Principia Mathematica (Motte Translation, 1848) — part 43 of 45

In the former case we must shorten the distance of the comets, lest we be obliged to allow that the smoke arising from their heads is propagated through such a vast extent of space, and with such a velocity of expansion, as will seem altogether incredible : in the latter case the whole light of both head and tail must be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself in splendor, especially when it emits a very large and lucid tail. If, therefore, under a less apparent diameter, it reflects more light, it must be much more illuminated by the sun, and therefore much nearer to it. So the comet tliat appeared Dec. 12 and 15, O.S. Amio 1679, at the time it emitted a very shining tail, whose splendor was equal to that of many stars like Jupiter, if their light were dilated and spread through so great a space, was, as to the mag- nitude of its nucleus, less than Jupiter (as Mr. Flamsted observed), and therefore was much nearer to the sun : nay, it was even less than Mercury. For on the 17th of that month, when it was nearer to the earth, it ap- peared to Cassini through a telescope of 35 feet a little less than the globe of Saturn. On the 8th of this month, in the morning. Dr. Halley saw the tail, appearing broad and very short, and as if it rose from the body of the sun itself, at that time very near its rising. Its form was like that of an extraordinary bright cloud ; nor did it disappear till the sun itself began to be seen above the horizon. Its splendor, therefore, exceeded the light of the clouds till the sun rose, and far surpassed that of all the stars together, as yielding only to the immediate brightness of the sun itself. Neither Mercury, nor Venus, nor the moon itself, are seen so near the rising sun. Imagine all this dilated light collected together, and to be crowded into the orbit of the comet's nucleus which was less than Mercury; by its splendor, thus increased, becoming so much more conspicuous, it will vastly exceed Mercury, and therefore must be nearer to the sun. On the 12th and 15th of the same month, this tail, extending itself over a much greater space, appeared more rare ; but its light was still so vigorous as to become visible when the fixed stars were hardly to be seen, and soon after to appear like a fiery beam shining in a wonderful manner. From its length, which was 40 or 50 degrees, and its breadth of 2 degrees, we may compute what the light of the whole must be.

This near approach of the comets to the sun is confirmed from the situ- tion they are seen in when their tails appear most resplendent ; for wheHi.

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the head passes by the sun, and lies hid under the solar rays, very bright and shining tails, like fiery beams, are said to issue from the horizon ; but afterwards, when the head begins to appear, and is got farther from the sun, that splendor always decreases, and turns by degrees into a paleness like to that of the milky way, but much more sensible at first ; after that vanishing gradually. Such was that most resplendent comet described by Aristotle, Lib. 1, Meteor. 6. " The head thereof could not be seen, because it set before the sun, or at least was hid under the sun's rays ; but the next day it was seen as well as might be ; for, having left the sun but a very little way, it set im-mediately after it ; and the scattered light of the head obscured by the too great splendour (of the tail) did not yet appear. But afterwards (says Aristotle), when the splendour of the tail was now dimin- ished (the head of), the comet recovered its native brightness. And the splendour of its tail reached now to a third part of the heavens (that is, to 60°). It appeared in the winter season, and, rising to Orion^s girdle, there vanished away." Two comets of the same kind are described by Justin^ Lib. 37, which, according to his account, '-'shined so bright, that the whole heaven seemed to be on fire ; and by their greatness filled up a fourth part of the heavens, and by their splendour exceeded that of the sun." By which last words a near position of these bright comets and the rising or setting sun is intimated (p. 494, 495). We may add to these the comet of the year 1101 or 1106, " the star of which was small and obscure (like that of 1 6S0) ; but the splendour arising from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Si??ieo7i, the monk of Durham. It appeared at the beginning of February about the evening in the south-west. From this and from the situation of the tail we may infer that the head was near the sun. Mattheio Paris says, '4t was about one cubit from the sun ; from the third [or rather the sixth] to the ninth hour sendins^ out a lono; stream of lio-ht." The comet of 1264, in July, or about the solstice, preceded the rising sun, sending out its beams with a great light towards the west as far as the middle of the heavens ; and at the beginning it ascended a little above the horizon : but as the sun went forwards it retired every day farther from the horizon, till it passed by the very middle of the heavens. It is said to have been at the beginning large and bright, having a large coma, which decayed from day to day. It is described in Append. Matth, Paris, Hist. Ang. after this manner : '^ An. Christi 1265, there appeared a comet so wonderful, that none then living had ever seen the like ; for, rising from the east with a great brightness, it extended itself with a great light as far as the middle of the hemisphere towards the west." The Latin original being somewhat barbarous and ob- scure, it is here subjoined. Ab oriente enim cum magno fulgore sur- gens, usque ad medium he??iispkcBrii versus occide7ite?n, omnia perlucidQ pertrahebat.

