The Mathematical Principles of Natural Philosophy (1846)/BookIII-Prop6

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 ΥQA, ΥQB, ΥQC, be three observed longitudes of the comet about the time of its first appearing, and ΥQF 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 ΥQG 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 ΥQG, 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 TΥ a right line drawn to the beginning of Aries. Set off the angle ΥTV equal to the angle ΥQF, 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 gΥ parallel to TΥ, and make the angle ΥgV equal to the angle ΥQG, this angle ΥgV 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 ${\displaystyle \scriptstyle {\sqrt {4}}}$ 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.

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 some times 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 them selves, 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 1, 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 4 ${\displaystyle \scriptstyle {\sqrt {4}}}$ (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 1, 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 semidiameter 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 71675½ 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 ${\displaystyle \scriptstyle {\sqrt {2}}}$ to 1, would by its diurnal motion describe 2432747 parts, and 101364½ 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 1216373½ parts, and the horary area will be 50682¼ 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.

 ${\displaystyle b}$ ${\displaystyle 2b}$ ${\displaystyle 3b}$ ${\displaystyle 4b}$ ${\displaystyle 5b}$ ${\displaystyle c}$ ${\displaystyle 2c}$ ${\displaystyle 3c}$ ${\displaystyle 4c}$ ${\displaystyle d}$ ${\displaystyle 2d}$ ${\displaystyle 3d}$ ${\displaystyle e}$ ${\displaystyle 2e}$ ${\displaystyle f}$

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 AH = a, - HS = p, ½p into - IS = q, ⅓q into + SK = r, ¼r into + SL = s, 15s 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 ${\displaystyle \scriptstyle ={\frac {AH-BI}{HI}}}$, ${\displaystyle \scriptstyle 2b={\frac {BI-CK}{IK}}}$, ${\displaystyle \scriptstyle 3b={\frac {CK-DL}{KL}}}$, &c., then ${\displaystyle \scriptstyle c={\frac {b-2b}{HK}}}$, ${\displaystyle \scriptstyle 2c={\frac {2b-3b}{IL}}}$, ${\displaystyle \scriptstyle 3c={\frac {3b-4b}{KM}}}$, &c., then ${\displaystyle \scriptstyle d={\frac {c-2c}{HL}}}$, ${\displaystyle \scriptstyle 2d={\frac {2c-3c}{IM}}}$, &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 Iμ and vertex μ. In Iμ produced take μO equal to one half of Iμ. Join OS, and produce it to ξ, so as Sξ may be equal to 2SO. Now, supposing a comet to revolve in the arc CBA, draw ξB, 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 μX touching the same arc in the vertex μ, and meeting EO in X, the curvilinear area AEXμA will be to the curvilinear area ACYμA as AE to AC; and, therefore, since the triangle ASE is to the triangle ASC in the same proportion, the whole area ASEXμA will be to the whole area ASCYμA as

AE to AC. But, because ξO 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 ASEXμA we add the triangle EXB, and from the sum subduct the triangle SEB, there will remain the area ASBXμA, equal to the area ASEXμA, and therefore in proportion to the area ASCYμA as AE to AC. But the area ASBYμA is nearly equal to the area ASBXμA; and this area ASBYμA is to the area ASCYμA 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 μ of the parabola, AE is to AC accurately in the proportion of the times.

SCHOLIUM.

If we join μξ cutting AC in δ, and in it take ξn in proportion to μB as 27MI to 16Mμ, 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 ξ, according as the point B is more or less distant from the principal vertex of the parabola than the point μ.

LEMMA IX.

The right lines Iμ and μM, and the length ${\displaystyle \scriptstyle {\frac {AI^{2}}{4S\mu }}}$, are equal among themselves.

For 4Sμ is the latus rectum of the parabola belonging to the vertex μ.

LEMMA X.

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

Cor. Therefore a comet, with that velocity which it hath in the height Sμ + ⅔Iμ, 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 Sμ + ⅓Iμ, 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 Iμ.

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 1), 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 1) 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 ${\displaystyle \scriptstyle {\frac {AI^{2}}{4SP}}}$. 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 Sμ, 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 ${\displaystyle \scriptstyle {\frac {AI^{2}}{4S\mu }}}$; that is, a space equal to the length Iμ or μM .   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, τ, three places of the earth in the orbis magnus; TA, tB, τC, 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 Tτ. 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 τC, 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 λ, and complete the parallelogram iI λμ. Take Iσ equal to 3Iλ; and through the sun S draw the obscure line σξ equal to 3Sσ + 3. Then, cancelling the letters A, E, C, I, from the point B towards the point ξ, draw the new obscure line BE, which may be to the former BE in the duplicate proportion of the distance BS to the quantity Sμ + ⅓. 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 τC. Join MN, cutting IO in O. Draw the rectangular parallelogram iIλμ, as before. In IA produced take ID equal to Sμ + ⅔. 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 β assumed at pleasure, find out the new points e, a, c, g; and ε, α, κ, γ. Then through G, g, and γ, draw the circumference of a circle G, cutting the right line τC in Z: and Z will he one place of the comet in the plane of the ecliptic. And in AC, ac, ακ, taking AF, af, αϕ, equal respectively to CG, cg, κγ; through the points F, f, and ϕ, draw the circumference of a circle F, 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 τZ, two places of the comet in its own orbit will be determined. Lastly, if (by Prop. XIX., Book 1) 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 Bξ 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, β, 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 43Tτ 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 τC 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 β, 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 τC 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.

