14. Sir ~ISAAC NEWTON~ applies these experiments yet farther,

and examines by them the general question concerning the absolute

plenitude of space. According to the Aristotelians, all space was

full without any the least vacuities whatever. DESCARTES embraced the

same opinion, and therefore supposed a subtile fluid matter, which

should pervade all bodies, and adequately fill up their pores. The

Atomical philosophers, who suppose all bodies both fluid and solid to

be composed of very minute but solid atoms, assert that no fluid, how

subtile soever the particles or atoms whereof it is composed should be,

can ever cause an absolute plenitude; because it is impossible that

any body can pass through the fluid without putting the particles of

it into such a motion, as to separate them, at least in part, from one

another, and so perpetually to cause small vacuities; by which these

Atomists endeavour to prove, that a vacuum, or some space empty of

all matter, is absolutely necessary to be in nature. Sir ~ISAAC

NEWTON~ objects against the filling of space with such a subtile

fluid, that all bodies in motion must be unmeasurably resisted by a

fluid so dense, as absolutely to fill up all the space, through which

it is spread. And lest it should be thought, that this objection might

be evaded by ascribing to this fluid such very minute and smooth parts,

as might remove all adhesion or friction between them, whereby all

resistance would be lost, which this fluid might otherwise give to

bodies moving in it; Sir ~ISAAC NEWTON~ proves, in the manner

above related, that fluids resist from the power of inactivity of their

particles; and that water and the air resist almost entirely on this

account: so that in this subtile fluid, however minute and lubricated

the particles, which compose it, might be; yet if the whole fluid was

as dense as water, it would resist very near as much as water does; and

whereas such a fluid, whose parts are absolutely close together without

any intervening spaces, must be a great deal more dense than water,

it must resist more than water in proportion to its greater density;

unless we will suppose the matter, of which this fluid is composed, not

to be endued with the same degree of inactivity as other matter. But if

you deprive any substance of the property so universally belonging to

all other matter, without impropriety of speech it can scarce be called

by this name.

15. Sir ~ISAAC NEWTON~ made also an experiment to try

in particular, whether the internal parts of bodies suffered any

resistance. And the result did indeed appear to favour some small

degree of resistance; but so very little, as to leave it doubtful,

whether the effect did not arise from some other latent cause[154].

CHAP. II.

Concerning the cause, which keeps in motion the primary planets.

SINCE the planets move in a void space and are free from resistance;

they, like all other bodies, when once in motion, would move on in a

straight line without end, if left to themselves. And it is now to be

explained what kind of action upon them carries them round the sun.

Here I shall treat of the primary planets only, and discourse of the

secondary apart in the next chapter. It has been just now declared,

that these primary planets move so about the sun, that a line extended

from the sun to the planet, will, by accompanying the planet in its

motion, pass over equal spaces in equal portions of time[155]. And

this one property in the motion of the planets proves, that they are

continually acted on by a power directed perpetually to the sun as a

center. This therefore is one property of the cause, which keeps the

planets in their courses, that it is a centripetal power, whose center

is the sun.

2. AGAIN, in the chapter upon centripetal forces[156] it was observ’d,

that if the strength of the centripetal power was suitably accommodated

every where to the motion of any body round a center, the body might

be carried in any bent line whatever, whose concavity should be every

where turned towards the center of the force. It was farther remarked,

that the strength of the centripetal force, in each place, was to be

collected from the nature of the line, wherein the body moved[157].

Now since each planet moves in an ellipsis, and the sun is placed in

one focus; Sir ~ISAAC NEWTON~ deduces from hence, that the

strength of this power is reciprocally in the duplicate proportion of

the distance from the sun. This is deduced from the properties, which

the geometers have discovered in the ellipsis. The process of the

reasoning is not proper to be enlarged upon here; but I shall endeavour

to explain what is meant by the reciprocal duplicate proportion. Each

of the terms reciprocal proportion, and duplicate proportion, has been

already defined[158]. Their sense when thus united is as follows.