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"In the year 1401 or 1402, the sun being got below the horizon, there appeared in the west a bright and shining comet, sending out a tail up- wards, in splendor like a flame of fire, and in form like a spear, darting its rays from west to east. When the sun was sunk below the horizon, by the lustre of its own rays it enlightened all the borders of the earth, not per- mitting the other stars to shew their light, or the shades of night to darken the air, because its light exceeded that of the others, and extended itself to the upper part of the heavens, flaming," <fcc.. Hist. Byzant. Due. Mich. Nepot. From the situation of the tail of this comet, and the time of its first appearance, we may infer that the head was then near the sun, and went farther from him every day ; for that comet continued three months. In the year 1527, Aitg. 11, about four in the morning, there was seen al- most throughout Europe a terrible comet in Leo, which continued flaming an hour and a quarter every day. It rose from the east, and ascended to the south and west to a prodigious length. It was most conspicuous to the north, and its cloud (that is, its tail) was very terrible ; having, according to the fancies of the vulgar, the form of an arm a little bent holding a sword of a vast magnitude. In the year 1618, in ^e end of November ^ there began a rumour, that there appeared about sun-rising a bright beam, which was the tail of a comet whose head was yet concealed within the brightness of the solar rays. On Nov. 24, and from that time, the comet itself appeared with a bright light, its head and tail being extremely re- splendent. The length of the tail, which was at first 20 or 30 deg., in- creased till December 9, when it arose to 75 deg., but with a light much more faint and dilute than at the beginning. In the year 1668, March 5, N. S., about 7 in the evening, P. Valent. Estanciits, being in Brazil, saw a comet near the horizon in the south-west. Its head was small, and scarcely discernible, but its tail extremely bright and refulgent, so that the reflection of it from the sea was easily seen by those who stood upon the shore. This great splendor lasted but three days, decreasing very remark- ably from that time. The tail at the beginning extended itself from west to south, and in a situation almost parallel to the horizon, appearing like a shining beam 23 deg. in length. Afterwards, the light decreasing, its magnitude increased till the comet ceased to be visible ; so that Cassini, at Bologna, saw it {Mar. 10, 11, 12) rising from the horizon 32 deg. in length. In Portugal it is said to have taken up a fourth part of the heavens (that is, 45 deg.), extending itself from west to east with a notable brightness; though the whole of it was not seen, because the head in this part of the world always lay hid below the horizon. From the increase of the tail it is plain that the head receded from the sun, and was nearest to it at the beginning, when the tail appeared brightest.

To all these we may add the comet of 1680, whose wonderful splendor at the conjunction of the head with the sun was above described. But so

554 THE SYSTEM OF THE WORLD.

great a splendor argues the comets of this kind to have really passed near the fountain of light, especially since the tails never shine so much in their opposition to the sun ; nor do we read that fiery beams have ever ap- peared there.

Lastly, the same thing is inferred (p. 466, 467) from the light of the heads increasing in the recess of the comets from the earth towards the sun, and decreasing in their return from the sun towards the earth ; for so the last comet of the year 1665 (by the observation of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee : yet the splendor of its head was daily increasing, till, being hid by the sun's rays, the comet ceased to ap- pear. The comet of the year 1683 (by the observation of the same He- velius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orbit in a day's time. But from that time its diurnal motion was continually upon the increase till Septemler 4, when it arose to about 5 degrees ; and therefore in all this interval of time the comet was approaching to the earth. Which is likewise proved from the diameter of its head measured with a microme- ter ; for, August the 6th, Hevelius found it only 6' 5", including the coma ; which, September 2, he observed 9' 7". And therefore its head appeared far less about the beginning than towards the end of its motion, though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Where- fore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its access towards the earth. The comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with their greatest velocity, and were therefore th«n in their perigees ; but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays ; and the greatest splendor of their tails a little more early, when yet nearer to the sun. The head of the former comet, according to the observations of Cysatus, Dec. 1, appeared greater than the stars of the first magnitude ; and, Dec. 16 (being then in its perigee), of a small magnitude, and the splendor or clearness was much diminished. Jan. 7, Kepler, being uncertain about the head, left oif observing. Dec. 12, the head of the last comet was seen and observed by Flamsted at the distance of 9 degrees from the sun, which a star of the third magnitude could hardly have been. December 15 and 17, the same appeared like a star of the third magnitude, its splendor being diminished by the bright clouds near the setting sun. Dec. 26, when it moved with the greatest swiftness, and was almost in its perigee, it was inferior to Os Pegasi, a star of the third magnitude. Jan. 3, it appeared like a star of the fourth ; Jan. 9, like a star of the fifth. Ja^i. 13, it disappeared, by reason of the

