receive no more light from our sun than our planets do from the fixed stars.

So far we have gone, without considering the obscuration which comets suffer from that plenty of thick smoke which encompasseth their heads, and through which the heads always shew dull, as through a cloud; for by how much the more a body is obscured by this smoke, by so much the more near it must be allowed to come to the sun, that it may vie with the planets in the quantity of light which it reflects. Whence it is probable that the comets descend far below the orb of Saturn, as we proved before from their parallax. But, above all, the thing is evinced from their tails, which must be owing either to the sun's light reflected by a smoke arising from them, and dispersing itself through the æther, or to the light of their own heads. In the former case, we must shorten the distance of the comets, lest we be obliged to allow that the smoke arising from their heads is propagated through such a vast extent of space, and with such a velocity and expansion as will seem altogether incredible; in the latter case, the whole light of both head and tail is to be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself in splendor, especially when it emits a very large and lucid tail. If, therefore, under a less apparent diameter, it reflects more light, it must be much more illuminated by the sun, and therefore much nearer to it; and the same argument will bring down the heads of comets sometimes within the orb of Venus, viz., when, being hid under the sun's rays, they emit such huge and splendid tails, like beams of fire, as sometimes they do; for if all that light was supposed to be gathered together into one star, it would sometimes exceed not one Venus only, but a great many such united into one.

Lastly; the same thing is inferred from the light of the heads, which increases in the recess of the comets from the earth towards the sun, and decreases in their return from the sun towards the earth; for so the comet of the year 1665 (by the observations of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee; but yet the splendor of its head was daily increasing, till, being hid under the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observations of the same Hevelius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orb in a day's time; but from that time its diurnal motion was continually upon the increase, till September 4, when it arose to about 5 degrees; and therefore, in all this interval of time, the comet was approaching to the earth. Which is likewise proved from the diameter of its head, measured with a micrometer; for, August 6, Hevelius found it only 6' 05'', including the coma, which, September 2 he observed to be 9' 07'', and therefore its head appeared far less about the beginning than towards the end of the motion; though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its access towards the earth. The comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with their greatest velocity, and were therefore then in their perigees; but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays; and the greatest splendor of their tails a little more early, when yet nearer to the sun. The head of the former comet (according to the observations of Cysatus), December 1, appeared greater than the stars of the first magnitude; and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12, the head of the latter comet was seen and observed by Mr. Flamsted, when but 9 degrees distant from the sun; which is scarcely to be done in a star of the third magnitude. December 15 and 17, it appeared as a star of the third magnitude, its lustre being diminished by the brightness of the clouds near the setting sun. December 26, when it moved with the greatest velocity, being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. January 3, it appeared as a star of the fourth. January 9, as one of the fifth. January 13, it was hid by the splendor of the moon, then in her increase. January 25, it was scarcely equal to the stars of the seventh magnitude. If we compare equal intervals of time on one side and on the other from the perigee, we shall find that the head of the comet, which at both intervals of time was far, but yet equally, removed from the earth, and should have therefore shone with equal splendor, appeared brightest on the side of the perigee towards the sun, and disappeared on the other. Therefore, from the great difference of light in the one situation and in the other, we conclude the great vicinity of the sun and comet in the former; for the light of comets uses to be regular, and to appear greatest when the heads move fastest, and are therefore in their perigees; excepting in so far as it is increased by their nearness to the sun.

COR. 1. Therefore the comets shine by the sun's light, which they reflect.

COR. 2. From what has been said, we may likewise understand why comets are so frequently seen in that hemisphere in which the sun is, and so seldom in the other. If they were visible in the regions far above Saturn, they would appear more frequently in the parts opposite to the sun; for such as were in those parts would be nearer to the earth, whereas the presence of the sun must obscure and hide those that appear in the hemisphere in which he is. Yet, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere; besides, without doubt, not a few, which have been hid by the light of the sun: for comets descending into our parts neither emit tails, nor are so well illuminated by the sun, as to discover themselves to our naked eyes, until they are come nearer to us than Jupiter. But the far greater part of that spherical space, which is described about the sun with so small an interval, lies on that side of the earth which regards the sun; and the comets in that greater part are commonly more strongly illuminated, as being for the most part nearer to the sun.

COR. 3. Hence also it is evident that the celestial spaces are void of resistance; for though the comets are carried in oblique paths, and sometimes contrary to the course of the planets, yet they move every way with the greatest freedom, and preserve their motions for an exceeding long time, even where contrary to the course of the planets. I am out in my judgment if they are not a sort of planets revolving in orbits returning into themselves with a perpetual motion; for, as to what some writers contend, that they are no other than meteors, led into this opinion by the perpetual changes that happen to their heads, it seems to have no foundation; for the heads of comets are encompassed with huge atmospheres, and the lowermost parts of these atmospheres must be the densest; and therefore it is in the clouds only, not in the bodies of the comets themselves, that these changes are seen. Thus the earth, if it was viewed from the planets, would, without all doubt, shine by the light of its clouds, and the solid body would scarcely appear through the surrounding clouds. Thus also the belts of Jupiter are formed in the clouds of that planet, for they change their position one to another, and the solid body of Jupiter is hardly to be seen through them; and much more must the bodies of comets be hid under their atmospheres, which are both deeper and thicker.

