Principia Mathematica (Motte Translation, 1848) — part 35 of 45

Cor. 9. The mean distance of the centres of the earth and moon in the syzygies of the moon is 60 mean semi-diameters of the earth, and a 10th part of one semi-diameter : and in the moon's quadratures the mean dis- tance of the same centres is 61 mean semi- diameters of the earth, subduct- ing one 30th part of one semi-diameter.

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

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

PROPOSITION XXXVIII. PROBLEM XIX.

To find the figure of the moon^s body.

If the moon^s body were fluid like our sea, the force of the earth to raise that fluid in the nearest and remotest parts would be to the force of the moon by which our sea is raised in the places under and opposite to the moon as the accelerative gravity of the moon towards the earth to the ac- celerative gravity of the earth towards the moon, and the diameter of the moon to the diameter of the earth conjunctly ; that is, as 39,788 to 1, and 100 to 365 conjunctly, or as 1081 to 100. Wherefore, since our sea, by the force of the moon, is raised to 8f feet, the lunar fluid would be raised by the force of the earth to 93 feet ; and upon this account the figure of the moon would be a spheroid, w^hose greatest diameter produced would pass through the centre of the earth, and exceed the diameters perpendicu- lar thereto by 186 feet. Such a figure, therefore, the moon affects, and must have put on from the beginning. Q,.E.I.

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

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OF NATURAL PHILOSOPHY.

455

LEMMA I.

If APEp represent the earth uniformly dense, marked with the centre C, the poles P, p, arid the equator AE; and if about tJie centre C, loith the radius OP, loe sup-pose the sphere Pape to he described^ and. QR to denote the plane on luhicJi a right line, draivn from, the centre of the sun to the centre of the earth, insists at right angles; and further suppose that the several particles of the whole exterior earth PapAPepE, without the height of the said sphere, endeavour to recede toivards this side and that side from the plaiie (^R, every particle by a force pro- portional to its distance from that plane ; I say, in the first place, that the whole force and efficacy of all the particles that are situate in AE, the circle of the equator, and disposed uniformly wi},kout the globe, encompassing the same after the manner of a ring, to wheel the earth about its centre, is to the whole force and efficacy of as many particles in that point A of the equator which is at the greatest distance from, the plane QR, to wheel the earth about its centre with a like circidar motion, as \ to 2. And that circular m^otion loill be performed about an axis lying in the common section of the equator and the plane Q,R.

For let there be described from the centre K, with the diameter IL, the semi-circle INL. Suppose the semi-circumference INL to be divided into innumerable equal parts, and from the several parts N to the diameter

I K Mli

IL let fall the sines NM. Then the sums of the squares of all the sines NM will be equal to the sums of the squares of the sines KM, and both sums together will be equal to the sums of the squares of as many semi- diameters KN ; and therefore the sum of the squares of all the sines NM will be but half so great as the sum of the squares of as many semi-diam- eters KN.

Suppose now the circumference of the circle AE to be divided into the like number of little equal parts, and from every such part P a perpen- dicular FG to be let fall upon the plane QR, as well as the perpendicular AH from the point A. Then the force by which the particle F recedes.

456 THE MATHEMATICAL PRINCIPLES [BoOK III.

from the plane QR will (by supposition) be as that perpendicular FG ; and this force multiplied by the distance CG will represent the power of the particle F to turn the earth round its centre. And, therefore, the power of a particle in the place F will be to the power of a particle in the place A as FG X GO to AH X HC; that is, as FC^ to AC^ : and therefore the whole power of all the particles F, in their proper places F, will be to the power of the like number of particles in the place A as the sum of all the FC^ to the sum of all the AC^, that is (by what we have demonstrated before), as 1 to 2. Q.E.D.

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

^ LEMMA II.

T'he same things still supposed, I say, in the second place, that the total force or power of all the particles situated every where about the sphere to turn the earth about the said axis is to the lohole force of the like number of particles, iiniforinly disposed round the whole circunfiference of the equator AE in the fashion of a ring, to turn the whole earth about with the like circular motion, as 2 to 5.

