Principia Mathematica (Motte Translation, 1848) — part 40 of 45

" Cassini ... 5

8

13

23

" Borelli, more ex-

actly ... 5^

8K

14

24K

After the invention of the micrometer : —

By Town ley . . . 5,51 '■ Flamsted . . . 5,31 More accurately by

the eclipses . . 5,578

8.78 8,85

8,876

13,47 13,98

14,159

24,72 24,23

24,903

THE SYSTEM OF THE WORLD. 517

And the periodic times of those satellities, by the observations of Mr. Flamsted, are 1^^ 18\ 28' 36" | S-^. 13'\ 17' 54" | 7\ 3^. 59' 36" | 16^*. 18^ 5' 13", as above.

And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968, accurately agreeing with the distances by observation.

Cassini assures us (p. 388, 389) that the same proportion is observed in the circum-saturnal planets. But a longer course of observations is 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 is revolved about the sun is demonstrated from the phases which it shews, and the proportion of its apparent diameters (p. 388, 389, and 390) ; for from its appearing full near conjunction with the sun, and gibbous in its quadratures, it is certain that it surrounds the sun.

And since its diameter appears about five times greater when in opposi- tion to the sun than when in conjunction therewith, and its distance from the earth is reciprocally as its apparent diameter, that distance will be about five times less when in opposition to than when in conjunction with the sun ; but in both cases its distance from the sun will be nearly about the same with the distance which is inferred from its gibbous appearance in the quadratures. And as it encompasses the sun at almost equal distan- ces, but in respect of the earth is very unequally distant, so by radii drawn to the sun it describes areas nearly uniform ; but by radii drawn to the earth, it is sometimes swift, sometimes stationary, and sometimes retrograde.

That Jupiter, in a higher orb than Mars, is likewise revolved about the sun, with a motion nearly equable, as well in distance as in the areas des- cribed, I infer thus.

Mr. Flamsted assured me, by letters, that all the eclipses of the inner- most satellite which hitherto have been well observed do agree with his theory so nearly, as never to differ therefrom by two minutes of time ; that in the outmost the error is little greater ; in the outmost but one, scarcely three times greater : that in the innermost but one the difference is indeed much greater, yet so as to agree as nearly with his computations as the moon does with the common tables ; and that he computes those eclipses only from the mean motions corrected by the equation of light dis- covered and introduced by Mr. Romer. Supposing, then, that the theory differs by a less error than that of 2' from the motion of the outmost sat- ellite as hitherto described, and taking as the periodic time 16'^. 18^. 5' 13" to 2' in time, so is the whole circle or 360° to the arc 1' 48", the error of Mr. Flamsted's computation, reduced to the satellite's orbit, will be less than 1' 48" ; that is, the longitude of the satellite, as seen from the centre of Jupiter, will be determined with a less error than 1' 48". But when

518 , THE SYSTEM OF THE WORLD.

the satellite 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 (as 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 dis- tance of Jupiter from the sun is determined ; which must, therefore, be the very same with that which the hypothesis exhibits. For that greatest error of 1' 48'' 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.

And 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.

For if Jupiter was viewed from the sun, it would never appear retro- grade 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. And the 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 ia Mars and Saturn nearly, but accurately enough in Jupiter. It may be alledged that the sun and planets are impelled by some other force equally and in the direction of parallel lines ; but by such a force (by Cor. YI -of the Laws of Motion) no change would happen in the situation of the planets one to another, nor any sensible effect follow : but our business is with the causes of sensible effects. Let iis, therefore, neglect every such force as imaginary and precarious, and of no use in the ph^enomena of the heavens ; and the whole remaining force by which Jupiter is impelled will be directed (by Prop. Ill, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tychoj we place the earth in the centre of the system, or the sun with Copernicus : and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus. have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables

THE SYSTEM OF THE WORLD. 519

agree best with the heavens. And in all the planets, in Jupiter and Mars, in 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. YI, Prop. IV) the centripetal circum-solar force throughout all the plane- tary regions decreases in the duplicate proportion of the distances from the sun. In examining this proportion, we are to use the mean distances, or the transverse semi-axes of the orbits (by Prop. XV). and to neglect those little fractions, which, in defining the orbits, may have arisen from the in- sensible errors of observation, or may be ascribed to other causes which we shall afterwards explain. And 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 951000, 519650, 152350, 100000, 72400, 3S806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 7239S, 38585 ; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, 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- 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 EphemerideSj Bullialdus, Hevelius, and Micciolus, 59 ; according to Flamsted, 59 1- ; according to Tycho, 5Q\ ; to Vendelin, 60 ; tO| Copernicus, 60} ; to Kir- Cher, 621 (p. 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment the horizontal parallax of the moon by about the like number of minutes ; that is, by about the 12th or 15th part of the whole parallax. Correct this error, and the distance will be- come 60 or 61 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 of the earth, and its periodic time in respect of the fixed stars 27^^. 7^ 43', as astronomers have determined it. And (by Cor. VI, Prop. IV) a body revolved in our air, near the surface of the earth supposed at rest, by means of a centripetal force which should be to the same force at the dis- tance of the moon in the reciprocal duplicate proportion of the distances from the centre of the earth, that is, as 3600 to 1, would (secluding the resistance of the air) complete a revolution in 1''. 24' 27".

