The revolution of the earth means that centripetal forces are actually directed to the bodies of the sun, of the earth, and other planets.

The moon revolves about our earth. A line drawn from the earth’s center to the moon coincides with the lunar cycles, from the moon’s velocity compared with its diameter.

The moon’s motion is slower when its diameter is less (and therefore is farther from earth). Its motion is faster when its diameter is greater.

The revolutions of Jupiter’s moons are more regular (p. 386). They draw concentric circles with Jupiter by equable motions.

Saturn’s moons revolve circularly.

Venus and Mercury revolve around the sun just like moons revolve around planets.

This is why they shine with a full face, when they are in those parts of their orbs which in respect of the earth lie beyond the sun when they appear half full, they are in those parts while over when horned, in those parts which lie between against the sun

the earth and the sun and sometimes they pass over the sun s disk, when between the eirth and the sun. directly interposed And Venus, with a motion almost uniform, describes an orb nearly cir cular and concentric with the sun.

Mercury has a more eccentric motion, makes remarkable ap and goes off again by turns but it is always swifter the sun, and therefore by a radius drawn to the sun still proaches to the sun, as it is near to

describes areas proportional to the times.

Lastly, that the earth describes around the sun, or the sun about the by a radius from the one to the other, areas exactly proportional to the times, is demonstrable from the apparent diameter of the sun com earth, pared with its apparent motion. These are astronomical experiments

from which it follows, by Prop. in the Book first of our Pn /triples, and their Corollaries (p. 213, 214). that there are centripetal forces actually directed (either accu rately or without considerable error) to the centres of the earth, of Jupi of S.iturn, and of the sun. In Mercury, Venus, Mars, and the lesser wheie are must planets, experiments wanting, the arguments from ter, analogy be allowed in their place.

That those forces (p. 212, 213, 214) decrease in the duplicate propor tion of the distances from the centre of every planet, appears by Cor. VI, for the periodic times of the satellites of Jupiter are Prop. IV, Book 1 one to another (p. 386, 387) in the sesquiplicate proportion of their distances from the centre of this planet. This proportion has been long ago observed in those satellites and Mr. Flamsted, who had often measured their distances from Jupiter by the micrometer, and by the eclipses of the satellites, wrote to me, that it holds

the accuracy that possibly can be discerned by our senses.

the dimensions of their orbits taken by the micrometer, and reduced to the mean distance of Jupiter from the earth, or from the sun, to all sent me together with the times of their revolutions, as follows

The greatest elongation of the satellitesfrorn the centre of Jupiter as seen from the sun.

the periodic times of those satellites, by the observations of Mr. Flamsted, are l 18 28 above. . And the distances thence computed are 5,578 8,878 accurately agreeing with the distances by observation. Cassini assures us 14,168 | | | 24,968, 388, 389) that the same proportion is observed But a longer course of observations is planets. (p. in the circum-saturnal required before we can have a certain and accurate theory of those planets. In the circum -solar planets, Mercury and Venus, the same proportion holds with great accuracy, according to the dimensions of their orbs, as determined by the observations of the best astronomers. That Mars which it and 390) ; gibbous in And is revolved about the sun is demonstrated from the phases shews, and the proportion of its apparent diameters for its from its (p. 388, 389, near conjunction with the sun, and certain that it surrounds the sun. appearing quadratures, it is fall since its diameter appears about five times greater tion to the sun than when in conjunction therewith, and when its in opposi

distance from reciprocally as its apparent diameter, that distance will be when in opposition to than when in conjunction with about both in cases its distance from the sun will be nearly about but the sun; distance which is inferred from its gibbous appearance the with same the the earth five is times less

And as it encompasses the sun at almost equal dist n- quadratures. of the earth is very unequally distant, so by radii drawn in but ces, respect but by radii drawn to the to the sun it describes areas nearly uniform sometimes it sometimes is swift, earth, stationary, and sometimes retrograde. in the ; That Jupiter, in a higher orb than Mars, is likewise revolved about the a motion nearly equable, as well in distance as in the areas des with sun, cribed, 1 infer thus.

Mr. Flamsted assured me, by letters, that all the eclipses of the inner satellite which hitherto have been well observed do agree with his most theory so nearly, as never to differ therefrom by two minutes of time in the outmost but one, that in the outmost the error is little greater ; ; scarcely three times greater ; that in the innermost but one the difference much greater, yet so as to agree as nearly with his computation? and that he computes those as the moon does with the common tables eclipses only from the mean motions corrected by the equation of light dis is indeed ; covered and introduced by Mr. Rower. Supposing, then, that the theory differs by a less error than that of 2 from the motion of the outmost sat hitherto described, and taking as the periodic time 16 18 h 5 2 in time, so is the whole circle or 360 to the arc 1 the error ol

Mr. Flamsted s computation, reduced to the satellite s orbit, will be less that is, the longitude of the satellite, as seen from tlie centre than 1 ellite as

of Jupiter ; will be determined with a less error than

is in the middle of the shadow, that longitude is the same with the heliocentric longitude of Jupiter and, therefore, the hypothesis which Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and the satellite ; fas to the motion of Jupiter) lately corrected by himself, rightly represents that longitude within a less error than 1 48" but by this longitude, to gether with the geocentric longitude, which is always easily found, the distance of Jupiter from the sun is determined which must, therefore, be the

For that greatest error very same with that which the hypothesis exhibits. of I that can happen in the heliocentric longitude is almost insensi ble, and quite to be neglected, and perhaps may arise from some yet undis covered eccentricity of the satellite

but since both longitude and distance are rightly determined, it follows of necessity that Jupiter, by radii drawn to the sun. describes areas so conditioned as the hypothesis requires, that is. proportional to the times.

The same thing may be concluded of Saturn from his satellite, by the observations of Mr. Huygens and Dr. Halley ; though a longer series of observations is yet wanting to confirm the thing, and to bring it under a sufficiently exact computation.

If Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always to go forward with a motion nearly uniform (p. 389). And from the very great inequality of its apparent geocentric motion, we infer (by Prop. Ill Cor. IV) that the force by which Jupiter is turned out of a rectilinear course, and made to revolve in an orb, is not directed to the centre of the earth. same argument holds good in Mars and in Saturn. Another centre of these forces is therefore to be looked for (by Prop. II and III, and the Corollaries of the latter), about which the areas described by radii inter vening may be equable and that this is the sun, we have proved already And the ; Mars and Saturn It may be nearly, but accurately enough in Jupiter. other force some are that and the sun equally impelled by alledged planets and in the direction of parallel lines but by such a force (by Cor. VI of in

Laws of Motion) no change would happen in the situation of the but our business is one to another, nor any sensible eifect follow planets with the causes of sensible effects. Let us, therefore, neglect every such the

force as imaginary and precarious, and of no use in the phenomena of the and the whole remaining force by which Jupiter is impelled will be directed (by Prop. Ill, Cor. I) to the centre of the sun. heavens The ; distances of the planets from the sun come out the same, whether, we place the earth in the centre of the system, or the sun with with Tycho, Copernicus : and we have already proved that these distances are true ir. Jupiter.

Kepler and Bullialdiis have, with great care (p. 388), determined the

from the sun and hence it is that their table.-? listances of the planets

And in all the planets, in Jupiter and Mars, agree best with the heavens. Saturn and the earth, as well as in Venus and Mercury, the cubes of their distances are as the squares of their periodic times and therefore (by Cor. 6, Prop. 4) the centripetal circum-solar force throughout all the planetary regions decreases in the duplicate proportion of the distances from the

In examining this proportion, we are to use the mean distances, or the transverse semi-axes of the orbits (by Prop. XV). arid to neglect those sun.

little fractions, which, in denning the orbits, may have arisen from the in may be ascribed to other causes which we sensible errors of observation, or And shall afterwards explain.

thus we shall always find the said propor tion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, from the sun, drawn from the observations of as tronomers, are, according to the computation of Kepler, as the numbers 95 LOGO, 519650, 152350, 100000, 72400, 3S806; by the computation of /iHllialdus, as the numbers 95419S, 522520, 152350, 100000, 72393, 38585 and from the periodic times they come out 953806, 520116, 152399, ; 100000, Their 38710. 72333, distances, according to and Kepler scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times, fall in between them. That the circum-terrestrial force likewise decreases in the duplicate pro Ktillwldus, portion of the distances, I infer thus. The mean distance of the moon from the centre of the earth, is, in semi- diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bidliuldus, Hevelius, and Ricciolns, 59 1 to Vendelin, 60 according to Tycho, 56

Flamsted, 59| 60 1 to Kir- to Copernicus, ; : 62i ( p 391, 392, 393).

Cut Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) cher, . to exceed those of the fixed stars, in the horizon, did thereby and that by about four or minutes five augment the horizontal parallax of the moon by about the like number of minutes that is, by about the 12th or 15th Correct this error, and the distance will be part of the whole parallax. 60 or 61 come semi-diameters of the earth, nearly agreeing with what ; others have determined. Let us, then, assume the mean distance of the moon 60 semi-diameters d its periodic time in respect of the fixed stars 27 7 h 43 of the earth, and . as astronomers have determined revolved in our air,

(by Cor. 6, Prop. 4) a body near the surface of the earth supposed at rest, by means of a centripetal force it. which should be to the same force at the dis moon in the reciprocal duplicate proportion of the distances from the centre of the earth, that is, as 3600 to 1, would (secluding the tance of the resistance of the air) complete a revolution in l Suppose the circumference of the earth to be h 24 123249600 Paris

has been determined by the late mensuration of the French (vide p. 406) then the sume body, deprived of its circular motion, and falling by the ; impulse of the same centripetal force as before, would, in one second of 15-^ Paris feet.

This we infer by a calculus formed upon Prop. XXXYI, and it agrees with what we observe in all bodies about the earth. For by the experi ments of pendulums, and a computation raised thereon, Mr. Hnygens has demonstrated that bodies falling by all that centripetal force with which time, describe it is) they are impelled near the surface of the earth, one second of time, describe 15 T ^ Paris feet. whatever nature (of do, in But if the earth is supposed to move, the earth and moon together (by IV of the Laws of Motion, and Prop. LVID will be revolved about Cor. common their Ana centre of gravity. same periodic the 27 time, 1 . 7 h 43

the moon (by Prop. LX) will in with the same circum terrestrial force diminished in the duplicate proportion of the distance, describe an orbit whose semi-diameter is to the semi-diameter of the former orbit, that is, to 60 semi-diameters of the and moon earth, as the body of the earth mean apparent diameter 31^

sum of both the bodies of the earth two mean proportionals between this sum and the that is, if we suppose the moon (on account of its to the first of 43|2, or as about 128 ) to be to 127. about And ^ of the earth, as 43 to therefore the semi-diameter of the orbit, that is, the distance between the centres of the moon and almost the same earth, will in this case be 60^ semi-diameters of the earth, with that assigned by Copernicus, which the Tychonic observations by no means disprove and, therefore, the duplicate proportion of the decrement I have neglected the increment in this distance. of the force holds ; good of the orbit which arises from the action but if that of the sun as inconsiderable ; subducted, the true distance will remain about 60|- semi- is diameters of the earth. But farther (p. 390) this proportion of the decrement of the forces is confirmed from the eccentricity of the planets, and the very slow motion for (by the Corollaries of Prop. XLV) in no other pro- of their ; apses once in every revolution descend to portion could the circum-solar planets their least and once ascend to their greatest distance from the sun, and the small error from the du distances remain immoveable. places of those of the apses considerable in a motion would produce plicate proportion ; A many enormous. innumerable revolutions, hardly any such motion ha& every revolution, but in

Some astronomers affirm that there is no such motion; others reckon it no greater than what been perceived in the orbs of the circum-solar planets. arise from the causes hereafter to be assigned, and is of no mo may ment easily in the present question.

We may even neglect the motion of the moon s apsis (p. 390, 391), which far greater than in the circum-solar planets, amounting in every revolu and from this motion it is demonstrable that the tion to three degrees is ; circum-terrestrial force decreases in no less than the duplicate, but far less than the triplicate proportion of the distance for if the duplicate propor ; tion was gradually changed into the triplicate, the motion of the apsis would thereby increase to infinity; and, therefore, by a very small muta tion, would exceed the motion of the moon s from the action of the circum-solar apsis. force, as we This slow motion arises shall afterwards explain.

But, secluding this cause, the apsis or apogeon of the moon will be fixed, and the duplicate proportion of the decrease of the circum-terrestrial force in different distances from the earth will accurately take place. Now has been established, we that this proportion among themselves forces of the several planets (p. may compare the 391). In the mean distance of Jupiter from the earth, the greatest elongation of the outmost satellite from Jupiter s centre (by the observations of Mr. Flamsted] is and therefore the distance of the satellite from the

centre of Jupiter is mean to the distance of Jupiter from tne centre of the 52012, but to the mean distance of Venus from the centre d and 224f d of the sun as 124 to 7234; and their periodic times are 16 sun as 124 to

and from hence (according we the squares of the times, impelled towards Jupiter wards the sun as 442 to to Cor. II, Prop. IV), dividing the distances by infer that the force 143 by which the satellite is the force by which Venus is impelled to and if we diminish the force by which the to is

satellite is impelled in the duplicate proportion of the distance 124 to 7234, we shall have the circum-jovial force in the distance of Venus from yW the sun to the circum-solar force by which Venus is impelled as to 143, or as 1 to 1100; wherefore at equal distances the circum-solar force 1100 times greater than the circum-jovial. And, by the like computation, from the periodic time of the satellite ot Saturn 15 22 h and its greatest elongation from Saturn, while that planet it follows that the distance of this is in its mean distance from us, 3 satellite from Saturn s centre is to the distance of Venus from the sun as 92| to 7234; and from thence that the absolute circum-solar force is 2360 is times greater than the absolute circum-saturnal. From the regularity of the heliocentric and irregularity of the geocen tric motions of Venus, of Jupiter, and the other planets, it is evident (by Cor. 4, Prop. 3) that the circum-terrestrial force, compared with the cir is very small.