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brightness of the moon, which was then in its increase. Jan. 25, it was scarcely equal to the stars of the seventh magnitude. If we take equal times on each hand of the perigee, the heads placed at remote distances would have shined equally before and after, because of their equal distances from the earth. That in one case they shined very bright, and in the other vanished, is to be ascribed to the nearness of the sun in the first case, and his distance in the other ; and from the great difference of the light in these two cases we infer its great nearness in the first of them ; for the light of the comets uses to be regular, and to appear greatest when their heads move the swiftest, and are therefore in their perigees ; except- ing in so far as it is increased by their nearness to the sun.

From these things I at last discovered why the comets frequent so much the region of the sun. If they were to be seen in the regions a great way beyond Saturn, they must appear oftener in those parts of the heavens that are opposite to the sun ; for those which are in that situation would be nearer to the earth, and the interposition of the sun would obscure the others : but, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere ; besides, without doubt, not a few which have been hid by the light of the sun ; for comets descending into our parts neither emit tails, nor are so well illuminated by the sun, as to discover them- selves to our naked eyes, till they are come nearer to us than Jupiter. But the far greater part of that spherical space, which is described about the sun with so small an interval, lies on that side of the earth which regards the sun, and the comets in that greater part are more strongly illuminated, as being for the most part nearer to the sun : besides, from the remarka- ble eccentricity of their orbits, it comes to pass that their lower apsides are much nearer to the sun than if their revolutions were performed in circles concentric to the sun.

Hence also we understand why the tails of the comets, while their heads are descending towards the sun, always appear short and rare, and are sel- dom said to have exceeded 15 or 20 deg. in length; but in the recess of the heads from the sun often shine like fiery beams, and soon after reach to 40, 50, 60, 70 deg. in length, or more. This great splendor and length of the tails arises from the heat which the sun communicates to the comet as it passes near it. And thence, I think, it may be concluded, that all the comets that have had such tails have passed very near the sun.

Hence also we may collect that the tails arise from the atmospheres of the heads (p. 487 to 488) : but we have had three several opinions about the tails of comets ; for some will have it that they are nothing else but the beams of the sun's light transmitted through the comets' heads, which they suppose to be transparent ; others, that they proceed from the refrac- tion which light suffers in passing from the comet's head to the earth ;

656 THE SYSTEM OF THE WORLD.

and, lastly, others, that they are a sort of clouds or vapour constantly rising from the comets' heads, and tending towards the parts opposite to the sun. The first is the opinion of such as are yet unacquainted with optics ; for the beams of the sun are not seen in a darkened room, but in consequence of the light that is reflected from them by the little particles of dust and smoke which are always flying about in the air ; and hence It is that in air impregnated with thick smoke they appear with greater brightness, and are more faintly and more difiicultly seen in a finer air; but in the heavens, where there is no matter to reflect the light, they are not to be seen at all. Light is not seen as it is in the beams, but as it is thence reflected to our eyes ; for vision is not made but by rays falling upon the eyes, and therefore there must be some reflecting matter in those parts where the tails of comets are seen ; and so the argument turns upon the third opinion ; for that reflecting matter can be no where found but in the place of the tail, because otherwise, since all the celestial spaces are -equally illuminated by the sun's light, no part of the heavens could appear with more splendor than another. The second opinion is liable to many difl&culties. The tails of comets are never seen variegated with those colours which ever use to be inseparable from refraction ; and the distinct transmission of the light of the fixed stars and planets to us is a demon- stration that the jBther or celestial medium is not endowed with any re- fractive power. For as to what is alledged that the fixed stars have been sometimes seen by the Egyptians environed with a coma or capillitium, because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds, as well as the radiation and scintillation of the fixed stars to the refractions both of the eyes and air ; for upon applying a tele- scope to the eye, those radiations and scintillations immediately disappear. By the tremulous agitation of the air and ascending vapours, it happens that the rays of light are alternately turned aside from the narrow space of the pupil of the eye ; but no such thing can have place in the much wider aperture of the object-glass of a telescope ; and hence it is that a scintillation is occasioned in the former case which ceases in the latter ; and this cessation in the latter case is a demonstration of the regular trans- mission of light through the heavens without any sensible refraction. But, to obviate an objection that may be made from the appearing of no tail in such comets as shine but with a faint light, as if the secondary rays were then too weak to affect the eyes, and for this reason it is that the tails of the fixed stars do not appear, we are to consider that by the means of telescopes the light of the* fixed stars may be augmented above an hundred fold and yet no tails are seen; that the light of the planets is yet more copious without any tail, but that comets are seen sometimes with huge tails when the light of their heads is but faint and dull ; for so it happened in the comet of the year 1680, when in the month of De-