 1680, Dec. 1221242629301681, Jan. 5910132530Feb. 25 Time Sun'sLongitude Comet's Appar. True. Longitude. Lat. N. h.   ″4.466.32½6.125.147.558.025.516.495.546.567.448.076.206.50 h.   ′   ″4.46.06.36.596.17.525.20.448.03.028.10.266.01.387.00.536.06.107.08.557.58.428.21.536.34.517.04.41 °   ′   ″♑   1.51.2311.06.4414.09.2616.09.2219.19.4320.21.0926.22.18♒   0.29.021.27.434.33.2016.45.3621.49.5824.46.5927.49.51 °   ′   ″♑   6.32.30♒   5.08.1218.49.2328.24.13♓   13.10.4117.38.20♈ 8.48.5318.44.0420.40.5025.59.48♉ 9.35.013.19.5115.13.5316.59.06 °   ′   ″8.28. 021.42.1325.23. 527.00.5228.09.5828.11.5326.15. 724.11.5623.43.5222.17.2817.56.3016.42.1816.04. 115.27. 3

To these you may add some observations of mine.

 1681, Feb. 2527Mar. 12579 Ap. Time. Comet's Longitude Lat. N. h.   ′8.308.1511. 08. 011.309.308.30 °   ′   ″♉   26.18.3527.04.3027.52.4228.12.4829.18. 0♊     0. 4. 00. 43. 4 °   ′   ″12.46.4612.36.1212.23.4012.19.3812.03.1611.57. 011.45.52
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 ζ), C a star of the sixth magnitude (Bayer's n) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, δ, 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 80712 parts, AC was 52¼ of those parts; BC, 5856; AD, 57512; BD, 82611; CD, 23⅔; AE, 2947; CE, 57½; DE, 491112; AI, 27712; BI, 5216; CI, 36712; DI, 53511; AK, 38⅔; BK, 43; CK, 3159; FK, 29; FB, 23; FC, 36¼; AH, 1867; DH, 5078; BN, 46512; CN, 31⅓; BL, 45512; NL, 3157. 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 16CD. 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.

 Thefixedstars. TheirLongitudes LatitudeNorth. Thefixedstars. TheirLongitudes LatitudeNorth. ABCEFGHIK °   ′   ″♉   26.41.5028.40.2327.58.3026.27.1728.28.3726.56. 827.11.4527.25. 227.42. 7 °   ′   ″12. 8.3611.17.5412.40.2512.52. 711.52.2214.4.5812.2. 111.53.1111.53.26 LMNZαβγδ °   ′   ″♉   29.33.3429.18.5428.48.2929.44.4829.52. 3♊   0. 8.230.40.101. 3.20 °   ′   ″12. 7.4812. 7.2012.31. 911.57.1311.55.4811.48.5311.55.1811.30.42

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

Friday, February 25, O.S. at 8½h. P. M. the distance of the comet in p from the star E was less than 313AE, and greater than 15AE, and therefore nearly equal to 314AE; 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 15pE.

The same night, at 9½h., the distance of the comet in P from the star E was greater than ${\displaystyle \scriptstyle {\frac {1}{4{\frac {1}{2}}}}}$AE, and less than ${\displaystyle \scriptstyle {\frac {1}{5{\frac {1}{4}}}}}$AE, and therefore nearly equal to ${\displaystyle \scriptstyle {\frac {1}{4{\frac {7}{8}}}}}$ of AE, or 839 AE. But the distance of the comet from the perpendicular let fall from the star A upon the right line PE was 45PE.

Sunday, February 27, 8¼h. 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, 11h . 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 ⅓CK, and a little less than ⅓CK + 18CR, and therefore = ⅓CK + 116CR, or 1645CK.

Wednesday, March 2, 8h. P. M. the distance of the comet in S from the star C was nearly 49FC; the distance of the star F from the right line CS produced was 124FC; 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, 11½h. P. M. when the comet was in T, the right line MT was equal to ½ML, 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, 9½h. P. M. the comet being in V, the right line Va produced did pass between B and F, cutting off, from BF towards F, 110 of BF, and was to the right line Vβ as 5 to 4. And the distance of the comet from the right line αβ was ½Vβ.

Wednesday, March 9, 8½h. P. M. the comet being in X, the right line γX was equal to ¼γδ and the perpendicular let fall from the star δ upon the right γX was 25 of γδ.

The same night, at 12h. the comet being in Y, the right line γY was equal to ⅓ of γδ, or a little less, as perhaps 516 of γδ; and a perpendicular let fall from the star δ on the right line γY was equal to about 16 or 17 γδ. 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.

Prom 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, Sμ 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 τZ 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° 20⅓′, 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. 8d.00h.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 16⅓ 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.fromsun. Longitudecomputed. Latitud.compu-ted. Longitudeobserved. Latitudeobserved DifLo. Dif.Lat. Dec. 12 2792 ♑  6°.32′ 8°.18½ ♑  6° 31½ 8°.26 +1 − 7½ 29 8403 ♓ 13°.13⅔ 28.°00 ♓ 13°.11¾ 28°.101⁄12 +2 −101⁄12 Feb.  5 16669 ♉ 17°.00 15.°29⅔ ♉ 16°.59⅞ 15°.27⅖ +0 −+ 2¼ Mar.  5 21737 ♉ 29°.19¾ 12.°04 ♉ 29°.206⁄7 12°. 3½ −1 −+  ½