PROPOSITION XL. THEOREM XX.

That the comets move in some of the conic sections, having their foci in the centre of the sun; and by radii drawn to the sun describe areas proportional to the times.

This proposition appears from Cor. 1, Prop. XIII, Book I, compared with Prop. VIII, XII, and XIII, Book III.

COR. 1. Hence if comets are revolved in orbits returning into themselves, those orbits will be ellipses; and their periodic times be to the periodic times of the planets in the sesquiplicate proportion of their principal axes. And therefore the comets, which for the most part of their course are higher than the planets, and upon that account describe orbits with greater axes, will require a longer time to finish their revolutions. Thus if the axis of a comet's orbit was four times greater than the axis of the orbit of Saturn, the time of the revolution of the comet would be to the time of the revolution of Saturn, that is, to 30 years, as (or 8) to 1, and would therefore be 240 years.

COR. 2. But their orbits will be so near to parabolas, that parabolas may be used for them without sensible error.

COR. 3. And, therefore, by Cor. 7, Prop. XVI, Book I, the velocity of every comet will always be to the velocity of any planet, supposed to be revolved at the same distance in a circle about the sun, nearly in the subduplicate proportion of double the distance of the planet from the centre of the sun to the distance of the comet from the sun's centre, very nearly. Let us suppose the radius of the orbis magnus, or the greatest semi-diameter of the ellipsis which the earth describes, to consist of 100000000 parts; and then the earth by its mean diurnal motion will describe 1720212 of those parts, and by its horary motion. And therefore the comet, at the same mean distance of the earth from the sun, with a velocity which is to the velocity of the earth as to 1, would by its diurnal motion describe 2432747 parts, and parts by its horary motion. But at greater or less distances both the diurnal and horary motion will be to this diurnal and horary motion in the reciprocal subduplicate proportion of the distances, and is therefore given.

COR. 4. Wherefore if the latus rectum of the parabola is quadruple of the radius of the orbis magnus, and the square of that radius is supposed to consist of 100000000 parts, the area which the comet will daily describe by a radius drawn to the sun will be parts, and the horary area will be parts. But, if the latus rectum is greater or less in any proportion, the diurnal and horary area will be less or greater in the subduplicate of the same proportion reciprocally.

LEMMA V.

To find a curve line of the parabolic kind which shall pass through any given number of points.

Let those points be A, B, C, D, E, F, &c., and from the same to any right line HN, given in position, let fall as many perpendiculars AH, BI, CK, DL, EM, FN, &c.

CASE 1. If HI, IK, KL, &c., the intervals of the points H, I, K, L, M, N, &c., are equal, take b, 2b, 3b, 4b, 5b, &c., the first differences of the perpendiculars AH, BI, CK, &c.; their second differences c, 2c, 3c, 4c, &c.; their third, d, 2d, 3d, &c., that is to say, so as AH - BI may be = b, BI - CK = 2b, CK - DL = 3b, DL + EM = 4b, - EM + FN = 5b, &c.; then b - 2b = c, &c., and so on to the last difference, which is here f. Then, erecting any perpendicular RS, which may be considered as an ordinate of the curve required, in order to find the length of this ordinate, suppose the intervals HI, IK, KL, LM, &c., to be units, and let , , into , into , into + SL = s, into + SM = t; proceeding, to wit, to ME, the last perpendicular but one, and prefixing negative signs before the terms HS, IS, &c., which lie from S towards A; and affirmative signs before the terms SK, SL, &c., which lie on the other side of the point S; and, observing well the signs, RS will be = a + bp + cq + dr + es + ft, + &c.

CASE 2. But if HI, IK, &c., the intervals of the points H, I, K, L, &c., are unequal, take b, 2b, 3b, 4b, 5b, &c., the first differences of the perpendiculars AH, BI, CK, &c., divided by the intervals between those perpendiculars; c, 2c, 3c, 4c, &c., their second differences, divided by the intervals between every two; d, 2d, 3d, &c., their third differences, divided by the intervals between every three; e, 2e, &c., their fourth differences, divided by the intervals between every four; and so forth; that is, in such manner, that b may be , 2b , 3b , &c., then c , 2c , 3c , &c., then d , 2d , &c. And those differences being found, let AH be = a, - HS = p, p into - IS = q, q into + SK = r, r into + SL = s, s into + SM = t; proceeding, to wit, to ME, the last perpendicular but one: and the ordinate RS will be = a + bp + cq + dr + es + ft, + &c.

COR. Hence the areas of all curves may be nearly found; for if some number of points of the curve to be squared are found, and a parabola be supposed to be drawn through those points, the area of this parabola will be nearly the same with the area of the curvilinear figure proposed to be squared: but the parabola can be always squared geometrically by methods vulgarly known.

LEMMA VI.

Certain observed places of a comet being given, to find the place of the same to any intermediate given time.

Let 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.

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 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 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.

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, 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.