Q For let IK be any lesser circle parallel to

^^^ the equator AE, and let LZ be any two equal

/ ^^ particles in this circle, situated without the / V\ sphere Fape ; and if upon the plane Q.R,

\ \ which is at right angles with a radius drawn I ] to the sun, we let fall the perpendiculars LM, ^^^ / / Im, the total forces by which these particles / //E recede from the plane QR will be propor- ^;^^//2 ^^'^^>// tional to the perpendiculars LM, Im. Let

-^^^ —-^^^ the right line hi be drawn parallel to the

plane Fapc, and bisect the same in X ; and through the point X. draw Nw parallel to the plane Q.R, and meeting the perpendiculars LM, Im, in N and n ; and upon the plane Q.R let fall the perpendicular XY. And the contrary forces of the particles L and I to wheel about the earth contrariwise are as LM X MC, and Im X mC ; that is, as LN X MC + NM X MC, and In X mC — 7im X mC ; or LN X MC + NM X MC, and LN X mC — NM X mC, and LN X Mttz — NM X MC + mC, the difference of the two, is the force of both taken together to turn the earth round. The affirmative part of this difference LN X Mm, or 2LN X NX, is to 2AH X HC, the force of two particles •of the same size situated in A, as LX^ to AC ; and the negative part NM

Book III.] of natural philosophy. 457

X MC + mC, or 2XY X CY, is to 2AH X HC, the force of the same two particles situated in A, as CX^ to AC^. And therefore the difference of the parts, that is, the force of the two particles L and I, taken together, to wheel the earth about, is to the force of two particles, equal to the former and situated in the place A, to turn in like manner the earth round, as LX^ — CX^ to AC 2. But if the circumference IK of the circle IK is supposed to be divided into an infinite number of little equal parts L, all the LX^ will be to the like number of IX ^ as 1 to 2 (by Lem. 1) ; and to the same number of AC^ as IX ^ to 2A02 ; and the same number of CX^ to as many AC^ as 2CX2 to 2AC^ Wherefore the united forces of all the particles in the circumference of the circle IK are to the joint forces of as many particles in the place A as IX ^ — 2CX2 to 2AC2 ; and therefore (by Lem. 1) to the united forces of as many particles in the cir- cumference of the circle AE as IX ^ — 2CX^ to AC^

Now if Pjt?, the diameter of the sphere, is conceived to be divided into an infinite number of equal parts, upon which a like number of circles IK are supposed to insist, the matter in the circumference of every circle IK will be as IX^ ; and therefore the force of that matter to turn the earth about will be as IX ^ into IX '^ — 2CX2 : and the force of the same matter, if it was situated in the circumference of ihe circle AE, would be as IX^ into AC^. And therefore the force of all the particles of the whole- matter situated without the sphere in the circumferences of all the circles is to the force of the like number of particles situated in the circumfer- ence of the greatest circle AE as all the IX ^ into IX- — 2CX- to as many IX ^ into AC^ ; that is, as all the AC^ — CX^ into AC^ — SCX^ to as many AC^ — CX^ into AC^ ; that is, as all the JlC —AAC^ X CX^ + 3CX'i to as many AC^ — AC^ x CX^ ; that is, as the whole fluent quantity, whose fluxion is AC* — 4AC2 x CX^ + SCX^, to the whole fluent quantity, whose fluxion is AC * — AC ^ X CX ^ ; and, there • fore, by the method of fluxions, as AC* X CX — fAC^ x CX^ -f |CX^ to AC* X CX— iAC2 X CX3 ; that is, if for CX we write the whole Cjp, or AC, as AAC^ to |AC^ ; that is, as 2 to 5. Q.E.D.

LEMMA m.