Ricciolus and Vendelin have severally tried to determine the sun s parallax from the moon s dichotomies observed by the telescope, and they agree cum-solar, it does not exceed half a minute. Kepler, from Ti/cho s observations and his own, found that the parallax ofo22 Mars

insensible, even in opposition to the sun, thing greater than the sun when that parallax is some

Flamsted attempted the same parallax with the micrometer in the perigeon position of Mars. But he never found it above 25". He thus concluded that the sun s parallax at most to be 10".

It follows that the distance of the moon from the earth bears no greater proportion to the distance of the earth from the sun than 29 to 10,000, nor to the distance of Venus from the sun than 29 to 7,233.

From which distances, together with the periodic times, it is easy to infer that the absolute circumsolar force is greater than the absolute circumterrestrial force at least 229,400 times.

Ricciolus and Vcitdelin observed that the sun’s parallax was less than half a minute.

It follows that the absolute circum-solar force exceeds the absolute circum-terrestrial force 8,500 times.

By the like computations, I discovered an analogy between the forces and the bodies of the planets. But we must first get the apparent diameters of the planets.

Flamsted measured the diameter of:

• Jupiter to 40-41" (micrometer).
• Saturn’s ring to 50"
• the sun to 32'13"

Huygens and Halley measures the diameter of Saturn’s ring as 4 to 9.

• To Gulletins, it was as 4 to 10
• To Hooke, (by a telescope of 60 feet) it was as 5 to 12

From the mean proportion, 5 to 12, the diameter of Saturn s body is around 21".

Because of the unequal refrangibility of light, all lucid points are dilated by the telescope, and in the focus of the object-glass possess a circular space whose as about the 50th part of the aperture of the glass.

Towards the circumference the light is so rare as hardly to move the sense.

But towards the middle, it is of greater density and is sensible enough. It makes a small lucid circle, whose breadth varies according to the splendor of the lucid point. But it is generally about the 3d, or 4th, or 5th part of the breadth of the whole.

the small circle Let represent the circle of the whole light; according ABD PQ of the denser and clearer light; C the centre of both; CA, CB, semi-di the ameters of the greater circle containing a right angle at C the diagonal of that square comprehended under these semi-diameters ; ; square; PG EGH an hyperbola with the centre C ACBE AB and asymptotes CA, CB BC, and meeting a perpendicular erected from any point P of the line the hyperbola in G, and the right lines AB, AE, in K and F : and the density of the light in any place P, will, by my computation, be as the FG, and therefore at the centre infinite, but near the circumference line And very small. the whole light within the small circle PQ, is to the to the trian- without as the area of the quadrilateral figure gle PKB. And we required to Hence move to te there light, begins to be less than what are to understand the small circle terminated, where FG, the density of the is CAKP PQ, the sense. was, that, at the distance of 191 382 feet, a fire of 3 feet in di in ameter, through a telescope of 3 feet, appeared to Mr. Picart of 14 and hence it is breadth, when it should have appeared only of it

that the brighter fixed stars appear through the telescope as of in or diameter, and that with a good full light but with a fainter light they appear to run out to a greater breadth. Hence, likewise, it was that He-

by diminishing the aperture of the telescope, did cut off a great part of the light towards the circumference, and brought the disk of the star to be more distinctly defined, which, though hereby diminished, did yet ap velius, But Mr. Hvyg etis, only by clouding the smoke, did so effectually extinguish this scattered that the fixed stars appeared as mere points, void of all sensible pear as of

eye-glass with a light, Hence in diameter. little was that Mr. Huygens, from the breadth of bodies whole light of the planets, reckoned their diameters than hav others measured them by the micrometer for the greater breadth. also it interposed to intercept the scattered light, which could not be seen before for the stronger light of the Lastly, planet, when the planet is hid, appears every way farther spread. from hence it is that the planets appear so small in the disk of the sun, For to Hevelius, Galletius, and Dr. being lessened by the dilated light. or and Venus appeared Halley, Mercury did not seem to exceed

to Horrox but 1 Mr. Crabtrie only 1 3 though by the men surations of Hevelius and Hu&enius without the sun s disk, it ought to have been seen at least 1 24". Thus the apparent diameter of the moon, which in 1 684, a few days both before and after the sun s eclipse, was in the eclipse itself did not measured at the observatory of Paris 31

and therefore the diameters of the planets when without the sun, and to be augmented when are to be diminished But seconds. the some errors seem to be less than usual in within it, by seem to exceed made by the micrometer. So from the diameter determined of the shadow, by the eclipses of the satellites, Mr. Flamsted of Jupiter was to the greatest elongation of the diameter that semi- found the mensurations that are Wherefore since that elongation is the outmost satellite as 1 to 24,903. 8 13 the diameter of Jupiter will be 39^-" and, rejecting the scattered , ; diameter found by the micrometer 40" or 41" will be reduced to and the diameter of Saturn 21" is to be diminished by the like cor light, the 39|-" 5 rection, and to be reckoned 20", or something But less. am (if I not mis taken) the diameter of the sun, because of its stronger light, is to be dimin ished something more, and to be reckoned about 32 or 32 6 1 . , bodies so different in magnitude should come so near to an analogy with their forces, is not without some mystery (p. 400). That It may be that the remoter planets, for want of heat, have not those me substances and ponderous minerals with which our earth abounds and that the bodies of Venus and Mercury, as they are more exposed to the tallic ; s heat, are also harder baked, and more compact. For, from the experiment of the burning-glass, we see that the heat in and this density increases in the recipro creases with the density of light from whence the cal duplicate proportion of the distance from the sun sun ; ; san s seasons. silver proved to be sevenfold its heat in our summer But with this heat our water boils and those heavy fluids, quick heat in Mercury is ; and the I have tried by the spirit of vitriol, gently evaporate, as and therefore there can be no fluids in Mercury but what thermometer are heavy, and able ; to bear a great heat, may be nourished. And why not, if God has and from which substances of great density placed different bodies at different distances from the sun, so as the denser bodies always possess the nearer places, and each body enjoys a degree of heat suitable to its condition, and proper for its nourishment? From this consideration it will best appear that the weights of all the planets are one to another as their forces. should be glad the diameters of the planets were more accurately and that may be done, if a lamp, set at a great distance, is made to shine through a circular hole, and both the hole and the light of the But I measured

so diminished that the spectrum may appear through the telescope like the then the planet, and may be defined by the same measure just diameter of the hole will be to its distance from the objective glass as the true diameter of the planet to its distance from us. The light of the lamp lamp are : may be diminished by the interposition either of pieces of cloth, or of smoked glass. Of kin to the analogy we have been describing, there is another observed between the forces and the bodies attracted (p. 395, 396, 397).

Since the action of the centripetal force upon the planets decreases in the duplicate proportion of the distance, and the periodic time increases in the sesquiplicate thereof, it is evident that the actions of the centripetal force, and therefore the periodic times, would be equal in equal planets at equal distances from the sun and in equal distances of unequal planets the total actions of the centripetal force would be as the bodies of the planets for if the actions were not proportional to the bodies to be moved, they could not equally retract these bodies from the tangents of their orbs in equal times nor could the motions of the satellites of Jupiter be so regular, if it was not that the circum-solar force was equally exerted upon Jupiter and proportion of their several weights. And the same be said of Saturn in respect of its satellites, and of our earth spect of the moon, as appears from Cor. II and III, Prop. LXV. therefore, at equal distances, the actions of the centripetal force are all its satellites in is to thing in re Arid, equal upon all the planets in proportion of their bodies, or of the quantities of matter in their several bodies; and for the same reason must be the same upon the particles of the same size of which the planet is composed for was greater upon some sort of particles than upon others than all ; if the action in proportion to their quantity of matter, it would be also greater or less upon the whole planets not in proportion to the quantity only, but like wise of the sort of the matter more copiously found in one and more sparingly in another. In such bodies as are found on our earth of very different sorts, I exam ined this analogy with great accuracy (p. 343, 344). If the action of the circum-terrestrial force to be moved, it will (by the Second velocity in equal times, and will is proportional to the bodies of Motion) move them with equal all bodies let fall to descend through Law make equal spaces in equal times, and all bodies hung by equal threads to vibrate in equal times. If the action of the force was greater, the times would be less ; if But that was less, these would be greater. has been long ago observed by others, that (allowance being made for the small resistance of the air) all bodies descend through equal spaces it in equal times and, by the help of pendulums, that equality of tim-es be distinguished to great exactness.

1 tried the thing in gold, silver, lead, glass, sand, common salt may wood, and wheat. I provided two equal wooden boxes. I filled the one with wood, and suspended an equal weight of gold (as exactly as I water, in the centre of oscillation of the other. of 11 feet, The boxes, hung by equal could) threads made a couple of pendulums perfectly equal in weight and fig to the resistance of the air and, placing the one by the other, I observed them to play together forwards and backwards for ure, and equally exposed : a long while, with And therefore (by Cor. 1 and VI, equal vibrations. Prop. XXIV. Book II) the quantity of matter in the gold was to the quan tity of matter in the wood as the action of the motive force upon same upon gold the one to the weight of the other. to the action of the all the wood ; that is, all the as the weight of And by these experiments, in bodies of the same weight, could have dis covered a difference of matter less than the thousandth part of the whole. Since the action of the centripetal force upon the bodies attracted is, at equal distances, proportional to the quantities of matter in those bodies, reason requires that it should be also proportional to the quantity of ter in the body attracting. mat For all action is mutual, and (p. 83, 93. by the Third Law of Motion) makes the bodies mutually to approach one to the other, and therefore must be the same in both bodies. It is true that we may consider one body as attracting, another as attracted; but this distinction is more mathematical than natural. The attraction is really common of either to other, and therefore of the same kind in both. And hence it is tracts Jupiter that the attractive force and the other planets is found in both. The sun at for the Jupiter attracts its satellites and, reason, the satellites act as well one upon another as upon Ju two terms. Two

same and all the planets mutually one upon another. piter, And though the mutual actions of two planets may be distinguished and considered as two, by which each attracts the other, yet, as those ac tions are intermediate, they do not make two but one operation between bodies may be mutually attracted each to the other by the contraction of a cord interposed. There is a double cause of action, to wit, the disposition of both bodies, as well as a double action in so far as the action is considered as upon two bodies ; but as betwixt two bodies but one single one. It is not one action by which the sun attracts but it is one ac Jupiter, and another by which Jupiter attracts the sun it is ; by which the sun and Jupiter mutually endeavour to approach each the other. By the action with which the sun attracts Jupiter, Jupiter and the sun endeavours to come nearer together (by the Third Law of Mo tion tion) ; and by the action with which Jupiter attracts the sun, likewise Ju- 527 and the sun endeavor to come nearer together. But the sun is not attracted towards Jupiter by a twofold action, nor Jupiter by a twofold but it is one single intermediate action, by which action towards the sun pitcr ; both approach nearer together. Thus iron draws the load-stone draws the iron other iron. (p. 93), as well as the load-stone for all iron in the : But neighbourhood of the load-stone draws the action betwixt the load-stone and iron is single, and considered as single by the philosophers. The action of iron upon the load-stone, is, indeed, the action of the load-stone betwixt itself and the and so it mani iron, by which both endeavour to come nearer together is : festly appears ; for if you remove the load-stone, the whole force of the iron almost ceases. it is that we are to conceive one single action to be ex two planets, arising from the conspiring natures of both and this action standing in the same relation to both, if it is proportional to the quantity of matter in the one, it will be also to the Tn this sense erted betwixt : proportional quantity of matter in the other. Perhaps it may be objected, that, to this according philosophy (p. 39S), bodies should mutually attract one another, contrary to the evidence of experiments in terrestrial bodies but I answer, that the experiments in terrestrial bodies come to no account for the attraction of all ; ; homogeneous spheres near their surfaces are (by Prop. LXXII) as their diameters. Whence a sphere of one foot in diameter, and of a like nature to the earth, would attract a small body placed near its surface with a force 20UOOOOO times less than the earth would do if placed near its surface; but so small a force could produce no sensible effect. If two such spheres were distant but by 1 of an inch, they would not, even in spaces void of 528 come together by the force of their mutual attraction in less s time and less spheres will come together at a rate yet slower, viz.. in the proportion of their diameters. Nay, whole mountains will not be sufficient to produce any sensible effect. A mountain of an hemispherical figure, three miles high, and six broad, will not, by its at traction, draw the pendulum two minutes out of the true perpendicular resistance, than a month j : and only in the great bodies of the planets that these forces are to be perceived, unless we may reason about smaller bodies in manner following. it is Let ABCD the globe of the earth cut by any plane (p. 93) represent The part into two parts ACB, and A CD. bearing upon the sustain part presses it with its whole weight; nor can the part ACB AC ACD ACD this pressure trary pressure. weights, that is, Motion ; not opposed by an equal con And therefore the parts equally press each other by their equally attract each other, according to the third Law of and continue unmoved, if it is and, if separated and let go, would fall towards each other with All which we may try and see in the velocities reciprocally as the bodies. load-stone, whose attracted part does not propel the part attracting, but is only stopped and sustained thereby. Suppose now that represents some small body on the earth s sur face then, because the mutual attractions of this particle, and of the re ACB : ACD of the earth towards each other, are equal, but the attraction of the particle towards the earth (or its weight) is as the matter of the particle (as we have proved by the experiment of the pendulums), maining part the attraction of the earth towards the particle will likewise be as the and therefore the attractive forces of all terres matter of the particle trial bodies will The forces (p. ; be as their several quantities of matter. 396), which are as the matter in terrestrial bodies of all forms, and therefore are not mutable with the forms, must be found in all sorts of bodies whatsoever, celestial as well as terrestrial, and be in all all there is no proportional to their quantities of matter, because among But in the celes difference of substance, but of modes and forms only. We have shewn that likewise proved thus. all the the action of the circum-solar force upon planets (reduced to equal that the action of the circum- distances) is as the matter of the planets tial bodies the same thing is ; jovial force upon the satellites of Jupiter observes the same law

and the same thing is to be said of the attraction of all the planets towards every that their attractive forces planet but thence it follows (by Prop. LXIX) : are as their several quantities of matter.