Suppose the planet moved in the orbit A B C (in fig. 93.) about the sun

in S. Then, when it is said, that the centripetal power, which acts on

the planet in A, bears to the power acting on it in B a proportion,

which is the reciprocal of the duplicate proportion of the distance S

A to the distance S B; it is meant that the power in A bears to the

power in B the duplicate of the proportion of the distance S B to the

distance S A. The reciprocal duplicate proportion may be explained

also by numbers as follows. Suppose several distances to bear to each

other proportions expressed by the numbers 1, 2, 3, 4, 5; that is, let

the second distance be double the first, the third be three times,

the fourth four times, and the fifth five times as great as the

first. Multiply each of these numbers by it self, and 1 multiplied by

1 produces still 1, 2 multiplied by 2 produces 4, 3 by 3 makes 9, 4

by 4 makes 16, and 5 by 5 gives 25. This being done, the fractions ¼,

1/9, 1/16, 1/25, will respectively express the proportion, which the

centripetal power in each of the following distances bears to the power

at the first distance: for in the second distance, which is double the

first, the centripetal power will be one fourth part only of the power

at the first distance; at the third distance the power will be one

ninth part only of the first power; at the fourth distance, the power

will be but one sixteenth part of the first; and at the fifth distance,

one twenty fifth part of the first power.

3. THUS is found the proportion, in which this centripetal power

decreases, as the distance from the sun increases, within the compass

of one planet’s motion. How it comes to pass, that the planet can be

carried about the sun by this centripetal power in a continual round,

sometimes rising from the sun, then descending again as low, and from

thence be carried up again as far remote as before, alternately rising

and falling without end; appears from what has been written above

concerning centripetal forces: for the orbits of the planets resemble

in shape the curve line proposed in § 17 of the chapter on these

forces[159].

4. BUT farther, in order to know whether this centripetal force

extends in the same proportion throughout, and consequently whether

all the planets are influenced by the very same power, our author

proceeds thus. He inquires what relation there ought to be between

the periods of the different planets, provided they were acted

upon by the same power decreasing throughout in the forementioned

proportion; and he finds, that the period of each in this case would

have that very relation to the greater axis of its orbit, as I have

declared above[160] to be found in the planets by the observations

of astronomers. And this puts it beyond question, that the different

planets are pressed towards the sun, in the same proportion to their

distances, as one planet is in its several distances. And thence in the

last place it is justly concluded, that there is such a power acting

towards the sun in the foresaid proportion at all distances from it.

5. THIS power, when referred to the planets, our author calls

centripetal, when to the sun attractive; he gives it likewise the

name of gravity, because he finds it to be of the same nature with

that power of gravity, which is observed in our earth, as will appear

hereafter[161]. By all these names he designs only to signify a power

endued with the properties before mentioned; but by no means would he

have it understood, as if these names referred any way to the cause of

it. In particular in one place where he uses the name of attraction, he

cautions us expressly against implying any thing but a power directing

a body to a center without any reference to the cause of it, whether

residing in that center, or arising from any external impulse[162].

6. BUT now, in these demonstrations some very minute inequalities in

the motion of the planets are neglected; which is done with a great

deal of judgment; for whatever be their cause, the effects are very

inconsiderable, they being so exceeding small, that some astronomers

have thought fit wholly to pass them by[163]. However the excellency

of this philosophy, when in the hands of so great a geometer as our

author, is such, that it is able to trace the least variations of

things up to their causes. The only inequalities, which have been

observed common to all the planets, are the motion of the aphelion and

the nodes. The transverse axis of each orbit does not always remain

fixed, but moves about the sun with a very slow progressive motion:

nor do the planets keep constantly the same plane, but change them,

and the lines in which those planes intersect each other by insensible

degrees. The first of these inequalities, which is the motion of the

aphelion, may be accounted for, by supposing the gravitation of the

planets towards the sun to differ a little from the forementioned

reciprocal duplicate proportion of the distances; but the second,

which is the motion of the nodes, cannot be accounted for by any

power directed towards the sun; for no such can give the planet any

lateral impulse to divert it from the plane of its motion into any new

plane, but of necessity must be derived from some other center. Where

that power is lodged, remains to be discovered. Now it is proved, as

shall be explained in the following chapter, that the three primary

planets Saturn, Jupiter, and the earth, which have satellites revolving

about them, are endued with a power of causing bodies, in particular

those satellites, to gravitate towards them with a force, which is

reciprocally in the duplicate proportion of their distances; and the

planets are in all respects, in which they come under our examination,

so similar and alike, that there is no reason to question, but they

have all the same property. Though it be sufficient for the present

purpose to have it proved of Jupiter and Saturn only; for these

planets contain much greater quantities of matter than the rest, and

proportionally exceed the others in power[164]. But the influence of

these two planets being allowed, it is evident how the planets come to

shift continually their planes: for each of the planets moving in a

different plane, the action of Jupiter and Saturn upon the rest will

be oblique to the planes of their motion; and therefore will gradually

draw them into new ones. The same action of these two planets upon

the rest will cause likewise a progressive motion of the aphelion; so

that there will be no necessity of having recourse to the other cause

for this motion, which was before hinted at[165]; viz, the gravitation

of the planets towards the sun differing from the exact reciprocal

duplicate proportion of the distances. And in the last place, the

action of Jupiter and Saturn upon each other will produce in their

motions the same inequalities, as their joint action produces in the

rest. All this is effected in the same manner, as the sun produces the

same kind of inequalities and many others in the motion of the moon and

the other secondary planets; and therefore will be best apprehended by

what shall be said in the next chapter. Those other irregularities in

the motion of the secondary planets have place likewise here; but are

too minute to be observable: because they are produced and rectified

alternately, for the most part in the time of a single revolution;

whereas the motion of the aphelion and nodes, which continually

increase, become sensible in a long series of years. Yet some of these

other inequalities are discernible in Jupiter and Saturn, in Saturn

chiefly; for when Jupiter, who moves faster than Saturn, approaches

near to a conjunction with him, his action upon Saturn will a little

retard the motion of that planet, and by the reciprocal action of

Saturn he will himself be accelerated. After conjunction, Jupiter will

again accelerate Saturn, and be likewise retarded in the same degree,

as before the first was retarded and the latter accelerated. Whatever

inequalities besides are produced in the motion of Saturn by the action

of Jupiter upon that planet, will be sufficiently rectified, by placing

the focus of Saturn’s ellipsis, which should otherwise be in the sun,

in the common center of gravity of the sun and Jupiter. And all the

inequalities in the motion of Jupiter, caused by Saturn’s action upon

him, are much less considerable than the irregularities of Saturn’s

motion[166].

7. THIS one principle therefore of the planets having a power, as well

as the sun, to cause bodies to gravitate towards them, which is proved

by the motion of the secondary planets to obtain in fact, explains

all the irregularities relating to the planets ever observed by

astronomers.

8. Sir ~ISAAC NEWTON~ after this proceeds to make an

improvement in astronomy by applying this theory to the farther

correction of their motions. For as we have here observed the planets

to possess a principle of gravitation, as well as the sun; so it will

be explained at large hereafter, that the third law of motion, which

makes action and reaction equal, is to be applied in this case[167];

and that the sun does not only attract each planet, but is it self

also attracted by them; the force, wherewith the planet is acted on,

bearing to the force, wherewith the sun it self is acted on at the same

time, the proportion, which the quantity of matter in the sun bears

to the quantity of matter in the planet. From the action between the

sun and planet being thus mutual Sir ISAAC NEWTON proves that the sun

and planet will describe about their common center of gravity similar

ellipsis’s; and then that the transverse axis of the ellipsis described

thus about the moveable sun, will bear to the transverse axis of the

ellipsis, which would be described about the sun at rest in the same

time, the same proportion as the quantity of solid matter in the sun

and planet together bears to the first of two mean proportionals

between this quantity and the quantity of matter in the sun only[168].