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° 20⅓′, as well as the time of the comet's being in perihelio, Dec. 8d.00h.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 fromthe sun. Longitude computed. Latitudecomputed. Errors inLong.     Lat. d.   h.   ′ ″Dec. 12.4.46.  21.6.37.  24.6.18.  26.5.20.  29.8. 3.  30.8.10.  Jan. 5.3.1.½9.7. 0.  10.6. 6.  13.7. 9.  25.7.59.  30.8.22.  Feb. 2.6.35.  5.7.4.½25.8.41.  Mar. 5.11.39. 280286107670008755768402186661101440110959113162120000145370155303160951166686202570216205 °   ′   ″♑ 6.29.25♒ 5.6.3018.48.2028.22.45♓ 13.12.4017.40.5♈ 8.49.4918.44.3620.41.026.0.21♉ 9.33.4013.17.4115.11.1116.58.5526.15.4629.18.35 °   ′   ″8.26.0 bor.21.43.2025.22.4027.1.3628.10.1028.11.2026.15.1524.12.5423.44.1022.17.3017.57.5516.42.716.4.1515.29.1312.48.015.5.40 ′   ″-3.5-1.42-1.3-1.28+1.59+1.45+0.56+0.32+0.10+0.33-1.20-2.10-2.42-0.41-2.49+0.35 ′   ″-2.0+1.7-0.25+0.44+0.12-0.33+0.8+0.58+0.18+0.2+1.25-0.11+0.14+2.0+1.10+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, 17h.2′, apparent time at London, the comet was in ♌ 29 deg. 51′, with 1 deg. 17′ 45″ latitude north.

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

November 10, 16h.31′, the comet was equally distant from two stars in ♌ which are σ and τ 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 σ was then in ♍ 14° 15′, with 1 deg. 41′ lat. north nearly, and τ in ♍ 17° 3½′, with 0 deg. 34′ lat. south; and the middle point between those stars was ♍ 15° 39¼′, with 0° 33½′ 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 7½′; 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 7d.23h.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. Longitudesobserved. Latitude Northobs. Longitudecomputed. Latitudecomputed. Errors inLong.   Lat. d.   h.   ′Nov. 3.16.475.15.3710.16.1816.17.0018.21.3420.17.023.17.5Dec. 12.4.4621.6.3724.6.1826.5.2129.8.330.8.10Jan. 5.6.1½9.7.710.6.613.7.925.7.5930.8.22Feb. 2.6.355.7.4½25.8.41Mar. 1.11.105.11.399.8.38 °   ′   ″♌   29.51.0♍   3.23.015.32. 0♑   6.32.30♒ 5. 8.1218.49.2328.24.13♓   13.10.4117.38. 0♈   8.48.5318.44. 420.40.5025.59.48♉   9.35. 013.19.5115.13.5316.59. 626.18.3527.52.4229.18. 0♊   0.43.4 °   ′   ″1.17.451.6. 00.27. 08.28. 021.42.1325.23. 527. 0.5228. 9.5828.11.5326.15. 724.11.5623.43.3222.17.2817.56.3016.42.1816. 4. 115.27. 312.46.4612.23.4012. 3.1611.45.52 °   ′   ″♌   29.51.22♍   3.24.3215.33. 2♎   8.16.4518.52.1528.10.36♏   13.22.42♑   6.31.20♒   5. 6.1418.47.3028.21.42♓   13.11.1417.38.27♈   8.48.5118.43.5120.40.2326. 0. 8♉   9.34.1113.18.2515.11.5916.59.1726.16.5927.51.4729.20.11♊   0.42.43 °   ′   ″1.17.32    N1. 6. 90.25. 70.53. 7    S1.26.541.53.352.29. 08.29. 6    N21.44.4225.23.3527. 2. 128.10.3828.11.3726.14.5724.12.1723.43.2522.16.3217.56. 616.40. 516. 2.1715.27. 012.45.2212.22.2812. 2.5011.45.35 ′   ″+0.22+1.32+1.2-1.10-1.58-1.53-2.31+0,33+0.7-0.2-0.13-0.27+0.20-0,49-1.23-1.54+0.11-1.36-0.55+2.11-0.21 ′   ″-0.13+0.9-1.53+1.6+2.29+0.30+1.9+0.40-0.16-0.10+0.21-0.7-0.56-0.24-2.13-1.54-0.3-1.24-1.12-0.26-0.17

The observations of this comet from the beginning to the end agree at 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 6h. in the morning at Rome (that is, 5h.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 5h.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 6h.30′ in the morning at Rome (that is, at 5h.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 5h.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 5h. in the morning (that is, at 5h.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 ψ; and the other is the outmost of the wing, Bayer's θ. 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 42½ deg. at 5h. in the morning (that is, at 9h.44′ in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30′ south.

Nov. 19, at 4½h. 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 5h. 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. 38½°, at 5 in the morning (that is, at 10h. at London), saw the comet above Spica ♍, and very nearly joined with it, the distance between them being about ¾ of one deg. And from these observations compared, I conclude, that at 9h.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 6h. in the morning at Venice (that is, 5h.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 7¼h. in the morning, ob served the comet in ♎ 27° 50′, with latitude 1° 16′ south; Cellius, in ♎ 28°; P. Ango at 5h. 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 5h. morning, at Ballasore, in the East Indies (that is, at 11h.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 5h.40′ (that is, at 5h. 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 5h. morning at Ballasore,ihe comet was observed in ♏ 1° 50′; and therefore at 5h. morning at London, the comet was ♏ 3° 5′ nearly. The same day, at 6½h. 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 5h. morning, at Nuremberg (that is, at 4½h. 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 1½°. 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 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22′ 2″.

Nov. 25, before sunrise, Montenari observed the comet in ♏ 17¾′ 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 ♏ 18⅓° 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 15½ 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 ½ 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 δ in the south wing of Aquila, and did terminate near the stars A, ω, b, in Bayer's tables. Therefore the end of the tail was in ♑ 19½°, with latitude about 34¼° north; Dec 11, it ascended to the head of Sagitta (Bayer's α, β), 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 42½° 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, 5h.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 β 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 47½°; 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 π in the breast of Andromeda on its right side, and the star μ 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 κ 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 8½°. 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.

