Proposition 39, Problem 20
To find the precession of the equinoxes.

The mean hourly motion of the nodes of the moon in a circular orbit was, for the nodes in the quadratures, 16″35‴16^iv36^v, and half of this, 8″17‴38^iv18^v, is (for the reasons explained above [at the end of corol. 2 to prop. 30]) the mean hourly motion of the nodes in such an orbit; and in a whole sidereal year the mean motion adds up to 20°11′46″ [see beginning of prop. 32]. Therefore, since in a year the nodes of the moon would, in such an orbit, move backward [or in antecedentia] through 20°11′46″; and since, if there were more moons, the motion of the nodes of each (by book 1, prop. 66, corol. 16) would be as the periodic times; it follows that if the moon revolved near the surface of the earth in the space of a sidereal day, the annual motion of the nodes would be to 20°11′46″ as a sidereal day of 23^h56^m is to the periodic time of the moon, 27^d7^h43^m—that is, as 1,436 to 39,343. And the same is true of the nodes of a ring of moons surrounding the earth, whether those moons do not touch one another, or whether they become liquid and take the form of a continuous ring, or finally whether that ring becomes rigid and inflexible.

Let us imagine therefore that this ring, as to its quantity of matter, is equal to all of the earth PapAPepE that lies outside of the globe Papea (as in the figure to lem. 2). This globe is to the earth that lies outside of it as aC² to AC² − aC², that is (since the earth’s smaller semidiameter PC or aC is to its greater semidiameter AC as 229 to 230), as 52,441 to 459. Hence, if this ring girded the earth along the equator and both together revolved about the diameter of the ring, the motion of the ring would be to the motion of the interior globe (by lem. 3 of this third book) as 459 to 52,441 and 1,000,000 to 925,275 jointly, that is, as 4,590 to 485,223; and so the motion of the ring would be to the sum of the motions of the ring and globe as 4,590 to 489,813. Hence, if the ring adheres to the globe and communicates to the globe its own motion with which its nodes or equinoctial points regress, the motion that will remain in the ring will be to its former motion as 4,590 to 489,813, and therefore the motion of the equinoctial points will be diminished in the same ratio. Therefore the annual motion of the equinoctial points of a body composed of the ring and the globe will be to the motion 20°11′46″ as 1,436 to 39,343 and 4,590 to 489,813 jointly, that is, as 100 to 292,369. But the forces by which the nodes of the moons [i.e., a ring of moons] regress (as I have explained above), and so by which the equinoctial points of the ring regress (that is, the forces 3IT in the figure to prop. 30), are—in the individual particles—as the distances of those particles from the plane QR, and it is with these forces that the particles recede from the plane; and therefore (by lem. 2), if the matter of the ring were scattered over the whole surface of the globe, as in the configuration PapAPepE, so as to constitute that exterior part of the earth, the total force and efficacy of all the particles to rotate the earth about any diameter of the equator, and thus to move the equinoctial points, would come out less than before in the ratio of 2 to 5. And hence the annual regression of the equinoxes would now be to 20°11′46″ as 10 to 73,092, and accordingly would become 9″56‴50^iv.

^bBut because of the inclination of the plane of the equator to the plane of the ecliptic, this motion must be diminished in the ratio of the sine 91,706 (which is the sine of the complement of 23½ degrees [or the cosine of 23½ degrees]) to the radius 100,000. And thus this motion will now become 9″7‴20^iv. This is the annual precession of the equinoxes that arises from the force of the sun.

Now, the force of the moon to move the sea was to the force of the sun as roughly 4.4815 to 1. And the force of the moon to move the equinoxes is to the force of the sun in this same proportion. And so the annual precession of the equinoxes that arises from the force of the moon comes out 40″52‴52^iv, and the total annual precession arising from both forces will be 50″00‴12^iv. And this motion of precession agrees with the phenomena. For, from astronomical observations, the precession of the equinoxes is more or less 50 seconds annually.

If the height of the earth at the equator exceeds its height at the poles by more than 17⅙ miles, its matter will be rarer at the circumference than at the center; and the precession of the equinoxes has to be increased because of that excess in height, and diminished because of the greater rarity.^b

We have now described the system of the sun, the earth, the moon, and the planets; something must still be said about comets.

Lemma 4
The comets are higher than the moon and move in the planetary regions.

Just as the lack of diurnal parallax requires that comets be located beyond the sublunar regions, so the fact that comets have an annual parallax is convincing evidence that they descend into the regions of the planets. For comets which move forward according to the order of the signs are all, toward the end of their visibility, either slower than normal or retrograde if the earth is between them and the sun, but swifter than they should be if the earth is approaching opposition. And conversely those comets that move contrary to the order of the signs are swifter than they should be, at the end of their visibility when the earth is between them and the sun, and slower than they should be or retrograde if the earth is on the opposite side of the sun. This happens principally as a result of the motion of the earth in its different positions [with respect to the comets], just as is the case for the planets, which, according as the motion of the earth is either in the same direction or in an opposite one, are sometimes retrograde, and sometimes seem to advance more slowly and at other times more swiftly. If the earth goes in the same direction as the comet and by its angular motion is carried about the sun so much more swiftly that a straight line continually drawn through the earth and the comet converges toward the regions beyond the comet, then the comet as seen from the earth will appear to be retrograde because of its slower motion; but if the earth is going more slowly, the motion of the comet (taking away the motion of the earth) becomes at least slower. But if the earth goes in a direction opposite to the comet’s motion, the motion of the comet will as a result appear speeded up. And from the acceleration or retardation or retrograde motion, the distance of the comet may be ascertained in the following way.

Let QA, QB, and QC be three observed longitudes of a comet at the beginning of its [visible] motion, and let QF be its last observed longitude, just as the comet ceases to be seen. Draw the straight line ABC, whose parts AB and BC placed between the straight lines QA and QB, and between the straight lines QB and QC, are to each other as the times between the first three observations. Let AC be produced to G, so that AG is to AB as the time between the first and the last observation is to the time between the first and the second observation, and let QG be joined. Then, if the comet moved uniformly in a straight line and the earth were either at rest or also moved forward in a straight line with uniform motion, the angle QG would be the longitude of the comet at the time of the last observation. Therefore, the angle FQG, which is the difference between the longitudes, arises from the inequality of the motions of the comet and of the earth. And this angle, if the earth and comet move in opposite directions, is added to the angle QG, and thus makes the apparent motion of the comet swifter; but if the comet is going in the same direction as the earth, this angle is subtracted from that same angle QG and makes the motion of the comet either slower or possibly retrograde, as I have just explained. Therefore this angle arises chiefly from the motion of the earth and on that account is rightly regarded as the parallax of the comet, ignoring, of course, any increase or decrease in it which could arise from the nonuniform motion of the comet in its own orbit. And the distance of the comet may be ascertained from this parallax in the following manner.

Let S represent the sun, acT the earth’s orbit, a the place of the earth 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 let T be a straight line drawn toward the beginning of Aries. Let angle TV be taken equal to angle QF, that is, equal to the longitude of the comet when the earth is in T. Let ac be drawn and produced to g, so that ag is to ac as AG to AC; then g will be the place which the earth would reach at the time of the last observation, with its motion uniformly continued in the straight line ac. And so if g is drawn parallel to T and the angle gV is taken equal to the angle QG, this angle gV will be equal to the longitude of the comet as seen from place g, and the angle TVg will be the parallax that arises from the transfer of the earth from place g to place T; and accordingly V will be the place of the comet in the plane of the ecliptic. And this place V is ordinarily lower than the orbit of Jupiter.

The same may be ascertained from the curvature of the path of comets. These bodies go almost in great circles as long as they move more swiftly, but at the end of their course, when that part of their apparent motion which arises from parallax has a greater proportion to the total apparent motion, they tend to deviate from such circles, and whenever the earth moves in one direction, they tend to go off in the opposite direction. Because this deviation corresponds to the motion of the earth, it arises chiefly from parallax, and its extraordinary quantity, according to my computation, has placed disappearing comets quite far below Jupiter. Hence it follows that when comets are closer to us, in their perigees and perihelions, they very often descend below the orbits of Mars and of the inferior planets.

The nearness of comets is confirmed also from the light of their heads. For the brightness of a heavenly body illuminated by the sun and going off into distant regions is diminished as the fourth power of the distance; that is, it is diminished as the square because of the increased distance of the body from the sun and diminished as the square again because of the diminished apparent diameter. Thus, if both the quantity of light [i.e., brightness] and the apparent diameter of the comet are given, its distance will be found by taking its distance to the distance of some planet directly in the ratio of diameter to diameter and inversely as the square root of the ratio of light to light. Thus, as observed by Flamsteed through a sixteen-foot telescope and measured with a micrometer, the least diameter of the comaa of the comet of the year 1682 equaled 2′0″, while the nucleus or star in the middle of the head occupied scarcely a tenth of this width and therefore was only 11″ or 12″ wide. But in the light and brilliance of its head it surpassed the head of the comet of the year 1680 and rivaled stars of the first or second magnitude. Let us suppose that Saturn with its ring was about four times brighter; then, because the light of the ring almost equals the light of the globe within it, and the apparent diameter of the globe is about 21″, so that the light of the globe and the ring together would equal the light of a globe whose diameter was 30″, it follows that the distance of the comet will be to the distance of Saturn as 1 to √4 inversely and 12″ to 30″ directly, that is, as 24 to 30 or as 4 to 5. Again, on the authority of Hevelius, the comet of April 1665 surpassed in its brilliance almost all the fixed stars, and even Saturn itself (that is, by reason of its far more vivid color). Indeed, this comet was brighter than the one which had appeared at the end of the preceding year and was comparable to stars of the first magnitude. The width of the comet’s coma was about 6′, but the nucleus, when compared with the planets by the aid of a telescope, was clearly smaller than Jupiter and was judged to be sometimes smaller than the central body of Saturn and sometimes equal to it. Further, since the diameter of the coma of comets rarely exceeds 8′ or 12′, and the diameter of the nucleus or central star is about a tenth or perhaps a fifteenth of the diameter of the coma, it is evident that such stars generally have the same apparent magnitude as the planets. Hence, since their light can often be compared to the light of Saturn and sometimes surpasses it, it is manifest that all the comets in their perihelions should be placed either below Saturn or not far above. Those who banish the comets almost to the region of the fixed stars are, therefore, entirely wrong; certainly in such a situation, they would not be illuminated by our sun any more than the planets in our solar system are illuminated by the fixed stars.

In treating these matters, we have not been considering the obscuring of comets by that very copious and thick smoke by which the head is surrounded, always gleaming dully as if through a cloud. For the darker the body is rendered by this smoke, the closer it must approach to the sun for the amount of light reflected from it to rival that of the planets. This makes it likely that the comets descend far below the sphere of Saturn, as we have proved from their parallax.

But this same result is, to the highest degree, confirmed from their tails. These arise either from reflection by the smoke scattered through the aether or from the light of the head. In the first case the distance of the comets must be diminished, since otherwise the smoke always arising from the head would be propagated through spaces far too great, with such a velocity and expansion as to be unbelievable. In the second case, all the light of both the tail and the coma must be ascribed to the nucleus of the head. Therefore, if we suppose that all this light is united and condensed within the disc of the nucleus, then certainly that nucleus, whenever it emits a very large and very bright tail, will far surpass in its brilliance even Jupiter itself. Therefore, if it has a smaller apparent diameter and is sending forth more light, it will be much more illuminated by the sun and thus will be much closer to the sun. By the same argument, furthermore, the heads ought to be located below the orbit of Venus, when they are hidden under the sun and emit tails both very great and very bright like fiery beams, as they do sometimes. For if all of that light were understood to be gathered together into a single star, it would sometimes surpass Venus itself, not to say several Venuses combined.