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 , and complete the parallelogram iI . Take equal to ; and through the sun S draw the obscure line equal to . Then, cancelling the letters A, E, C, I, from the point B towards the point , draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity . 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 . Join MN, cutting IO in O. Draw the rectangular parallelogram , as before. In IA produced take ID equal to . 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 assumed at pleasure, find out the new points e, a, c, g; and , , , . Then through G, g, and , draw the circumference of a circle , cutting the right line in Z: and Z will be one place of the comet in the plane of the ecliptic. And in AC, ac, , making AF, af, , equal respectively to CG, cg, ; through the points F, f, and , draw the circumference of a circle , 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 , two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book I) 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 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, , 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 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 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 , 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 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.

Time. Sun's Longitude. Comet's

Appar. True. Longitude. Lat. N.

h. '' h. ' '' ° ' '' ° ' '' ° ' ''

1680, Dec. 12 4.46 4.46. 0 ♑ 1.51.23 ♑ 6.32.30 8.28. 0

21 6.36.59 11.06.44 ♒ 5.08.12 21.42.13

24 6.12 6.17.52 14.09.26 18.49.23 25.23. 5

26 5.14 5.20.44 16.09.22 28.24.13 27.00.52

29 7.55 8.03.02 19.19.43 ♓ 13.10.41 28.09.58

30 8.02 8.10.26 20.21.09 17.38.20 28.11.53

1681, Jan. 5 5.51 6.01.38 26.22.18 ♈ 8.48.53 26.15. 7

9 6.49 7.00.53 ♒ 0.29.02 18.44.04 24.11.56

10 5.54 6.06.10 1.27.43 20.40.50 23.43.52

13 6.56 7.08.55 4.33.20 25.59.48 22.17.28

25 7.44 7.58.42 16.45.36 ♉ 9.35. 0 17.56.30

30 8.07 8.21.53 21.49.58 13.19.51 16.42.18

Feb. 2 6.20 6.34.51 24.46.59 15.13.53 16.04. 1

5 6.50 7.04.41 27.49.51 16.59.06 15.27. 3

To these you may add some observations of mine.

Ap. Time. Comet's

Longitude. Lat. N.

h. ' ° ' '' ° ' ''

1681, Feb. 25 8.30 ♉︎ 26.18.35 12.46.46

27 8.15 27.04.30 12.36.12

Mar. 1 11. 0 27.52.42 12.23.40

2 8. 0 28.12.48 12.19.38

5 11.30 29.18. 0 12.03.16

7 9.30 ♊︎ 0. 4. 0 11.57. 0

9 8.30 0.43. 4 11.45.52

These observations were made by a telescope of 7 feet, with a micrometer and threads placed in the focus of the telescope; by which instruments we determined the positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ), C a star of the sixth magnitude (Bayer's ) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, , , , , other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places of the comet in the observations above set down; and, reckoning the distance AB of parts, AC was of those parts; BC, ; AD, ; BD, ; CD, ; AE, ; CE, ; DE, ; AI, ; BI, ; CI, ; DI, ; AK, ; BK, 43; CK, ; FK, 29; FB, 23; FC, ; AH, ; DH, ; BN, ; CN, ; BL, ; NL, . HO was to HI as 7 to 6, and, produced, did pass between the stars D and E, so as the distance of the star D from this right line was . LM was to LN as 2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.

Mr. Pound has since observed a second time the positions of those fixed stars amongst themselves, and collected their longitudes and latitudes according to the following table.

The fixed stars. Their Longitudes. Latitute North. The fixed stars. Their Longitudes. Latitute North.

° ' '' ° ' '' ° ' '' ° ' ''

A ♉︎ 26.41.50 12. 8.36 L ♉︎ 29.33.34 12. 7.48

B 28.40.23 11.17.54 M 9.18.54 2. 7.20

C 27.58.30 12.40.25 N 28.48.29 2.31. 9

E 26.27.17 12.52. 7 Z 29.44.48 11.57.13

F 28.28.37 11.52.22 29.52. 3 11.55.48

G 26.56. 8 12. 4.58 ♊︎ 0. 8.23 11.48.56

H 27.11.45 12. 2. 1 0.40.10 11.55.18

I 27.25. 2 11.53.11 1. 3.20 11.30.42

K 27.42. 7 11.53.26

The positions of the comet to these fixed stars were observed to be as follow:

Friday, February 25, O.S. at P.M. the distance of the comet in p from the star E was less than , and greater than , and therefore nearly equal to ; and the angle ApE was a little obtuse, but almost right. For from A, letting fall a perpendicular on pE, the distance of the comet from that perpendicular was .

The same night, at , the distance of the comet in P from the star E was greater than , and less than , and therefore nearly equal to of AE, or . But the distance of the comet from the perpendicular let fall from the star A upon the right line PE was .

Sunday, February 27, P. M. the distance of the comet in Q, from the star O was equal to the distance of the stars O and H; and the right line QO produced passed between the stars K and B. I could not, by reason of intervening clouds, determine the position of the star to greater accuracy.

Tuesday, March 1, 11^h. P. M. the comet in R lay exactly in a line between the stars K and C, so as the part CR of the right line CRK was a little greater than , and a little less than , and therefore = , or .