But that the atmospheres of comets may furnish a supply of vapour great enough to fill so immense spaces, we may easily understand from the rarity of our own air; for the air near the surface of our earth possesses a space 850 times greater than water of the same weight; and therefore a cylinder of air 850 feet high is of equal weight with a cylinder of water of the same breadth, and but one foot high. But a cylinder of air reaching to the top of the atmosphere is of equal weight with a cylinder of water about 33 feet high: and, therefore, if from the whole cylinder of air the lower part of 850 feet high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 feet high: and from thence (and by the hypothesis, confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere, and that the force of gravity is reciprocally as the square of the distance from the centre of the earth) raising a calculus, by Cor. Prop. XXII, Book II, I found, that, at the height of one semi-diameter of the earth, reckoned from the earth's surface, the air is more rare than with us in a far greater proportion than of the whole space within the orb of Saturn to a spherical space of one inch in diameter; and therefore if a sphere of our air of but one inch in thickness was equally rarefied with the air at the height of one semi-diameter of the earth from the earth's surface, it would fill all the regions of the planets to the orb of Saturn, and far beyond it. Wherefore since the air at greater distances is immensely rarefied, and the coma or atmosphere of comets is ordinarily about ten times higher, reckoning from their centres, than the surface of the nucleus, and the tails rise yet higher, they must therefore be exceedingly rare; and though, on account of the much thicker atmospheres of comets, and the great gravitation of their bodies towards the sun, as well as of the particles of their air and vapours mutually one towards another, it may happen that the air in the celestial spaces and in the tails of comets is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapour is abundantly sufficient to produce all the appearances of the tails of comets; for that they are, indeed, of a very notable rarity appears from the shining of the stars through them. The atmosphere of the earth, illuminated by the sun's light, though but of a few miles in thickness, quite obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor. Nor is the brightness of the tails of most comets ordinarily greater than that of our air, an inch or two in thickness, reflecting in a darkened room the light of the sun-beams let in by a hole of the window-shutter.

The tails, therefore, that rise in the perihelion positions of the comets will go along with their heads into far remote parts, and together with the heads will either return again from thence to us, after a long course of years, or rather will be there rarefied, and by degrees quite vanish away; for afterwards, in the descent of the heads towards the sun, new short tails will be emitted from the heads with a slow motion; and those tails by degrees will be augmented immensely, especially in such comets as in their perihelion distances descend as low as the sun's atmosphere; for all vapour in those free spaces is in a perpetual state of rarefaction and dilatation; and from hence it is that the tails of all comets are broader at their upper extremity than near their heads. And it is not unlikely but that the vapour, thus perpetually rarefied and dilated, may be at last dissipated and scattered through the whole heavens, and by little and little be attracted towards the planets by its gravity, and mixed with their atmosphere; for as the seas are absolutely necessary to the constitution of our earth, that from them, the sun, by its heat, may exhale a sufficient quantity of vapours, which, being gathered together into clouds, may drop down in rain, for watering of the earth, and for the production and nourishment of vegetables; or, being condensed with cold on the tops of mountains (as some philosophers with reason judge), may run down in springs and rivers; so for the conservation of the seas, and fluids of the planets, comets seem to be required, that, from their exhalations and vapours condensed, the wastes of the planetary fluids spent upon vegetation and putrefaction, and converted into dry earth, may be continually supplied and made up; for all vegetables entirely derive their growths from fluids, and afterwards, in great measure, are turned into dry earth by putrefaction; and a sort of slime is always found to settle at the bottom of putrefied fluids; and hence it is that the bulk of the solid earth is continually increased; and the fluids, if they are not supplied from without, must be in a continual decrease, and quite fail at last. I suspect, moreover, that it is chiefly from the comets that spirit comes, which is indeed the smallest but the most subtle and useful part of our air, and so much required to sustain the life of all things with us.

The atmospheres of comets, in their descent towards the sun, by running out into the tails, are spent and diminished, and become narrower, at least on that side which regards the sun; and in receding from the sun, when they less run out into the tails, they are again enlarged, if Hevelius has justly marked their appearances. But they are seen least of all just after they have been most heated by the sun, and on that account then emit the longest and most resplendent tails; and, perhaps, at the same time, the nuclei are environed with a denser and blacker smoke in the lowermost parts of their atmosphere; for smoke that is raised by a great and intense heat is commonly the denser and blacker. Thus the head of that comet which we have been describing, at equal distances both from the sun and from the earth, appeared darker after it had passed by its perihelion than it did before; for in the month of December it was commonly compared with the stars of the third magnitude, but in November with those of the first or second; and such as saw both appearances have described the first as of another and greater comet than the second. For, November 19, this comet appeared to a young man at Cambridge, though with a pale and dull light, yet equal to Spica Virginis; and at that time it shone with greater brightness than it did afterwards. And Montenari, November 20, st. vet. observed it larger than the stars of the first magnitude, its tail being then 2 degrees long. And Mr. Storer (by letters which have come into my hands) writes, that in the month of December, when the tail appeared of the greatest bulk and splendor, the head was but small, and far less than that which was seen in the month of November before sun-rising; and, conjecturing at the cause of the appearance, he judged it to proceed from there being a greater quantity of matter in the head at first, which was afterwards gradually spent.

We have said, that comets are a sort of planets revolved in very eccentric orbits about the sun; and as, in the planets which are without tails, those are commonly less which are revolved in lesser orbits, and nearer to the sun, so in comets it is probable that those which in their perihelion approach nearer to the sun ate generally of less magnitude, that they may not agitate the sun too much by their attractions. But as to the transverse diameters of their orbits, and the periodic times of their revolutions, I leave them to be determined by comparing comets together which after long intervals of time return again in the same orbit. In the mean time, the following Proposition may give some light in that inquiry.

PROPOSITION XLII. PROBLEM XXII.

To correct a comet's trajectory found as above.