9. ABOVE, where I shewed how to find a cube, that should bear any

proportion to another cube[169], the lines F T and T S are two mean

proportionals between E F and F G; and counting from E F, F T is called

the first, and F S the second of those means. In numbers these mean

proportionals are thus found.

[Illustration]

Suppose A and B two numbers, and it be required to find C the first,

and D the second of the two mean proportionals between them. First

multiply A by it self, and the product multiply by B; then C will be

the number which in arithmetic is called the cubic root of this last

product; that is, the number C being multiplied by it self, and the

product again multiplied by the same number C, will produce the product

above mentioned. In like manner D is the cubic root of the product

of B multiplied by it self, and the produce of that multiplication

multiplied again by A.

10. IT will be asked, perhaps, how this correction can be admitted,

when the cause of the motions of the planets was before found by

supposing the sun the center of the power, which acted upon them: for

according to the present correction this power appears rather to be

directed to their common center of gravity. But whereas the sun was

at first concluded to be the center, to which the power acting on the

planets was directed, because the spaces described round the sun in

equal times were found to be equal; so Sir ~ISAAC NEWTON~

proves, that if the sun and planet move round their common center of

gravity, yet to an eye placed in the planet, the spaces, which will

appear to be described about the sun, will have the same relation to

the times of their description, as the real spaces would have, if the

sun were at rest[170]. I farther asserted, that, supposing the planets

to move round the sun at rest, and to be attracted by a power, which

every where should act with degrees of strength reciprocally in the

duplicate proportion of the distances; then the periods of the planets

must observe the same relation to their distances, as astronomers find

them to do. But here it must not be supposed, that the observations of

astronomers absolutely agree without any the least difference; and the

present correction will not cause a deviation from any one astronomer’s

observations, so much as they differ from one another. For in Jupiter,

where this correction is greatest, it hardly amounts to the 3000^{th}

part of the whole axis.

11. UPON this head I think it not improper to mention a reflection made

by our excellent author upon these small inequalities in the planets

motions; which contains under it a very strong philosophical argument

against the eternity of the world. It is this, that these inequalities

must continually increase by slow degrees, till they render at length

the present frame of nature unfit for the purposes, it now serves[171].

And a more convincing proof cannot be desired against the present

constitution’s having existed from eternity than this, that a certain

period of years will bring it to an end. I am aware this thought of

our author has been represented even as impious, and as no less than

casting a reflection upon the wisdom of the author of nature, for

framing a perishable work. But I think so bold an assertion ought to

have been made with singular caution. For if this remark upon the

increasing irregularities of the heavenly motions be true in fact,

as it really is, the imputation must return upon the asserter, that

this does detract from the divine wisdom. Certainly we cannot pretend

to know all the omniscient Creator’s purposes in making this world,

and therefore cannot undertake to determine how long he designed it

should last. And it is sufficient, if it endure the time intended by

the author. The body of every animal shews the unlimited wisdom of its

author no less, nay in many respects more, than the larger frame of

nature; and yet we see, they are all designed to last but a small space

of time.

12. THERE need nothing more be said of the primary planets; the motions

of the secondary shall be next considered.

CHAP. III.

Of the motion of the MOON and the other SECONDARY PLANETS.

THE excellency of this philosophy sufficiently appears from its

extending in the manner, which has been related, to the minutest

circumstances of the primary planets motions; which nevertheless

bears no proportion to the vast success of it in the motions of the

secondary; for it not only accounts for all the irregularities, by

which their motions were known to be disturbed, but has discovered

others so complicated, that astronomers were never able to distinguish

them, and reduce them under proper heads; but these were only to be

found out from their causes, which this philosophy has brought to

light, and has shewn the dependence of these inequalities upon such

causes in so perfect a manner, that we not only learn from thence in

general, what those inequalities are, but are able to compute the

degree of them. Of this Sir ~IS. NEWTON~ has given several

specimens, and has moreover found means to reduce the moon’s motion so

completely to rule, that he has framed a theory, from which the place

of that planet may at all times be computed, very nearly or altogether

as exactly, as the places of the primary planets themselves, which is

much beyond what the greatest astronomers could ever effect.