Finally, the same thing may be ascertained from the light of the heads, which increases as comets recede from the earth toward the sun and decreases as they recede from the sun toward the earth. Thus the latter comet of 1665 (according to the observations of Hevelius), from the time when it began to be seen, was always decreasing in its apparent motion and therefore had already passed its perigee; but the splendor of its head nevertheless increased from day to day until the comet, concealed by the sun’s rays, ceased to be visible. The comet of 1683 (also according to the observations of Hevelius) at the end of July, when it was first sighted, was moving very slowly, advancing about 40′ or 45′ in its orbit each day. From that time its daily motion kept increasing continually until 4 September, when it came to about 5°. Therefore, in all this time the comet was approaching the earth. This is gathered also from the diameter of the head, as measured with a micrometer, since Hevelius found it to be on 6 August only 6′5″ including the coma, but on 2 September 9′7″. Therefore the head appeared far smaller at the beginning than at the end of the motion; yet at the beginning the head showed itself far brighter in the vicinity of the sun than toward the end of its motion, as Hevelius also reports. Accordingly, in all this time, because of its receding from the sun, it decreased with respect to its light, notwithstanding its approach to the earth.

The comet of 1618, about the middle of December, and that of 1680, about the end of the same month, were moving very swiftly and therefore were then in their perigees. Yet the greatest splendor of their heads occurred about two weeks earlier, when they had just emerged from the sun’s rays, and the greatest splendor of their tails occurred a little before that, when they were even nearer to the sun. The head of the first of these comets, according to the observations of [Johann Baptist] Cysat, seemed on 1 December to be greater than stars of the first magnitude, and on 16 December (being now in its perigee) it had failed little in magnitude, but very much in the splendor or clarity of its light. On 7 January Kepler, being uncertain about its head, brought his observing to an end. On 12 December the head of the second of these comets was sighted, and was observed by Flamsteed at a distance of 9° from the sun, a thing which would scarcely have been possible in a star of the third magnitude. On 15 and 17 December it appeared as a star of the third magnitude, since it was diminished by the brightness of clouds near the setting sun. On 26 December, moving with the greatest speed and being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. On 3 January it appeared as a star of the fourth magnitude, on 9 January as a star of the fifth magnitude, and on 13 January it disappeared from view, as a result of the splendor of the crescent moon. On 25 January it scarcely equaled stars of the seventh magnitude. If equal times are taken on both sides of the perigee (before and after), then the head, being placed at those times in distant regions, ought to have shone with equal brilliance because of its equal distances from the earth, but it appeared brightest in the region [on the side of the perigee] toward the sun and disappeared on the other side of the perigee. Therefore from the great difference of light in these two situations, it is concluded that there is a great nearness of the sun and the comet in the first of these situations. For the light of comets tends to be regular and be greatest when the heads move most swiftly, and accordingly are in their perigees, except insofar as this light becomes greater in the vicinity of the sun.

COROLLARY 1. Therefore comets shine by the sun’s light reflected from them.

COROLLARY 2. From what has been said it will also be understood why comets appear so frequently in the region of the sun. If they were visible in the regions far beyond Saturn, they would have to appear more often in the parts of the sky opposite to the sun. For those that were in these parts would be nearer to the earth; and the sun, being in between, would obscure the others. Yet in running through the histories of comets, I have found that four or five times more have been detected in the hemisphere toward the sun than in the opposite hemisphere, besides without doubt not a few others which the sun’s light hid from view. Certainly, in their descent to our regions comets neither emit tails nor are so brightly illuminated by the sun that they show themselves to the naked eye so as to be discovered before they are closer to us than Jupiter itself. But by far the greater part of the space described about the sun with so small a radius is situated on the side of the earth that faces the sun, and comets are generally more brightly illuminated in that greater part, since they are much closer to the sun.

COROLLARY 3. Hence also it is manifest that the heavens are lacking in resistance. For the comets, following paths that are oblique and sometimes contrary to the course of the planets, move in all directions very freely and preserve their motions for a very long time even when these are contrary to the course of the planets. Unless I am mistaken, comets are a kind of planet and revolve in their orbits with a continual motion. For there seems to be no foundation for the allegation of some writers, basing their argument on the continual changes of the heads, that comets are meteors. The heads of comets are encompassed with huge atmospheres, and the atmospheres must be denser as one goes lower. Therefore, it is in these clouds, and not in the very bodies of the comets, that those changes are seen. Thus, if the earth were viewed from the planets, it would doubtless shine with the light of its own clouds, and its solid body would be almost hidden beneath the clouds. Thus, the belts of Jupiter are formed in the clouds of that planet, since they change their situation relative to one another, and the solid body of Jupiter is seen with greater difficulty through those clouds. And the bodies of comets must be much more hidden beneath their atmospheres, which are both deeper and thicker.

Proposition 40, Theorem 20
Comets move in conics having their foci in the center of the sun, and by radii drawn to the sun, they describe areas proportional to the times.

This is evident by corol. 1 to prop. 13 of the first book compared with props. 8, 12, and 13 of the third book.

COROLLARY 1. Hence, if comets revolve in orbits, these orbits will be ellipses, and the periodic times will be to the periodic times of the planets as the ³/2 powers of their principal axes. And therefore comets, for the most part being beyond the planets and on that account describing orbits with greater axes, will revolve more slowly. For example, if the axis of the orbit of a comet were four times greater than the axis of the orbit of Saturn, the time of a revolution of the comet would be to the time of a revolution of Saturn (that is, to 30 years) as 4√4 (or 8) to 1, and accordingly would be 240 years.

COROLLARY 2. But these orbits will be so close to parabolas that parabolas can be substituted for them without sensible errors.

COROLLARY 3. And therefore (by book 1, prop. 16, corol. 7) the velocity of every comet will always be to the velocity of any planet, [considered to be] revolving in a circle about the sun, very nearly as the square root of twice the distance of the planet from the center of the sun to the distance of the comet from the center of the sun. Let us take the radius of the earth’s orbit (or the greatest semidiameter of the ellipse in which the earth revolves) to be of 100,000,000 parts; then the earth will describe by its mean daily motion 1,720,212 of these parts, and by its hourly motion 71,675½ parts. And therefore the comet, at the same mean distance of the earth from the sun, and having a velocity that is to the velocity of the earth as √2 to 1, will describe by its daily motion 2,432,747 of these parts, and by its hourly motion 101,364½ parts. But at greater or smaller distances, both the daily and the hourly motion will be to this daily and hourly motion as the square root of the ratio of the distances inversely, and therefore is given.

COROLLARY 4. Hence, if the latus rectum of a parabola is four times greater than the radius of the earth’s orbit, and if the square of that radius is taken to be 100,000,000 parts, the area that the comet describes each day by a radius drawn to the sun will be l,216,373½ parts, and in each hour that area will be 50,682¼ parts. But if the latus rectum is greater or smaller in any ratio, then the daily and hourly area will be greater or smaller, as the square root of that ratio.

Lemma 5
To find a parabolic curve that will pass through any number of given points.

Let the points be A, B, C, D, E, F, . . . , and from them to any straight line HN, given in position, drop the perpendiculars AH, BI, CK, DL, EM, FN, . . . .

CASE 1. If the intervals HI, IK, KL, . . . between the points H, I, K, L, M, N are equal, take the first differences b, b₂, b₃, b₄, b₅, . . . of the perpendiculars AH, BI, CK, . . . ; the second differences c, c₂, c₃, c₄, . . . ; the third differences d, d₂, d₃, . . . ; that is, in such a way that AH − BI = b, BI − CK = b₂, CK − DL = b₃, DL + EM = b₄ −EM + FN = b₅, . . . , then b − b₂ = c, . . . , and go on in this way to the last difference, which here is f. Then, if any perpendicular RS is erected, which is to be an ordinate to the required curve, in order to find its length, suppose each of the intervals HI, IK, KL, LM, . . . to be unity, and let AH be equal to a, −HS = p, ½p × (−IS) = q, ⅓q × (+SK) = r, ¼r × (+SL) = s, ⅕s × (+SM) = t, proceeding, that is, up to the penultimate perpendicular ME, and prefixing negative signs to the terms HS, IS, . . . , which lie on the same side of the point S as A, and positive signs to the terms SK, SL, . . . , which lie on the other side of the point S. Then if the signs are observed exactly, RS will be = a + bp + cq + dr + es + ft + . . . .

CASE 2. But if the intervals HI, IK, . . . between the points H, I, K, L, . . . are unequal, take b, b₂, b₃, b₄, b₅, . . . , the first differences of the perpendiculars AH, BI, CK, . . . divided by the intervals between the perpendiculars; take c, c₂, c₃, c₄, . . . , the second differences divided by each two intervals; d, d₂, d₃, . . . , the third differences divided by each three intervals; e, e₂, . . . , the fourth differences divided by each four intervals, and so on—that is, in such a way that , . . . , and then , . . . , and afterward . . . . When these differences have been found, let AH be equal to a, −HS = p, p × (−IS) = q, q × (+SK) = r, r × (+SL) = s, s × (+SM) = t, proceeding, that is, up to the penultimate perpendicular ME; then the ordinate RS will be = a + bp + cq + dr + es + ft + . . . .

COROLLARY. Hence the areas of all curves can be found very nearly. For if several points are found of any curve which is to be squared [i.e., any curve whose area is desired] and a parabola is understood to be drawn through them, the area of this parabola will be very nearly the same as the area of that curve which is to be squared. Moreover, a parabola can always be squared geometrically by methods which are very well known.

Lemma 6
From several observed places of a comet, to find its place at any given intermediate time.

Let HI, IK, KL, LM represent the times between the observations (in the figure to lem. 5), HA, IB, KC, LD, ME five observed longitudes of the comet, and HS the given time between the first observation and the required longitude. Then, if a regular curve ABCDE is understood to be drawn through the points A, B, C, D, E, and if the ordinate RS is found by the above lemma, RS will be the required longitude.

By the same method the latitude at a given time is found from five observed latitudes.

If the differences of the observed longitudes are small, say only 4 or 5 degrees, three or four observations would suffice for finding the new longitude and latitude. But if the differences are greater, say 10 or 20 degrees, five observations must be used.

Lemma 7
To draw a straight line BC through a given point P, so that the parts PB and PC of that line, cut off by two straight lines AB and AC, given in position, have a given ratio to each other.

From that point P draw any straight line PD to either of the straight lines, say AB, and produce PD toward the other straight line AC as far as E, so that PE is to PD in the given ratio. Let EC be parallel to AD; and if CPB is drawn, PC will be to PB as PE to PD. Q.E.F.

Lemma 8
Let ABC be a parabola with focus S. Let the segment ABCI be cut off by the chord AC (which is bisected at I), let its diameter be Iμ, and let its vertex be μ. On Iμ produced, take μO equal to half of Iμ. Join OS and produce it to ξ, so that Sξ is equal to 2SO. Then, if a comet B moves in the arc CBA, and if ξB is drawn cutting AC in E, I say that the point E will cut off from the chord AC the segment AE very nearly proportional to the time.