As the parts of the earth mutually attract one another, so do those of and its satellites were brought together, and doubt they would continue mutually to without formed into one globe, before. attract one another as And, on the other hand, if the body of was broke into more globes, to be sure, these would no less attract. all the planets. Jupiter If Jupiter 3ne another than they do the satellites now. From 529 these attractions it is that the bodies of the earth and all the planets effect a spherical figure, and But we have their parts cohere, and are not dispersed through the aether. before proved that these forces arise from the universal nature of matter 398), and that, therefore, the force of any whole globe is made up of (p. And from thence it follows (by Cor. the several forces of all its parts. Prop. 74) that the force of every particle decreases in the duplicate proportion of the distance from that particle and (by Prop. 73

that the force of an entire globe, reckoning from the surface outwards, decreases in the duplicate, but, reckoning inwards, in the sim ple proportion of the distances from the centres, if the matter of the globe and LXXV) And though be uniform. centre towards the surface, the matter of the globe, reckoning from the not uniform (p. 398, 399), yet the decrease in is the duplicate proportion of the distance outwards would (by Prop. LXXVI) take place, provided that difformity is similar in places round about at And two sucli globes will (by the same equal distances from the centre. one the with attract other a force decreasing in the duplicate Proposition) distance their of the centres. between, proportion Wherefore the absolute force of every globe is as the quantity of matter which the globe contains but the motive force by which every globe is attracted towards another, and which, in terrestrial bodies, we commonly call their weight, is as the content under the quantities of matter in both ; globes applied to the square of the distance between their centres (by Cor. IV, Prop. LXXVI), to which force the quantity of motion, by which each globe in a given time will be carried towards the other, is proportional. And the accelerative force, by which every globe according to its quantity is attracted towards another, is as the quantity of matter in that other globe applied to the square of the distance between the centres of the two (by Cor. II, Prop. LXXVI)= to which force, the velocity by which the attracted globe will, in a given time, be carried towards the other is of matter And from these principles well understood, it will be now easy to determine the motions of the celestial bodies among themselves. From comparing the forces of the planets one with another, we have above seen that the circum-solar does more than a thousand times exceed proportional. the rest but by the action of a force so greab it is unavoidable but that bodies within, nay, and far beyond, the bounds of the planetary system must descend directly to the sun, unless by other motions they are impelled all ; all towards other parts such bodies nor is our earth to be excluded from the number of for certainly the planets, and subject moon is a body of the same nature with the same attractions with the other planets, seeing it is by the circum-terrestrial force that it is retained in its orbit. But that the earth and moon are equally attracted towards the sun, we have above proved ; to the we have likewise before proved that all bodies are subject to530

the said common laws of attraction. to be deprived of its circular Nay, supposing any of those bodies motion about the sun, by having its distance from the sun, we may find (by Prop. would in its descent arrive at the sun XXXVI) ; in what space of time to wit, in half that periodic it time in .vhich the body might be revolved at one half of its former distance; or in a space of time that is to the periodic time of the 1 to 4</2; as as planet that Venus in its descent would arrive at the sun in the space of 40 days, Jupiter in the space of two years and one month, and the earth and moon together in the space of 66 days and 19 hours. But, since no such thing happens, it must needs be, that those bodies are moved towards other parts To hinder such a (p. 75), nor is every motion sufficient for this purpose. And hence descent, a due proportion of velocity is required. the depends force of the argument drawn from the retardation of the motions of the Unless the circum-solar force decreased in the duplicate ratio of planets. their increasing slowness, the excess thereof would force those bodies to de scend to the sun for instance, if the motion (c&teris was retarded paribns) by one half, the planet would be retained in its orb by one fourth of the former circum-solar force, and by the excess of the other three fourths would descend to the sun. And therefore the planets (Saturn, ; Jupiter, Mars, Venus, and Mercury) are not really retarded in their perigees, nor become really stationary, or regressive with slow motions. All these are but apparent, and the absolute motions, by which the planets continue to revolve in their orbits, are always direct, and nearly equable. But that such motions are performed about the sun, we have already proved and ; therefore the sun, as the centre of the absolute motions, is quiescent. For we can by no means allow quiescence to the earth, lest the planets in their perigees should indeed be truly retarded, and become truly stationary and want of motion should descend to the sun. But regressive, and so for farther drawn ; since the planets (Venus, Mars, Jupiter, and the rest) by radi: sun describe regular orbits, and areas (as W C have shewn) T to the nearly and to sense proportional to the times, it follows (by Prop. III. and Cor. Ill, Prop. LXV) that the sun is moved with no notable force, unless all the planets are equally moved with, according to their several quantities of matter, in parallel lines, and so the whole sys tem is transferred in right lines. Reject that translation of the whole perhaps w ith such as T will be almost quiescent in the centre thereof. If the system, and the sun gun was revolved about the earth, and carried the other planets round about itself, the earth ought to attract the sun with a great force, but the cir cum-solar planets with no force producing any sensible effect, which is Add to this, that if hitherto the earth, contrary to Cor. Ill, Prop. LXV. because of the gravitation of its parts, has been placed by most authors in the lowermost region of the universe now, for better reason, the sun pos ; sessed of a centripetal force exceeding our terrestrial gravitation a thousand

times and more, ought to be depressed into the lowermost place, and to be And thus the true disposition of the held for the centre of the system. whole system will be more fully and more exactly understood. Because the fixed stars are quiescent one in respect of another (p. 401, one system of bodies 4U2), we may consider the sun, earth, and planets, as carried hither and thither by various motions among themselves; and the common centre of gravity of all (by Cor. IT of the Laws of Motion) will in which either be quiescent, or move uniformly forward in a right line

whole system will likewise move uniformly forward in right lines. But this is an hypothesis hardly to be admitted and, therefore, setting it arfide, that common centre will be quiescent= and from it the sun is never case the

far The common removed. centre of gravity of the sun and Jupiter falls and though all the planets were placed towards the same parts from the sun with Jupiter the common centre of the sun and all of them would scarcely recede twice as far from the sun s centre on the surface of the sun ; ; and, therefore, though the sun, according to the various situation of the planets, is variously agitated, and always wandering to and fro with a slow libration, yet it never recedes one entire diameter of its own body from the quiescent centre of the whole system. But from the weights of the sun and planets above determined, and the situation of all among them selves, their common centre of gravity may be found and, this being given, motion of ; the sun s place to any supposed time may be obtained. About the sun thus librated the other planets are revolved in elliptic orbits (p 403), and, by radii drawn to the sun, describe areas nearly pro If the sun was quiscent he portional to the times, as is explained in Prop. LXV. did not and the other act one escent, mutually planets upon another, their orbits would be elliptic, and the areas exactly proportional to the times (by But the actions of the planets amonir Prop. XIII). themselves, compared with the actions of the sun on the planets, are of no moment, and produce no sensible errors. And those errors are less in rev Prop. XI, and Cor. 1, olutions about the sun agitated in the manner but now described than if those revolutions were made about the sun quiescent (by Prop. LXV1, and Cor. Prop. LXVIll), especially if the focus of every orbit is placed in the common centre of gravity of all the lower included planets; viz., the focus Mercury in the centre of the sun the focus of the orbit of of the orbit of Venus : common centre of gravity of Mercury and the sun the focus of the orbit of thp earth in the common centre of gravity of Venus, Mer in the and so of the rest. And by this means the foci of the cury, and the sun crbits of all the planets, except Saturn, will not be sensibly removed from the centre of the sun, nor will the focus of the orbit of Saturn recede sensi ; And bly from the common centre of gravity of Jupiter and the sun. therefore astronomers are not far from the truth, when they reckon the sun s centre the common focus of all the planetary orbits. In Saturn itselfTHE SYSTEM CF THE W )RLD. the error thence arising docs not exceed 1 45 And if its orbit, by placing the focus thereof in the common centre of gravity of Jupiter and the sun, ghall happen to agree better with the phenomena, from thence all that we . have said will be farther confirmed. sun was quiescent, and the planets did not act one on another, the aphelions and nodes of their orbits would likewise (by Prop. 1, XI, and Cor. If the XIU) Prop. be quiescent. And the longer axes of their elliptic orbits would (by Prop. XV) be as the cubic roots of the squares of their periodic times and therefore from the given periodic times would be also given. But those times are to be measured not from the equinoctial points, which : Put the semi-axis are rnoveable, but from the first star of Aries. of the 100000, and the semi-axes of the orbits of Saturn, Jupiter, Mars, Venus, and Mercury, from their periodic times, will come out 953806, 520116, 152399, 72333, 38710 respectively. But from the sun s earth’s orbit motion every semi-axis is increased (bv Prop. LX) by about one third of the distance of the sun s centre from the common centre of gravity of the sun and planet (p. 405, 406.) And from the actions of the exterior planets on the interior, the periodic times of the interior are something and their aphelions protracted, though scarcely by any sensible quantity

VI and VII, Prop. LXVI)by very slow motions on the like account the periodic times of all, espe are transferred (by Cor. in conset/ue/ttia. And planets, will be prolonged by the actions of the somets, if any such there are, without the orb of Saturn, and the aphe But from lions of all will be thereby carried forwards in consequent-la. of the cially exterior the progress of the aphelions the regress of the nodes follows (by Cor. XI, XIII, Prop. 1 jXVI). And if the plane of the ecliptic is quiescent, the regress of the nodes (by Cor. XVI, Prop. LX.VI) will be to the progress of *he aphelion in every orbit as the regress of the nodes of the moon s orbit to the progress of its apogeon nearly, that is, as about 10 to 21. But as tronomical observations seem to confirm a very slow progress of the aphe lions, it is and a regress of the nodes in respect of the fixed stars. And hence probable that there are comets in the regions beyond the planets, which, revolving in very eccentric orbs, quickly fly through their perihelion parts, and, by an exceedingly slow motion in their aphelions, spend almost their whole time in the regions beyond the planets plain more

as we shall afterwards ex at large. The at the planets thus revolved about the sun (p. 413, 41.4, 415) may or as satellites same time carry others revolving about themselves moons, as appears by Prop. LXVI. But from the action of the sun our moon the earth, de velocity, and, by a radius drawn to scribe an area greater for the time it must have its orbit less curve, and therefore approach nearer to the earth in the syzygies than in the quadratures, except in so far as the must move with greater

motion of eccentricity hinders those effects. Per the eccentricity is

greatest when the moon s apogeon is in the syzygies, and hence it is that the is in the quadratures and least when the same

swifter and nearer to us, but the apogeon moon slower and But farther; the farther from us, in the syzygies than in the quadratures. perigeon moon is apogeon has a progressive and the nodes a regressive motion, both unequa is more swiftly progressive in its syzygies, more in its quadratures, and by the excess of its progress above slowly regressive its regress is yearly transferred in coiisequentia ; but the nodes are quies ble.

For the apogeon cent in their syzygies, and most swiftly regressive in their quadratures. But farther, still, the greatest latitude of the moon is greater in its quadra and the mean motion swifter in the aphelion of tures than in its syzygies

More inequalities in the moon s motion have not hitherto been taken notice of by astronomers but all these follow from our principles in Cor. 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, Prop. 46, and are known really to exist in the heavens.

And this may seen in that most ingenious, and if I mistake not, of all, the the earth than in its perihelion. : most acccurate, hypothesis of Mr. Horrnx, which Mr. Flamsted has fitted but the astronomical hypotheses are to be corrected in the motion of the nodes for the nodes admit the greatest equation or pros- to the heavens

thaphaeresis in their octants, and this inequality is most conspicuous when the moon is in the nodes, and therefore also in the octants and hence it ; was that Tycho, and others after him, referred inequality to the octants of the moon, and made it menstrual; but the reasons by us addu ced prove that it ought to be referred to the octants of the nodes, and to be this made annual. Beside those inequalities taken notice of by astronomers (p. 414, 445, 447,) there are yet some others, by which the moon s motions are so dis turbed, that hitherto by no law could they be reduced to any certain regu For the velocities or horr.ry motions of the apogee and nodes of the moon, and their equations, as well as the differs ice betwixt the greatest lation. eccentricity in the syzygies and the least in the rrdratures, and that ine quality which we call the variation, in the progress of the year are augmented and diminished r.sarly

(by Cor. 14, Prop. 46) in the triplicate ratio

Beside that, the variation is mutable in the duplicate ratio of the time between the quadratures (by Cor. of the sun s apparent diameter. Cor. XVI, Prop. LXVI); and all those inequali II, Lem. X, are something greater in that part of the orbit which respects the sun than in the opposite part, but by a difference that is scarcely or not at all I and and ties perceptible. By a computation (p. 422), which for brevity s sake I do not describe, 1 that the area which the moon by a radius drawn to the earth also find describes in the several equal moments of time is nearly as the number 237 T, and versed sine of the double distance of the sum of the moon frour.

the nearest quadrature in a circle whose radius is unity and therefore that the square of the moon s distance from the earth is as that sum divided by the horary motion of the moon. Thus it is when the variation in

but if the variation is greater or less, the octants is in its mean quantity

Let that versed sine must be augmented or diminished in the same ratio. ; how f exactly the distances thus found will agree with jie moon s apparent diameters. From the motions of our moon we may derive the motions of themoon* astronomers try or satellites of Jupiter and Saturn (p. 413); for the mean motion of the nodes of the outmost satellite of Jupiter is to the mean motion of the nodes moon in a proportion compounded of the duplicate proportion of the periodic time of the earth about the sun to the periodic time of Jupiter about the sun, and the simple proportion of the periodic time of the sat of our ellite Gor. about Jupiter to the periodic time of our 16, Prop. 46)

moon about and therefore those nodes, the earth (by hun in the space of a The mean dred years, are carried 8 24 backwards, or in atitecedeutia. motions of the nodes of the inner satellites are to the (mean) motion of (the nodes of) the outmost as their periodic times to the periodic time of by the same corollary, and are thence given. And the motion of the apsis of every satellite in consequentia is to the motion of its nodes in a/ttecedentia, as the motion of the apogee of our moon to the motion of i s this, nodes (by the same Corollary), and is thence given. The greatest equa tions of the nodes and line of the apses of each satellite are to the greatest equations of the nodes and the line of the apses of the moon respectively as the motion of the nodes and line of the apses of the satellites in the time of one resolution of the first equations to the motion of the nodes and apogeon of the moon in the time of one revolution of the last equa The variation of a satellite seen from Jupiter is to the variation tions.