The same things still supposed, I say, in the third place, that the mo- tion of the v'hole earth about the axis above- nafned arising from the TTiotions of all the particles, ivill be to the motion of the aforesaid ring about the same axis in a proportion compounded of the proportion of tlie meatier in the earth to the meatier in the ring ; and the proportion of three squares of the quadrantal arc of any circle to two sqitares of its diameter, that is, in the proportion of the matter to the matter ^ and of the number 925275 to the number 1000000. For the motion of a cylinder revolved about its quiescent axis is to the

458 THE MATHEMATICAL PRINCIPLES [BoOK III.

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

HYPOTHESIS n.

If the other 'parts of the earth loere taken aioay^ and the remainins^ ring was carried alone about the sun in the orbit of the earth by the annual motiojt, ivhile by the diurnal 'motion it tvas in the niean ti'me revolved about its oivn axis inclined to the plane of the ecliptic by an angle of 23 1 degrees, the 'motion of the equinoctial points would be the sa'me, whether the i^ing loere fluid^ or lohether it consisted of a hard and rigid 'matter.

PROPOSITION XXXIX. PROBLEM XX.

Tofmd the precession of the equinoxes. The middle horary motion of the moon's nodes in a circular orbit, when the nodes are in the quadratures, was 16" 35'" 16''\ 36^. ; the half of which, 8" 17'" 3S'\ IS"", (for the reasons above explained) is the mean ho- rary motion of the node,s in such an orbit, which motion in a whole side- real year becomes 20° 11' 46". Because, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11' 46" in antecedentia ; and, if there were more moons, the motion of the nodes of every one (by Cor. 16, Prop. LXVI, Book 1) would be as its periodic time ; if upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11' 46" as 23^. 56', the sidereal day, to 27^. 7^. 43', the periodic time of our moon, that is, as 1436 to 39343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other^ or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible.

Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth Pa;?APe;;E, which lies without the sphere Fape (see fig. Lem. 11) ; and because this sphere is to that exterior earth as aC- to AC 2 — aC^, that is (seeing PC or aC the least semi-diameter of the earth is to AC the greatest semi-diameter of the same as 229 to 230), as 52441 to 459 ; if this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of

Book IIL] ' of natural philosophy. 459

the ring (by Lem. Ill) would be to the motion of the inner sphere as 459 to 52441 and 1000000 to 925275 conjunctly, that is, as 4590 to 485223; and therefore the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 to 489813. Wherefore if the ring ad- heres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 to 489813; upon which account the motion of the equinoctial points will be diminished in the same propor- tion. Wherefore the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° IT 46" as 1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369. But the forces by which the nodes of a number of moons (as we explained above), and therefore .by which the equinoctial points of the ring reotde (that is, the forces 3IT, in fig. Prop. XXX), are in the several particles as the distances of those particles from the plane Q.R ; and by these forces the particles recede from that plane : and therefore (by Lem. II) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PopAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5. Wherefore the annual regress of the equinoxes now would be to 20° 11' 46'' as 10 to 73092 ; that is, would be 9" 56'" 50'\

But because the plane of the equator is inclined to that of the ecliptic, this motion is to be diminished in the proportion of the sine 91706 (which is the co-sine of 23^ deg.) to the radius 100000 ; and the remain- ing motion will now be 9" 7'" 20'"". which is the annual precession of the equinoxes arising from the force of the sun.

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

If the height of the earth at the equator exceeds its height at the poles by more than 17} miles, the matter thereof will be more rare near the surface than at the centre ; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity.

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

460 THE MATHEMATICAL PRINCIPLES [BoOK IIL

LEMMA IV.

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

planets.