For join EO, cutting the parabolic arc ABC in Y, and draw μX so as to touch the same arc in the vertex μ and meet EO in X; then the curvilinear area AEXμA will be to the curvilinear area ACYμA as AE to AC. And thus, since triangle ASE is in the same ratio to triangle ASC as the ratio of those curvilinear areas, the total area ASEXμA will be to the total area ASCYμA as AE to AC. Moreover, since ξO is to SO as 3 to 1, and EO is in the same ratio to XO, SX will be parallel to EB; and therefore, if BX is joined, the triangle SEB will be equal to the triangle XEB. Thus, if the triangle EXB is added to the area ASEXμA and from that sum the triangle SEB is taken away, there will remain the area ASBXμA equal to the area ASEXμA, and thus it will be to the area ASCYμA as AE to AC. But the area ASBYμA is very nearly equal to the area ASBXμA, and the area ASBYμA is to the area ASCYμA as the time in which the arc AB is described to the time of describing the total arc AC. And thus AE is to AC very nearly in the ratio of the times. Q.E.D.

COROLLARY. When point B falls upon the vertex μ of the parabola, AE is to AC exactly in the ratio of the times.

Scholium
If μξ is joined, cutting AC at δ, and if ξn, which is to μB as 27MI to 16Mμ, is taken in this line, then when Bn is drawn it will cut the chord AC more nearly in the ratio of the lines than before. But the point n is to be taken so as to lie beyond point ξ if point B is more distant than point μ from the principal vertex of the parabola; and contrariwise if B is less distant from that vertex.

Lemma 9
The straight lines Iμ and μM and the length are equal to one another.

For 4Sμ is the latus rectum of a parabola, extending to the vertex μ.

Lemma 10
Let Sμ be produced to N and P, so that μN is one-third of μI, and so that SP is to SN as SN to Sμ. Then, in the time in which a comet describes the arc AμC, it would—if it moved forward always with the velocity that it has at a height equal to SP—describe a length equal to the chord AC.

For if the comet were to move forward in the same time uniformly in the straight line that touches the parabola at μ, and with the velocity that it has in μ, then the area that it would describe by a radius drawn to point S would be equal to the parabolic area ASCμ. And hence the space determined by the length described along the tangent and the length Sμ would be to the space determined by the lengths AC and SM as the area ASCμ to the triangle ASC, that is, as SN to SM. Therefore, AC is to the length described along the tangent as Sμ to SN. But the velocity of the comet at the height SP is (by book 1, prop. 16, corol. 6) to its velocity at the height Sμ as the square root of the ratio of SP to Sμ inversely, that is, in the ratio of Sμ to SN; hence the length described in the same time with this velocity will be to the length described along the tangent as Sμ to SN. Therefore, since AC and the length described with this new velocity are in the same ratio to the length described along the tangent, they are equal to each other. Q.E.D.

COROLLARY. Therefore, in that same time, the comet, with the velocity that it has at the height Sμ + ⅔Iμ, would describe the chord AC very nearly.

Lemma 11
Suppose a comet, deprived of all motion, to be let fall from the height SN or Sμ + ⅓Iμ, so as to fall toward the sun, and suppose this comet to be urged toward the sun always by that force, uniformly continued, by which it is urged at the beginning. Then in half of the time in which the comet describes the arc AC in its orbit, it would—in this descent toward the sun—describe a space equal to the length Iμ.

For by lem. 10, in the same time in which the comet describes the parabolic arc AC, it will—with the velocity that it has at the height SP—describe the chord AC; and hence (by book 1, prop. 16, corol. 7), revolving by the force of its own gravity, it would—in that same time, in a circle whose semidiameter was SP—describe an arc whose length would be to the chord AC of the parabolic arc in the ratio of 1 to √2. And therefore, falling from the height SP toward the sun with the weight that it has toward the sun at that height, it would in half that time (by book 1, prop. 4, corol. 9) describe a space equal to the square of half of that chord, divided by four times the height SP, that is, the space . Thus, since the weight of the comet toward the sun at the height SN is to its weight toward the sun at the height SP as SP to Sμ, the comet—falling toward the sun with the weight that it has at the height SN—will in the same time describe the space , that is, a space equal to the length Iμ or Mμ. Q.E.D.

Proposition 41, Problem 21
To determine the trajectory of a comet moving in a parabola, from three given observations.

Having tried many approaches to this exceedingly difficult problem, I devised certain problems [i.e., propositions] in book 1 which are intended for its solution. But later on, I conceived the following slightly simpler solution.

Let three observations be chosen, distant from one another by nearly equal intervals of time. But let that interval of time when the comet moves more slowly be a little greater than the other, that is, so that the difference of the times is to the sum of the times as the sum of the times to more or less six hundred days, or so that the point E (in the figure to lem. 8) falls very nearly on the point M and deviates from there toward I rather than toward A. If such observations are not at hand, a new place of the comet must be found by the method of lem. 6.

Let S represent the sun; T, t, and τ three places of the earth in its orbit; TA, tB, and 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 that the comet could describe in that total time [V + W] with the velocity that it has in the mean distance of the earth from the sun (which length is to be found by the method of book 3, prop. 40, corol. 3); and let tV be a perpendicular to the chord Tτ. In the mean observed longitude tB, let the point B be taken anywhere at all for the place of the comet in the plane of the ecliptic, and from there toward the sun S draw line BE so as to be to the sagitta tV as the contenta of SB and St² to the cube of the hypotenuse 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 point E (by lem. 7 of this third book) draw the straight line AEC so that its parts AE and EC, terminated in the straight lines TA and τC, are to each other as the times V and W. Then A and C will be the places of the comet in the plane of the ecliptic in the first and third observations very nearly, provided that B is its correctly assumed place in the second observation.

Upon AC, bisected in I, erect a perpendicular Ii. Through point B let a line Bi be imagined,^b drawn parallel to AC. Let Si be a line imagined as cutting AC at λ, and complete the parallelogram iIλμ. Take Iσ equal to 3Iλ, and through the sun S draw the dotted line σξ equal to 3Sσ + 3iλ. And after deleting the letters A, B, C, and I, let a new imagined line BE be drawn from the point B toward the point ξ so that it is to the former line BE as the square of the distance BS to the quantity Sμ + ⅓iλ. And through point E again draw the straight line AEC according to the same rule as before, that is, so that its parts AE and EC are to each other as the times V and W between observations. Then A and C will be the places of the comet more exactly.

Upon AC, bisected in I, erect the perpendiculars AM, CN, and IO, so that, of these perpendiculars, AM and CN are the tangents of the latitudes^c in the first and third observations (to the radii TA and τC). Join MN, cutting IO in O. Construct the rectangle iIλμ as before. On IA produced, take ID equal to Sμ + ⅔iλ. Then on MN, toward N, take MP so that it is to the length X found above as the square root of the ratio of the mean distance of the earth from the sun (or of the semidiameter of the earth’s orbit) to the distance OD. If point P falls upon point N, then 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 point P does not fall upon point N, then on the straight line AC take CG equal to NP, in such a way that points G and P lie on the same side of the straight line NC.

Using the same method by which points E, A, C, and G were found from the assumed point B, find from other points b and β (assumed in any way whatever) the new points e, a, c, g, and ε, α, κ, γ. Then if the circumference of circle Ggγ is drawn through G, g, and γ, cutting the straight line τC in Z, Z will be a place of the comet in the plane of the ecliptic. And if on AC, ac, and ακ, there are taken AF, af, and αφ, equal respectively to CG, cg, and κγ, and if the circumference of a circle Ffφ is drawn through points F, f, and φ, cutting the straight line AT in X, then point X will be another place of the comet in the plane of the ecliptic. At the points X and Z, erect the tangents of the latitudes of the comet (to the radii TX and τZ), and two places of the comet in its orbit will be found. Finally (by book 1, prop. 19), let a parabola with focus S be described through those two places; this parabola will be the trajectory of the comet. Q.E.I.

The demonstration of this construction follows from the lemmas, since the straight line AC is cut in E in the ratio of the times, by lem. 7, as required by lem. 8; and since BE, by lem. 11, is that part of the straight line BS or Bξ which lies in the plane of the ecliptic between the arc ABC and the chord AEC; and since MP (by lem. 10, corol.) is the length of the chord of the arc that the comet must describe in its orbit between the first observation and the third, and therefore would be equal to MN, provided that B is a true place of the comet in the plane of the ecliptic.

But it is best not to choose the points B, b, and β any place whatever, but to take them as close to true as possible. If the angle AQt, at which the projection of the orbit described in the plane of the ecliptic cuts the straight line tB, is known approximately, imagine the straight line AC drawn at that angle so that it is to ⁴/3Tτ as the square root of the ratio of SQ to St. And by drawing the straight line SEB, so that its part EB is equal to the length Vt, point B will be determined, which may be used the first time around. Then, after deleting the straight line AC and drawing AC anew according to the preceding construction, and after additionally finding the length MP, take point b on tB according to the rule that if TA and τC cut each other in Y, the distance Yb is to the distance YB in a ratio compounded of the ratio of MP to MN and the square root of the ratio of SB to Sb. And the third point β will have to be found by the same method, if it is desired to repeat the operation for the third time. But by this method two operations would, for the most part, be sufficient. For if the distance Bb happens to be very small, then after the points F, f and G, g have been found, the straight lines Ff and Gg (when drawn) will cut TA and τC in the required points X and Z.

EXAMPLE. Let the comet of 1680 be proposed as the example. The following table shows its motion as observed by Flamsteed and as calculated by him from these observations, and corrected by Halley on the basis of the same observations.

To these add certain observations of my own.

These observations were made with a seven-foot telescope, and a micrometer the threads of which were placed in the focus of the telescope; and with these instruments we determined both the positions of the fixed stars in relation to one another and the positions of the comet in relation to 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 the star of the sixth magnitude in the heel of the same foot (Bayer’s n) and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, and δ other smaller stars in the same foot. And let p, P, Q, R, S, T, V, and X be the places of the comet in the observations described above; and, the distance AB being reckoned at 80⁷/12 parts, AC was 52¼ parts, BC 58⅚, AD 57⁵/12, BD 82⁶/11, CD 23⅔, AE 29⁴/7, CE 57½, DE 49¹¹/12, AI 27⁷/12, BI 52⅙, CI 36⁷/12, DI 53⁵/11, AK 38⅔, BK 43, CK 31⁵/9, FK 29, FB 23, FC 36¼, AH 18⁶/7, DH 50⅞, BN 46⁵/12, CN 31⅓, BL 45⁵/12, NL 31⁵/7. HO was to HI as 7 to 6 and, when produced, passed between stars D and E in such a way that the distance of star D from this straight line was ⅙CD. LM was to LN as 2 to 9 and, when produced, passed through star H. Thus the positions of the fixed stars in relation to one another were determined.

Finally our fellow countryman Pound again observed the positions of these fixed stars in relation to one another and recorded their longitudes and latitudes, as in the following table.

I observed the positions of the comet in relation to these fixed stars as follows.

On Friday, 25 February (O.S.), at 8^h30^m P.M., the distance of the comet, which was at p, from star E was less than ³/13 AE, and greater than ⅕AE, and thus was approximately ³/14AE; and the angle ApE was somewhat obtuse, but almost a right angle. For if a perpendicular were dropped from A to pE, the distance of the comet from that perpendicular was ⅕pE.