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cember it was scarcely equal in light to the stars of the second magnitude, and yet emitted a notable tail, extending to the length of 40°, 50°, 60°. or 70°, and upwards ; and afterwards, on the 27th and 28th of January, the head appeared but as a star of the seventh magnitude ; but the tail (as was said above), with a light that was sensible enough, though faint, was stretched out to 6 or 7 degrees in length, and with a languishing light that was more difficultly seen, even to 12° and upwards. But on the 9th and 10th of Fehruary, when to the naked eye the head appeared no more, I saw through a telescope the tail of 2° in length. But farther ; if the tail was owing to the refraction of the celestial matter, and did deviate from the opposition of the sun, according as the figure of the heavens re- quires, that deviation, in the same places of the heavens, should be always directed towards the same parts : but the comet of the year 16S0, Decem- ber 28^^. 8]''. P. M. at London, was seen in Pisces, 8° 41', with latitude north 28° 6', while the sun was in Capricorn 18° 26'. And the comet of the year 1577, December 29, was in Pisces 8° 41', with latitude north 28° 40' ; and the sun, as before, in about Capricorn 18° 26'. In both cases the situation of the earth was the same, and the comet appeared in the same place of the heavens ; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of 4|- de- grees, whereas in the latter there was (according to the observation of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that the phaeno- mena of the tails of comets must be derived from some reflecting matter. That vapours sufficient to fill such immense spaces may arise from the comet's atmospheres, may be easily understood from what follows.

It is well known that the air near the surface of our earth possesses a space about 1200 times greater than water of the same weight ; and there- fore a cylindric column of air 1200 feet high is of equal weight with a cylinder of water of the same breadth, and but one foot high. But a cylinder of air reaching to the top of the atmosphere is of equal weight with a cylinder of water about 33 feet high ; and therefore if from the whole cylinder of air the lower part of 1200 feet high is taken away, the remaining upper part will be of equal weight with a cylinder of water 32 feet high. Wherefore at the height of 1200 feet, or two furlongs, the weight of the incumbent air is less, and consequently the rarity of the compressed air greater, than near the surface of the earth in the ratio of 33 to 32. And, having this ratio, we may compute the rarity of the air in all places whatsoever (by the help of Cor. Prop. XXII, Book II), sup- posing the expansion thereof to be reciprocally proportional to its compres- sion ; and this proportion has been proved by the experiments of Hooke and others. The result of the computation I have set down in the follow-

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THE SYSTEM OF THE WORLD.

ing table, in the first column of which you have the height of the air in miles, whereof 4000 make a semi-diameter of the earth : in the second the compression of the air, or the incumbent weight ; in the third its rarity or expansion, supposing gravity to decrease in the duplicate ratio of the distances from the earth's centre. And the Latin numeral characters are here used for certain numbers of ciphers, as O.xvii 1224 for 0,000000000000000001224, and 2mm xv for 26956000000000000000.

AIR'S

Height.

Compression.

Expansion.

0

33

1

5

17,8515

1,8486

10

9,6717

3,4151

20

2.852

11,571

40

0',2525

136,83

400

O.xvii 1224

26956 XV

4000

Olcv. 4465

73907 cii

40000

0,cxcii 1628

20263 clxxxix

400000

0,ccx 7895

41798 ccvii

4000000

OjCcxii 9878

33414 ccix

Infinite.