Wednesday, March 2, 8^h. P. M. the distance of the comet in S from the star C was nearly ; the distance of the star F from the right line CS produced was ; and the distance of the star B from the same right line was five times greater than the distance of the star F; and the right line NS produced passed between the stars H and I five or six times nearer to the star H than to the star I.

Saturday, March 5, P. M. when the comet was in T, the right line MT was equal to , and the right line LT produced passed between B and F four or five times nearer to F than to B, cutting off from BF a fifth or sixth part thereof towards F: and MT produced passed on the outside of the space BF towards the star B four times nearer to the star B than to the star F. M was a very small star, scarcely to be seen by the telescope; but the star L was greater, and of about the eighth magnitude.

Monday, March 7, P. M. the comet being in V, the right line produced did pass between B and F, cutting off, from BF towards F, of BF, and was to the right line as 5 to 4. And the distance of the comet from the right line was .

Wednesday, March 9, P. M. the comet being in X, the right line was equal to ; and the perpendicular let fall from the star upon the right was of .

The same night, at 12^h. the comet being in Y, the right line was equal to of , or a little less, as perhaps of ; and a perpendicular let fall from the star on the right line was equal to about or . But the comet being then extremely near the horizon, was scarcely discernible, and therefore its place could not be determined with that certainty as in the foregoing observations.

From these observations, by constructions of figures and calculations, I deduced the longitudes and latitudes of the comet; and Mr. Pound, by correcting the places of the fixed stars, hath determined more correctly the places of the comet, which correct places are set down above. Though my micrometer was none of the best, yet the errors in longitude and latitude (as derived from my observations) scarcely exceed one minute. The comet (according to my observations), about the end of its motion, began to decline sensibly towards the north, from the parallel which it described about the end of February.

Now, in order to determine the orbit of the comet out of the observations above described, I selected those three which Flamsted made, Dec. 21, Jan. 5, and Jan. 25; from which I found St of 9842,1 parts, and Vt of 455, such as the semi-diameter of the orbis magnus contains 10000. Then for the first observation, assuming tB of 5657 of those parts, I found SB 9747, BE for the first time 412, 9503, 413, BE for the second time 421, OD 10186, X 8528,4, PM 8450, MN 8475, NP 25; from whence, by the second operation, I collected the distance tb 5640; and by this operation I at last deduced the distances TX 4775 and 11322. From which, limiting the orbit, I found its descending node in ♋, and ascending node in ♑ 1° 53'; the inclination of its plane to the plane of the ecliptic 61° 20, the vertex thereof (or the perihelion of the comet) distant from the node 8° 38', and in ♐ 27° 43', with latitude 7° 34' south; its latus rectum 236,8; and the diurnal area described by a radius drawn to the sun 93585, supposing the square of the semi-diameter of the orbis magnus 100000000; that the comet in this orbit moved directly according to the order of the signs, and on Dec. 8^d. 00^h. 04' P. M. was in the vertex or perihelion of its orbit. All which I determined by scale and compass, and the chords of angles, taken from the table of natural sines, in a pretty large figure, in which, to wit, the radius of the orbis magnus (consisting of 10000 parts) was equal to inches of an English foot.

Lastly, in order to discover whether the comet did truly move in the orbit so determined, I investigated its places in this orbit partly by arithmetical operations, and partly by scale and compass, to the times of some of the observations, as may be seen in the following table:—

The Comet's

Dist. from sun. Longitude computed. Latitud. computed. Longitude observed. Latitute observed. Dif. Lo. Dif. Lat.

Dec. 12 2792 ♑ 6°.32' 8°.18 ♑ 6°.31 8°.26 +1 - 7

29 8403 ♓ 13 .13 28. 00 ♓ 13.11 28 .10 +2 -10

Feb. 5 16669 ♉ 17 .00 15. 29 ♉ 16 .59 15 .27 +0 + 2

Mar. 5 21737 29 .19 12. 4 29 .20 12 . 3 -1 +

But afterwards Dr. Halley did determine the orbit to a greater accuracy by an arithmetical calculus than could be done by linear descriptions; and, retaining the place of the nodes in ♋ and ♑ 1° 53' and the inclination of the plane of the orbit to the ecliptic 61° , as well as the time of the comet's being in perihelion, Dec. 8^d. 00^h. 04' he found the distance of the perihelion from the ascending node measured in the comet's orbit 9° 20', and the latus rectum of the parabola 2430 parts, supposing the mean distance of the sun from the earth to be 100000 parts; and from these data, by an accurate arithmetical calculus, he computed the places of the comet to the times of the observations as follows:—

The Comet's

True time. Dist. from the sun. Longitude computed. Latitude computed. Errors in

Long. Lo. Lat. Lat.

d. h. ' '' ° ' '' ° ' '' ' '' ' ''