On the same night at 9^h30^m, the distance of the comet (which was at P) from star E was greater than AE and less than AE, and thus was very nearly AE, or ⁸/39AE. And the distance of the comet from a perpendicular dropped from star A to the straight line PE was ⅘PE.

On Sunday, 27 February, at 8^h15^m P.M., the distance of the comet (which was at Q) from star O equaled the distance between stars O and H; and the straight line QO, produced, passed between stars K and B. Because of intervening clouds, I could not determine the position of this straight line more exactly.

On Tuesday, 1 March, at 11^h P.M., the comet (which was at R) lay exactly between stars K and C; and the part CR of the straight line CRK was a little greater than ⅓CK and a little smaller than ⅓CK + ⅛CR, and thus was equal to ⅓CK + ¹/16CR, or ¹⁶/45CK.

On Wednesday, 2 March, at 8^h P.M., the distance of the comet (which was at S) from star C was very close to ⁴/9FC. The distance of star F from the straight line CS, produced, was ¹/24FC, and the distance of star B from that same straight line was five times greater than the distance of star F. Also, the straight line NS, produced, passed between stars H and I and was five or six times nearer to star H than to star I.

On Saturday, 5 March, at 11^h30^m P.M. (when the comet was at T), the straight line MT was equal to ½ML, and the straight line LT, produced, passed between B and F four or five times closer to F than to B, cutting off from BF a fifth or sixth part of it toward F. And MT, produced, passed outside the space BF on the side of star B and was four times closer to star B than to star F. M was a very small star that could scarcely be seen through the telescope, and L was a greater star, of about the eighth magnitude.

On Monday, 7 March, at 9^h30^m P.M. (when the comet was at V), the straight line Vα, produced, passed between B and F, cutting off ¹/10BF from BF on the side of F, and was to the straight line Vβ as 5 to 4. And the distance of the comet from the straight line αβ was ½Vβ.

On Wednesday, 9 March, at 8^h30^m P.M. (when the comet was at X), the straight line γX was equal to ¼γδ, and a perpendicular dropped from star δ to the straight line γX was ⅖γδ.

On the same night at 12^h (when the comet was at Y), the straight line γY was equal to ⅓γδ or a little smaller, say ⁵/16γδ, and a perpendicular dropped from star δ to the straight line γY was equal to about ⅙ or ¹/7γδ. But the comet could scarcely be discerned because of its nearness to the horizon, nor could its place be determined so surely as in the preceding observations.

From observations of this sort, by constructions of diagrams, and by calculations, I found the longitudes and latitudes of the comet, and from the corrected places of the fixed stars our fellow countryman Pound corrected the places of the comet, and these corrected places are given above. I used a crudely made micrometer, but nevertheless the errors of longitudes and latitudes (insofar as they come from my observations) scarcely exceed one minute. Moreover, the comet (according to my observations) at the end of its motion began to decline noticeably toward the north from the parallel which it had occupied at the end of February.

Now, in order to determine the orbit of the comet, I selected—from the observations hitherto described—three that Flamsteed made, on 21 December, 5 January, and 25 January. From these observations I found St to be of 9,842.1 parts, and Vt to be of 455 parts (10,000 such parts being the semidiameter of the earth’s orbit). Then for the first operation, assuming tB to be of 5,657 parts, I found SB to be of 9,747, BE the first time 412, Sμ 9,503, iλ 413; BE the second time 421, OD 10,186, X 8,528.4, MP 8,450, MN 8,475, NP 25. Hence for the second operation I reckoned the distance tb to be 5,640. And by this operation I found at last the distance TX to be 4,775 and the distance τZ to be 11,322. In determining the orbit from these distances, I found the descending node in 1°53′ and the ascending node in 1°53′, and the inclination of its plane to the plane of the ecliptic to be 61°20⅓′. I found that its vertex (or the perihelion of the comet) was 8°38′ distant from the node and was in 27°43′ with a latitude 7°34′ S; and that its latus rectum was 236.8, and that the area described each day by a radius drawn to the sun was 93,585, supposing the square of the semidiameter of the earth’s orbit to be 100,000,000; and I found that the comet had advanced in this orbit in the order of the signs, and was on 8 December 0^h4^m A.M. in the vertex of the orbit or the perihelion. I made all these determinations graphically by a scale of equal parts and by chords of angles, taken from the table of natural sines, constructing a fairly large diagram, that is, one in which the semidiameter of the earth’s orbit (of 10,000 parts) was equal to 16⅓ inches of an English foot.

Finally, in order to establish whether the comet moved truly in the orbit thus found, I calculated—partly by arithmetical and partly by graphical operations—the places of the comet in this orbit at the times of certain observations, as can be seen in the following table.

^dLater, our fellow countryman Halley determined the orbit more exactly by an arithmetical calculation than could be done graphically [lit. by the descriptions of lines]; and while he kept the place of the nodes in 1°53′ and in 1°53′ and the inclination of the plane of the orbit to the ecliptic 61°20⅓′, and also the time of the perihelion of the comet 8 December 0^h4^m, he found the distance of the perihelion from the ascending node (measured in the orbit of the comet) to be 9°20′, and the latus rectum of the parabola to be 2,430 parts, the mean distance of the sun from the earth being 100,000 parts. And by making the same kind of arithmetical calculation exactly (using these data), he calculated the places of the comet at the times of the observations, as follows.

This comet also appeared in the preceding November and was observed by Mr. Gottfried Kirch at Coburg in Saxony on the fourth, sixth, and eleventh days of this month (O.S.); and from its positions with respect to the nearest fixed stars (observed with sufficient accuracy, sometimes through a two-foot telescope and sometimes through a ten-foot telescope), from the difference of the longitudes of Coburg and London, eleven degrees, and from the places of the fixed stars observed by our fellow countryman Pound, our own Halley has determined the places of the comet as follows.

On 3 November 17^h2^m, apparent time at London, the comet was in 29°51′ with latitude 1°17′45″ N.

On 5 November 15^h58^m, the comet was in 3°23′ with latitude 1°6′ N.

On 10 November 16^h31^m, the comet was equally distant from the stars σ and τ (Bayer) of Leo; it had not yet reached the straight line joining these stars, but was not far from it. In Flamsteed’s catalog of stars, σ then was in 14°15′ with latitude about 1°41′ N, while τ was in 17°3½′ with latitude 0°34′ S. And the midpoint between these stars was 15°39¼′ with latitude 0°33½′ N. Let the distance of the comet from that straight line be about 10′ or 12′; then the difference of the longitudes of the comet and that midpoint will be 7′ and the difference of the latitudes roughly 7½′. And thus the comet was in 15°32′ with roughly latitude 26′ N.

The first observation of the position of the comet in relation to certain small fixed stars was more than exact enough. The second also was exact enough. In the third observation, which was less exact, there could have been an error of six or seven minutes, but hardly a greater one. And the longitude of the comet in the first observation, which was more exact than the others, being computed in the parabolic orbit mentioned above, was 29°30′22″, its latitude 1°25′7″ N, and its distance from the sun 115,546.

Further, Halley noted that a remarkable comet had appeared four times at intervals of 575 years—namely, in September after the murder of Julius Caesar; in A.D. 531 in the consulship of Lampadius and Orestes; in February A.D. 1106; and toward the end of 1680—and that this comet had a long and remarkable tail (except that in the year of Caesar’s death the tail was less visible because of the inconvenient position of the earth); and he set out to find an elliptical orbit whose major axis would be 1,382,957 parts, the mean distance of the earth from the sun being 10,000 parts, that is, an orbit in which a comet might revolve in 575 years. Then he computed the motion of the comet in this elliptical orbit with the following conditions: the ascending node in 2°2′, the inclination of the plane of the orbit to the plane of the ecliptic 61°6′48″, the perihelion of the comet in this plane in 22°44′25″, the equated time of the perihelion 7 December 23^h9^m, the distance of the perihelion from the ascending node in the plane of the ecliptic 9°17′35″, and the conjugate axis 18,481.2. The places of this comet, as deduced from observations as well as calculated for this orbit, are displayed in the following table [page 912].

The observations of this comet from beginning to end agree no less with the motion of a comet in the orbit just described than the motions of the planets generally agree with planetary theories, and this agreement provides proof that it was one and the same comet which appeared all this time and that its orbit has been correctly determined here.^d

^eIn this table we have omitted the observations made on 16, 18, 20, and 23 November as being less exact. Yet the comet was observed at these times also.^e In fact, [Giuseppe Dionigi] Ponteo and his associates, on 17 November (O.S.) at 6^h A.M. in Rome, that is, at 5^h10^m London time, using threads applied to the fixed stars, observed the comet in 8°30′ with latitude 0°40′ S. Their observations may be found in the treatise which Ponteo published about this comet. [Marco Antonio] Cellio, who was present and sent his own observations in a letter to Mr. Cassini, saw the comet at the same hour in 8°30′ with latitude 0°30′ S. At the same hour Gallet in Avignon (that is, at 5^h42^m A.M. London time) saw the comet in 8° with null latitude; at which time, according to the theory, the comet was in 8°16′45″ with latitude 0°53′7″ S.

On 18 November at 6^h30^m A.M. in Rome (that is, at 5^h40^m London time) Ponteo saw the comet in 3°30′ with latitude 1°20′ S; Cellio saw it in 13°30′ with latitude 1°00′ S. Moreover, Gallet at 5^h30^m A.M. in Avignon saw the comet in 13°00′ with latitude l°00′ S. And the Reverend Father Ango at the College of La Flèche in France, at 5^h A.M. (that is, at 5^h9^m London time), saw the comet midway between two small stars, of which one is the middle star of three in a straight line in the southern hand of Virgo, Bayer’s ψ, and the other is the outermost star of the wing, Bayer’s ϑ. Thus the comet was then in 12°46′ with latitude 50′ S. On the same day at Boston in New England, at a latitude of 42½°, at 5^h A.M. (that is, 9^h44^m London time), the comet was seen near 14° with latitude 1°30′ S, as I was informed by the distinguished Halley.

On 19 November at 4^h30^m A.M. in Cambridge, the comet (according to the observation of a certain young man) was about 2 degrees distant from Spica Virginis toward the northwest. And Spica was in 19°23′47″ with latitude 2°1′59″ S. On the same day at 5^h A.M. at Boston in New England, the comet was 1 degree distant from Spica Virginis, the difference of latitudes being 40 minutes. On the same day on the island of Jamaica, the comet was about 1 degree distant from Spica. On the same day Mr. Arthur Storer, at the Patuxent River, near Hunting Creek in Maryland, which borders on Virginia, at latitude 38½°, at 5^h A.M. (that is, 10^h London time), saw the comet above Spica Virginis and almost conjoined with Spica, the distance between them being about ¾ of a degree. And comparing these observations with one another, I gather that at 9^h44^m in London the comet was in 18°50′ with latitude roughly l°25′ S. And by the theory the comet was then in 18°52′15″ with latitude 1°26′54″ S.f

On 20 November, Mr. Geminiano Montanari, professor of astronomy in Padua, at 6^h A.M. in Venice (that is, 5^h10^m London time), saw the comet in 23° with latitude l°30′ S. On the same day at Boston the comet was distant from Spica by 4 degrees of longitude eastward and so was approximately in 23°24′.