As the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a con- vincing proof of their descending into the regions of the planets ; for all the comets which move in a direct course according to the order of the signs, about the end of their appearance become more than ordinarily slow or retrograde, if the earth is between them and the sun; and more than ordinarily swift, if the earth is approaching to a heliocentric opposition with them ; whereas, on the other hand, those which move against the or- dei%)f the signs, towards the end of their appearance appear swifter than they ought to be, if the earth is between them and the sun ; and slower, and perhaps retrograde, if the earth is in the other side of its orbit. And these appearances proceed chiefly from the diverse situations which the earth acquires in the course of its motion, after the same manner as it hap- pens to the planets, which appear sometimes retrograde, sometimes more sloAvly, and sometimes more swiftly, progressive, according as the motion of the earth falls in with that of the planet, or is directed the contrary way. If the earth move the same way with the comet, but, by an angular motion about the sun, so much swifter that right lines drawn from the earth to the comet converge towardS' the parts beyond the comet, the comet seen from the earth, because of its slower motion, will appear retrograde ; and even if the earth is slower than the comet, the motion of the earth beins: subducted, the motion of the comet will at least appear retarded ; but if the earth tends the contrary way to that of the comet, the motion of the comet will from thence appear accelerated ; and from this apparent acceleration, or retardation, or regressive motion, the distance of the comet may be in-

G J] c B A. ferred in this manner. Let TQ,A,

TQB, raC, be three observed lon- o:itudes of the comet about the time of its first appearing, andXQF its last observed longitude before its disappearing. Draw the right line ABC, whose parts AB, BO, inter- "q^ cepted between the right lines QA

and QB, QB and Q.C, may be one to the other as the two times between the three first observations. Produce AC to G, so as AG may be to AB as the time between the first and last observation to the time between ihe first and second ; and join Q.G. Now if the comet did move uniformly in a ridit line, and the earth either stood still, or was likewise carried for- wards in a right line by an uniform motion, the angle TQG would be the

Book III.] of natural philosophy. 461

longitude of the comet at the time of the last observation. The angle, therefore, FQG, which is the difference of the longitude, proceeds from the inequality of the motions of the comet and the earth ; and this angle, if the earth and comet move contrary ways, is added to the angle TQ,G, and accelerates the apparent motion of the comet ; but if the comet move the same way with the earth, it is subtracted, and either retards the motion^of the comet, or perhaps renders it retrograde, as we have but now explained. This angle, therefore, proceeding chiefly from the motion of the earth, is justly to be esteemed the parallax of the comet ; neglecting, to wit, some little increment or decrement that may arise from the unequal motion of the comet in its orbit ; and from this parallax we thus deduce the distance of the comet. Let S represent the sun, acT v the orbis magnus, a the earth's place in the first observation, c the place of the earth in the third observation, T the place of the earth in the last observation, and TT a right line drawn to the beginning of Aries. Set off the angle TTV equal to the angle xaF, that IS, equal to the longitude of the comet at the time when the earth is in T ; join ac, and produce it to g, so as ag- may be to ac as AG to AC ; and g will be the place at which the earth would have arrived in the time of the last observation, if it had con- tinued to move uniformly in the right line ac. Wherefore, if we draw gT parallel to Tx, and make the angle TgY equal to the angle TQ,G, this angle TgY will be equal to the longitude of the comet s«en from the place g, and the angle TYg will be the parallax which arises from the earth's being transferred from the place g into the place T ; and therefore V v^^ill be the place of the comet in the plane of the ecliptic. And this place Y is commonly lower than the orb of Jupiter.

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

462 THE MATHEMATICAL PKINCIPLES [BoOK TIL

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

So far we have gone, without considering the obscuration which comets suffer from that plenty of thick smoke which encompasseth their heads, and throuo-h which the heads always shew dull, as through a cloud ; for by