0,ccxii 6041

54622 ccix

But from this table it appears that the air, in proceeding upwards, is rarefied in such manner, that a sphere of that air which is nearest to the earth, of but one inch in diameter, if dilated with that rarefaction which it would have at the height of one semi-diameter of the earth, would fill all the planetary regions as far as the sphere of Saturn, and a great way be- yond ; and at the height of ten semi-diameters of the earth would fill up more space than is contained in the whole heavens on this side the fixed stars, according to the preceding computation of their distance. And though, by reason of the far greater thickness of the atmospheres of comets, and the great quantity of the circum-solar centripetal force, it may happen that ih.Q air in the celestial spaces, and in the tails of comets, is not so vastly rarefied, yet from this computation it is plain that a very small quantity of air and vapour is abundantly sufficient to produce all the ap- pearances of the tails of comets ; for that they are indeed of a very notable rarity appears from the shining of the stars through them. The atmos- phere of the earth, illuminated by the sun's light, though but of a few miles in thickness, obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the 'immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor.

Kepler ascribes the ascent of the tails of comets to the atmospheres of their heads, and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets' tails ; and without any great incongruity we may suppose that, in so free spaces, so fine a matter as that of the asther may yield to the action

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of the rays of the sun's light; though those rays are not able sensibly to move the gross substances in our parts, which are clogged with so palpable a re- sistance. Another author thinks that there may be a sort of particles of matter endowed with a principle of levity as well as others are with a power of gravity ; that the matter of the tails of comets may be of the former sort, and that its ascent from the sun may be owing to its levity ; but, considering the gravity of terrestrial bodies is as the matter of the bodies, and therefore can be neither more nor less in the same quantity of matter, I am inclined to believe that this ascent may rather proceed from the rarefaction of the matter of the comets' tails. The ascent of smoke in a chimney is owing to the impulse of the air with which it is entangled. The air rarefied by heat ascends, because its specific gravity is diminished, and in its ascent carries alono- with it the smoke with which it is eno^ao^ed.

O DO

And why may not the tail of a comet rise from the sun after the same manner ? for the sun's rays do not act any way upon the mediums which they pervade but by reflection and refraction ; and those reflecting parti- cles heated by this action, heat the matter of the aether which is involved with them. That matter is rarefied by the heat which it acquires, and because by this rarefaction the specific gravity, with which it tended towards the sun before, is diminished, it will ascend therefrom like a stream, and carry along with it the reflecting particles of which the tail of the comet is composed ; the impulse of the sun's light, as we have said, pro- moting the ascent.

But that the tails of comets do arise from their heads (p. 488), and tend towards the parts opposite to the sun, is farther confirmed from the laws which the tails observe ; for, lying in the planes of the comets' orbits which pass through the sun, they constantly deviate from the opposition of the sun towards the parts which the comets' heads in their progress along those ^ orbits have left; and to a spectator placed in those planes they appear in the parts directly opposite to the sun ; but as the spectator recedes from those planes, their deviation begins to appear, and daily becomes greater. And the deviation, cceter is paribus, appears less when the tail is more ob- . lique to the orbit of the comet, as well as when the head of the comet ap- proaches nearer to the sun .; especially if the angle of deviation is estimated near the head of the comet. Farther ; the tails which have no deviation appear straight, but the tails which deviate are likewise bended into a cer- tain curvature ; and this curvature is greater when the deviation is greater, and is more sensible when the tail, ccsteris paribus, is longer ; for in the shorter tails the curvature is hardly to be perceived. And the angle of deviation is less near the comet's head, but greater towards the other end of the tail, and that because the lower side of the tail regards the parts from which the deviation is made, and which lie in a right line drawn out infinitely from the sun through the comet's head. And the tails that are

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THE SYSTEM OF THE WORLD.