On 21 November, Ponteo and his associates at 7^h15^m A.M. observed the comet in 27°50′ with latitude 1°16′ S, Cellio in 28°, Ango at 5^h A.M. in 27°45′, Montanari in 27°51′. On the same day on the island of Jamaica the comet was seen near the beginning of Scorpio and had roughly the same latitude as Spica Virginis, that is, 2°2′. On the same day at 5^h A.M. at Balasore in the East Indies (that is, at 11^h20^m the preceding night, London time) the comet was distant 7°35′ eastward from Spica Virginis. It was in a straight line between Spica and the scale [or pan of the Balance] and so was in 26°58′ with latitude roughly 1°11′ S, and after 5 hours and 40 minutes (that is, at 5^h A.M. London time) was in 28°12′ with latitude 1°16′ S. And by the theory the comet was then in 28°10′36″ with latitude 1°53′35″ S.

On 22 November, the comet was seen by Montanari in 2°33′, while at Boston in New England it appeared in approximately 3°, with about the same latitude as before, that is, 1°30′. On the same day at 5^h A.M. at Balasore the comet was observed in 1°50′, and so at 5^h A.M. in London the comet was approximately in 3°5′. On the same day at London at 6^h30^m A.M. our fellow countryman Hooke saw the comet in approximately 3°30′, on a straight line that passes between Spica Virginis and the heart of Leo, not exactly indeed, but deviating a little from that line toward the north. Montanari likewise noted that a line drawn from the comet through Spica passed, on this day and the following days, through the southern side of the heart of Leo, there being a very small interval between the heart of Leo and this line. The straight line passing through the heart of Leo and Spica Virginis cut the ecliptic in 3°46′, at an angle of 2°51′. And if the comet had been located in this line in 3°, its latitude would have been 2°26′. But since the comet, by the agreement of Hooke and Montanari, was at some distance from this line toward the north, its latitude was a little less. On the 20th, according to the observation of Montanari, its latitude almost equaled the latitude of Spica Virginis and was roughly 1°30′; and by the agreement of Hooke, Montanari, and Ango, the latitude was continually increasing and so now (on the 22d) was sensibly greater than 1°30′. And the mean latitude between the limits now established, 2°26′ and 1°30′, will be roughly 1°58′. The tail of the comet, by the agreement of Hooke and Montanari, was directed toward Spica Virginis, declining somewhat from that star—southward according to Hooke, northward according to Montanari; and so that declination was hardly perceptible, and the tail, being almost parallel to the equator, was deflected somewhat northward from the opposition of the sun.

On 23 November (O.S.) at 5^h A.M. at Nuremberg (that is, at 4^h30^m London time) Mr. [Johann Jacob] Zimmermann saw the comet in 8°8′ with latitude 2°31′ S, determining its distances from the fixed stars.

On 24 November before sunrise the comet was seen by Montanari in 12°52′ on the northern side of a straight line drawn through the heart of Leo and Spica Virginis, and so had a latitude a little less than 2°38′. This latitude (as we have said), according to the observations of Montanari, Ango, and Hooke, was continually increasing, and so it was now (on the 24th) a little greater than 1°58′, and at its mean magnitude can be taken as 2°18′ without perceptible error. Ponteo and Gallet would have the latitude decreased now, and Cellio and the observer in New England would have it retained at about the same magnitude, namely 1 or 1½ degrees. The observations of Ponteo and Cellio are rather crude, especially those that were made by taking azimuths and altitudes, and so are those of Gallet; better are the ones that were made by means of the positions of the comet in relation to fixed stars by Montanari, Hooke, Ango, and the observer in New England, and sometimes by Ponteo and Cellio. On the same day at 5^h A.M. at Balasore, the comet was observed in 11°45′, and so at 5^h A.M. at London it was nearly in 13°. And by the theory the comet was at that time in 13°22′42″.

On 25 November before sunrise Montanari observed the comet approximately in 17¾°. And Cellio observed at the same time that the comet was in a straight line between the bright star in the right thigh of Virgo and the southern scale of Libra, and this straight line cuts the path of the comet in 18°36′. And by the theory the comet was at that time approximately in 18⅓°.

Therefore these observations agree with the theory insofar as they agree with one another, and by such agreement they prove that it was one and the same comet that appeared in the whole time from the 4th of November to the 9th of March. The trajectory of this comet cut the plane of the ecliptic twice and therefore was not rectilinear. It cut the ecliptic not in opposite parts of the heavens, but at the end of Virgo and at the beginning of Capricorn, at points separated by an interval of about 98 degrees; and thus the course of the comet greatly deviated from a great circle. For in November its course declined by at least 3 degrees from the ecliptic toward the south, and afterward in December verged from the ecliptic 29 degrees toward the north: the two parts of its orbit, in which the comet tended toward the sun and returned from the sun, declining from each other by an apparent angle of more than 30 degrees, as Montanari observed. This comet moved through nine signs, namely from the last degree of Leo to the beginning of Gemini, besides [that part of] the sign of Leo through which it moved before it began to be seen; and there is no other theory according to which a comet may travel over so great a part of the heaven with a motion according to some rule. Its motion was extremely nonuniform. For about the 20th of November it described approximately 5 degrees per day; then, with a retarded motion between 26 November and 12 December, that is, during 15½ days, it described only 40 degrees; and afterward, with its motion accelerated again, it described about 5 degrees per day until its motion began to be retarded again. And the theory that corresponds exactly to so nonuniform a motion through the greatest part of the heavens, and that observes the same laws as the theory of the planets, and that agrees exactly with exact astronomical observations cannot fail to be true.

Furthermore, it seemed appropriate to show the trajectory that the comet described and the actual tail that it projected in different positions, as in the accompanying figure, in the plane of the trajectory; in this figure, ABC denotes the trajectory of the comet, D the sun, DE the axis of the trajectory, DF the line of nodes, GH the intersection of the sphere of the earth’s orbit with the plane of the trajectory, I the place of the comet on 4 November 1680, K its place on 11 November, L its place on 19 November, M its place on 12 December, N its place on 21 December, O its place on 29 December, P its place on 5 January of the following year, Q its place on 25 January, R its place on 5 February, S its place on 25 February, T its place on 5 March, and V its place on 9 March. I used the following observations in determining the tail.

On 4 and 6 November the tail was not yet visible. On 11 November the tail, which had now begun to be seen, was observed through a ten-foot telescope to be no more than half a degree long. On 17 November the tail was observed by Ponteo to be more than 15 degrees long. On 18 November the tail was seen in New England to be 30 degrees long and directly opposite to the sun, and it was extended out to the star ♂ [i.e., the planet Mars], which was then in 9°54′. On 19 November, in Maryland, the tail was seen to be 15 or 20 degrees long. On 10 December the tail (according to the observations of Flamsteed) was passing through the middle of the distance between the tail of Serpens (the Serpent of Ophiuchus) and the star δ in the southern wing of Aquila and terminated near the stars A, ω, b in Bayer’s tables. Therefore the end of the comet’s tail was in 19½° with a latitude of about 34¼° N. On 11 December the tail was rising as far as the head of Sagitta (Bayer’s α, β), terminating in 26°43′, with a latitude of 38°34′ N. On 12 December the tail was passing through the middle of Sagitta and did not extend very much further, terminating in 4°, with a latitude of about 42½° N.

These things are to be understood of the length of the brighter part of the tail. For when the light was fainter and the sky perhaps clearer, on 12 December at 5^h40^m in Rome, the tail was observed by Ponteo to extend to 10 degrees beyond the uropygium of Cygnus [i.e., the rump of the Swan], and its side toward the northwest terminated 45 minutes from this star. Moreover, in those days the tail was 3 degrees wide near its upper end, and so the middle of it was 2°15′ distant from that star toward the south, and its upper end was in 22° with a latitude of 61° N. And hence the tail was about 70 degrees long.

On 21 December the tail rose almost to Cassiopeia’s Chair, being equally distant from β and Schedar [= α Cassiopeiae] and having a distance from each of them equal to their distance from each other, and so terminating in 24° with a latitude of 47½°. On 29 December the tail was touching Scheat, which was situated to the left of it, and exactly filled the space between the two stars in the northern foot of Andromeda; it was 54 degrees long; accordingly it terminated in 19° with a latitude of 35°. On 5 January the tail touched the star π in the breast of Andromeda on the right side and the star μ in the girdle on the left side, and (according to our observations) was 40 degrees long; but it was curved, and its convex side faced to the south. Near the head of the comet, the tail made an angle of 4 degrees with the circle passing through the sun and the head of the comet; but near the other end, it was inclined to that circle at an angle of 10 or 11 degrees, and the chord of the tail contained an angle of 8 degrees with that circle. On 13 January the tail was visible enough between Alamech and Algol [= β Persei], but it ended in a very faint light toward the star κ in Perseus’s side. The distance of the end of the tail from the circle joining the sun and the comet was 3°50′, and the inclination of the chord of the tail to that circle was 8½ degrees. On 25 and 26 January the tail shone with a faint light to a length of 6 or 7 degrees; and, a night or so later, when the sky was extremely clear, it attained a length of 12 degrees and a little more, with a light that was very faint and scarcely to be perceived. But its axis was directed exactly toward the bright star in the eastern shoulder of Auriga, and accordingly declined from the opposition of the sun toward the north at an angle of 10 degrees. Finally on 10 February, my eyes armed [with a telescope], I saw the tail to be 2 degrees long. For the fainter light mentioned above was not visible through the glasses. But Ponteo writes that on 7 February he saw the tail with a length of 12 degrees. On 25 February and thereafter, the comet appeared without a tail.

Whoever considers the orbit just described and turns over in his mind the other phenomena of this comet will without difficulty agree that the bodies of comets are solid, compact, fixed, and durable, like the bodies of planets. For if comets were nothing other than vapors or exhalations of the earth, the sun, and the planets, this one ought to have been dissipated at once during its passage through the vicinity of the sun. For the heat of the sun is as the density of its rays, that is, inversely as the square of the distance of places from the sun. And thus, since the distance of the comet from the center of the sun on 8 December, when it was in its perihelion, was to the distance of the earth from the center of the sun as approximately 6 to 1,000, the heat of the sun on the comet at that time was to the heat of the summer sun here on earth as 1,000,000 to 36, or as 28,000 to 1. But the heat of boiling water is about three times greater than the heat that dry earth acquires in the summer sun, as I have found [by experiment]; and the heat of incandescent iron (if I conjecture correctly) is about three or four times greater than the heat of boiling water; and hence the heat that dry earth on the comet would have received from the sun’s rays, when it was in its perihelion, would be about two thousand times greater than the heat of incandescent iron. But with so great a heat, vapors and exhalations, and all volatile matter, would have to have been consumed and dissipated at once.

Therefore the comet, in its perihelion, received an immense heat at [i.e., when near] the sun, and it can retain that heat for a very long time. For a globe of incandescent iron, one inch wide, standing in the air would scarcely lose all its heat in the space of one hour. But a larger globe would preserve its heat for a longer time in the ratio of its diameter, because its surface (which is the measure according to which it is cooled by contact with the surrounding air) is smaller in that ratio with respect to the quantity of hot matter it contains. And so a globe of incandescent iron equal to this earth of ours—that is, more or less 40,000,000 feet wide—would scarcely cool off in as many days, or about 50,000 years. Nevertheless, I suspect that the duration of heat is increased in a smaller ratio than that of the diameter because of some latent causes, and I wish that the true ratio might be investigated by experiments.