Dec. 12. 4.46. 28028 ♑︎ 6.29.25 8.26. 0 bor. -3. 5 -2. 0

21. 6.37. 61076 ♒︎ 5. 6.30 21.43.20 -1.42 +1. 7

24. 6.18. 70008 18.48.20 25.22.40 -1. 3 -0.25

26. 5.20. 75576 28.22.45 27. 1.36 -1.28 +0.44

29. 8. 3. 84021 ♓︎ 13.12.40 28.10.10 +1.59 +0.12

30. 8.10. 86661 17.40. 5 28.11.20 +1.45 -0.33

Jan. 5. 6. 1. 101440 ♈︎ 8.49.49 26.15.15 +0.56 +0. 8

9. 7. 0. 110959 18.44.36 24.12.54 +0.32 +0.58

10. 6. 6. 113162 20.41. 0 23.44.10 +0.10 +0.18

13. 7. 9. 120000 26. 0.21 22.17.30 +0.33 +0. 2

25. 7.59. 145370 ♉︎ 9.33.40 17.57.55 -1.20 +1.25

30. 8.22. 155303 13.17.41 16.42. 7 -2.10 -0.11

Feb. 2. 6.35. 160951 15.11.11 16. 4.15 -2.42 +0.14

5. 7. 4. 166686 16.58.55 15.29.13 -0.41 +2. 0

25. 8.41. 202570 26.15.46 12.48. 0 -2.49 +1.10

Mar. 5.11.39. 216205 29.18.35 12. 5.40 +0.35 +2.14

This comet also appeared in the November before, and at Coburg, in Saxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, on the 6th and 11th O. S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes with a two feet, and sometimes with a ten feet telescope; from the difference of longitudes of Coburg and London; 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:—

Nov. 3, 17^h. 2', apparent time at London, the comet was in ♌ 29 deg. 51', with 1 deg. 17' 45'' latitude north.

November 5, 15^h. 58' the comet was in ♍ 3° 23', with 1° 6' lat. north.

November 10, 16^h. 31', the comet was equally distant from two stars in ♌, which are and in Bayer; but it had not quite touched the right line that joins them, but was very little distant from it. In Flamsted's catalogue this star was then in ♍︎ 14° 15', with 1 deg. 41' lat. north nearly, and in ♍ 17° with 0 deg. 34' lat. south; and the middle point between those stars was ♍ 15° , with 0° lat. north. Let the distance of the comet from that right line be about 10' or 12'; and the difference of the longitude of the comet and that middle point will be 7'; and the difference of the latitude nearly ; and thence it follows that the comet was in ♍ 15° 32', with about 26' lat. north.

The first observation from the position of the comet with respect to certain small fixed stars had all the exactness that could be desired; the second also was accurate enough. In the third observation, which was the least accurate, there might be an error of 6 or 7 minutes, but hardly greater. The longitude of the comet, as found in the first and most accurate observation, being computed in the aforesaid parabolic orbit, comes out ♌ 29° 30' 22'', its latitude north 1° 25' 7'', and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Cæsar's death, at which time, by reason of the inconvenient situation of the earth, the tail was not so conspicuous), set himself to find out an elliptic orbit whose greater axis should be 1382957 parts, the mean distance of the earth from the sun containing 10000 such; in which orbit a comet might revolve in 575 years; and, placing the ascending node in ♋ 2° 2', the inclination of the plane of the orbit to the plane of the ecliptic in an angle of 61° 6' 48'', the perihelion of the comet in this plane in ♐ 22° 44' 25'', the equal time of the perihelion December 7^d. 23^h. 9', the distance of the perihelion from the ascending node in the plane of the ecliptic 9° 17' 35'', and its conjugate axis 18481,2, he computed the motions of the comet in this elliptic orbit. The places of the comet, as deduced from the observations, and as arising from computation made in this orbit, may be seen in the following table.

True time. Longitude observed. Latit'de North obs. Longitude comp. Latitude computed. Errors in

Long. Lo. Lat. Lat.

d. h. ' '' ° ' '' ° ' '' ° ' '' ' '' ' '' ' ''