2. THE first thing demonstrated of these secondary planets is, that

they are drawn towards their respective primary in the same manner

as the primary planets are attracted by the sun. That each secondary

planet is kept in its orbit by a power pointed towards the center of

the primary planet, about which the secondary revolves; and that the

power, by which the secondaries of the same primary are influenced,

bears the same relation to the distance from the primary, as the power,

by which the primary planets are guided, does in regard to the distance

from the sun[172]. This is proved in the satellites of Jupiter and

Saturn, because they move in circles, as far as we can observe, about

their respective primary with an equable course, the respective primary

being the center of each orbit: and by comparing the times, in which

the different satellites of the same primary perform their periods,

they are found to observe the same relation to the distances from

their primary, as the primary planets observe in respect of their mean

distances from the sun[173]. Here these bodies moving in circles with

an equable motion, each satellite passes over equal parts of its orbit

in equal portions of time; consequently the line drawn from the center

of the orbit, that is, from the primary planet, to the satellite, will

pass over equal spaces along with the satellite in equal portions of

time; which proves the power, by which each satellite is held in its

orbit, to be pointed towards the primary as a center[174]. It is also

manifest that the centripetal power, which carries a body in a circle

concentrical with the power, acts upon the body at all times with the

same strength. But Sir ~ISAAC NEWTON~ demonstrates that, when

bodies are carried in different circles by centripetal powers directed

to the centers of those circles, then, the degrees of strength of

those powers are to be compared by considering the relation between

the times, in which the bodies perform their periods through those

circles[175]; and in particular he shews, that if the periodical times

bear that relation, which I have just now asserted the satellites

of the same primary to observe; then the centripetal powers are

reciprocally in the duplicate proportion of the semidiameters of the

circles, or in that proportion to the distances of the bodies from the

centers[176]. Hence it follows that in the planets Jupiter and Saturn,

the centripetal power in each decreases with the increase of distance,

in the same proportion as the centripetal power appertaining to the

sun decreases with the increase of distance. I do not here mean that

this proportion of the centripetal powers holds between the power of

Jupiter at any distance compared with the power of Saturn at any other

distance; but only in the change of strength of the power belonging to

the same planet at different distances from him. Moreover what is here

discovered of the planets Jupiter and Saturn by means of the different

satellites, which revolve round each of them, appears in the earth by

the moon alone; because she is found to move round the earth in an

ellipsis after the same manner as the primary planets do about the sun;

excepting only some small irregularities in her motion, the cause of

which will be particularly explained in what follows, whereby it will

appear, that they are no objection against the earth’s acting on the

moon in the same manner as the sun acts on the primary planets; that

is, as the other primary planets Jupiter and Saturn act upon their

satellites. Certainly since these irregularities can be otherwise

accounted for, we ought not to depart from that rule of induction so

necessary in philosophy, that to like bodies like properties are to

be attributed, where no reason to the contrary appears. We cannot

therefore but ascribe to the earth the same kind of action upon the

moon, as the other primary planets Jupiter and Saturn have upon their

satellites; which is known to be very exactly in the proportion

assigned by the method of comparing the periodical times and distances

of all the satellites which move about the same planet; this abundantly

compensating our not being near enough to observe the exact figure of

their orbits. For if the little deviation of the moon’s orbit orbit

from a true permanent ellipsis arose from the action of the earth upon

the moon not being in the exact reciprocal duplicate proportion of the

distance, were another moon to revolve about the earth, the proportion

between the periodical times of this new moon, and the present,

would discover the deviation from the mentioned proportion much more

manifestly.