Further, it should be noted that in December, when the comet had just become hot at the sun, it was emitting a far larger and more splendid tail than it had done earlier in November, when it had not yet reached its perihelion. And, universally, the greatest and brightest tails all arise from comets immediately after their passage through the region of the sun. Therefore the heating up of the comet is conducive to a great size of its tail, and from this I believe it can be concluded that the tail is nothing other than extremely thin vapor that the head or nucleus of the comet emits by its heat.

There are indeed three opinions about the tails of comets: that the tails are the brightness of the sun’s light propagated through the translucent heads of comets; that the tails arise from the refraction of light in its progress from the head of the comet to the earth; and finally that these tails are a cloud or vapor continually rising from the head of the comet and going off in a direction away from the sun. The first opinion is held by those who are not yet instructed in the science of optics. For beams of sunlight are not seen in a dark room except insofar as the light is reflected from particles of dust and smoke always flying about through the air, and for this reason in air darkened with thicker smoke the beams of sunlight appear brighter and strike the eye more strongly, while in clearer air these beams are fainter and are perceived with greater difficulty, but in the heavens, where there is no matter to reflect these beams of sunlight, they cannot be seen at all. Light is not seen insofar as it is in the beam, but only to the degree that it is reflected to our eyes; for vision results only from rays that impinge upon the eyes. Therefore some reflecting matter must exist in the region of the tail, since otherwise the whole sky, illuminated by the light of the sun, would shine uniformly.

The second opinion is beset with many difficulties. The tails are never variegated in color, and yet colors are generally the inseparable concomitants of refractions. The light of the fixed stars and the planets which is transmitted to us is distinct [i.e., clearly defined]; this demonstrates that the celestial medium is empowered with no refractive force. It is said that the Egyptians sometimes saw the fixed stars surrounded by a head of hair, but this happens very rarely, and so it must be ascribed to some chance refraction by clouds. The radiation and scintillation of the fixed stars also should be referred to refractions both by the eyes and by the tremulous air, since they disappear when these stars are viewed through telescopes. By the tremor of the air and of the ascending vapors it happens that rays are easily turned aside alternately from the narrow space of the pupil of the eye but not at all from the wider aperture of the objective lens of a telescope. Thus it is that scintillation is generated in the former case while it ceases in the latter; and the cessation of scintillation in the latter case demonstrates the regular transmission of light through the heavens without any sensible refraction. And to counter the argument that tails are not generally seen in comets when their light is not strong enough, for the reason that the secondary rays do not then have enough force to affect the eyes, and that this is why the tails of the fixed stars are not seen, it should be pointed out that the light of the fixed stars can be increased more than a hundred times by means of telescopes, and yet no tails are seen. The planets also shine with more light, but they have no tails; and often comets have the greatest tails when the light of their heads is faint and exceedingly dull. For such was the case for the comet of 1680; in December, at a time when the light from its head scarcely equaled stars of the second magnitude, it was emitting a tail of notable splendor as great as 40, 50, 60, or 70 degrees in length and more. Afterward, on 27 and 28 January, the head appeared as a star of only the seventh magnitude, but the tail extended to 6 or 7 degrees in length with a very faint light that was sensible enough; and with a very dim light, which could scarcely be seen, it stretched out as far as 12 degrees or a little further, as was said above. But even on 9 and 10 February, when the head had ceased to be seen by the naked eye, the tail—when I viewed it through a telescope—was 2 degrees long. Further, if the tail arose from refraction by celestial matter, and if it deviated from the opposition of the sun in accordance with the form of the heavens, then, in the same regions of the heavens, that deviation ought always to take place in the same direction. But the comet of 1680, on 28 December at 8^h30^m P.M. London time, was in 8°41′ with a latitude of 28°6′ N, the sun being in 18°26′. And the comet of 1577, on 29 December, was in 8°41′ with a latitude of 28°40′ N, the sun again being in approximately 18°26′. In both cases the earth was in the same place and the comet appeared in the same part of the sky; yet in the former case the tail of the comet (according to my observations and those made by others) was declining by an angle of 4½ degrees from the opposition of the sun toward the north, but in the latter case (according to the observations of Tycho) the declination was 21 degrees toward the south. Therefore, since refraction by the heavens has been rejected, the remaining possibility is to derive the phenomena of comets’ tails from some matter that reflects light.

Moreover, the laws which the tails of comets observe prove that these tails arise from the heads and ascend into regions turned away from the sun. For example, if the tails lie in planes of the comets’ orbits which pass through the sun, they always deviate from being directly opposite the sun and point toward the region which the heads, advancing in those orbits, have left behind. Again, to a spectator placed in those planes, the tails appear in regions directly turned away from the sun; while for observers not in those planes, the deviation gradually begins to be perceived and appears greater from day to day. Furthermore, other things being equal, the deviation is less when the tail is more oblique to the orbit of the comet, and also when the head of the comet approaches closer to the sun, especially if the angle of deviation is taken near the head of the comet. And besides, the tails that do not deviate appear straight, while those that do deviate are curved. Again, this curvature is greater when the deviation is greater, and more sensible when the tail, other things being equal, is longer; for in shorter tails the curvature is scarcely noticed. Then, too, the angle of deviation is smaller near the head of the comet and larger near the other extremity of the tail; and thus the convex side of the tail faces the direction from which the deviation is made and which is along a straight line drawn from the sun through the head of the comet indefinitely. Finally, the tails that are more extended and wider and that shine with a more vigorous light are a little more resplendent on their convex sides and are terminated by a less indistinct limit than on their concave sides. For all these reasons, then, the phenomena of the tail depend on the motion of the head and not on the region of the sky in which the head is seen; and therefore these phenomena do not come about through refraction by the heavens, but arise from the head supplying the matter. For as in our air the smoke of any ignited body seeks to ascend and does so either perpendicularly (if the body is at rest) or obliquely (if the body is moving sideways), so in the heavens, where bodies gravitate toward the sun, smoke and vapors must ascend with respect to the sun (as has already been said) and move upward either directly, if the smoking body is at rest, or obliquely, if the body by advancing always leaves the places from which the higher parts of the vapor have previously ascended. And the swifter the ascent of the vapor, the less the obliquity, namely in the vicinity of the sun and near the smoking body. Moreover, as a result of this difference in obliquity, the column of vapor will be curved; and since the vapor on that side of the column in the direction of the comet’s motion is a little more recent [i.e., more recently exhaled], so also the column will be somewhat more dense on that same side, and therefore will reflect light more abundantly and will be terminated by a less indistinct limit. I add nothing here concerning sudden and uncertain agitations of the tails, nor concerning their irregular shapes (which are sometimes described), because either these effects may arise from changes in our air and the motions of the clouds that may obscure those tails in one part or the other; or, perhaps, these effects may arise because some parts of the Milky Way may be confused with the tails as they pass by and may be considered as if they were parts of the tails.

Moreover, the rarity of our own air makes it understandable that vapors sufficient to fill such immense spaces can arise from the atmospheres of comets. For the air near the surface of the earth occupies a space about 850 times greater than water of the same weight, and thus a cylindrical column of air 850 feet high has the same weight as a foot-high column of water of the same width. Further, a column of air rising to the top of our atmosphere is equal in weight to a column of water about 33 feet high; and therefore if the lower part, 850 feet high, of the whole air column is taken away, the remaining upper part will be equal in weight to a column of water 32 feet high. And hence (by a rule confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere and that gravity is inversely as the square of the distance of places from the center of the earth), by making a computation using the corollary of prop. 22, book 2, I found that air, at a height above the surface of the earth of one terrestrial semidiameter, is rarer than here on earth in a far greater ratio than that of all space below the orbit of Saturn to a globe described with a diameter of one inch. And thus a globe of our air one inch wide, with the rarity that it would have at the height of one terrestrial semidiameter, would fill all the regions of the planets as far out as the sphere of Saturn and far beyond. Accordingly, since still higher air becomes immensely rare and since the comag or atmosphere of a comet is (as reckoned from the center) about ten times higher than the surface of the nucleus is, and the tail then ascends even higher, the tail will have to be exceedingly rare. And even if, because of the much thicker atmosphere of comets and the great gravitation of bodies toward the sun and the gravitation of the particles of air and vapors toward one another, it can happen that the air in the celestial spaces and in the tails of comets is not so greatly rarefied, it is nevertheless clear from this computation that a very slight quantity of air and vapors is abundantly sufficient to produce all those phenomena of the tails. For the extraordinary rarity of the tails is also evident from the fact that stars shine through them. The terrestrial atmosphere, shining with the light of the sun, by its thickness of only a few miles obscures and utterly extinguishes the light not only of all the stars but also of the moon itself; yet the smallest stars are known to shine, without any loss in their brightness, through the immense thickness of the tails, which are likewise illuminated by the light of the sun. Nor is the brightness of most cometary tails generally greater than that of our air reflecting the light of the sun in a beam, one or two inches wide, let into a dark room.

The space of time in which the vapor ascends from the head to the end of the tail can more or less be found by drawing a straight line from the end of the tail to the sun and noting the place where this straight line cuts the trajectory. For if the vapor has been ascending in a straight line away from the sun, then the vapor that is now in the end of the tail must have begun to ascend from the head at the time when the head was in that place of intersection. But the vapor does not ascend in a straight line away from the sun, but rather ascends obliquely, since the vapor retains the motion of the comet which it had before its ascent and this motion is compounded with the motion of its own ascent. And therefore the solution of the problem will be nearer the true one if the straight line that cuts the orbit is drawn parallel to the length of the tail, or rather (because of the curvilinear motion of the comet) if it diverges from the line of the tail. In this way I found that the vapor that was in the end of the tail on 25 January had begun to ascend from the head before 11 December and thus had spent more than forty-five days in its total ascent. But all of the tail that appeared on 10 December had ascended in the space of those two days that had elapsed after the time of the perihelion of the comet. The vapor, therefore, rose most swiftly at the beginning of its ascent, in the vicinity of the sun, and afterward proceeded to ascend with a motion always retarded by the vapor’s own gravity; and as the vapor ascended, it increased the length of the tail. The tail, however, as long as it was visible, consisted of almost all the vapor which had ascended from the comet’s head since the time of the comet’s perihelion; and that vapor which was the first to ascend, and which composed the end of the tail, did not disappear from view until its distance both from the sun which illuminated it and from our eyes became too great for it to be seen any longer. Hence it happens, also, that in other comets which have short tails, those tails do not rise up with a swift and continual motion from the heads of the comets and soon disappear, but are permanent columns of vapors and exhalations (propagated from the heads by a very slow motion that lasts many days) which, by sharing in the motion that the heads had at the beginning of the exhalations of the vapors, continue to move along through the heavens together with the heads. And hence again it may be concluded that the celestial spaces are lacking in any force of resisting, since in them not only the solid bodies of the planets and comets but also the rarest vapors of the tails move very freely and preserve their extremely swift motions for a very long time.