Nov. 3.16.47 ♌ 29.51. 0 1.17.45 ♌ 29.51.22 1.17.32 N +0.22 -0.13

5.15.37 ♍ 3.23. 0 1. 6. 0 ♍ 3.24.32 1. 6. 9 +1.32 +0. 9

10.16.18 15.32. 0 0.27. 0 15.33. 2 0.25. 7 +1. 2 -1.53

16.17.00 ♎ 8.16.45 0.53. 7 S

18.21.34 18.52.15 1.26.54

20.17. 0 28.10.36 1.53.35

23.17. 5 ♏ 13.22.42 2.29. 0

Dec. 12. 4.46 ♑ 6.32.30 8.28. 0 ♑ 6.31.20 8.29. 6 N -1.10 +1. 6

21. 6.37 ♒ 5. 8.12 21.42.13 ♒ 5. 6.14 21.44.42 -1.58 +2.29

24. 6.18 18.49.23 25.23. 5 18.47.30 25.23.35 -1.53 +0.30

26. 5.21 28.24.13 27. 0.52 28.21.42 27. 2. 1 -2.31 +1. 9

29. 8. 3 ♓ 13.10.41 28. 9.58 ♓ 13.11.14 28.10.38 +0.33 +0.40

30. 8.10 17.38. 0 28.11.53 17.38.27 28.11.37 +0. 7 -0.16

Jan. 5. 6. 1 ♈ 8.48.53 26.15. 7 ♈ 8.48.51 26.14.57 -0. 2 -0.10

9. 7. 1 18.44. 4 24.11.56 18.43.51 24.12.17 -0.13 +0.21

10. 6. 6 20.40.50 23.43.32 20.40.23 23.43.25 -0.27 -0. 7

13. 7. 9 25.59.48 22.17.28 26. 0. 8 22.16.32 +0.20 -0.56

25. 7.59 ♉ 9.35. 0 17.56.30 ♉ 9.34.11 17.56. 6 -0.49 -0.24

30. 8.22 13.19.51 16.42.18 13.18.28 16.40. 5 -1.23 -2.13

Feb. 2. 6.35 15.13.53 16. 4. 1 15.11.59 16. 2.17 -1.54 -1.54

5. 7. 4 16.59. 6 15.27. 3 16.59.17 15.27. 0 +0.11 -0. 3

25. 8.41 26.18.35 12.46.46 26.16.59 12.45.22 -1.36 -1.24

Mar. 1.11.10 27.52.42 12.23.40 27.51.47 12.22.28 -0.55 -1.12

5.11.39 29.18. 0 12. 3.16 29.20.11 12. 2.50 +2.11 -0.26

9. 8.38 ♊ 0.43. 4 ♊.45.52 ♊ 0.42.43 11.45.35 -0.21 -0.17

The observations of this comet from the beginning to the end agree as perfectly with the motion of the comet in the orbit just now described as the motions of the planets do with the theories from whence they are calculated; and by this agreement plainly evince that it was one and the same comet that appeared all that time, and also that the orbit of that comet is here rightly defined.

In the foregoing table we have omitted the observations of Nov. 16, 18, 20, and 23, as not sufficiently accurate, for at those times several persons had observed the comet. Nov. 17, O. S. Ponthæus and his companions, at 6^h. in the morning at Rome (that is, 5^h. 10' at London), by threads directed to the fixed stars, observed the comet in ♎ 8° 30', with latitude 0° 40' south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini, saw the comet at the same hour in ♎ 8° 30', with latitude 0° 30' south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5^h. 42' morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16' 45'', and its latitude was 0° 53' 7'' south.

Nov. 18, at 6^h. 30' in the morning at Rome (that is, at 5^h. 40' at London), Ponthæus observed the comet in ♎ 13° 30', with latitude 1° 20' south; and Cellius in ♎ 13° 30', with latitude 1° 00' south. But at 5^h. 30' in the morning at Avignon, Galletius saw it in ♎ 13° 00' with latitude 1° 00' south. In the University of La Fleche, in France, at 5^h. in the morning (that is, at 5^h. 9' at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ; and the other is the outmost of the wing, Bayer's . Whence the comet was then in ♎ 12° 46' with latitude 50' south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of deg. at 5^h. in the morning (that is, at 9^h. 44' in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30' south.

Nov. 19, at at Cambridge, the comet (by the observation of a young man) was distant from Spica ♍ about 2° towards the north west. Now the spike was at that time in ♎ 19° 23' 47'', with latitude 2° 1' 59'' south. The same day, at 5^h. in the morning, at Boston in New England, the comet was distant from Spica ♍ 1°, with the difference of 40' in latitude. The same day, in the island of Jamaica, it was about 1° distant from Spica ♍. The same day, Mr. Arthur Storer, at the river Patuxent, near Hunting Creek, in Maryland, in the confines of Virginia, in lat. ^°, at 5 in the morning (that is, at 10^h. at London), saw the comet above Spica ♍, and very nearly joined with it, the distance between them being about of one deg. And from these observations compared, I conclude, that at 9^h. 44' at London, the comet was in ♎ 18° 50', with about 1° 25' latitude south. Now by the theory the comet was at that time in ♎ 18° 52' 15'', with 1° 26' 54'' lat. south.

Nov. 20, Montenari, professor of astronomy at Padua, at 6^h. in the morning at Venice (that is, 5^h. 10' at London), saw the comet in ♎ 23°, with latitude 1° 30' south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24' nearly.

Nov. 21, Ponthæus and his companions, at in the morning, observed the comet in ♎ 27° 50', with latitude 1° 16' south; Cellius, in ♎ 28°; P. Ango at 5^h. in the morning, in ♎ 27° 45'; Montenari in ♎ 27° 51'. The same day, in the island of Jamaica, it was seen near the beginning of ♏, and of about the same latitude with Spica ♍, that is, 2° 2'. The same day, at 5^h. morning, at Ballasore, in the East Indies (that is, at 11^h. 20' of the night preceding at London), the distance of the comet from Spica ♍ was taken 7° 35' to the east. It was in a right line between the spike and the balance, and therefore was then in ♎ 26° 58', with about 1° 11' lat. south; and after 5^h. 40' (that is, at 5^h. morning at London), it was in ♎ 28° 12', with 1° 16' lat. south. Now by the theory the comet was then in ♎ 28° 10' 36'', with 1° 53' 35'' lat. south.