Operation 1. Assume that position of the plane of the trajectory which was determined according to the preceding proposition; and select three places of the comet, deduced from very accurate observations, and at great distances one from the other. Then suppose A to represent the time between the first observation and the second, and B the time between the second and the third; but it will be convenient that in one of those times the comet be in its perigeon, or at least not far from it. From those apparent places find, by trigonometric operations, the three true places of the comet in that assumed plane of the trajectory; then through the places found, and about the centre of the sun as the focus, describe a conic section by arithmetical operations, according to Prop. XXI., Book 1. Let the areas of this figure which are terminated by radii drawn from the sun to the places found be D and E; to wit, D the area between the first observation and the second, and E the area between the second and third; and let T represent the whole time in which the whole area D + E should be described with the velocity of the comet found by Prop. XVI., Book 1.

Oper. 2. Retaining the inclination of the plane of the trajectory to the plane of the ecliptic, let the longitude of the nodes of the plane of the trajectory be increased by the addition of 20 or 30 minutes, which call P. Then from the aforesaid three observed places of the comet let the three true places be found (as before) in this new plane; as also the orbit passing through those places, and the two areas of the same described between the two observations, which call d and e; and let t be the whole time in which the whole area d + e should be described.

Oper. 3. Retaining the longitude of the nodes in the first operation, let the inclination of the plane of the trajectory to the plane of the ecliptic be increased by adding thereto 20′ or 30′, which call Q. Then from the aforesaid three observed apparent places of the comet let the three true places be found in this new plane, as well as the orbit passing through them, and the two areas of the same described between the observation, which call δ and ε; and let τ be the whole time in which the whole area δ + ε should be described.

Then taking C to 1 as A to B; and G to 1 as D to E; and g to 1 as d to e; and γ to 1 as δ to ε; let S be the true time between the first observation and the third; and, observing well the signs + and -, let such numbers m and n be found out as will make 2G - 2C, = mG - mg + nG - nγ; and 2T - 2S = mT - mt + . And if, in the first operation, I represents the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of either node, then I + nQ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K + mP the true longitude of the node. And, lastly, if in the first, second, and third operations, the quantities R, r, and ρ, represent the parameters of the trajectory, and the quantities 1L, 1l, 1λ, the transverse diameters of the same, then R + mr - mR + - nR will be the true parameter, and ${\displaystyle \scriptstyle {\frac {1}{L+ml-mL+n\lambda -nL}}}$ will be the true transverse diameter of the trajectory which the comet describes; and from the transverse diameter given the periodic time of the comet is also given.   Q.E.I.   But the periodic times of the revolutions of comets, and the transverse diameters of their orbits, cannot be accurately enough determined but by comparing comets together which appear at different times. If, after equal intervals of time, several comets are found to have described the same orbit, we may thence conclude that they are all but one and the same comet revolved in the same orbit; and then from the times of their revolutions the transverse diameters of their orbits will be given, and from those diameters the elliptic orbits themselves will be determined.

To this purpose the trajectories of many comets ought to be computed, supposing those trajectories to be parabolic; for such trajectories will always nearly agree with the phænomena, as appears not only from the parabolic trajectory of the comet of the year 1680, which I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13′ 55″; the inclination of the orbit to the plane of the ecliptic 21° 18′ 40″; the distance of its perihelion from the node, estimated in the comet's orbit, 49° 27′ 30″, its perihelion in ♌ 8° 40′ 30″, with heliocentric latitude south 16° 01′ 45″; the comet to have been in its perihelion November 24d.11h.52′ P.M. equal time at London, or 13h.8′ at Dantzick, O. S.; and that the latus rectum of the parabola was 410286 such parts as the sun's mean distance from the earth is supposed to contain 100000. And how nearly the places of the comet computed in this orbit agree with the observations, will appear from the annexed table, calculated by Dr. Halley.

 Appar. Timeat Dantzick. The observed Distances of the Comet from The observed Places. The Placescomputed inthe Orb. December 00.°00.′00″ 00.°00.′00″ 00.°00.′00″ 00.d.00.h.00′ The Lion's heart 46.24.20 Long. ♎ 7.01.00 ♎ 7. 1.29 3.18.29½ The Virgin's spike 22.52.10 Lat. S. 21.39. 0 21.38.50 4.18. 1½ The Lion's heart 46. 2.45 Long. ♎ 6.15. 0 ♎ 6.16. 5 The Virgin's spike 23.52.40 Lat. S. 22.24. 0 22.24. 0 7.17.48 The Lion's heart 44.48. 0 Long. ♎ 3. 6. 0 ♎ 3. 7.33 The Virgin's spike 27.56.40 Lat. S. 25.22. 0 25.21.40 17.14.43 The Lion's heart 53.15.15 Long. ♌ 2.56. 0 ♌ 2.56. 0 Orion's right shoulder 45.43.30 Lat. S. 49.25. 0 49.25. 0 19. 9.25 Procyon 35.13.50 Long. ♊ 28.40.30 ♊ 28.43. 0 Bright star of Whale's jaw 52.56. 0 Lat. S. 45.48. 0 45.46. 0 20. 9.53½ Procyon 40.49. 0 Long. ♊ 13.03. 0 ♊ 13. 5. 0 Bright star of Whale's jaw 40.04. 0 Lat. S. 39.54. 0 39.53. 0 21. 9. 9½ Orion's right shoulder 26.21.25 Long. ♊ 2.16. 0 ♊ 2.18.30 Bright star of Whale's jaw 29.28. 0 Lat. S. 33.41. 0 33.39.40 22. 9. 0 Orion's right shoulder 29.47. 0 Long. ♉ 24.24. 0 ♉ 24.27. 0 Bright star of Whale's jaw 20.29.30 Lat. S. 27.45. 0 27.46. 0 26. 7.58 The bright star of Aries 23.20. 0 Long. ♉ 9. 0. 0 ♉ 9. 2.28 Aldebaran 26.44. 0 Lat. S. 12.36. 0 12.34.13 27. 6.45 The bright star of Aries 20.45. 0 Long. ♉ 7. 5.40 ♉ 7. 8.45 Aldebaran 28.10. 0 Lat. S. 10.23. 0 10.23.13 28. 7.39 The bright star of Aries 18.29. 0 Long. ♉ 5.24.45 ♉ 5.27.52 Palilicium 29.37. 0 Lat. S. 8.22.50 8.23.37 31. 6.45 Andromeda's girdle 30.48.10 Long. ♉ 2. 7.40 ♉ 2. 8.20 Palilicium 32.53.30 Lat. S. 4.13. 0 4.16.25 Jan. 1665 Andromeda's girdle 25.11. 0 Long. ♈ 28.24.47 ♈ 28.24. 0 7. 7.37½ Palilicium 37.12.25 Lat. N. 0.54. 0 0.53. 0 13. 7. 0 Andromeda's head 28. 7.10 Long. ♈ 27. 6.54 ♈ 27. 6.39 Palilicium 38.55.20 Lat. N. 3. 6.50 3. 7.40 24. 7.29 Andromeda's girdle 20.32.15 Long. ♈ 26.29.15 ♈ 26.28.50 Palilicium 40. 5. 0 Lat. N. 5.25.50 5.26. 0 Feb. Long. ♈ 27. 4.46 ♈ 27.24.55 7. 8.37 Lat. N. 7. 3.29 7. 3.15 22. 8.46 Long. ♈ 28.29.46 ♈ 28.29.58 Lat. N. 8.12.36 8.10.25 March Long. ♈ 29.18.15 ♈ 29.18.20 1. 8.16 Lat. N. 8.36.26 8.36.12 7. 8.37 Long. ♉ 0. 2.48 ♉ 0. 2.42 Lat. N. 8.56.30 8.56.56