longer and broader, and shine with a stronger light, appear more resplendent and more exactly defined on the convex than on the concave side. Upon which accounts it is plain that the phasnomena of the tails of comets de- pend upon the motions of their heads, and by no means upon the places of the heavens in which their heads are seen ; and that, therefore, the tails of the comets do not proceed from the refraction of the heavens, but from their own heads, which furnish the matter that forms the tail ; for as in our air the smoke of a heated body ascends either perpendicularly, if the body is at rest, or obli{|uely if the body is moved obliquely, so in the heavens, where all the bodies gravitate towards the sun, smoke and vapour must (as we have already said) ascend from the sun, and either rise perpen- dicularly, if the smoking body is at rest, or obliquely, if the body, in the progress of its motion, is always leaving those places from which the upper or higher parts of the vapours had risen before. And that obliquity will be less where the vapour ascends with more velocity, to wit, near the smoking body, when that is near the sun ; for there the force of the sun by which the vapour ascends is stronger. But because the obliquity is varied, the column of vapour will be incurvated ; and because the vapour in the preceding side is something more recent, that is, has ascended something more lately from the body, it will therefore be something more dense on that side, and must on that account reflect more light, as well as be better defined ; the vapour on the other side languishing by degrees, and vanish- ing out of sight.

But it is none of our present business to explain the causes of the ap- pearances of nature. Let those things which we have last said be true or false, we have at least made out, in the preceding discourse, that the rays of light are directly propagated from the tails of comets in right lines through the heavens, in which those tails appear to the spectators wherever placed ; and consequently the tails must ascend from the heads of the comets towards the parts opposite to the sun. And from this principle we may __ determine anew the limits of their dis-

tances in manner following. Let S rep- resent the sun, T the earth, STA the elongation of a comet from the sun, and ATB the apparent length of its tail; and because the light is propagated from the extremity of the tail in the direction of the right line TB, that extremity must lie somewhere in the line TB. Suppose it in D, and join DS cutting Tx4l in C. Then, because the tail is al- ways stretched out towards the parts nearly opposite to the sun, and therefore

THE SYSTEM OF THE WORLD. 561

the sun, the head of the comet, and the extremity of the tail, lie in a right line, the comet's head will be found in C. Parallel to TB draw SA, meet- ing the line TA in A, and the comet's head C must necessarily be found between T and A, because the extremity of the tail lies somewhere in the infinite line TB ; and all the lines SD which can possibly be drawn from the point S to the line TB must cut the line TA somewhere between T and A. Wherefore the distance of the comet from the earth cannot exceed the interval TA, nor its distance from the sun the interval SA beyond, or ST on this side the sun. For instance : the elongation of the comet of 16S0 from the sun, Dec. 12, was 9°, and the length of its tail 35° at least. If, therefore, a triangle TSA is made, whose angle T is equal to the elon- gation 9°, and angle A equal to ATB, or to the length of the tail, viz., 35°, then SA will be to ST, .that is, the limit of the greatest possible distance of the comet from the sun to the semi -diameter of the orbis magnus, as the sine of the angle T to the sine of the angle A, that is, as about 3 to 11. And therefore the comet at that time was less distant from the sun than by j\ of the earth's distance from the sun, and consequently either was within the orb of Mercury, or between that orb and the earth. Again, Dec. 21, the elongation of the comet from the sun was 32|°, and the length of its tail 70°. Wherefore as the sine of 32|° to the sine of 70°, that is, as 4 to 7, so was the limit of the comet's distance from the sun to the dis- tance of the earth from the sun, and consequently the comet had not then got without the orb of Venus. Dec. 28, the elongation of the comet from the sun was 55°, and the length of its tail 56° ; and therefore the limit of the comet's distance from the sun was not yet equal to the distance of the earth from the same, and consequently the comet had not then got without the earth's orbit. But from its parallax we find that its egress from the orbit happened about Jan. 5, as well as that it had descended far within the orbit of Mercury. Let us suppose it to have been in its perihelion Dec. the 8th, when it was in conjunction with the sun ; and it will follow that in the journey from its perihelion to its exit out of the earth's orbit it had spent 28 days ; and consequently that in the 26 or 27 days fol- lowing, in which it ceased to be farther seen by the naked eye, it had scarcely doubled its distance from the sun ; and by limiting the distances of other comets by the like arguments, we come at last to this conclu- sion,— that all comets, during the time in which they are visible by us, are within the compass of a spherical space described about the sun as a centre, with a radius double, or at most triple, of the distance of the earth from the sun.

And hence it follows that the comets, during the whole time of their appearance unto us, being within the sphere of activity of the circum- solar force, and therefore agitated by the impulse of that force, will (by Cor. 1, Prop. XII, Book I, for the same reason as the planets) be made ta

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562 THE SYSTEM OF THE WORLD.