Suppose the circumference of the earth to be J 23249600 Paris feet, as

520 THE SYSTEM OF THE WORLD.

has been determined by the late mensuration of the French (vide p. 406) ; then the same body, deprived of its circular motion, and falling by the impulse of the same centripetal force as before, would, in one second of time, describe ISy^^- 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. Huygens has demonstrated that bodies falling by all that centripetal force with which (of whatever nature it is) they are impelled near the surface of the earth, do, in one second of time, describe 15^^ Paris feet.

But if the earth is supposed to move, the earth and moon together (by Cor. lY of the Laws of Motion, and Prop. LYII) will be revolved about their common centre of gravity. And the moon (by Prop. LX) will in the same periodic time, 27^. 7'\ 43', 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 earth, as the sum of both the bodies of the earth and moon to the first of two mean proportionals between this sum and the body of the earth ; that is, if we suppose the moon (on account of its mean apparent diameter 31 1') to be about -^\ oi the earth, as 43 to

/ 42 -f 43|-, or as about 128 to 127. And therefore the semi-diameter of the orbit, that is, the distance between the centres of the moon and earth, will in this case be 60| semi-diameters of the earth, almost the same with that assigned by Copernicus, which the Tychonic observations by no means disprove ; and, therefore, the duplicate proportion of the decrement of the force holds good in this distance. I have neglected the increment of the orbit which arises from the action of the sun as inconsiderable ; but if that is subducted, the true distance will remain about 60|- semi- diameters of the earth.

But farther (p. 390) ; this proportion of the decrement of the forces is conJSirmed from the eccentricity of the planets, and the very slow motion of their apses ; for (by the Corollaries of Prop. XLY) in no other pro- portion could the circum-solar planets once in every revolution descend to their least and once ascend to their greatest distance from the sun, and the places of those distances remain immoveable. A small error from the du- plicate proportion would produce a motion of the apses considerable in every revolution, but in many enormous.

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

THE SYSTEM OF THE WORLD. 521

We may even neglect the motion of the moon's apsis (p. 390, 391), which is far greater than in the circum-solar planets, amounting in every revolu- tion to three degrees ; and from this motion it is demonstrable that the 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 apsis. This slow motion arises from the action of the circum-solar force, as we 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 that this proportion has been established, we may compare the forces of the several planets among themselves (j3. 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 8' 13" : and therefore the distance of the satellite from the centre of Jupiter is to the mean distance of Jupiter from the centre of the sun as 124 to 52012, but to the mean distance of Yenus from the centre of the sun as 124 to 7234; and their periodic times are 16f''. and 224|'^- ; and from hence (according to Cor. II, Prop. IV), dividing the distances by the squares of the times, we infer that the force by which the satellite is impelled towards Jupiter is to the force by which Yenus is impelled to- wards the sun as 442 to 143 ; and if we diminish the force by which the satellite is impelled in the duplicate proportion of the distance 124 to 7234, we shall have the circum-jovial force in the distance of Yenus from the sun to the circum-solar force by which Yenus is impelled as jV^ to 143, or as 1 to 1100; wherefore at equal distances the circum-solar force is 1100 times greater than the circum-jovial.

And, by the like computation, from the periodic time of the satellite of Saturn IS'^. 22'^ and its greatest elongation from Saturn, while that planet is in its mean distance from us, 3' 20", it follows that the distance of this satellite from Saturn's centre is to the distance of Yenus from the sun as 92| to 7234 ; and from thence that the absolute circum-solar force is 2360 times greater than the absolute circum-saturnal.

Prom the regularity of the heliocentric and irregularity of the geocen- tric motions of Yenus, of Jupiter, and the other planets, it is evident (by Cor. lY, Prop. Ill) that the circum-terrestrial force, compared with the cir- cum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's par- allax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler J from Tychd's observations and his own, found the parallax of

522

THE SYSTEM OF THE WORLD.

Mars insensible, even in opposition to the sun, when that parallax is some- thing greater than the sun's.

Flamsted attempted the same parallax with the micrometer in the peri- geon position of Mars, but never found it above 25" ; and thence conclud- ed the sun's parallax at most 10".

Whence 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 10000 ; nor to the distance of Yenus from the sun than 29 to 7233.