Book III.] of natural philosophy. 463

how much the more a body is obscured by this smoke, by so much the more near it must be allowed to come to the sun, that it may vie with the plan- ets in the quantity of light which it reflects. Whence it is probable that the comets descend far below the orb of Saturn, as we proved before from their parallax. But, above all, the thing is evinced from their tails, which must be owing either to the sun's light reflected by a smoke arising from them, and dispersing itself through the aether, or to the light of their own heads. In the former case, we must shorten the distance of the comets, lest we be obliged to allow that the smoke arising from their heads is propagated through such a vast extent of space, and with such a velocity and expansion as will seem altogether incredible ; in the latter case, the whole light of both head and tail is to be ascribed to the central nucleus. But, then, if we suppose all this light to be united and condensed within the disk of the nucleus, certainly the nucleus will by far exceed Jupiter itself in splendor, especially when it emits a very large and lucid tail. If, therefore, under a less apparent diameter, it reflects more light, it must be much more illuminated by the sun, and therefore much nearer to it ; and the same argument will bring down the heads of comets sometimes within the orb of Yen us, viz., when, being hid under the sun's rays, they emit such huge and splendid tails, like beams of fire, as sometimes they do ; for if all that light was supposed to be gathered together into one star, it would sometimes exceed not one Yenus only, but a great many such united into one.

Lastly ; the same thing is inferred from the light of the heads, which increases in the recess of the comets from the earth towards the sun, and decreases in their return from the sun towards the earth ; for so the comet of the year 1665 (by the observations of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee ; but yet the splendor of its head was daily in- creasing, till, being hid under the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observations of the same Hevelius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orb in a day's time ; but from that time its diurnal motion was continually upon the increase, till Septefnber 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 like- wise proved from the diameter of its head, measured vfith a micrometer ; for, August 6, Hevelius found it only 6' 05", including the coma, which, September 2, he observed to be 9' 07", and therefore its head appeared far less about the beginning than towards the end of the motion ; though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelitrs declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased

464 THE MATHEMATICAL PRINCIPLES [BoOK IIL

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 velo- city, and were therefore then in their perigees : but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays : and the greatest splendor of their tails a little more early, when yet nearer to the sun. The head of the former comet (according to the observations of Cysatiis), December 1, appea-red greater than the stars of the first magnitude: and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12, the head of the latter comet was seen and observed by Mr. Flamsted, when but 9 degrees distant from the sun ; which is scarcely to be done in a star of the third magnitude. De- cember 15 and 17, it appeared as a star of the third magnitude, its lustre being diminished by the brightness of the clouds near the setting sun. December 26, when it moved with the greatest velocity, being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. January 3, it appeared as a star of the fourth. January 9, as one of the fifth. January 1 3, it was hid by the splendor of the moon, then in her increase. January 25, it was scarcely equal to the stars of the seventh magnitude. If we compare equal intervals of time on one side and on the other from the perigee, we shall find that the head of the comet, which at both intervals of time was far, but yet equall}^, removed from the earth, and should have therefore shone with equal splendor, ap- peared brightest on the side of the perigee towards the sun, and disap- peared on the other. Therefore, from the great difference of light in the one situation and in the other, we conclude the great vicinity of the sun and comet in the former ; for the light of comets uses to be regular, and to appear greatest when the heads move fastest, and are therefore in their perigees ; excepting in so far as it is increased by their nearness to the sun.

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

Cor. 2. From what has been said, we may likewise understand wliy comets are so frequently seen in that hemisphere in which the sun is, and so seldom in the other. If they were visible in the regions far above Saturn, they would appear more frequently in the parts opposite to the sun • for such as were in those parts would be nearer to the earth, whereas the presence of the sun must obscure and hide those that appear in the hemisphere in which he is. Yet, looking over the history of comets, I find that four or five times more have been seen in the hemisphere towards the sun than in the opposite hemisphere ; besides, without doubt, not a few which have been hid by the light of the sun : for comets descending

Book III.] of natural philosophy. 465

into our parts neither emit tails, nor are so well illuminated by the sun, as to discover themselves to our naked eyes, until they are come nearer to us than Jupiter. But the far greater part of that spherical space, which is described about the sun with so small an interval, lies on that side of the earth which regards the sun ; and the comets in that greater part are commonly more strongly illuminated, as being for the most part nearer to the sun.

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

PROPOSITION XL. THEOREM XX.

That the comets m^ove in some of the conic sections, having theii' foci

in the centre of the sun ; and by radii draion to the sun describe

areas proportional to the times.

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