Nov. 22, the comet was seen by Montenari in ♏ 2° 33'; but at Boston in New England, it was found in about ♏ 3°, and with almost the same latitude as before, that is, 1° 30'. The same day, at 5^h. morning at Ballasore, the comet was observed in ♏ 1° 50'; and therefore at 5^h. morning at London, the comet was in ♏ 3° 5' nearly. The same day, at in the morning at London, Dr. Hook observed it in about ♏ 3° 30', and that in the right line which passeth through Spica ♍ and Cor Leonis; not, indeed, exactly, but deviating a little from that line towards the north. Montenari likewise observed, that this day, and some days after, a right line drawn from the comet through Spica passed by the south side of Cor Leonis at a very small distance therefrom. The right line through Cor Leonis and Spica ♍ did cut the ecliptic in ♍ 3° 46' at an angle of 2° 51'; and if the comet had been in this line and in ♏ 3°, its latitude would have been 2° 26'; but since Hook and Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been something less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica ♍, that is, about 1° 30'. But by the agreement of Hook, Montenari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30'; and, taking a mean between the extreme limits but now stated, 2° 26' and 1° 30', the latitude will be about 1° 58'. Hook and Montenari agree that the tail of the comet was directed towards Spica ♍, declining a little from that star towards the south according to Hook, but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the equator, deviated a little from the opposition of the sun towards the north.

Nov. 23, O. S. at 5^h. morning, at Nuremberg (that is, at at London), Mr. Zimmerman saw the comet in ♏ 8° 8', with 2° 31' south lat. its place being collected by taking its distances from fixed stars.

Nov. 24, before sun-rising, the comet was seen by Montenari in ♏ 12° 52' on the north side of the right line through Cor Leonis and Spica ♍, and therefore its latitude was something less than 2° 38'; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58'; and, taking the mean quantity, may be reckoned 2° 18', without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or ^°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by Ponthæus and Cellius. The same day, at 5^h. morning, at Ballasore, the comet was observed in ♏ 11° 45'; and, therefore, at 5^h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22' 42''.

Nov. 25, before sunrise, Montenari observed the comet in ♏ ^° nearly; and Cellius observed at the same time that the comet was in a right line between the bright star in the right thigh of Virgo and the southern scale of Libra; and this right line cuts the comet's way in ♏ 18° 36'. And, by the theory, the comet was in ♏ ^° nearly.

From all this it is plain that these observations agree with the theory, so far as they agree with one another; and by this agreement it is made clear that it was one and the same comet that appeared all the time from Nov. 4 to Mar. 9. The path of this comet did twice cut the plane of the ecliptic, and therefore was not a right line. It did cut the ecliptic not in opposite parts of the heavens, but in the end of Virgo and beginning of Capricorn, including an arc of about 98°; and therefore the way of the comet did very much deviate from the path of a great circle; for in the month of Nov. it declined at least 3° from the ecliptic towards the south; and in the month of Dec. following it declined 29° from the ecliptic towards the north; the two parts of the orbit in which the comet descended towards the sun, and ascended again from the sun, declining one from the other by an apparent angle of above 30°, as observed by Montenari. This comet travelled over 9 signs, to wit, from the last deg. of ♌ to the beginning of ♊, beside the sign of ♌, through which it passed before it began to be seen; and there is no other theory by which a comet can go over so great a part of the heavens with a regular motion. The motion of this comet was very unequable; for about the 20th of Nov. it described about 5° a day. Then its motion being retarded between Nov. 26 and Dec. 12, to wit, in the space of days, it described only 40°. But the motion thereof being afterwards accelerated, it described near 5° a day, till its motion began to be again retarded. And the theory which justly corresponds with a motion so unequable, and through so great a part of the heavens, which observes the same laws with the theory of the planets, and which accurately agrees with accurate astronomical observations, cannot be otherwise than true.

And, thinking it would not be improper, I have given a true representation of the orbit which this comet described, and of the tail which it emitted in several places, in the annexed figure; protracted in the plane of the trajectory. In this scheme ABC represents the trajectory of the comet, D the sun, DE the axis of the trajectory, DF the line of the nodes, GH the intersection of the sphere of the orbis magnus with the plane of the trajectory, I the place of the comet Nov. 4, Ann. 1680; K the place of the same Nov. 11; L the place of the same Nov. 19; M its place Dec. 12; N its place Dec. 21; O its place Dec. 29; P its place Jan. 5 following; Q its place Jan. 25; R its place Feb. 5; S its place Feb. 25; T its place March 5; and V its place March 9. In determining the length of the tail, I made the following observations.