In February, the beginning of the year 1665, the first star of Aries, which I shall hereafter call γ, was in ♈ 28° 30′ 15″, with 7° 8′ 58″ north lat.; the second star of Aries was in ♈ 29° 17′ 18″, with 8° 28′ 16″ north lat.; and another star of the seventh magnitude, which I call A, was in ♈ 28° 24′ 45″, with 8° 28′ 33″ north lat. The comet Feb. 7d.7h.30′ at Paris (that is, Feb. 7d.8h.30′ at Dantzick) O. S. made a triangle with those stars γ and A, which was right-angled in γ; and the distance of the comet from the star γ was equal to the distance of the stars γ and A, that is, 1° 19′ 46″ of a great circle; and therefore in the parallel of the latitude of the star γ it was 1° 20′ 26″. Therefore if from the longitude of the star γ there be subducted the longitude 1° 20′ 26″, there will remain the longitude of the comet ♈ 27° 9′ 49″. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0′, nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59′ 24″. I place it in ♈ 27° 4′ 46″, taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4′ or 5′ to the north; but he had done better to have made it 7° 3′ 29″, the difference of the latitudes of the comet and the star γ being equal to the difference of the longitude of the stars γ and A.

February 22d.7h.30′ at London, that is, February 22d. 8h.46′ at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15′ 57″; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4′; and therefore the comet was in ♈ 28° 29′ 46″, with 8° 12′ 36″ north lat.

March 1, 7h at London, that is, March 1, 8h.16′ at Dantzick. the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1′. Wherefore if to the longitude of the first star in Aries we add 1′, and 8′ 10″ to its latitude, we shall have the longitude of the comet ♈ 29° 18′, with 8° 36′ 26″ north lat.

March 7, 7h.30′ at Paris (that is, March 7, 8h.37′ at Dantzick), from the observations of M. Auzout, the distance of the comet from the second star in Aries was equal to the distance of that star from the star A, that is, 52,′ 29″; and the difference of the longitude of the comet and the second star in Aries was 45′ or 46′, or, taking a mean quantity, 45′ 30″; and therefore the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the latitude of the comet 8° 55′ 30″. And, by farther correcting this irregularity, the latitude may become 8° 56, or 8° 57′.

This comet was also seen March 9, and at that time its place must have been in ♉ 0° 18′, with 9° 3½' north lat. nearly.

This comet appeared three months together, in which space of time it travelled over almost six signs, and in one of the days thereof described almost 20 deg. Its course did very much deviate from a great circle, bending towards the north, and its motion towards the end from retrograde became direct; and, notwithstanding its course was so uncommon, yet by the table it appears that the theory, from beginning to end, agrees with the observations no less accurately than the theories of the planets usually do with the observations of them: but we are to subduct about 2′ when the comet was swiftest, which we may effect by taking off 12″ from the angle between the ascending node and the perihelion, or by making that angle 49° 27′ 18″. The annual parallax of both these comets (this and the preceding) was very conspicuous, and by its quantity demonstrates the annual motion of the earth in the orbis magnus.

This theory is likewise confirmed by the motion of that comet, which in the year 1683 appeared retrograde, in an orbit whose plane contained almost a right angle with the plane of the ecliptic, and whose ascending node (by the computation of Dr. Halley) was in ♍ 23° 23′; the inclination of its orbit to the ecliptic 83° 11′; its perihelion in ♊ 25° 29′ 30″; its perihelion distance from the sun 56020 of such parts as the radius of the orbis magnus contains 100000; and the time of its perihelion July 2d.3h.50′. And the places thereof, computed by Dr. Halley in this orbit, are compared with the places of the same observed by Mr. Flamsted, in the following table:—