From which distances, together with the periodic times, by the method above explained, it is easy to infer that the absolute circum-solar force is greater than, the absolute circum-terrestrial force at least 229400 times.

And though we were only certain, from the observations of Ricciolus and Vendelin, that the sun's parallax was less than half a minute, yet from this it will follow that the absolute circum-solar force exceeds the absolute circum-terrestrial force S500 times.

By the like computations I happened to discover an analogy, that is ob- served between the forces and the bodies of the planets ; but, before I ex- plain this analogy, the apparent diameters of the planets in their mean distances from the earth must be first determined.

Mr. Flamsted (p. 3S7), by the micrometer, measured the diameter of Jupiter 40" or 41"; the diameter of Saturn's ring 50" ; and the diameter of the sun about 32' 13" (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9 ; according to Galletius, as 4 to 10 ; and according to Hooke (by a telescope of 60 feet), as 5 to 12. And from the mean proportion, 5 to 12, the diameter of Saturn's body is in- ferred about 21".

Such as we have said are the apparent magnitudes ; but, because of the unequal refrangibility of light, all lucid points are dilated by the tele- scope, and in the focus of the object-glass possess a circular space whose breadth is about the 50th part of the aperture of the glass.

It is true, that towards the circumference the light is so rare as hardly to move the sense ; but towards the middle, wliere 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 is generally about the 3d, or 4th, or 5th part of the breadth of the whole.

Let ABD represent the circle of the whole light ; PQ, the small circle of the denser and clearer light ; C the centre of both ; CA, CB, semi-di- ameters of the greater circle containing a right angle at C ; ACBE the square comprehended under these semi-diameters : AB the diagonal of that square ; EGH an hyperbola with the centre C and asymptotes CA, CB ; PG a perpendicular erected from any point P of the line BC, and meeting the hyperbola in G, and the right lines AB, AE, in K and F : and the

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523

density of the light in any place P, will, by my computation, be as the line FG, and therefore at the centre infinite, but near the circumference very small. And the whole light within the small circle PQ, is to the whole without as the area of the quadrilateral figure CAKP to the trian-

gle PKB. And we are to understand the small circle PQ, to be there terminated, where FG, the density of the light, begins to be less than what is required to move the sense.

Hence it was, that, at the distance of 191382 feet, a fire of 3 feet in di- ameter, through a telescope of 3 feet, appeared to Mr. Picart of 8" in breadth, when it should have appeared only of 3" 14'" ; and hence it is that the brighter fixed stars appear through the telescope as of 5" or 6" in 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- xelhts, 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- pear as of b" or 6'' in diameter. But M^. Huygens^ only by clouding the eye-glass with a little smoke, did so effectually extinguish this scattered light, that the fixed stars appeared as mere points, void of all sensible breadth. Hence also it was that Mr. Hvygens^ from the breadth of bodies interposed to intercept the whole light*of the planets, reckoned their diam- eters greater than others have measured them by the micrometer ; for the

524 THE SYSTEM OF THE WORLD.

scattered liglit; wMch could not be seen before for the stronger light of the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletiiis, and Dr. Halley, Mercury did not seem to exceed 12" or 15' : and Venus appeared to Mr. Crahtrie only 1' 3" ; to Horrox but V 12'' ; though by the men- surations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1' 24". Thus the apparent diameter of the moon, which in 16S4, a few days both before and after the sun's eclipse, was measured at the observatory of Paris 31' 30", in the eclipse itself did not seem to exceed 30' or 30' 05" ; and therefore the diameters of the planets are to be diminished when without the sun, and to be augmented when within it, by some seconds. But the errors seem to be less than usual in the mensurations that are made by the micrometer. So from the diameter of the shadow, determined by the eclipses of the satellites, Mr. Flamsted found that the semi-diameter of Jupiter was to the greatest elongation of the outmost satellite as 1 to 24,903. Wherefore since that elongation is 8' 13", the diameter of Jupiter will be 39|" : and, rejecting the scattered light, the diameter found by the micrometer 40" or 41" will be reduced to 39^" ; and the diameter of Saturn 21" is to be diminished by the like cor- rection, and to be reckoned 20", or something less. But (if I am 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 '.

That bodies so different in magnitude should comedo near to an analogy with their forces, is not without some mystery (p. 400).

It may be that the remoter planets, for want of heat, have not those me- tallic substances and ponderous minerals with which our earth abounds ; and that the bodies of Venus and Mercm-y, as they are more exposed to the gun's heat, are also harder baked, and more compact.

For, from the experiment of the burning-glass, we see that the heat in- creases with the density of light ; and this density increases in the recipro- cal duplicate proportion of the distance from the sun : from whence the san's heat in Mercury is proved to be sevenfold its heat in our summer seasons. But with this heat our water boils ; and those heavy fluids, quick- silver and the spirit of vitriol, gently evaporate, as I have tried by the thermometer ; and therefore there can be no fluids in Mercury but what are heavy, and able to bear a great heat, and from which substances of great density may be nourished.