moon same proportion as the whole motions of their nodes respectively, during the times in which the satellite and our moon (after ami parting from) arc revolved (again) to the sun, by the same Corollary of our in the therefore in the outmost satellite the variation does not exceed

From the small quantity of those inequalities, and the slowness of the motions, it happens that the motions of the satellites are found to be so all motion to the regular, that the more modern astronomers either deny them to affirm be or nodes, very slowly regressive. While the planets are thus revolved in orbits about remote mean time they make their several rotations about their Mars in 24f, h the sun in 26 days; Jupiter in 9 h 56 proper axes; Venus in 23 h and that in planes not much inclined to the plane of the (P. 404). centres, in the

and according to the order of the signs, as astronomers determine the from spots or macula? that by turns present themselves to our sight in h their bodies; and there is a like revolution of our earth performed in 24 ecliptic, find those motions are neither accelerated 535 nor retarded the actions of the centripetal forces, as appears by Cor. 22, Prop. fore of all others they are the most equable and most tion of time; but those LXVI fit revolutions are to be reckoned and there ; for the mensura equable not from some fixed star= for as the position of the un equably varied, the revolutions of those planets their return to the sun, but to planets to the sun is from sun to sun are rendered unequable. In like equable manner is the moon revolved about in respect of the fixed space of a sidereal mean motion of the stars, viz., in month so that moon in its orbit

its axis by a motion most 27 J 7 h 43 that is, in the

motion this diurnal is equal to the upon which account the same face of the moon always respects the centre about which this mean motion is performed, that is, the exterior focus of the moon s orbit nearly and hence : ; arises a deflection of the moon s face from the earth, sometimes towards the east, and other times towards the west, according to the position of the it and this deflection is equal to the equation of the respects focus which moon s orbit, this is the ; or to the difference betwixt its moon s libration in a libration in latitude arising mean and longitude= but it is true motions; and likewise affected with from the inclination of the moon s axis to the plane of the orbit in which the moon is revolved about the earth for that axis retains the same position to the fixed stars nearly, and hence the ; poles present themselves to our view by turns, as we may understand from the example of the motion of the earth, whose poles, by reason of the incli nation of its axis to the plane of the ecliptic, are by turns illuminated by the sun. To fixed stars, determine exactly the position of the moon and the variation of axis to the s problem worthy of an this position, is a astronomer. By reason of the diurnal revolutions of the planets, the matter which they contain endeavours to recede from the axis of this motion and hence the fluid parts rising higher towards the equator than about the poles ; would lay the solid parts about the equator under water, if those did not rise also (p. 405, 409) upon which account the planets are parts something thicker about the equator than about the poles and their equi (p. 405),

noctial points (p. 413) thence become regressive and their axes, by a motion of nutation, twice in every revolution, librate towards their eclip tics, and twice return again to their former inclination, as is explained in

Cor. 18, Prop. 46 and hence it is that Jupiter, viewed through very long telescopes, does not appear altogether round (p. 409). but having its diameter that lies parallel to the ecliptic something longer than that which is drawn from north to south. And from the diurnal motion and the attractions (p. 415, 418) of the Bun and moon our sea ought twice to rise and twice to fall every day, as well lunar as solar (by Cor. 19, 20, Prop. 46), and the greatest height of the water to happen before the sixth hour of either day and after the 12th hour preceding.

By the slowness of the diurnal motion the flood is retracted to the 12th hour and by the force of the motion of reciprocation it is protracted and deferred till a time nearer to the sixth hour. But till that time is more certainly determined by the pheno mena, choosing the middle between those extremes, why may we not conjecture the greatest height of the water to happen at the third hour ? for thus the water will rise all that time in which the force of the lumi naries to raise is less : viz., it is greater, from the ninth and from the third to the the appulse of each as above the horizon and will fall all that time in to the third ninth when which their force hour when that force it is less. The hours I is greater, reckon from luminary to the meridian of the place, as well under and by the hours of the lunar day I understand the

24th parts of that time which the moon spends before it comes about again by its apparent diurnal motion to the meridian of the place which it left the day before.

But the two motions which the two luminaries guished, but will make a certain mixed motion. raise will not appear distin In the conjunction or op will of the luminaries their be forces position conjoined, and bring on the In flood the the sun will raise the waters and ebb. greatest quadratures which the moon dcpresseth. and depress the waters which the moon raiseth ; and from the difference of their forces the smallest of all tides will follow. And because (as experience tells us) the force of the moon is greater than that of the sun, the greatest height of the water will happen about the third lunar hour.

Out of the syzygies and quadratures the greatest tide which by the single force of the moon ought to fall out at the third lunar hour, and by the single force of the sun at the third solar hour, by the compounded forces of both must fall out in an intermediate time that ap proaches nearer to the third hour of the moon than to that of the sun: and, therefore, while the moon is passing from the syzygies to the quadra tures, during which time the third hour of the sun precedes the third of the moon, the greatest tide will precede the third lunar hour, and that by the greatest interval a little after the octants of the moon and by like intervals the greatest tide will follow the third lunar hour, while the moon is passing from the quadratures to the syzygies. ; the effects of the luminaries depend upon their distances from the for when they are less distant their effects are greater, and when distant their effects are less, and that in the triplicate proportion of But earth more ; Therefore it is that the sun in the winter time, their apparent diameters. a greater effect, and makes the tides in the has in then its perigee, being and those in the quadratures something less, ies something greater, syzyii and every month the moon, panbiis, than in the summer season vhile in the perigee, raiseth greater tides than at the distance of 15 days cre/m.<? K forc or after, tides nighest when it is in its apogee. Whence it 537 comes to pasa that two do not follow one the other in two immediately succeeding syzygies. The of either luminary doth likewise depend upon its declination from the equator for if the luminary was placed at the pole, would constantly attract all the parts of the waters, without any inten effect or distance it ; sion or remission of its action, and could cause no reciprocation of motion and, therefore, as the luminaries decline from the equator towards either ; pole, they will by degrees lose their force, and on this account will excite than in the equinoctial syzygies. But in the lesser tides in the solstitial quadratures they will raise greater tides than in the quadratures about the equinoxes because the effect of the moon, then situated in the therefore the greatest tides equator, most exceeds the effect of the sun solstitial ; ; out in those syzygies. and the least in those quadratures, which happen about the time of both equinoxes and the greatest tide in the syzygies is always succeeded by the least tide in the quadratures, as we lind by expe fall ; rience. But because the sun less distant is from the earth in winter than cornes to pass that the greatest and least tides more fre summer, quently appear before than after the vernal equinox, and more frequently in it after than before the autumnal. Moreover, the effects cf che KN lumina depend upon the latitudes of places. Let AjoEP represent the earth on all ries sides covered centre; P, p, C with deep waters: its poles; AE its the equa P any place without the equator: the parallel of the place Del the correspondent parallel OD the other side tor: F/ : of the equator; L the rlnoe which the moon possessed three hours before h the opposite place K, k, the place of the earth directly under it the places at 90 degrees distance CH, Ch, the greatest heights of the sea H

from the centre of the earth the axes H/?,, ; K/r, an ellipsis and is CK, C&, described, the least heights and if with and by the revolution rf that

a spheroid HPK//jt?A* is formed, this sphe about its longer axis roid will nearly represent the figure of the sea; and CF, C/, CD, Cd, will But farther if in the said revo in the places F,/, D, d. represent the sea ellipsis : lution of the ellipsis any point N describes the circle NM, cutting the Dr/ in any places R, T, and the equator AE in S, CN will parallels F/, of the sea in all those places R, S, T, situated in this represent the height in the diurnal revolution of any place F the greatest Wherefore circle. third hour after the appulse of the moon to the at the F. in be will flood ? meridian above the horizon ; and afterwards the greatest ebb in Q, at the moon and then the greatest flood inf. third hour after the setting of the

at the third horizon Lour after the appulse of the rnoon to the meridian under tht and. lastly, the greatest ebb in Q. at the third hour after the of the moon; and the latter flood iny" will be less than the preced rising in flood For the whole sea is divided into two huge and hemis ing ,

pherical floods, one in the hemisphere KH/rC on the north side, the other in the opposite hemisphere KH/cC, whicli we may therefore call the north ern and the southern floods the other, come by turns : these floods being always opposite the one to meridians of all places after the interval to the of twelve lunar hours and, seeing the northern countries partake more of the northern flood, and the southern countries more of the southern flood, thence arise tides alternately greater and less in all places without ; the equator in Avhich the luminaries rLe and set. But the greater tide will happen when the moon declines towards the vertex of the place, about the third hour after the -appulse of the moon to the meridian above the horizon and when the moon changes its declination, that which was the ; changed into a lesser and the greatest difference of the floods will fall out about the times of the solstices, especially if the ascending node of the moon is about the first of Aries. So the morning greater tide will be ; tides in winter exceed those of the evening, and those of the morning in summer at Plymouth ; the evening tides exceed by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Qvleptvss and Stitrnnj.

But the motions which we have been describing suffer some alteration from that force of reciprocation which the waters [having once received] retain a little while by their vis iiisita ; whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should cease. This power of retaining the impressed motion lessens the and makes those tides which immediately the syzygies greater, and those which follow next after the difference of the alternate tides, succeed after And hence it is that the alternate tides at l 1y month quadratures less.

and Bristol do not differ much more one from the other than by the height of a foot, or of 15 inches; and that the greatest tides all at those ports but the third after the syzygies.

Besides, all the motions are retarded in their passage through shallow channels, so that the greatest tides of all, in some strai s and mouths are not the first of rivers, are the fourth, or even the fifth, after the syzygies. It may also happen that the greatest tide may be the fourth or fifth after the syzygies, or fall out yet later, because the motions of the sea are retarded in passing through shallow places towards the shores= for so the tide arrives at the western coast of Ireland at the third lunar hour, and an hour or two after at the ports in the southern coast of the same island also at the islands Cftssiterides, ; as then successively at commonly Sorliti^s Palrnonth. Plymouth, Portland, the isle of Wight, Winchester, Dover, the mouth of the Thames, arid London Btidge spending twelve hours in

But farther; the propagation of the tides may he obstructed channels of the ocean itself, when they are not of depth enough, of the this passage. even by the happens at the third lunar hour in the Canary islands for the flood

and at all those western coasts that lie towards the Atlantic ocean, as of Ire land, France, Spain, and all Africa, to the Cape of Good Hope, except some shallow places, where it is impeded, and falls out later and in the in ; Gibraltar, where, by reason of a motion propagated from the Mediterranean sea, it flows sooner. But, passing from those coasts over straits of the breadth of the ocean to the coasts of America, the flood arrives first at the most eastern shores of Brazil, about the fourth or fifth lunar hour; then at the mouth of the river of the Amazons at the sixth hour, but at the neighbouring islands at the fourth hour afterwards at the islands of ; Bermudas hour, and at port St. An^nstin in Florida at therefore the tide is propagated through the ocean at the seventh seven and a half. And with a slower motion than moon

and it should be according to the course of the this retardation is very necessary, that the sea at the between Brazil and same time New France, and rise at the Canary islands, may and on the coasts of Europe and Africa, and vice versa = for the sea can fall And it is probable that not rise in one place but by falling in another. the Pacific sea is agitated by the same laws for in the coasts of Chili and Peru the highest flood is said to happen at the third lunar hour. But

with what velocity it Japan, the Philippine yet learned. Farther; it is thence propagated to the eastern coasts of and other islands adjacent to China, I have not may happen (p. 418) that the tide the ocean through different channels towards the may same be propagated from port, and may pass quicker through some channels than through others, in which case the same tide, divided into two or more succeeding one another, may compound new motions of different kinds. Let us suppose one tide to be divided into two equal tides, the former whereof precedes the other by the space of six hours, and happens at the third or twenty-seventh hour from the appulse of the moon to the meridian of the port. If the moon at the time of this appulse to the meridian was in the equator, every six hours alternately there would arise equal floods, which, meeting with as many equal ebbs, would so balance one the other, that, for that day, the water w ould stag r moon then declined from the equator, the and remain quiet.

would be alternately greater and less, as was said; and from hence two greater and two lesser tides would be alternately propa But the two greater floods would make the gated towards that port. nate, If the tides in the ocean greatest height of the waters to fall out in the middle time betwixt both, and the & Greater and lesser floods would make the waters to rise to a mean height in the middle time between them; and in the middle time between 540 the two lesser floods the waters would rise to their least height. Thus the space of twenty-four hours the waters would come, but once riot twice, in only to their greatest, and once only to their least height and their great est height, if the moon declined towards the elevated pole, would happen

at the sixth or thirtieth hour after the appulse of the moon to the meridian its declination, this flood would be changed and when the moon changed into an ebb.

Of all which we have an example in the port of Batsham, in the kingdom of Tunquin. in the latitude of 20 50 north.