 1683Eq. time. Sun's place Comet'sLong. com. Lat. Nor.comput. Comet'sLong. obs'd Lat.Nor.observ'd Diff.Long. Diff.Lat. d.    h.    ′July 13.12.5515.11.1517.10.2023.13.4025.14.531.9.4231.14.55Aug. 2.14.564.10.496.10.99.10.2615.14.116.15.1018.15.4422.14.4423.15.5226.16. 2 °    ′    ″♌ 1.02.302.53.124.45.4510.38.2112.35.2818.09.2218.21.5320.17.1622.02.5023.56.4526.50.52♍ 2.47.133.48. 25.45.339.35.4910.36.4813.31.10 °    ′    ″♋ 13.05.4211.37.4810. 7. 65.10.273.27.53♊ 27.55. 327.41. 725.29.3223.18.2020.42.2316 7.573.30.480.43. 7♉ 24.52.5311. 7.147. 2.18♈ 24.45.31 °    ′    ″29.28.1329.34. 029.33.3028.51.4224.24.4726.22.5226.16.5725.16.1924.10.4922.17. 520. 6.3711.37.339.34.165.11.15South.5.16.588.17. 916.38. 0 °    ′    ″♋ 13. 6.4211.39.4310. 8.405.11.303.27. 0♊ 27.54.2427.41. 825.28.4623.16.5520.40.3216. 5.553.26.180.41.55♉ 24.49. 511.07.127. 1.17♈ 24.44.00 °    ′    ″29.28.2029.34.5029.34. 028.50.2828.23.4026.22.2526.14.5025.17.2824.12.1922.49. 520. 6.1011.32. 19.34.135. 9.11South5.16.588.16.4116.38.20 ′    ″+ 1.00+ 1.55+ 1.34+ 1.03- 0.53- 0.39+ 0. 1- 0.46- 1.25- 1.51- 2. 2- 4.30- 1.12- 3.48- 0. 2- 1. 1- 1.31 ′    ″+ 0.07+ 0.50+ 0.30- 1.14-1. 7- 0.27- 2. 7+ 1. 9+ 1.30+ 2. 0- 0.27- 5.32- 0. 3- 2. 4-0. 3- 0.28+ 0.20

This theory is yet farther confirmed by the motion of that retrograde comet which appeared in the year 1682. The ascending node of this (by Dr. Halley's computation) was in ♉ 21° 16′ 30″; the inclination of its orbit to the plane of the ecliptic 17° 56′ 00″; its perihelion in ♒ 2° 52′ 50″; its perihelion distance from the sun 58328 parts, of which the radius of the orbis magnus contains 100000; the equal time of the comet's being in its perihelion Sept. 4d.7h.39′. And its places, collected from Mr. Flamsted's observations, are compared with its places computed from our theory in the following table:—

 1682App. Time. Sun's place Comet'sLong. comp. Lat. Nor.comp. Com. Long.observed. Lat.Nor.observ. Diff.Long. Diff.Lat. d.    h.    ′Aug. 19.16.3820.15.3821. 8.2122. 8. 829.08.2030. 7.45Sept. 1. 7.334. 7.225. 7.328. 7.169. 7.26 °    ′    ″♍ 7. 0. 77.55 528.36.149.33.5516.22.4017.19.4119.16. 922.11.2823.10.2926. 5.5827. 5. 9 °    ′    ″♌ 18.14 2824.46.2329.37.15♍ 6.29.53♎ 12.37.5415 36. 120.30.5325.42. 027. 0.4629.58.44♏ 0.44.10 °    ′    ″25.50. 726.14.4226.20. 326. 8.4218.37.4717.26.4315.13. 012.23.4811.33.089.26.468.49.10 °    ′    ″♌ 18.14.4024.46.2229.38.02♍ 6.30. 3♎ 12.37.4915.35.1820.27. 425.40.5826.59.2429.58.45♏ 0.44. 4 °    ′    ″25.49.5526.12.5226.17.3726. 7.1218.34. 517.27.1715. 9.4912.22. 011.33.519.26.438.48.25 ′    ″- 0.12+ 0. 1- 0.47- 0.10+ 0. 5+ 0.43+ 3.49+ 1. 2+ 1.22- 0.1+ 0. 6 ′    ″+ 0.12+ 1.50+ 2.26+ 1.30+ 3.42- 0.34+ 3.11+ 1.48- 0.43+ 0. 3+ 0.45

This theory is also confirmed by the retrograde motion of the comet that appeared in the year 1723. The ascending node of this comet (according to the computation of Mr. Bradley, Savilian Professor of Astronomy at Oxford) was in ♈ 14° 16′. The inclination of the orbit to the plane of the ecliptic 49° 59′. Its perihelion was in ♉ 12° 15′ 20″. Its perihelion distance from the sun 998651 parts, of which the radius of the orbis magnus contains 1000000, and the equal time of its perihelion September 16d 16h.10′. The places of this comet computed in this orbit by Mr. Bradley, and compared with the places observed by himself, his uncle Mr. Pound, and Dr. Halley, may be seen in the following table.

 1723Eq. Time. Comet'sLong. obs. Lat. Nor.obs. Comet'sLon. com. Lat.Nor.comp. Diff.Lon. Diff.Lat. d.    h.    ′Oct. 9.8. 510.6.2112.7.2214.8.5715.6.3521.6.2222. 6.2424.8. 229.8.5630.6.20Nov. 5.5.538.7. 614.6.2020.7.45Dec. 7.6.45 °    ′    ″♒ 7.22.156.41.125.39.584.59.494.47.414. 2.323.59. 23.55.293.56.173.58. 94.16.304.29.365. 2.165.42.208. 4.13 °    ′    ″5. 2. 07.44.1311.55. 014.43.5015.40.5119.41.4920. 8.1220.55.1822.20.2722.32.2823.38 3324. 4.3024.48.4625.24.4526.54.18 °    ′    ″♒ 7.21.266.41.425.40.195. 0.374.47.454. 2.213.59.103.55.113.56.423.58.174.16.234.29.545. 2.515.43.138. 3.55 °    ′    ″5. 2 477.43.1811.54.5514.44. 115.40.5519.42. 320. 8.1720.55. 922.20.1022.32.1223.38. 724. 4.4024.48.1625.25.1726.53.42 ″+ 49- 50- 21- 48- 4+ 11- 8+ 18- 25- 8+ 7- 18- 35- 53+ 18 ″- 47+ 55+ 5- 11- 4- 14- 5+ 9+ 17+ 16+ 26- 10+ 30- 32+ 36

From these examples it is abundantly evident that the motions of comets are no less accurately represented by our theory than the motions of the planets commonly are by the theories of them; and, therefore, by means of this theory, we may enumerate the orbits of comets, and so discover the periodic time of a comet's revolution in any orbit; whence, at last, we shall have the transverse diameters of their elliptic orbits and their aphelion distances.