The ascent of the tails of comets from the atmospheres of the heads and the movement of the tails in directions away from the sun are ascribed by Kepler to the action of rays of light that carry the matter of the tail along with them. And it is not altogether unreasonable to suppose that in very free [or empty] spaces, the extremely thin upper air should yield to the action of the rays, despite the fact that gross substances in the very obstructed regions here on earth cannot be sensibly propelled by the rays of the sun. Someone else believes that there can be particles with the property of levity as well as gravity and that the matter of the tails levitates and through its levitation ascends away from the sun. But since the gravity of terrestrial bodies is as the quantity of matter in the bodies and thus, if the quantity of matter remains constant, cannot be intended and remitted [or increased and decreased], I suspect that this ascent arises rather from the rarefaction of the matter of the tails. Smoke ascends in a chimney by the impulse of the air in which it floats. This air, rarefied by heat, ascends because of its diminished specific gravity and carries along with it the entangled smoke. Why should the tail of a comet not ascend away from the sun in the same manner? For the sun’s rays do not act on the mediums through which they pass except in reflection and refraction. The reflecting particles, warmed by this action, will warm the aethereal upper air in which they are entangled. This will become rarefied on account of the heat communicated to it; and because its specific gravity, with which it was formerly tending toward the sun, is diminished by this rarefaction, it will ascend and will carry with it the reflecting particles of which the tail is composed. This ascent of the vapors is also increased by the fact that they revolve about the sun and endeavor by this action to recede from the sun, while the atmosphere of the sun and the matter of the heavens are either completely at rest or revolve more slowly only by the motion that they have received from the rotation of the sun.

These are the causes of the ascent of tails of comets in the vicinity of the sun, where the orbits are more curved, and the comets are within the denser (and, on that account, heavier) atmosphere of the sun and soon emit extremely long tails. For the tails which arise at that point, by conserving their motion and meanwhile gravitating toward the sun, will move about the sun in ellipses as the heads of the comets do; and by that motion they will always accompany the heads and will very freely adhere to them. For the gravity of the vapors toward the sun will no more cause the tails to fall afterward from the heads toward the sun than the gravity of the heads can cause them to fall from the tails. By their common gravity they will either fall simultaneously and together toward the sun or will be simultaneously retarded in their ascent; and therefore this gravity does not hinder the tails and heads of comets from very easily acquiring (whether from the causes already described or any others whatsoever), and afterward very freely preserving, any position in relation to one another.

The tails that are formed when comets are in their perihelia will therefore go off into distant regions together with their heads, and either will return to us from there together with the heads after a long series of years or rather, having been rarefied there, will disappear by degrees. For afterward, in the descent of the heads toward the sun, new little tails should be propagated from the heads with a slow motion, and thereupon should be immeasurably increased in the perihelia of those comets which descend as far as the atmosphere of the sun. For vapor in those very free spaces becomes continually rarefied and dilated. For this reason it happens that every tail at its upper extremity is broader than near the head of the comet. Moreover, it seems reasonable that by this rarefaction the vapor—continually dilated—is finally diffused and scattered throughout the whole heavens, and then is by degrees attracted toward the planets by its gravity and mixed with their atmospheres. For just as the seas are absolutely necessary for the constitution of this earth, so that vapors may be abundantly enough aroused from them by the heat of the sun, which vapors either—being gathered into clouds—fall in rains and irrigate and nourish the whole earth for the propagation of vegetables, or—being condensed in the cold peaks of mountains (as some philosophize with good reason)—run down into springs and rivers; so for the conservation of the seas and fluids on the planets, comets seem to be required, so that from the condensation of their exhalations and vapors, there can be a continual supply and renewal of whatever liquid is consumed by vegetation and putrefaction and converted into dry earth. For all vegetables grow entirely from fluids and afterward, in great part, change into dry earth by putrefaction, and slime is continually deposited from putrefied liquids. Hence the bulk of dry earth is increased from day to day, and fluids—if they did not have an outside source of increase—would have to decrease continually and finally to fail. Further, I suspect that that spirit which is the smallest but most subtle and most excellent part of our air, and which is required for the life of all things, comes chiefly from comets.

In the descent of comets to the sun, their atmospheres are diminished by running out into tails and (certainly in that part which faces toward the sun) are made narrower; and, in turn, when comets are receding from the sun, and when they are now running out less into tails, they become enlarged, if Hevelius has correctly noted their phenomena. Moreover, these atmospheres appear smallest when the heads, after having been heated by the sun, have gone off into the largest and brightest tails, and the nuclei are surrounded in the lowest parts of their atmospheres by smoke possibly coarser and blacker. For all smoke produced by great heat is generally coarser and blacker. Thus, at equal distances from the sun and the earth, the head of the comet which we have been discussing appeared darker after its perihelion than before. For in December it was generally compared to stars of the third magnitude, but in November to stars of the first magnitude and the second magnitude. And those who saw both describe the earlier appearance as a greater comet. For a certain young man of Cambridge, who saw this comet on 19 November, found its light, however leaden and pale, to be equal to Spica Virginis and to shine more brightly than afterward. And on 20 November (O.S.) the comet appeared to Montanari greater than stars of the first magnitude, its tail being 2 degrees long. And Mr. Storer, in a letter that came into our hands, wrote that in December, at a time when the largest and brightest tail was being emitted, the head of the comet was small and in visible magnitude was far inferior to the comet which had appeared in November before sunrise. And he conjectured that the reason for this was that in the beginning the matter of the head was more copious and had been gradually consumed.

It seems to pertain to the same point that the heads of other comets that emitted very large and very bright tails appeared rather dull and very small. For on 5 March 1668 (N.S.) at 7^h P.M., the Reverend Father Valentin Stansel, in Brazil, saw a comet very close to the horizon toward the southwest with a very small head that was scarcely visible, but with a tail so shining beyond measure that those who were standing on the shore easily saw its appearance reflected from the sea. In fact it had the appearance of a brilliantly shining torch with a length of 23 degrees, verging from west to south and almost parallel to the horizon. But so great a splendor lasted only three days, decreasing noticeably immediately afterward; and meanwhile, as its splendor was decreasing, the tail was increasing in size. Thus in Portugal the tail is said to have occupied almost a quarter of the sky—that is, 45 degrees—stretched out from west to east with remarkable splendor, and yet not all of the tail was visible, since in those regions the head was always hidden below the horizon. From the increase of the size of the tail and the decrease of the splendor, it is manifest that the head was receding from the sun and had been nearest to the sun at the beginning of its visibility, as was the case for the comet of 1680. And in the Anglo-Saxon Chronicle, one reads about a similar comet of 1106, “of which the star was small and dim ^h(as was that of 1680),^h but the splendor that came out of it stretched out extremely bright and like a huge torch toward the northeast” as Hevelius also has it from Simeon the Monk of Durham. This comet appeared at the beginning of February, and thereafter was seen at about evening toward the southwest. And from this and from the position of the tail it is concluded that the head was near the sun. “Its distance from the sun,” says Matthew of Paris, “was about one cubit, as from the third hour (more correctly, the sixth) until the ninth hour it emitted a long ray from itself.” Such also was that fiery comet described by Aristotle (Meteor. 1.6), “whose head, on the first day, was not seen because it had set before the sun, or at least was hidden under the sun’s rays; but on the following day, it was seen as much as it could be. For it was distant from the sun by the least possible distance, and soon set. Because of the excessive burning (of the tail, that is), the scattered fire of the head did not yet appear, but as time went on,” says Aristotle, “since (the tail) was now flaming less, the comet’s own face came back to (the head). And it extended its splendor as far as a third of the sky (that is, to 60 degrees). Moreover, it appeared in the winter (in the 4th year of the 101st Olympiad) and, ascending up to Orion’s belt, vanished there.”

The comet of 1618 which emerged out of the sun’s rays with a very large tail seemed to equal stars of the first magnitude, or even to surpass them a little, but a number of greater comets have appeared which had shorter tails. Some of these are said to have equaled Jupiter, others Venus or even the moon.

We said that comets are a kind of planet revolving about the sun in very eccentric orbits. And just as among the primary planets (which have no tails) those which revolve in smaller orbits closer to the sun are generally smaller, so it seems reasonable also that the comets which approach closer to the sun in their perihelia are for the most part smaller, since otherwise they would act on the sun too much by their attraction. I leave the transverse diameters of the orbits and the periodic times of revolution of the comets to be determined by comparing comets that return in the same orbits after long intervals of time. Meanwhile the following proposition may shed some light on this matter.

Proposition 42, Problem 22
To correct a comet’s trajectory that has been found [by the method of prop. 41].

OPERATION 1. Assume the position of the plane of the trajectory, as found by prop. 41, and select three places of the comet which have been determined by very accurate observations and which are as greatly distant from one another as possible; let A be the time between the first and second observations, and B the time between the second and third. The comet should be in its perigee in one of these places, or at least not far from perigee. From these apparent places find, by trigonometric operations, three true places of the comet in that assumed plane of the trajectory. Then through those places thus found, describe a conic about the center of the sun as focus, by arithmetical operations made along the lines of prop. 21, book 1; and let D and E be areas of the conic which are bounded by radii drawn from the sun to those places—namely, D the area between the first and second observations, and E the area between the second and third. And let T be the total time in which the total area D + E should be described by the comet, with the velocity as found by prop. 16, book 1.

OPERATION 2. Let the longitude of the nodes of the plane of the trajectory be increased by adding 20 or 30 minutes (which can be called P) to that longitude; but keep constant the inclination of that plane to the plane of the ecliptic. Then from the three aforesaid observed places of the comet, let three true places of the comet be found in this new plane (as in oper. 1); and also the orbit passing through those places, two of its areas (which can be called d and e) described between observations, and the total time t in which the total area d + e should be described.

OPERATION 3. Keep constant the longitude of the nodes in the first operation, and let the inclination of the plane of the trajectory to the plane of the ecliptic be increased by adding 20 or 30 minutes (which can be called Q) to that inclination. Then from the aforesaid three observed apparent places of the comet, let three true places be found in this new plane; and also the orbit passing through those places, two of its areas (which can be called δ and ϵ) described between observations, and the total time τ in which the total area δ + ϵ should be described.

Now take C so as to be to 1 as A to B, and take G to 1 as D to E, and g to 1 as d to e, and γ to 1 as δ to ϵ, and let S be the true time between the first and third observations; and carefully observing the signs + and −, seek the numbers m and n, by the rule that 2G − 2C = mG − mg + nG − nγ; and 2T − 2S = mT − mt + nT − nτ. And if, in the first operation, I designates the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of either node, I + nQ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K + mP will be the true longitude of the node. And finally if in the first, second, and third operations, the quantities R, r, and ρ designate the latera recta of the trajectory, and the quantities the transverse diameters [or latera transversa] respectively, R + mr − mR + nρ − nR will be the true latus rectum, and will be the true transverse diameter of the trajectory that the comet describes. And given the transverse diameter, the periodic time of the comet is also given. Q.E.I.

But the periodic times of revolving comets, and the transverse diameters [latera transversa] of their orbits, will by no means be determined exactly enough except by the comparison with one another of comets that appear at diverse times. If several comets are found, after equal intervals of times, to have described the same orbit, it will have to be concluded that all these are one and the same comet revolving in the same orbit. And then finally from the times of their revolutions the transverse diameters of the orbits will be given, and from these diameters the elliptical orbits will be determined.