In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate when the moon declines to the north, they begin to fluw ; and ebb, not twice, as in flood This other ports, but once only every day ; and the happens at the setting, and the greatest ebb at the rising of the moon. tide increaseth with the declination of the moon till the seventh or eighth day then for the seventh or eighth day following it decreaseth at the same rate as it had increased before, and ceaseth when the moon ; After which the flood is immediately changed changeth its declination. into an ebb and thenceforth the ebb happens at the setting and the flood at the rising of the moon, till the moon again changes its declination. There are two inlets from the ocean to this port; one more direct and short ; between the island Hainan and the coast of QuanttiHg, a province of of China ; the other round about between the same island and the coast Cochim ; and through the shorter passage the tide is sooner propagated Batsham.

to

In the channels of rivers the influx and reflux depends upon the current of the rivers, which obstructs the ingress of the waters from the sea. and promotes their egress to the sea, making the ingress later and slower, and the egress sooner arid faster; and hence it is that the reflux is of longer duration that the influx, especially far up the rivers, where the force of the is less. So Sturmy tells us, that in the river Avon, three miles below sea and without doubt Bristol, the water flows only five hours, but ebbs seven the difference is yet greater above Bristol, as at Carcs/iam or the Bath. ; This difference does likewise depend upon the quantity of the flux and re for the more vehement motion of the sea near the syzygies of the flux ; luminaries more easily overcoming the resistance of the rivers, will make the ingress of the water to happen sooner and to continue longer, and will therefore diminish this difference. But while the moon is approaching to the syzygies, the rivers will be more plentifully filled, their currents being obstructed by the greatness of the tides, and therefore will something more retard the reflux of the sea a little after than a little before the syzygies. Upon which account the slowest tides of all will not happen in the syzy- but precede them a little and I observed above that the tides before the sy/ygies were also retarded by the force of the sun and from both ^ies,

causes conjoined the retardation of the tides will be both greater and sooner

All which I find to be so, by the tide-tables which before the syzygies. Ftamsted has composed from a great many observations. By the laws we have been describing, the times of the tides but the greatness of th-e tides are governed depends upon the greatness of the ; Let seas. EAUB the oval figure of the seas, CA C represent the centre of the earth, the longer semi-axis of this oval, OB the shorter insisting at right angles the middle point between A and B, and or eCf upon the former, D EOF the angle at the centre of the earth, subtended by the breadth of the sea that terminates in the shores E, F, or e,f. Now, supposing that the point A D the middle between the in the middle points E, F, and the point if the difference of the heights CA, CB, represent the quantity of the tide in a very deep sea surrounding the whole earth, is in between the points e,/, the excess of the or OF will represent the height CA above the height quantity of the tide in the middle of the sea EF, terminated by the shores E, F and the excess of the height Ce above the height Cf will nearly OE ; represent the quantity of the tide on the shores/" of the same sea. Whence it appears that the tides are far less in the middle of the sea than at the EF and that the tides at the shores are nearly as the (p. 451, 452), breadth of the sea not exceeding a quadrantal arc. And hence it is that near the equator, where the sea between Africa and America is narrow, shores ; the tides are far less than towards either side in the temperate zones, w here the seas are extended wider or on almost all the shores of the Pacific sea r ; ; as well towards America as towards China,, and within as well as without and that in islands in the middle of the sea they scarcely rise higher than two or three feet, but on the shores of great continents are three or four times greater, and above, especially if the motions propagated the tropics ; from the ocean are by degrees contracted into a narrow space, and the water, to fill and empty the bays alternately, is forced to flow and ebb with great violence through shallow places as Plymouth and Chepstow Bridge in Michael and town of Avranches in Aor- England) at the mount of ; >S*/. mcihdy, and at Cambaia and Peyn. in the East Indies. In which places. 642 the sea, hurried in and out with great violence, sometimes lays the shores under water, sometimes leaves them dry, for many miles. Nor is the force of the influx and efflux to be broke till it has raised or depressed the water to forty or fifty feet and more. Thus also -long and shallow straits that with mouths wider and deeper than the rest of their chan nel (such as those about Britain and the Magellanic Straits at the east ern entry) will have a greater flood and ebb, or will more intend and remit open to the sea Or higher and be depressed lower. said that the Pacific sea in its reflux sometimes retreats two miles, and gets out of sight of those that stand on shore. Whence in these places the floods will be also higher but in deepei their course, the coast of and therefore will South America it rise is ; waters the velocity of influx and efflux is always Nor in such places ascent and descent is so too. ascend to more than six, eight, or ten feet. in the following (419. P 420), is and therefore the ocean tlu known to quantity of the ascent I manner compute Let S represent the sun, earth The less, the T the moon, PAGB the moon s orbit. In SP take SK equal to ST and SL to SK in the duplicate ratio of SK to SP. Parallel to PT draw LM ; and, supposing the mean quantity of the circum-solar force directed towards the earth to be represented \j the distance or SK, SL will represent the quantity thereof directed ST LM But that towards the moon. force is compounded of the parts SM, that part of which is represented by TJVI, do disturb the motion of the moon (as appears from Prop. LXVI, and its In so far as the earth and moon are revolved about their Corollaries) LM and of which the force common forces. ; SM centre of gravity, the earth will be liable to the action of the like But we may refer the sums as well of the forces as of the motions TM and moon, and represent the sums of the forces by the lines in mean The force its them. are to which LM, ML, quan proportional be revolved in an orbit, about tity, is to the force by which the moon may to the PT in the duplicate ratio of the moon s the earth quiescent, at the distance s periodic time about the nun earth about earth to the time the periodic h d the in that Cor. is, XVII, Prop. LXVI) duplicate ratio of 27 7 (by : . . The force by or as 1000 to 178725, or 1 to 178f f. 43 to 365 d 6 h 9 which the moon may be revolved in its orb about the earth in rest, at the distance PT of 60| semi-diameters of the earth, is to the force by which it may revolve in the same time at the distance of 60 semi- diameters as 60i to 60 and this force is to the force of gravity with us as 1 to 60 X 60 nearly and therefore the mean force ML is to the force of gravity at the surface of the earth as 1 X 60| to 60 X 60 X 178 f, or 1 to . . ; ; ;THE SYSTEM OF THF. WORLD. Whence 638092,6. of the lines moon If TM, TM will the force And Ml,. 543 be ulso given from the proportion these are the forces of the sun, by which the motions are disturbed. s from the moon s orbit (p. 449V we descend to the earth s surface, those diminished in the ratio of the distances 60| and 1 and will then become 3S604600 times less than the therefore the force forces will be ; LM But of gravity. force this force acting equally every where upon the earth, will scarcely effect any change on the motion of the sea, and there fore may be neglected in the explication of that motion. The other force I M, where the sun in places quantity of the force force of gravity. Suppose now is ML, and vertical, or therefore but in their nadir, is triple the less than the 12868200 times AUBE to represent the spherical surface of the enrth, the surface of the water overspreading it, C the centre of both, A the place to winch the sun is vertical, B the place opposite I), E. places at 90 degrees distance from the former ; ACEwz/A a right angled cylmdric in any place is force TM canal passing through the earth s centre. The as the distance of the place from the plane DE, on which a line fr^m to C insists at right angles, and A J) therefore in the part of the ca is represented by EC no quantity, but in the nal which ini is of other part AClk is as the gravity at the several heights for in ; descending towards the centre of the earth, gravity LXX1II) height TM ; is (by where every Prop- / 7; -pi ; the as and therefore the force drawing the water upwards will diminish its gravity in the leg AC//J of the canal in a given ratio : upon which account the water will ascend in this nor will leg, till its defect of gravity is supplied by its in an equilibrium till its total gravity greater height becomes equal to the total gravity in EC/m, the other leg of the canal. Because the gravity of every particle is as its distance from the earth s : centre, the it rest weight of the whole water in either leg will increase in the and therefore the height of the water in the duplicate ratio of the height leg AC/A* will be to the height thereof in the leg C/wE in the subdupli- ; number 12868201 to 12808200, or in the ratio of the number 25623053 to the number 25623052, and the height of the water But in the leg EC/ra to the difference of the heights, as 25623052 to 1. the height in the lea= EC/m is of 19615800 Pa rift feet, as hits been lately cate ratio of the 544 found by the mensuration of the French ; and, therefore, by the preceding analogy, the difference of the heights comes out 9} inches of the Paris A and the sun s force will make the height of the sea at to exceed the height of the same at by 9 inches. And though the water of the canal ACE/??7/,= be supposed to be frozen into a hard and solid consistence, foot ; E yet the heights thereof at still A and E, and all other intermediate places, would remain the same. Let Act (in the following figure) represent that excess of height of 9 inches at A, and hf the excess of height at any other place h; and upon let fall the perpendicular /G, meeting the globe of the earth in F and because the distance of the sun ib so great that all the right lines DC : TM in drawn thereto may be considered as parallel, the force will be to the same force in the place A as the sine FG And, therefore, since those forces tend to any place to the radius / AC. the sun in the direction of par allel lines, they will generate the parallel heights F/ An, in the same ratio and there ; fore the figure of the water Ylfaeb will be a spheroid made by ellipsis ab. the revolution of an about And height fh its longer axis the perpendicular will be to the ob lique height to or as FG ratio of FG to that AC, is, F/ as/G AC : to /C, and there fore the height fh is to the the duplicate height Art in double the of sine versed in the ratio of the And hence to the double the radius, and is thence given. the earth we about sun the of several moments of the apparent revolution at any waters the of descent and may infer the proportion of the ascent that ascent of diminution the of well as as given place under the equator, the sun s from or of latitude the from whether and angle DC/ to places arising that on account of the latitude of places, the ascent and ratio of the descent of the sea is in all places diminished in the duplicate the ascent s sun the of declination, account on and co-sines of latitude ratio of the in the diminished is the under duplicate and descent equator And in places without the equator the half sum v)-sine of declination. ascents (that is, the mean ascent) is diminished and of the descent, declination ; viz., ; evening morning nearly in the same ratio. Let S and L the forces of the sun and moon respectively represent the and at their mean distances from the earth; sun of the the versed sines of double the R placed in the equator, radius ; T and V complementsTHE SYSTEM and moon s declinations diameters of the sun and to OJ THE WORLD. any given time moon ; D 545 E and F and moan apparent the G be their appa and, supposing rent diameters to that given time, their forces to raise the tides under the : VG the equator will be, in VG 2RE -TTT 3 2R1) TF 3 –, L 3 syzygies-^ TF 3 ^ 1, -f to 3 ^ 3 S; in the quadratures, 3 S.

And if the same ratio is likewise observed under the parallels, from observations accurately made in our northern climates we may determine the proportion of the forces L and S and then by means of this rule predict the quantities of the tides to every syzygy and ; quadrature. At the mouth of the river Avon, three miles below Bristol (p. 450 to 453), in spring and autumn, the whole ascent of the water in the conjunc opposition of the luminaries (by the observation of Sturnty) is Because the apparent di feet, but in the quadratures only 25. ameters of the luminaries are not here determined, let us assume them in tion or about 45 quantities, as well as the moon s declination in the equinoctial and the versed sine of quadratures in its mean quantity, that is, 23| mean their ; will be 1082, supposing the radius to be 1000. But the declinations of the sun in the equinoxes and of the moon in the syzy- of double the complements gies are of no quantity, and the versed sines double its complement Whence are each 2000.
L S in the /cUOU those forces become feet, or of tiplying the extremes and the means, farther I ; S and in the syzygies, quadrature^ respectively proportional to the heights of the tides of 45 and 25 But L + remember to and 5 paces. we have 5L And, + mul- therefore, 15138 5S = TxTr L< have been told that in summer the ascent of the sea in the syzygies is to the ascent thereof in the quadratures as about In the solstices themselves it is probable that the proportion may 5 to 4. be something less, as about 6 to 5 then the proportion 5|S [for Till we can more us assume L is ; whence it L + L would follow that S : I, -S : : is 6 :

5]. certainly determine the proportion from observation, let and since the heights of the tides are as the forces = 5^S ; which excite them, and the force of the sun is able to raise the tides to the height of nine inches, the moon s force will be sufficient to raise the same And if we allow that this height may be to the height of four feet. doubled, or perhaps tripled, by that force of reciprocation which we observe in the motion of the waters, and by which their motion once be ^un 35 is kept 546 some time, there will be force enough we really find in the ocean. for up to generate all that quantity of tides which Thus we have seen that these forces are sufficient to move the sea. But. so far as I can observe, they will not be able to produce any other effect for since the weight of one grain in 4000 is not sensible on our earth and the sun s force to move the tides is sensible in the nicest balance

than the force of gravity arid the sum of the forces of both moon and sun, exceeding the sun s force only in the ratio of 6^ to 1, is still 2032890 times less than the force of gravity it is evident that both forces 12868200 less