And why not, if God has 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.

THE SYSTEM OF THE WORLD. 525

But I should be glad the diameters of the planets were more accurately measured ; 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 lamp are so diminished that the spectrum may appear through the telescope just like the planet, and may be defined by the same measure : then the 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 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 sesquipli- cate thereof, it is evident that the actions of the centripetal force, and therefore the periodic times, would be equal in equal planets at equal dis- tances 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 all its satellites in proportion of their several weights. And the same thing is to be said of Saturn in respect of its satellites, and of our earth in re- spect of the moon," as appears from Cor. II and 111, Prop. LXV. And, therefore, at equal distances, the actions of the centripetal force are 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 all the particles of the same size of which the planet is composed ; for if the action was greater upon some sort of particles than upon others than 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 diiFerent sorts, I exam- ined this analogy with great accuracy (p. 343, 344).

If the action of the circum-terrestrial force is proportional to the bodies to be moved, it will (by the Second Law of Motion) move them with equal velocity in equal times, and will make all bodies let fall to descend through 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 that was less, these would be greater.

But it has been long ago observed by others, that (allowance being made for the small resistance of the air) all bodies descend through equal spaces

526 THE SYSTEM OF THE WORLD.

in equal times ; and, by the help of pendulums, that equality of times may be distinguished to great exactness.

I tried the thing in gold, silver, lead, glass, sand, common salt wood, water, and wheat. I provided tAVO equal wooden boxes. I filled the one with wood, and suspended an equal weight of gold (as exactly as I could) in the centre of oscillation of the other. The boxes, hung by equal threads of 11 feet, made a couple of pendulums perfectly equal in weight and fig- ure, and equally exposed to the resistance of the air : and, placing the one by the other, I observed them to play together forwards and backwards for a long while, with equal vibrations. And therefore (by Cor. 1 and YI, 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 all the gold to the action of the same upon all the wood ; that is, as the weight of the one to the weight of the other.

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 mat- ter in the body attracting.

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 that the attractive force is found in both. The sun at- tracts Jupiter and the other planets ; Jupiter attracts its satellites ; and, for the same reason, the satellites act as well one upon another as upon Ju- piter, and all the planets mutually one upon another.

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 two terms. Two 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 it is but one single one. It is not one action by which the sun attracts Jupiter, and another by which Jupiter attracts the sun : but it is one ac- tion 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) ; and by the action with which Jupiter attracts the sun, likewise Ju-

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527

piter 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 action towards the sun ; but it is one single intermediate action, by which both approach nearer together.

Thus iron draws the load-stone (p. 93), as well as the load-stone draws the iron : for all iron in the neighbourhood of the load-stone draws other iron. But the action betwixt the load-stone and iron is single, and is 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 iron, by which both endeavour to come nearer together : and so it mani- festly appears ; for if you remove the load-stone, the whole force of the iron almost ceases.

In this sense it is that we are to conceive one single action to be ex- erted betwixt 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 proportional to the quantity of matter in the other.

Perhaps it may be objected, that, according to this philosophy (p. 398), all 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 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 20000000 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 I of an inch, they would not, even in spaces void of

528 THE SYSTEM OF THE WORLD.

resistance^ come together by the force of their mutual attraction in less than a month's time ; and less spheres will come together at a rate yet slower, viz.j 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 ; and it is 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.

Let ABCD (p. 93) represent the globe of the earth cut by any plane AC into two parts ACB, and A CD. The part ACB bearing upon the part ACD presses it with its whole weight ; nor can the part ACD sustain this pressure and continue unmoved, if it is not opposed by au equal con- trary pressure. And therefore the parts equally press each other by their weights, that is, equally attract each other, according to the third Law of Motion ; and, if separated and let go, would fall towards each other with velocities reciprocally as the bodies. All which we may try and see in the load-stone, whose attracted part does not propel the part attracting, but is only stopped and sustained thereby.

Suppose now that ACB represents some small body on the earth's sur- face; then, because the mutual attractions of this particle, and of the re- maining part 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), the attraction of the earth towards the particle will likewise be as the matter of the particle ; and therefore the attractive forces of all terres- trial bodies will be as their several quantities of matter.

The forces (p. 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 proportional to their quantities of matter, because among all there is no difference of substance, but of modes and forms only. But in the celes- tial bodies the same thing is likewise proved thus. We have shewn that the action of the circum-solar force upon all the planets (reduced to equal distances) is as the matter of the planets ; that the action of the circum- 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 planet : but thence it follows (by Prop. LXIX) that their attractive forces are as their several quantities of matter.