To this end, therefore, the trajectories of several comets should be calculated on the hypothesis that they are parabolic. For such trajectories will always agree very nearly with the phenomena. This is clear not only from the parabolic trajectory of the comet of 1680, which I compared above with the observations, but also from the trajectory of that remarkable comet which appeared in 1664 and 1665 and was observed by Hevelius. He calculated the longitudes and latitudes of this comet from his own observations, but not very accurately. From the same observations our own Halley calculated the places of this comet anew, and then finally he determined the trajectory of the comet from the places thus calculated. And he found its ascending node in 21°13′55″, the inclination of its orbit to the plane of the ecliptic 21°18′40″, the distance of its perihelion from the node in the orbit 49°27′30″. The perihelion in 8°40′30″ with heliocentric latitude 16°1′45″ S. The comet in its perihelion on 24 November, 11^h52^m P.M. mean time [lit. equated time] at London, or 13^h8^m (O.S.) at Gdansk, and the latus rectum of the parabola 410,286, the mean distance of the earth from the sun being 100,000. How exactly the calculated places of the comet in this orbit agree with the observations will be evident from the following table calculated by Halley [p. 932].

In February, in the beginning of 1665, the first star of Aries, which I shall from here on call γ, was in 28°30′15″ with latitude 7°8′58″ N. The second star of Aries was in 29°17′18″ with latitude 8°28′16″ N. And a certain other star of the seventh magnitude, which I shall call A, was in 28°24′45″ with latitude 8°28′33″ N. And on 7 February at 7^h30^m Paris time (that is, 7 February at 8^h30^m Gdansk time) (O.S.), the comet made a right triangle with those stars γ and A, with the right angle at γ. And the distance of the comet from the star γ was equal to the distance between the stars γ and A, that is, 1°19′46″ along a great circle, and therefore it was 1°20′26″ in the parallel of the latitude of the star γ. Therefore, if the longitude 1°20′26″ is taken away from the longitude of the star γ, there will remain the longitude of the comet 27°9′49″. Auzout, who had made this observation, put the comet in roughly 27°0′. And from the diagram with which Hooke delineated its motion, it was then in 26°59′24″. Taking the mean, I have put it in 27°4′46″. From the same observation, Auzout took the latitude of the comet at that time to be 7° and 4′ or 5′ toward the north. He would have put it more correctly at 7°3′29″, since the difference of the latitudes of the comet and of the star γ was equal to the difference of the longitudes of the stars γ and A.

On 22 February at 7^h30^m in London (that is, 22 February at 8^h46^m Gdansk time), the distance of the comet from the star A, according to Hooke’s observation (which he himself delineated in a diagram) and also according to Auzout’s observations (delineated in a diagram by Petit), was a fifth of the distance between the star A and the first star of Aries, or 15′57″. And the distance of the comet from the line joining the star A and the first star of Aries was a fourth of that same fifth part, that is, 4′. And hence the comet was in 28°29′46″ with latitude 8°12′36″ N.

On 1 March at 7^h0^m at London (that is, 1 March at 8^h16^m Gdansk time), the comet was observed near the second star of Aries, the distance between them being to the distance between the first and second stars of Aries, that is, to 1°33′, as 4 to 45 according to Hooke, or as 2 to 23 according to [Gilles François] Gottigniez. Accordingly, the distance of the comet from the second star of Aries was 8°16″ according to Hooke, or 8′5″ according to Gottigniez; or, taking the mean, was 8′10″. And according to Gottigniez the comet had now just gone beyond the second star of Aries by about a space of a fourth or a fifth of the course completed in one day, that is, roughly 1′35″ (and Auzout agrees well enough with this), or a little less according to Hooke, say 1′. Therefore, if 1′ is added to the longitude of the first star of Aries, and 8′10″ to its latitude, the longitude of the comet will be found to be 29°18′, and its latitude 8°36′26″ N.

On 7 March at 7^h30^m in Paris (that is, 7 March at 8^h37^m Gdansk time), the distance of the comet from the second star of Aries, according to Auzout’s observations, was equal to the distance of the second star of Aries from the star A, that is, 52′29″. And the difference between the longitudes of the comet and of the second star of Aries was 45′ or 46′ or, taking the mean, 45′30″. And therefore the comet was in 0°2′48″. From the diagram of Auzout’s observations that Petit constructed, Hevelius determined the latitude of the comet to be 8°54′. But the engraver curved the path of the comet irregularly toward the end of its motion, and Hevelius corrected the irregular curving in a diagram of Auzout’s observations drawn by Hevelius himself, and thus made the latitude of the comet 8°55′30″. And by correcting the irregularity a little more, the latitude can come out to be 8°56′, or 8°57′.

This comet was also seen on 9 March and then must have been located in 0°18’ with latitude roughly 9°3½′ N.

This comet was visible for three months in all, during which time it passed through about six signs, completing about 20 degrees in each day. Its path deviated considerably from a great circle, being curved northward; and toward the end, its motion changed from retrograde to direct. And notwithstanding so unusual a path, the theory agrees with the observations from beginning to end no less exactly than theories of the planets tend to agree with observations of them, as will be clear upon examination of the table. Nevertheless, roughly 2 minutes must be subtracted when the comet was swiftest, and this will result by taking away 12 seconds from the angle between the ascending node and the perihelion, or by making that angle 49°27′18″. The annual parallax of each of the two comets (both this one and the previous one) was quite pronounced, and as a result it gave proof of the annual motion of the earth in its orbit.

The theory is confirmed also by the motion of the comet that appeared in 1683. It had a retrograde motion in an orbit whose plane contained almost a right angle with the plane of the ecliptic. Its ascending node (by Halley’s calculation) was in 23°23′; the inclination of its orbit to the ecliptic 83°11′; its perihelion in 25°29′30″; its perihelial distance from the sun 56,020, the radius of the earth’s orbit being taken at 100,000, and the time of its perihelion 2 July 3^h50^m. And the places of the comet in this orbit, as calculated by Halley and compared with the places observed by Flamsteed, are displayed in the following table.

The theory is confirmed also by the motion of the retrograde comet that appeared in 1682. Its ascending node (by Halley’s calculation) was in 21°16′30″. The inclination of the orbit to the plane of the ecliptic 17°56′0″. Its perihelion in 2°52′50″. Its perihelial distance from the sun 58,328, the radius of the earth’s orbit being 100,000. And the perihelion 4 September 7^h39^m mean [lit. equated] time. And the places calculated from Flamsteed’s observations and compared with the places calculated by the theory are shown in the following table.

^aThe theory is confirmed also by the retrograde motion of the comet that appeared in 1723. Its ascending node (by the calculation 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 in 12°15′20″. Its perihelial distance from the sun 998,651, the radius of the earth’s orbit being 1,000,000, and the mean [lit. equated] time of the perihelion being 16 September 16^h10^m. And the places of the comet in this orbit, as calculated by Bradley and compared with the places observed by himself and his uncle Mr. Pound, and by Mr. Halley, are shown in the following table.^a

By these examples it is more than sufficiently evident that the motions of comets are no less exactly represented by the theory that we have set forth than the motions of planets are generally represented by planetary theories. And therefore the orbits of comets can be calculated by this theory, and the periodic time of a comet revolving in any orbit whatever can then be determined, and finally the transverse diameters [lit. latera transversa] of their elliptical orbits and their aphelian distances will become known.

The retrograde comet that appeared in 1607 described an orbit whose ascending node (according to Halley’s calculation) was in 20°21′; the inclination of the plane of its orbit to the plane of the ecliptic was 17°2′; its perihelion was in 2°16′; and its perihelial distance from the sun was 58,680, the radius of the earth’s orbit being 100,000. And the comet was in its perihelion on 16 October at 3^h50^m. This orbit agrees very closely with the orbit of the comet that was seen in 1682. If these two comets should be one and the same, this comet will revolve in a space of seventy-five years and the major axis of its orbit will be to the major axis of the earth’s orbit as ∛(75 × 75) to 1, or roughly 1,778 to 100. And the aphelial distance of this comet from the sun will be to the mean distance of the earth from the sun as roughly 35 to 1. And once these quantities are known, it will not be at all difficult to determine the elliptical orbit of this comet. What has just been said will be found to be true if the comet returns hereafter in this orbit in a space of seventy-five years. The other comets seem to revolve in a greater time and to ascend higher.

But because of the great number of comets, and the great distance of their aphelia from the sun, and the long time that they spend in their aphelia, they should be disturbed somewhat by their gravities toward one another, and hence their eccentricities and times of revolutions ought sometimes to be increased a little and sometimes decreased a little. Accordingly, it is not to be expected that the same comet will return exactly in the same orbit, and with the same periodic times. It is sufficient if no greater changes are found to occur than those that arise from the above-mentioned causes.

And hence a reason appears why comets are not restricted to the zodiac as planets are, but depart from there and are carried with various motions into all regions of the heavens—namely, for this purpose, that in their aphelia, when they move most slowly, they may be as far distant from one another as possible and may attract one another as little as possible. And this is the reason why comets that descend the lowest, and so move most slowly in their aphelia, should also ascend to the greatest heights.

The comet that appeared in 1680 was distant from the sun in its perihelion by less than a sixth of the sun’s diameter; and because its velocity was greatest in that region and also because the atmosphere of the sun has some density, the comet must have encountered some resistance and must have been somewhat slowed down and must have approached closer to the sun; and by approaching closer to the sun in every revolution, it will at length fall into the body of the sun. But also, in its aphelion, when it moves most slowly, the comet can sometimes be slowed down by the attraction of other comets and as a result fall into the sun. So also fixed stars, which are exhausted bit by bit in the exhalation of light and vapors, can be renewed by comets falling into them and then, kindled by their new nourishment, can be taken for new stars. Of this sort are those fixed stars that appear all of a sudden, and that at first shine with maximum brilliance and subsequently disappear little by little. Of such sort was the star that Cornelius Gemma saw in Cassiopeia’s Chair on 9 November 1572; it was shining brighter than all the fixed stars, scarcely inferior to Venus in its brilliance. But he did not see it at all on 8 November, when he was surveying that part of the sky on a clear night. Tycho Brahe saw this same star on the 11th of that month, when it shone with the greatest splendor; and he observed it decreasing little by little after that time, and he saw it disappearing after the space of sixteen months. In November, when it first appeared, it equaled Venus in brightness. In December, somewhat diminished, it equaled Jupiter. In January 1573 it was less than Jupiter and greater than Sirius, and it became equal to Sirius at the end of February and the beginning of March. In April and May it was equal to stars of the second magnitude; in June, July, and August, to stars of the third magnitude; in September, October, and November, to stars of the fourth magnitude; in December and in January 1574, to stars of the fifth magnitude; and in February, to stars of the sixth magnitude; and in March, it vanished from sight. Its color at the start was clear, whitish, and bright; afterward it became yellowish, and in March of 1573 reddish like Mars or the star Aldebaran, while in May it took on a livid whiteness such as we see in Saturn, and it maintained this color up to the end, yet all the while becoming fainter. Such also was the star in the right foot of Serpentarius, the beginning of whose visibility was observed by the pupils of Kepler in 1604, on 30 September (O.S.); they saw it exceeding Jupiter in its light, although it had not been visible at all on the preceding night. And from that time it decreased little by little and in the space of fifteen or sixteen months vanished from sight. It was when such a new star appeared shining beyond measure that Hipparchus is said to have been stimulated to observe the fixed stars and to put them into a catalog. But fixed stars that alternately appear and disappear, and increase little by little, and are hardly ever brighter than fixed stars of the third magnitude, seem to be of another kind and, in revolving, seem to show alternately a bright side and a dark side. And the vapors that arise from the sun and the fixed stars and the tails of comets can fall by their gravity into the atmospheres of the planets and there be condensed and converted into water and humid spirits, and then—by a slow heat—be transformed gradually into salts, sulphurs, tinctures, slime, mud, clay, sand, stones, corals, and other earthy substances.