together are 500 times less than what is required sensibly to increase diminish the weight of any body in a balance. And, therefore, they will not sensibly move any suspended body nor will they produce any sensible * ; on pendulums, barometers, bodies swimming in stagnant water, or eifect in In the atmosphere, indeed, they will excite the like statical experiments. such a flux and reflux as they do in the sea, but with so small a motion that no sensible wind will be thence produced. if the effects of both moon and sun in raising the tides would forces effect of (by Cor. moon the moon s 46. Now XIV, Prop. LXVI) be to is moon 5| is moon compared with sun compared with its magnitude to 1 But ; the and the the ratio of 31 1 to 32^, or of 45 to to be increased in the ratio of the effect s in directly, and in the triplicate ratio of the diameter inversely. force of the 454), as well as their magnitudes. the effect of the sun as about diameter less than the sun the force of the (p. were equal among themselves, their absolute as their apparent diameters, its in Whence the magnitude will be to the force of the the ratio compounded of 5-^- to 1, and 5^ to 1. the triplicate of 45 to 46 inversely, that is, in the ratio of about And therefore the moon, in respect of the magnitude of its body, has an absolute centripetal force greater than the sun in respect of the magnitude of its body in the ratio to 5 T \ to 1, and is therefore more dense in the same ratio. In the time of 27 1 . 7 h 43 . , in which the moon makes its revolution about the earth, a planet may be revolved about the sun at the distance of 18.95 1 diameters of the sun from the sun s centre, supposing the mean apparen and in the same time the moon may be diameter of the sun to be 32} volved about the earth at rest, at the distance of 30 of the earth s diame r"- ; ters. If in both cases the number of diameters was the same, the absolute circum-terrestrial force would (by Cor. II, Prop. LXXll) be to the absolute circum-solar force as the magnitude of the earth to the magnitude of the tun. of 30 Because the number of the earth s diameters is greater in the ratio of the will earth be less in the triplicate of that body in the ratio of 3|| to 1. Wherefore the earth s force, for the to 18,954, the ratio, that is, magnitude of its body, is to the sun s force, for the magnitude of its body, and consequently the earth s density to the sun s will be IL as 3f f to 1 the same ratio. 5J S to I, 23 or as Since, then, the moon s density will 16. Wh. veforc since the to magnitude as about I to 4l, the 547 moon s density is to the sun s density as be to the earth s density as 5 r \ to 3f {, the moon s magnitude is to the earth s moon s absolute centripetal force will be about I to 29, and the quantity to the earth s absolute centripetal force as of matter in the And moon to the quantity of matter in the earth in the same- common centre of gravity of the earth and moon is more exactly determined than hitherto has been done; from the knowledge of which AVC may now infer the moon s distance from the earth with greater ratio. hence the But I would rather wait till the proportion of the bodies of the moon and earth one to the other is more exactly defined from the phae nomena of the tides, hoping that in the mean time the circumference of the earth may be measured from more distant stations than any body has yet accuracy. employed for this purpose. Thus I have given an account of the system of the planets. As to the fixed stars, the smallness of their annual parallax proves them to be re moved immense distances from the system of the planets= that this than one minute is most certain and from thence it follows parallax that the distance of the fixed stars is above 360 times greater than the distance of Saturn from ;he sun. Such as reckon the earth one of the and the of fixed the sun one stars, may remove the fixed stars to planets, yet greater distances by the following arguments= from the annual motion to is less ; of the earth there would happen an apparent transposition of the fixed one in respect of another, almost equal to their double parallax= but the greater and nearer stars, in respect of the more remote, which are only stars, seen by the telescope, have not hitherto been observed to have the least motion. If we should suppose that motion to be but less than 20", the distance of the nearer fixed stars would exceed the mean distance of Saturn by above 2000 times. or Again= the disk of Saturn, which is only in diameter, receives but about ^+++- –^.^ of the sun s light; for so much less is that disk than the whole spherical surface of the orb of Saturn. 17" 18" Now if we suppose Saturn to rellec* about { of this light, the whole light ^ e illuminated hemisphere will be about T ^ ^Wo o"^~ ^ whole light emitted from the sun s hemisphere= and, therefore, since light is rarefied in the duplicate ratio of the distance from the luminous body, if reflected from the sun was its 10000 v/42 times more distant than Saturn, it would yet ap pear as lucid as Saturn now does without its ring, that is, something more Let us, therefore, suppose lucid than a fixed star of the first magnitude. that the distance from which the sun would shine as a fixed star exceeds that of Saturn by about 100,000 times, and its apparent diameter will be 7 V 16 vi . and its parallax arising from the annual motion of the earth

and so great will be the distance, the apparent diameter, and the parallax of the fixed stars of the first magnitude, in bulk and light equal to our sun.

Some may imagine that a great part of the light of the fixed stars intercepted and lost in its passage through so vast spaces, and upon that account pretend to place the fixed stars at nearer distances; but at this rate the remoter stars could be scarcely seen. Suppose, for example, that of the light perish in its passage from the nearest fixed stars to us then | will twice perish in its passage through a double space, thrice through a

Therefore, the fixed stars that are at a double triple, and so forth. ; distance wHl be 16 times more obscure, viz., 4 times more obscure on ac count of the diminished apparent diameter and, again, 4 times more on account of the lost light. And, by the same argument, the fixed stars at a distance will be 9 X 4. X 4, or 144 times more obscure; and those triple ; at a quadruple distance will be 16 X 4 X 4 X 4, or 1024 times more ob scure= but so great a diminution of light is no ways consistent with the phenomena and with that hypothesis which places the fixed stars at differ ent distances. Tne fixed stars being, therefore, at such vast distances from one another neither attract each other sensibly, nor be attracted by (p. 460, 461), can our sun. But the comets must unavoidably be acted on by the circum for as the comets were placed by astronomers above the moon, because they were found to have no diurnal parallax, so their annual parallax is a convincing proof of their descending into the regions of the For all the comets which move in a direct course, according to planets. solar force

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 heliocen tric opposition with them. Whereas, on the other hand, those which move against the order of the signs, towards the end of their appearance, appear swifter than they ought to be if the earth is between them and the sun ; and slower, and perhaps retrograde, if the earth is in the other side of its crbit. This is occasioned by the motion of the earth in different situa tions. If the earth go the same way with the comet, with a swifter motion, the comet becomes retrograde; if with a slower motion, the comet becomes slower, however and if the earth move the contrary way, it be comes swifter and by collecting the differences between the slower and swifter motions, and the sums of the more swift and retrograde motions, and comparing them with the situation and motion of the earth from, ; ; whence they arise, I found, by means of this parallax, that the distances of the comets at the time they cease to be visible to the naked eye are always less than the distance of Saturn, and generally even less than the distance of Jupiter. The same comets (p. thing 462). be collected from the curvature of the may These motion continues swift ; way of the bodies go on nearly in great circles while their but about the end of their course, when that part 549 of their apparent motion which arises from the parallax bears a greater proportion to their whole apparent motion, they commonly deviate from those circles and ; and when the earth goes to one side, they deviate to the corresponding with the motion of the earth, must arise chiefly from the parallax and the quantity there of is so considerable, as, by my computation, to place the disappearing other ; this deflection, because of its ; Whence it follows, that, when comets a good deal lower than Jupiter. to in nearer us their they approach perigees and perihelions, they often de scend below the orbits of Mars and the inferior planets. Moreover, this nearness of the cornets is confirmed by the annual paral lax of the orbit, in so far as the same is- pretty nearly collected by the The method supposition that the comets move uniformly in right lines. of collecting the distance of a comet according to this hypothesis from four observations (first attempted by Kepler, and perfected by Dr. Wallis is well known and the comets reduced to and Sir Christopher Wren) through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by that of the year 16 4 be Kepler, passed between the sun and the earth low the orbit of Mars; and that in 1680 below the orbit of Mercury, as this regularity generally pass : its motion was defined by Sir Christopher Wren rind others. By a like Hevelius placed all the comets about which we have rectilinear hypothesis, any observations below the orbit of Jupiter. It is a false notion, there fore, and contrary to astronomical calculation, which some have enter tained, who, from the regular motion of the comets, either remove them into the regions of the fixed stars, or deny the motion of the earth where : as their motions cannot be reduced to perfect regularity, unless we suppose them to pass through the regions near the earth in motion and these are ; the arguments drawn from the parallax, so far as it can be determined without an exact knowledge of the orbits and motions of the comets. The is farther confirmed from the light of 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 near approach of the comets their heads (p. 463, 465) ; ; and in another duplicate the sun Hence it proportion on account of the decrease of the apparent diameter. may be inferred, that Saturn being at a double distance, and having its apparent diameter nearly half of that of Jupiter, must appear about (5 of the increase of the distance from ; I times more obscure and that, if its distance were 4 times greater, its would 256 be times less and therefore would be hardly perceivable light to the naked eye. But now the comets often equal Saturn s light, without ; ; him their apparent diameters. So the comet of the year Dr. Hooke s 1668, according observations, equalled in brightness the of a fixed star of the first light magnitude and its head, or the star ID exceeding in to ; 550 the middle of the coma, appeared, through a telescope oi 15 feet, as lucid but the diameter of the head was only 25" as Saturn near the horizon ; almost the same with the diameter of a circle equal to Saturn and his ring. The coma or hair surrounding the head was about ten times that is, as broad; namely, 4 min. Again the least diameter of the hair of the comet of the year 1682, observed by Mr. Flamsted with a tube of 16 feet and measured with the micrometer, was 2 but the nucleus, or star in ; ; the middle, scarcely possessed the tenth part of this breadth, and was or 12" broad; but the light and clearness of its head therefore only 11 exceeded that of the year 1680, and was equal to that of the stars of the first or second magnitude. Moreover, the comet of the year 1665, in April, as Hevelws informs us, exceeded almost all the fixed stars in splendor, arid for this comet itself, as being of a much more vivid colour was more lucid than that which appeared at the end of the foregoing year and was compared to the stars of the first magnitude. The diameter of but the nucleus, compared with the planets by the coma was about 6 even Saturn ; ; means of a telescope, was plainly less than Jupiter, and was sometime*? thought less, sometimes equal to the body of Saturn within the ring. To this breadth add that of the ring, and the whole face of Saturn will be twice as great as that of the comet, with a light not at all more intense and therefore the comet was nearer to the sun than Saturn. From the ; proportion of the nucleus to the whole head found by these observations, and from its breadth, which seldom exceeds 8 or 12 it appears that thi- ; , Btars of the comets are most commonly but that their light as the planets of Saturn, and sometimes exceeds it. ; of the same apparent magnitude be compared oftentimes with that may And hence it is certain that in their can scarcely be greater than that of Saturn. At perihelia their distances twice that distance, the light would be four times less, which besides by its dim paleness would be much inferior to the light of Saturn as the light of of Saturn is to the splendor Jupiter but this difference would be easily ten times At a distance observed. greater, their bodies must be greattr as : than that of the sun that of Saturn. exceed the sun ;. ; but their light would be 100 times fainter than And at distances still greater, their bodies would far but, being in such dark regions, they must be no longer So impossible is it to place the comets in the middle regions be the sun and fixed stars, accounting the sun as one of the fixed stars: tween for certainly they would receive no more light there from the sun than visible. w<? do from the greatest of the fixed stars. So far we have gone without considering that obscuration which comets suffer from that plenty of thick smoke which encompasseth their heads. and through which the heads always shew dull as through a cloud for by how much the more a body is obscured by this smoke, by so much th.2 more near it must be allowed to come to the sun, that it may vie with the ; 651 whence it is probable planets in the quantity of light which it reflects that the comets descend far below the orbit 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 from a srnoke arising from them, and dispersing itself through the aether, or to the light uf their own headt. we must shorten the distance of the comets, lest we be smoke arising from their heads is propagated vast extent of space, and with such a velocity of expansion, In the former case obliged to allow that the through such a 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 jus nucleus, certainly the nucleus will by far exceed Jupiter itself in splendor, when it emits a very large and lucid tail. If, therefore, under a less especially apparent diameter, it reflects more light, it must be much more illuminated by the sun, and therefore much nearer to it. So the comet that appeared Dec. }2 and 15, O.S. Anno 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. Flaw sled observed), and therefore was much nearer to the sun : For on the 17th of that month, when nay, it it was even was nearer less than Mercury. the earth, it ap less than the globe to peared to Cassini through a telescope of 35 feet a little of Saturn. On the 8th of this month, in the morning, Dr. ffalfey saw the tail, appearing broad and very short, and as if it rose from the body of the 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 sun itself, at that ; the clouds till the sun rose, and far surpassed that of all the stars together, immediate brightness of the sun itself. Neither as yielding only to the Mercury, nor Venus, nor the moon Imagine all itself, are seen so near the rising sun. together, and to be crowded into nucleus which was less than Mercury by its this dilated light collected the orbit of the comet s ; splendor, thus increased, becoming so much more conspicuous, exceed Mercury, and therefore must be nearer to the sun. it will On vastly the 12th and 15th of the same month, this tail, extending itself over a much greater 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 space, appeared 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- like a fiery tion they are seen in when their tails appear most resplendent; for when the head passes by the sun, and lies hid under the solar rays, very bright ta Is, like fiery beams, are said to issue from the horizon; but afterwards, when the head begins to appear, and is got farther from the and shining 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 The head thereof could not be seen, because Aristotle, Lib. 1, Meteor. 6. " before the sun, or at least was hid under the sun s rays but the next it was seen as well as might be for, having left the sun but a very day little way, it set immediately after it and the scattered light of the head it set ; ; ; 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 of tail its now to a third the reached of heavens splendour part (that is, to 60). It vanished appeared in the winter season, and, rising to Orion s girdle, there Two comets of the same kind are described by Justin, away." 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 We may add to these the comet of (p. 494, 495). setting sun is intimated the year 1101 or 1106, " the star of which was small and obscure (like that but the splendour arising from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Simeon, the monk of Durham. It appeared at the beginning of February about the of ] 6SO) ; evening in the south-west. From this and from the situation of the infer that the head was near the sun. Matthew Paris says, we may tail "it was about one cubit from the sun from the third [or rather the sixth] to the ninth hour sending out a long stream of light." 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 the middle of the heavens. It is said to have been at the beginning very by a and It large bright, having large coma, which decayed from day to day. described in Append. Matth, Paris, Hist. Aug. after this manner ^Au. 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 is : ; \uth a great light as far as the middle of the hemisphere towards the west." The Latin original being somewhat barbarous and ob- extended gcure, " :i s, itself it is here subjoined. usque ad pertrahcbai. medium Ah oriente enim cum tnaguo fulgore sur- omuia per lucid* hcmisp]icerii versus occideutcm, 553 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 "In When the sun was sunk below the horizon, by the rays from west to east. 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 air, because its light exceeded that of the others, and extended itself to the upper part of the heavens, flaming," &c., Hist. Byzaut. Due. Mich. From the situation of the tail of this comet, and the time of its Nepot. the first appearance, we may infer that the went farther from him every day ; head was then near the sun, and comet continued three months. for that In the year 1527, Aug. 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 the 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 On Nov. 24, and from that time, the comet brightness of the solar rays. with a bright light, its head and tail being extremely re appeared The splendent. length of the tail, which was at first 20 or 30 dog., in itself 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. Volent. Estaucius, being in Brazil, saw a comet near the horizon in the south-west. Its head was small, and discernible, but and tail its scarcely refulgent, so that the extremely bright 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 from that time. The tail at the beginning extended itself from west ably 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 Cassiid, at Bologna^ saw it (Mar. 10, 11, 12) rising from the horizon 32 deg. in In Portugal it is said to have taken up a fourth part of the length. 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 creased ; From the increase of part of the world always lay hid below the horizon. the tail it is plain that the head receded from the sun. and was nearest to at the beginning, when the tail appeared brightest. To all these we may add the comet of 1680, whose wonderful splendor at the Hut so conjunction of the head with the sun was above described. it 554 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 nor do we read that fiery beams have ever ap their opposition to the sun j peared there. Lastly, the same thing is inferred (p. 466 407) 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 1 665 (by the observation of Hevelius]^ from the ; it was first seen, was always losing of its apparent motion, and 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 The comet of the year (683 (by the observation of the same He- pear. time that therefore : ? jelius), 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 I3ut from that time its diurnal motion was continually upon the time. increase till Septe/uber 4, when it arose to about 5 degrees ; and therefore 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 including the in all this interval of time the 5", ; And therefore its head which, September 2, he observed 9 appeared far less about the beginning than towards the end of its motion, coma 7". ; 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, The it decreased in splendor, notwithstanding its access towards the earth. comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with their greatest velocity, and were therefore then in their perigees but the greatest splendor of their heads was seen two weeks before, when they had just got and the greatest splendor of their tuild a little clear of the sun s rays : : more early, according when yet nearer to the sun. the stars of the first The head of the former comet, 1, appeared greater than 16 Dec. magnitude= and, (being then in its perigee), to the observations of Cysattis, Dec. a small magnitude, and the splendor or clearness was much diminished. Dec. 7, Kepler, being uncertain about the head left off observing. of the head the last Flamxted comet and observed at was seen the 12, by )i Jan. ? 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 fan. 9, like magnitude. a star of the fifth. appeared like a star of the fourth Jan. 13. it disappeared, by reason of Jait. 3, it : tb<~ 555 Jan. 25, it was brightness of the moon, which was then in its increase. the If to the stars of seventh we take equal scarcely equal magnitude. times on each hand of the perigee, the heads placed at remote distances would have shined equally before and frjin the earth. other vanished, after, because of their equal distances one case they shined very bright, and in the be ascribed to the nearness cf the sun in the first case, That is to in 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 ; increased by their nearness to the sun. From thee things I at last discovered why the comets frequent so much If they were to be seen in the regions a great way the region of the sun. ing in so fur as it is Saturn, they must appear oftener in these 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 beyond ; 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 th-3 besides, without doubt, not a few which have been opposite hemisphere hid by the light of the sun fur comets descending into our parts neither ; ; 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 emit tails, the far greater part of that spherical space, which is described about the sun with so small an interval, lies en 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 ble eccentricity of their orbits, it are much to the sun comes to : besides, from the remarka pass that their lower apsides 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 This great splendor and length to 40, 50, 60, 70 deg. in length, or more. of the tails arises from the heat which the sun communicates to the comet as it passes near And it. thence, I think, it comets that have had such may be concluded, that all the 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 tails : the tails of comets ; the beams of the sun for some will have it that they are nothing else but 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 s fi and, lastly, others, that they are a sort of clouds or vapour constantly rising from the cornets 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 difficultly seen in a finer air; but in the heavens, where there is no matter to reflect the light, they are 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 and so the argument turns upon parts where the tails of comets are seen not to be seen at ; ; for that reflecting matter can be no where found but in the third opinion the the place of 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 The of comets are never seen variegated with thos-e and the distinct be inseparable from refraction to use colours which ever to and us is a demon the fixed stars of the of transmission planets light medium not endowed with any re is aether or celestial the stration that difficulties. tails ; For as to what is alledged that the fixed stars have been seen sometimes 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 fractive power. stars to the refractions both of the eyes scope to the eye, those radiations and and air ; for upon applying a tele scintillations immediately disappear. the tremulous agitation of the air and ascending vapours, it happens that the rays of light are alternately turned aside from the narrow space but no such thing can have place in the much of the pupil of the eye 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 By 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 and for this reason it is that rays were then too weak to affect the eyes, 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 without any tail, but that comets are seen sometimes yet more copious 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- ; 557 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 But on the 9th that was more difficultly seen, even to 12 and upwards. ; and 10th of February, when to the naked eye the head appeared no more, 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 I : from the opposition of the sun, according as the figure of the heavens re be always quires, that deviation, in the same places of the heavens, should but the comet of the year 1680, Decem directed towards the same parts : 28 M. London, was seen in Pisces, 8 41 with latitude SJ And the comet of north 28 6 while the sun was in Capricorn 18 26 the year 1577, December 29, was in Pisces 8 41 with latitude north In both 2S D 40 and the sun, as before, in about Capricorn 18 26 cases the situation of the earth was the same, and the comet appeared in ber 1 1 . P. . at , . , , . ; the same place of the heavens yet in the former case the tail of the comet observations as well (as by the observations of others) deviated by my 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 ; Tycht] a deviation of 21 degrees towards the south. The refraction, heavens being thus disproved, it remains that the phaeno- therefore, of the meria of the tails of comets must be derived from some reflecting matter. sufficient to fill such immense spaces may arise from the That vapours comet s atmospheres, may be easily understood from what follows. known that the air near the surface of our earth possesses a It is well 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 ; But a cylinder of water of the same breadth, and but one foot high. 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 is less, and consequently the rarity of the weight of the incumbent air compressed air greater, than near the surface of the earth in the ratio of 33 to 32. And, having this ratio, we may compute in all places whatsoever (by the help of Cor. Prop. the rarity of the air XXII, Book II), sup posing the expansion thereof to be reciprocally proportional to its compres and this proportion has been proved by the experiments of Hooke sion ; and others. The result of the computation I have set down in the follow-THE SYSTEM CF THE WORLD. 558 first column of which you have the height o the air in whereof 4000 m:ike a semi-diameter of the earth; in the second the ing table, in the miles, 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 compression of the air, or ; are here used for certain numbers of IMJ00000000000000001224, and 26950 xv AlR s Height. ciphers, as 0,xvii 1224 for for 26956000000000000000, 559 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 Another author thinks that there may be a sort of particles sistance. 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 but, considering the gravity of terrestrial and therefore can be neither more nor bodies, am matter, I may bodies ; same quantity of rather may proceed from The ascent of smoke in less in the inclined to believe that this ascent the rarefaction of the matter of the comets be owing to its levity as the matter of the is tails. 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 along with it the smoke with which it is engaged. a is chimney owing to the /Vnd 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 heated by this action, heat the matter of the aether which is involved ; cles with them. That matter is by the heat which rarefied 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 towards arise from their heads (p. 488), and tend the sun, is farther confirmed from the laws lying in the planes of the comets orbits which the parts opposite to which the observe tails ; for, 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 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. orbits ; ; And the deviation, c&teris paribits, 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 when the paribus, is longer; for in the be And the angle of hardly perceived. less near the comet s head, but greater towards the other end and that because the lower side of the tail regards the parts sensible shorter tails the curvature deviation of the is tail, tail, cccteris to is made, and which lie in a right line drawn out from the sun through the comet s head. And the tails that are from which the deviation infinitely is 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 ; it is plain that the phenomena of the tails of comet? de of their heads, and by no means upon the places of the motions pend upon and that, therefore, the tailg of the heavens in which their heads are seen which accounts ; 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, body heavens, where all must if the body is moved obliquely, so in the the bodies gravitate towards the sun, smoke and vapour at rest, or obliquely if the is (as we have already said) ascend from the sun, and either rise perpen if the body, in the dicularly, if the smoking body is at rest, or obliquely, of its motion, is always leaving those places from which the upper progress And that obliquity will or higher parts of the vapours had risen before. 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 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 the vapour on the other side languishing by degrees, and vanish defined more lately ; ing out of sight. none of our present business to explain the causes of the ap Let those things which we have last said be true or pearances of nature. have at least made out, in the preceding discourse, that the rays false, we But it is 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- <~ manner tances in resent the sun, following. Let the earth, T S rep- STA the elongation of a comet from the sun, and the apparent length of its tail; ATB and because the light is propagated from the extremity of the tail in the direction of the right, line TB, that extremity must somewhere lie Suppose it TA in C. ways in the line TB. D, and join DS cutting Then, because the tail is al in