That retrograde comet which appeared in the year 1607 described an orbit whose ascending node (according to Dr. Halley's computation) was in ♉ 20° 21′; and the inclination of the plane of the orbit to the plane of the ecliptic 17° 2′; whose perihelion was in ♒ 2° 16′; and its perihelion distance from the sun 58680 of such parts as the radius of the orbis magnus contains 100000; and the comet was in its perihelion October 16d.3h.50′; which orbit agrees very nearly with the orbit of the comet which was seen in 1682. If these were not two different comets, but one and the same, that comet will finish one revolution in the space of 75 years; and the greater axis of its orbit will be to the greater axis of the orbis magnus as ${\displaystyle \scriptstyle {\sqrt {^{3}:75\times 75}}}$ to 1, or as 1778 to 100, nearly. And the aphelion distance of this comet from the sun will be to the mean distance of the earth from the sun as about 35 to 1; from which data it will be no hard matter to determine the elliptic orbit of this comet. But these things are to be supposed on condition, that, after the space of 75 years, the same comet shall return again in the same orbit. The other comets seem to ascend to greater heights, and to require a longer time to perform their revolutions.

But, because of the great number of comets, of the great distance of their aphelions from the sun, and of the slowness of their motions in the aphelions, they will, by their mutual gravitations, disturb each other; so that their eccentricities and the times of their revolutions will be sometimes a little increased, and sometimes diminished. Therefore we are not to expect that the same comet will return exactly in the same orbit, and in the same periodic times: it will be sufficient if we find the changes no greater than may arise from the causes just spoken of.

And hence a reason may be assigned why comets are not comprehended within the limits of a zodiac, as the planets are; but, being confined to no bounds, are with various motions dispersed all over the heavens; namely, to this purpose, that in their aphelions, where their motions are exceedingly slow, receding to greater distances one from another, they may suffer less disturbance from their mutual gravitations: and hence it is that the comets which descend the lowest, and therefore move the slowest in their aphelions, ought also to ascend the highest.

The comet which appeared in the year 1680 was in its perihelion less distant from the sun than by a sixth part of the sun's diameter; and because of its extreme velocity in that proximity to the sun, and some density of the sun's atmosphere, it must have suffered some resistance and retardation; and therefore, being attracted something nearer to the sun in every revolution, will at last fall down upon the body of the sun. Nay, in its aphelion, where it moves the slowest, it may sometimes happen to be yet farther retarded by the attractions of other comets, and in consequence of this retardation descend to the sun. So fixed stars, that have been gradually wasted by the light and vapours emitted from them for a long time, may be recruited by comets that fall upon them; and from this fresh supply of new fuel those old stars, acquiring new splendor, may pass for new stars. Of this kind are such fixed stars as appear on a sudden, and shine with a wonderful brightness at first, and afterwards vanish by little and little. Such was that star which appeared in Cassiopeia's chair; which Cornelius Gemma did not see upon the 8th of November, 1572, though he was observing that part of the heavens upon that very night, and the sky was perfectly serene; but the next night (November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor. Tycho Brahe saw it upon the 11th of the same month, when it shone with the greatest lustre; and from that time he observed it to decay by little and little; and in 16 months' time it entirely disappeared. In the month of November, when it first appeared, its light was equal to that of Venus. In the month of December its light was a little diminished, and was now become equal to that of Jupiter. In January 1573 it was less than Jupiter, and greater than Sirius; and about the end of February and the beginning of March became equal to that star. In the months of April and May it was equal to a star of the second magnitude; in June, July, and August, to a star of the third magnitude; in September, October, and November, to those of the fourth magnitude; in December and January 1574 to those of the fifth; in February to those of the sixth magnitude; and in March it entirely vanished. Its colour at the beginning was clear, bright, and inclining to white; afterwards it turned a little yellow; and in March 1573 it became ruddy, like Mars or Aldebaran: in May it turned to a kind of dusky whiteness, like that we observe in Saturn; and that colour it retained ever after, but growing always more and more obscure. Such also was the star in the right foot of Serpentarius, which Kepler's scholars first observed September 30, O.S. 1604, with a light exceeding that of Jupiter, though the night before it was not to be seen; and from that time it decreased by little and little, and in 15 or 16 months entirely disappeared. Such a new star appearing with an unusual splendor is said to have moved Hipparchus to observe, and make a catalogue of, the fixed stars. As to those fixed stars that appear and disappear by turns, and increase slowly and by degrees, and scarcely ever exceed the stars of the third magnitude, they seem to be of another kind, which revolve about their axes, and, having a light and a dark side, shew those two different sides by turns. The vapours which arise from the sun, the fixed stars, and the tails of the comets, may meet at last with, and fall into, the atmospheres of the planets by their gravity, and there be condensed and turned into water and humid spirits; and from thence, by a slow heat, pass gradually into the form of salts, and sulphurs, and tinctures, and mud, and clay, and sand, and stones, and coral, and other terrestrial substances.