Nov. 4 and 6, the tail did not appear; Nov. 11, the tail just begun to shew itself, but did not appear above deg. long through a 10 feet telescope; Nov. 17, the tail was seen by Ponthæus more than 15° long; Nov. 18, in New-England, the tail appeared 30° long, and directly opposite to the sun, extending itself to the planet Mars, which was then in ♍, 9° 54'; Nov. 19. in Maryland, the tail was found 15° or 20° long; Dec. 10 (by the observation of Mr. Flamsted), the tail passed through the middle of the distance intercepted between the tail of the Serpent of Ophiuchus and the star in the south wing of Aquila, and did terminate near the stars A, , b, in Bayer's tables. Therefore the end of the tail was in ♑ ^°, with latitude about ^° north; Dec. 11, it ascended to the head of Sagitta (Bayer's , ), terminating in ♑ 26° 43' with latitude 38° 34' north; Dec. 12, it passed through the middle of Sagitta, nor did it reach much farther; terminating in ♒ 4°, with latitude ^° north nearly. But these things are to be understood of the length of the brighter part of the tail; for with a more faint light, observed, too, perhaps, in a serener sky, at Rome, Dec. 12, 5^h. 40', by the observation of Ponthæus, the tail arose to 10° above the rump of the Swan, and the side thereof towards the west and towards the north was 45' distant from this star. But about that time the tail was 3° broad towards the upper end; and therefore the middle thereof was 2° 15' distant from that star towards the south, and the upper end was ♓ in 22°, with latitude 61° north; and thence the tail was about 70° long; Dec. 21, it extended almost to Cassiopeia's chair, equally distant from and from Schedir, so as its distance from either of the two was equal to the distance of the one from the other, and therefore did terminate in ♈ 24°, with latitude ^°; Dec. 29, it reached to a contact with Scheat on its left, and exactly filled up the space between the two stars in the northern foot of Andromeda, being 54° in length; and therefore terminated in ♉ 19°, with 35° of latitude; Jan. 5, it touched the star in the breast of Andromeda on its right side, and the star of the girdle on its left; and, according to our observations, was 40° long; but it was curved, and the convex side thereof lay to the south; and near the head of the comet it made an angle of 4° with the circle which passed through the sun and the comet's head; but towards the other end it was inclined to that circle in an angle of about 10° or 11°; and the chord of the tail contained with that circle an angle of 8°. Jan. 13, the tail terminated between Alamech and Algol, with a light that was sensible enough; but with a faint light it ended over against the star in Perseus's side. The distance of the end of the tail from the circle passing through the sun and the comet was 3° 50'; and the inclination of the chord of the tail to that circle was ^°. Jan. 25 and 26, it shone with a faint light to the length of 6° or 7°; and for a night or two after, when there was a very clear sky, it extended to the length of 12°, or something more, with a light that was very faint and very hardly to be seen; but the axis thereof was exactly directed to the bright star in the eastern shoulder of Auriga, and therefore deviated from the opposition of the sun towards the north by an angle of 10°. Lastly, Feb. 10, with a telescope I observed the tail 2° long; for that fainter light which I spoke of did not appear through the glasses. But Ponthæus writes, that, on Feb. 7, he saw the tail 12° long. Feb. 25, the comet was without a tail, and so continued till it disappeared.

Now if one reflects upon the orbit described, and duly considers the other appearances of this comet, he will be easily satisfied that the bodies of comets are solid, compact, fixed, and durable, like the bodies of the planets; for if they were nothing else but the vapours or exhalations of the earth, of the sun, and other planets, this comet, in its passage by the neighbourhood of the sun, would have been immediately dissipated; for the heat of the sun is as the density of its rays, that is, reciprocally as the square of the distance of the places from the sun. Therefore, since on Dec. 8, when the comet was in its perihelion, the distance thereof from the centre of the sun was to the distance of the earth from the same as about 6 to 1000, the sun's heat on the comet was at that time to the heat of the summer-sun with us as 1000000 to 36, or as 28000 to 1. But the heat of boiling water is about 3 times greater than the heat which dry earth acquires from the summer-sun, as I have tried; and the heat of red-hot iron (if my conjecture is right) is about three or four times greater than the heat of boiling water. And therefore the heat which dry earth on the comet, while in its perihelion, might have conceived from the rays of the sun, was about 2000 times greater than the heat of red-hot iron. But by so fierce a heat, vapours and exhalations, and every volatile matter, must have been immediately consumed and dissipated.

This comet, therefore, must have conceived an immense heat from the sun, and retained that heat for an exceeding long time; for a globe of iron of an inch in diameter, exposed red-hot to the open air, will scarcely lose all its heat in an hour's time; but a greater globe would retain its heat longer in the proportion of its diameter, because the surface (in proportion to which it is cooled by the contact of the ambient air) is in that proportion less in respect of the quantity of the included hot matter; and therefore a globe of red hot iron equal to our earth, that is, about 40000000 feet in diameter, would scarcely cool in an equal number of days, or in above 50000 years. But I suspect that the duration of heat may, on account of some latent causes, increase in a yet less proportion than that of the diameter; and I should be glad that the true proportion was investigated by experiments.

It is farther to be observed, that the comet in the month of December, just after it had been heated by the sun, did emit a much longer tail, and much more splendid, than in the month of November before, when it had not yet arrived at its perihelion; and, universally, the greatest and most fulgent tails always arise from comets immediately after their passing by the neighbourhood of the sun. Therefore the heat received by the comet conduces to the greatness of the tail: from whence, I think I may infer, that the tail is nothing else but a very fine vapour, which the head or nucleus of the comet emits by its heat.