stretched out towards the parts nearly opposite to the sun, and there! ore 561 the sun, the head of the comet, and the extremity of the Parallel to line, the comet s head will be found in C. tail, lie in a right TB draw SA, TA head meet C must necessarily be found between and A, because the extremity of the tail lies somewhere in the and all the lines SI) which can possibly be drawn from infinite line ing the line in A, arid the comet s T TB S the point and A. TB TA. nor gation 9, and angle SA SA beyond, or the elongation of the comet of and the length of its tail 35 at least. For instance 12, TSA therefore, a triangle then TA somewhere between T line distance from the sun the interval its on this side the sun. 16SO from the sun, Dec. If, must cut the Wherefore the distance of the comet from the earth cannot exceed the interval ST ; to the line A equal was 9, made, whose angle is to : ATB, T is equal to the elon or to the length of the tail, viz., 35, the limit of the greatest possible distance of the comet from the sun to the semi -diameter of the oj-bis ma gnus, as will be to ST, that is, T to the sine of the angle A, that is, as about 3 to the sine of the angle 11. And therefore the comet at that time was less distant from the sun than by T 3 T 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 32f and the length , 70. Wherefore as the sine of 3^| 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 of its tail Dec. 28, the elongation of the comet from got without the orb of Venus. and therefore the limit of the sun was 55, and the length of its tail 56 ; 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. orbit But from happened about Jan. its 5, as parallax we find that its egress from the well as that it had descended far within Let us suppose it to have been in its perihelion the orbit of Mercury. 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 ; 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 and by limiting the distances scarcely doubled its distance from the sun it ; ; of other comets by the like arguments, we that all comets, during the time in sion, come at last to this conclu 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 it follows that the comets, during the whole time of their unto us, being within the sphere of activity of the circum appearance 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 tc 36 hence 562 move in conic sections that have one focus in by radii drawn and the centre of the sun, to the sun, to describe areas proportional to the times for ; an immense distance, and will govern the far bodies of motions beyond the orbit of Saturn. There are three hypotheses about comets (p. 466) for some will have it that force is to propagated ; that they are generated and perish as often as they appear and vanish others, that they come from the regions of the rixed stars, and are seen by ; us in their passage through the system of our planets and, lastly, others, that they are bodies perpetually revolving about the sun in very eccentric ; In the orbits. the comets, according to their different vel first case, cities, move in conic sections of all sorts; in the second, they will describe hyperbolas, and in either of the two will frequent indifferently all quar will of the heavens, as well those about the poles as those towards the in the third, their motions will be performed in ellipses very ec ecliptic ters ; to parabolas. But (if the law of the will not much decline from the plane of their orbits observed) planets the ecliptic; and, so far as I could hitherto observe, the third case obtains; and very nearly approaching centric, is comets do, indeed, chiefly frequent the zodiac, and scarcely ever And that they move in orbits attain to a heliocentric latitude of 40. for the very nearly parabolical, I infer from their velocity ; for the velocity with which a parabola is described is every where to the velocity with which a comet or planet may be revolved about the sun in a circle at the same dis tance in the subduplicate ratio of 2 to 1 (by Gor. VII, Prop. XVI) and, ; computation, the velocity of comets is found to be much about I examined the thing by inferring nearly the velocities from the same. the distances, and the distances both from the parallaxes and the phaeno- rnena of the tails, and never found the errors of excess or defect in the ve by my than what might have arose from the errors in the dis locities greater tances collected after that manner. But ing that follows. Supposing the radius of the nrbis I likewise made use of the reason magiius to be divided into 1000 of the following table represent column parts: the distance of the vertex of the parabola from the sun s centre, expressed in col. 2, will pass by those parts and a comet in the times expressed from its perihelion to the surface of the spheie which is described about the sun as a centre with the radius of the orbis magnus ; and in the times expressed in col. 3, 4, and 5, it will double, triple, and quadruple, let the numbers in the first : that its distance from f.l:o sun TABLE i he dis- innoe of a comet s tiprihclion from Siin tip. the d c^-n- L 563564

The ingress 01 a comet into the sphere of the orbis magnus, or its from the same, happens at the time of its elongation from the sun, egress in col. 1, against its diurnal motion. So in the comet of 1681. expressed Jan. O.S. the apparent diurnal motion in 4, its orbit was about 3 5 , and the corresponding elongation 71 J and the comet had acquired this elon from about in Ae evening. the sun Jan. six 4, Again, in the year gation that then Nov. motion of the diurnal the comet 1680, 11, appeared was ; about 4| and the corresponding elongation 79f happened Now. ; 10, a before midnight. Now at the times named these comets had arrived at an equal distance from the sun with the earth, and the earth was then little But the first table is fitted to the earth s mean its perihelion. distance from the sun assumed of 1000 parts and this distance is greater by such an excess of space as the earth might describe by its annual motion almost in ; To reduce the in one day s time, or the comet by its motion in 16 hours. comet to this mean distance of 1000 parts, we add those 16 hours to the former time, and subduct them from the latter and thus the former be ; comes Jan. 4 d 10 the latter Nov. 10, about six in the morn from the tenor and progress of the diurnal motions it appears . ing. But afternoon 1 . ; that both comets were in conjunction with the sun between Dec. 7 and Dec. h 8 and from thence to Jan. 4 d 10 afternoon on one side, and to Nov. . . ; 6 h of the morning on the other, there are about 28 days. And so many days (by Table 1) the motions in parabolic trajectories do require. 10 . . But though we have hitherto considered those comets as two, yet, from the coincidence of their perihelions and agreement of their velocities, it is probable that in effect they were but one and the same and if so, the ; comet must have either been a parabola, or at least a conic section very little differing from a parabola, and at its vertex almost in For (by Tab. 2) the distance of the contact with the surface of the sun. orbit of this comet from the earth, Nov. 10, was about 360 parts, and Jan. 4, about From which distances, together with its longitudes and latitudes, 630. we infer the distance of the places in which the comet was at those times to have been about 280 comet s orbit, : the half of which, viz., 140, is an ordinate to the cutting off a portion of its axis nearly equal to the radius magnus, that is, to 1000 parts. And, therefore, dividing the of ordinate 140 by 1000, the segment of the axis, we find the the square latu$ rectum 19, 16, or in a round number 20 the fourth part whereof, But the 5, is the distance of the vertex of the orbit from the sun s centre. of the or bis ; time corresponding to the distance of 5 parts in Tab. 1 is 27 d 16 h 7 . . . Ir. which time, if the comet moved in a parabolic orbit, it would have been carried from its perihelion to the surface of the sphere of the orbis mag* nus described with the radius 1000, and would have spent the double of h d that time, viz., 55 8| in the whole course of its motion within that h d sphere and so in fact it did for from Nov. 10 6 of the morning, thf time of the comet s ingress into the sphere of the orbis OOC magnns, to Jan.. 10 h afternoon, the time of its egress from the same, there are 55 16 h The small difference of 7 u in this rude way of computing is to be neg lected, and perhaps may arise from the comet s motion being some small 4 1 (1 . . . . . it must have been if the true orbit in which it was car was an ellipsis. The middle time between its ingress and egress was December S d 2 of the morning and therefore at this time the comet matter slower, as ried 1 . . ; ought to have been in before sunrising, Dr. its perihelion. Halley (as we And said) accordingly that very day, just tail short and broad, but- saw the From the position very bright, rising perpendicularly from the horizon. of the tail it is certain that the comet had then crossed over the ecliptic, and got into north latitude, and therefore had passed by its perihelion, which lay on the other side of the ecliptic, though it had not yet come into conjunction with the sun and the comet [see more of this famous comet, p. 475 to 486] being at this time between its perihelion and its conjunc ; tion with the sun, must have been in its perihelion a few hours before; from the sun it must have been carried with great for in so near a distance velocity, and have apparently described almost half a degree every hour. By like computations I find that the comet of 1618 entered the sphere of the orbis maxims December 7, towards sun-setting but its conjunc ; tion with the sun was Nov. preceding comet ; for 9, or 10, about from the size of the tail of this, in to the preceding, it is probable equal almost into a contact with the sun. which this was the last. The 28 days intervening, as in the wtrch it was that this comet likewise did come Four comets were seen that year of second, which made its first appearance October 31, in the neighbourhood of the rising sun, and was soon after hid s rays, 1 suspect to have been the same with the fourth, under the sun which emerged out of the sun s rays about Nov. 9. To these we may add the comet of 1607, which entered the sphere of the orbis mi^-tnis Sept. 14, O.S. and arrived at its perihelion distance from the sun about October Its perihelion distance subtended an apparent 19, 35 days intervening. angle at the earth of about 23 degrees, and was therefore of 390 parts. And to this number of parts about 34 days correspond in Tab. 1 Far . ther ; the comet of 1665 entered the sphere of the orbis nta^tnts about March about April 16, 30 days intervening. an Its perihelion distance subtended angle at the earth of about seven and corresponding to this number and 122 of therefore was degrees, parts 17, and came to its perihelion : Again the comet of 1 682 entered Aug. 11, and arrived at its perihe lion about Sep. 16, being then distant from the sun by about 350 parts, to which, in Tab. I, belong 33^ days. Lastly that memorable comet of carried through the circum-polar in 1472 was which Regiomontanus, with such rapidity as to describe 40 of our northern parts hemisphere of parts, in Tab. 1, we find 30 days. the sphere of the orbis magnus about degrees in one day, entered the sphere of the orbis magnus Jan 21, abonl the time that it was passing by the pole, and, hastening from them* towards the sun, was hid under the sun s rays about the end of Feb. , whence it is probable that 30 days, or a few more, were spent between its ingress into the sphere of the orbis magnus and its Nor did perihelion. this comet truly move with more velocity than other comets, but owed the greatness of its apparent velocity to its passing by the earth at a near distance. It appears, then, that the velocity of comets (p. 471), so far as it can be determined by these rude ways of computing, is that very velocity with which parabolas, or ellipses near to parabolas, ought to be described; and therefore the distance between a comet and the sun being given, the velocity of the comet is nearly given. And hence arises this problem. 4 The PROBLEM. relation betwixt the velocity of a comet and its distance from the sun s centre being given, the comet s trajectory is required. If this problem was resolved, we should thence have a method of deter mining the trajectories of comets to the greatest accuracy for if that re lation be twice assumed, and from thence the trajectory be twice computed, and the error of each trajectory be found from observations, the assumption may be corrected by the Rule of False, and a third trajectory thence : may be found that will exactly agree with the observations. And bv deter mining the trajectories of comets after this method, we may come" at last, to a more exact knowledge of the parts through which those bodies travel, of the velocities with which they are carried, what sort of trajectories they describe, and what are the true magnitudes and forms of their tails accord ing to the various distances of their heads from the sun whether, after certain intervals of time, the same comets do return again, and in what ; But hhe problem ma? be periods they complete their several revolutions. resolved by determining, first, the hourly motion of a comet to a ffiven time from three or more observations, and then deriving the trajectory from motion. this And thus the invention of the trajectory, depending on one ob servation, and its hourly motion at the time of this observation, will either confirm or disprove itself; for the conclusion that is drawn from the mo tion only of an hour or two and a false hypothesis, will never agree with the motions of the comets from beginning to end. The method of fh* whole computation is this.

Lemma I.

To cut two right lines OR, TP, given in, position, by a third right line RP, so as TRP may be a right angle ; and, if another right line SP is drawn to any given point S, the solid contained under this line SP and the square of the right line OR terminated at a given point O, 5 may be of a given magnitude. done by linear description thus. Let the given magnitude of the 2 be x N from any point r of the right line OR erect the per It is M solid : TP pendicular rp meeting M line Sq equal to in p. X N 2 ^ Then through the point Sp draw the In like manner draw three or more right lines . 2 &c. and a regular line q2q3q, drawn through all the points S2q, in the point P, from which the y2q3q, &c., will cut the right line per is to be let fall. Q.E.F. pendicular ; S3<7, TP PR TP as found by the By trigonometry thus. Assuming the right line preceding method, the perpendiculars TR, SB, in the triangles TPR, TPS, and the side SP in the triangle SBP, as well as the will be thence given ; M error 2 X N ^r^ Let SP. + this error, suppose D, be to a new error, sup to the error error 2p2q 3p3q 2p3p ; or as the error 2p2q pose E, as the and this new error added to or subducted from the H- to the error 2pP D ; TP + E. The inspection of the give the correct length we add whether are to to or and if at any time subtract shew figure will for a farther the be use there should correction, operation may be repeated length TP, will ; 668 By arithmetic thus. Let us suppose the thing done, and let TP -f- e be the TP as found out by delineation and thence correct length of the right line : OR. BP, and SP, the correct lengths of the lines BP + and ^/SP 2 e, + 2BPe + ee 2 QRa = M N 2 ee, + M N 2TR 2 <fcc., 2 co-efficients ^- ^ SP, 2 M N 2 X 3 e + series, 3TR M N 66 2 x l BP 3TR 2 F i

, ppj, 3 gp> and carefully observing the -f- -p6 tor x M N ~ 2 SR 2 + ^ op~j the given SB 2 2 Tppl" signs, wo QR4 2SP~J F find F + ^ e -f ee 0, and e

• YT= H G. Whence, neglecting the very small G. If the error e 2 e 2 term op we have SP 2 M N 2TR X Tp -^- i TP F F putting F, ^^e. M N 20RX = Whence, by the method of converging TR OR will be ^, e comes out equal = G to ^ is not despicable, take e. jj And it is solving the to be observed that here a general method is hinted at for intricate sort of problems, as well by trigonometry as by more arithmetic, without those perplexed computations and resolutions of affected equations which hitherto have been in use.

Lemma II.

cut three right lines given in position by a fourth right line that shall pass through a point assigned in any of the three, and so as its intercepted parts shall be in a given ratio one to the other. To D to BC Let AB, AC, BC, be the right lines given in position, and suppose draw Parallel to be the given point in the line AC. meeting DG AB in G ; GF to BG FG to BG. and, taking 1)E as will be to in the given ratio, Q.E.F. draw FDE ; and FD

By all the trigonometry thus. In the triangle angles and the side are given, and from thence its sides are found and from remaining the given ratios the lines and are also given. CD ; GF BE Lemma 3

Find and represent hy a linear description the hourly motion of to any given time. a comet From observations of the best credit, let three longitudes of the comet be given, and, supposing ATR, RTB, to be their differences, let the hourly motion be required to the time of the middle observation draw the right line ARB, so as its intercepted parts By Lem AR, RB, may b< and if we suppose a body in the whole time to describe the whole line AB with an equal motion, and to be in the mean time viewed from the place T, the apparent motion of that as the times between the observations ; body about the point R will be nearly the same with that of the comet at the time of the observation TR. The same more accurately. Let Ta, T6, be two longitudes given at a greater distance on one sftle and on the other and by Lem,. II draw the right line aRb so as its inter cepted parts aR, Rft may be as the times between the observations aTR, RTA. Suppose this to cut the lines TA, TB, in D and E and because the error ; ; of the inclination between TRa increases nearly in the duplicate ratio of the time the observations, draw FRG, so as either the angle may be DRF to the angle ARF, or the line DF to the line of the whole time between the observations the observations line AB IB, and use the aTB line thus found FG in place of the found above. be convenient that the angles ATR, RTB, aTA, BT6, be nc than of ten or fifteen degrees, the times corresponding no greater than It will less A in the duplicate ratio to the whole time between AF,5~0

and the longitude^ taken when the comet jnoves with the greatest velocity for thus the errors of the observation \s will bear a less proportion to the differences of the longitudes. of eight or twelve days, Lemma To find It is to t FG done by taking in the line hgonometry is a comet the longitudes of the times, and drawing the IV. lines to times. any given the distances Rr, Rp, proportional The way of working by Tr, Tp. manifest. Lemma To find V. the latitudes. On TF, TR, TG, as radiuses, at right angles erect F/, RP, Gg-, tan of the and parallel to fg draw PH. The per observed latitudes gents will be the tangents of the sought latitudes PH, pw, pendiculars rp, meeting ; to Tr and Tp as radiuses. PROBLEM Prow, the assumed ratio of the velocity to I. determine the trajectory oj a comet. Let S represent the sun at e^ual distances its trajectory, so ; p, P, o5 ? ; /, as T, r three places of the earth in } many as the distances interposed betwixt place and place answer to the motion of one hour ; O, may pr, PR, wp, perpendiculars let fall on the vestige of the trajectory in this the plane of the ecliptic, and rRp Join S/?, SP, Sc5, SR, ST, tr, plane. TR its orbit corresponding places of the comet in TR, rp, TP , and let tr, -p, meet in same point O, or the error will be in the angles rOR, ROp, are given, lemmas premised will nearly converge to the considerable. By the as well as the ratios pr to //;, PR to TR, and wp rr to rp. lie figure Tr likewise given both in magnitude and position, together with the dis ST, and the angles STR, PTR, STP. Let us assume the velocity of the comet in the place P to be to the velocity of a planet revolved is tance about the sun in a have a line circle, at the same distance SP, as V to 1 and ; we shall be determined, of this condition, that the space /?w, described by the comet in two hours, may be to the space V X tr (that is.

to which the earth describes in the same time multiplied by the to the space number V) in the subduplicate ratio of ST, the distance of the earth from the sun, to SP, the distance of the comet from the sun and that the space pP, described by the comet in the first hour, may be to the space Pw, de ; scribed by the comet in the second hour, as the velocity in p to the velocity in P that is, in the subduplicate ratio of the distance SP to the distance ; + Sp ; for in this whole work I neglect 2Sp to SP small fractions that can produce no sensible error. In the first place, then, as mathematicians, in the resolution of affected S/7, or in the ratio of equations, are wont, for the first essay, to assume the root by conjecture, as I analytical operation, I judge of the sought distance TR so, in this best can by conjecture. Then, by Lem. II. I draw rp, first supposing / R equal to Rp, and again (after the ratio of SP to Sp is discovered) so as rR may be to Rp as 2SP to SP Sp, and I find the ratios of the lines Let be to V X tr as to pi** and pw, rp, and OR, one to the other. + M because the square of shall have, ex aquo, OR X SP 2 to is p<*> OR 2 to M equal to the given triangles STP, PTR, to be PR, will be given, by Lem. OR V X the square of 2 as solid ST to M X ST; now placed 2 tr as ; ST to SP, we SP, and therefore the solid whence (supposing the same plane) TR, TP, SP, in the by delineation in a rude and hasty way then by a new delineation with greater care and, lastly, by an arithmetical computation. Then I proceed to determine the position I. All this I do, first ; ; of the lines rp, pti, with the greatest accuracy, together with the nodes and inclination of the plane Spti to the plane of the ecliptic and in that ; the trajectory in which a body let go from the place in the direction of the given right line jf?c5 would be carried with i velo plane Spti P I describe V X city that is to the velocity of the earth as pti to PROBLEM To correct the assumed tr.

ratio of the velocity and the trajectory thence found.

Take an observation of the comet about the end of its appearance, or any other observation at a very great distance from the observations used before, and find the intersection of a right line drawn to the comet, in that observation with the plane Sjow, as well as the comet s place in its trajectory to the time of the observation.

If that intersection happens in this place, it is a proof that the trajectory was rightly determined if other new number V is to be assumed, and a new trajectory to be found then tlu place of tke comet in this trajectory to the time of that pro- batory observation, and the intersection of a right line drawn to the comet with the plane of the trajectory, are to be determined as before and by wise, a

comparing the variation of the error with the variation of the other quan tities, we may conclude, by the Rule of Three, how far those other quantities ought to be varied or corrected, so as the error may become as And by means of these corrections we may have the small as possible.

trajectory exactly, providing the observations was founded were trajectory upon which the computation err much in the assumption we did, the operation is to be repeated determined