3. BY the number of satellites, which move round Jupiter and Saturn,

the power of each of these planets is measured in a great diversity

of distance; for the distance of the outermost satellite in each of

these planets exceeds several times the distance of the innermost. In

Jupiter the astronomers have usually placed the innermost satellite

at a distance from the center of that planet equal to about 5⅔ of

the semidiameters of Jupiter’s body, and this satellite performs its

revolution in about 1 day 18½ hours. The next satellite, which revolves

round Jupiter in about 3 days 13⅕ hours, they place at the distance

from Jupiter of about 9 of that planet’s semidiameters. To the third

satellite, which performs its period nearly in 7 days 3¾ hours, they

assign the distance of about 14⅖ semidiameters. But the outermost

satellite they remove to 25⅓ semidiameters, and this satellite makes

its period in about 16 days 16½ hours[177]. In Saturn there is still

a greater diversity in the distance of the several satellites. By the

observations of the late ~CASSINI~, a celebrated astronomer

in France, who first discovered all these satellites, except one known

before, the innermost is distant about 4½ of Saturn’s semidiameters

from his center, and revolves round in about 1 day 21⅓ hours. The next

satellite is distant about 5¾ semidiameters, and makes its period in

about 2 days 17⅔ hours. The third is removed to the distance of about

8 semidiameters, and performs its revolution in near 4 days 12½ hours.

The fourth satellite discovered first by the great HUYGENS, is near

18⅔ semidiameters, and moves round Saturn in about 15 days 22⅔ hours.

The outermost is distant 56 semidiameters, and makes its revolution in

about 79 days 7⅘ hours[178]. Besides these satellites, there belongs

to the planet Saturn another body of a very singular kind. This is a

shining, broad, and flat ring, which encompasses the planet round.

The diameter of the outermost verge of this ring is more than double

the diameter of Saturn. ~HUYGENS~, who first described this

ring, makes the whole diameter thereof to bear to the diameter of

Saturn the proportion of 9 to 4. The late reverend Mr. POUND makes the

proportion something greater, viz. that of 7 to 3. The distances of the

satellites of this planet Saturn are compared by ~CASSINI~ to

the diameter of the ring. His numbers I have reduced to those above,

according to Mr. POUND’s proportion between the diameters of Saturn and

of his ring. As this ring appears to adhere no where to Saturn, so the

distance of Saturn from the inner edge of the ring seems rather greater

than the breadth of the ring. The distances, which have here been

given, of the several satellites, both for Jupiter and Saturn, may be

more depended on in relation to the proportion, which those belonging

to the same primary planet bear one to another, than in respect to the

very numbers, that have been here set down, by reason of the difficulty

there is in measuring to the greatest exactness the diameters of the

primary planets; as will be explained hereafter, when we come to treat

of telescopes[179]. By the observations of the forementioned Mr. POUND,

in Jupiter the distance of the innermost satellite should rather be

about 6 semidiameters, of the second 9-½, of the third 15, and of

the outermost 26⅔[180]; and in Saturn the distance of the innermost

satellite 4 semidiameters, of the next 6¼, of the third 8¾, of the

fourth 20⅓, and of the fifth 59[181]. However the proportion between

the distances of the satellites in the same primary is the only thing

necessary to the point we are here upon.

4. BUT moreover the force, wherewith the earth acts in different

distances, is confirmed from the following consideration, yet more

expresly than by the preceding analogical reasoning. It will appear,

that if the power of the earth, by which it retains the moon in her

orbit, be supposed to act at all distances between the earth and moon,

according to the forementioned rule; this power will be sufficient to

produce upon bodies, near the surface of the earth, all the effects

ascribed to the principle of gravity. This is discovered by the

following method. Let A (in fig. 94.) represent the earth, B the moon,

B C D the moon’s orbit, which differs little from a circle, of which A

is the center. If the moon in B were left to it self to move with the

velocity, it has in the point B, it would leave the orbit, and proceed

right forward in the line B E, which touches the orbit in B. Suppose

the moon would upon this condition move from B to E in the space of

one minute of time. By the action of the earth upon the moon, whereby

it is retained in its orbit, the moon will really be found at the end

of this minute in the point F, from whence a straight line drawn to A

shall make the space B F A in the circle equal to the triangular space

B E A; so that the moon in the time wherein it would have moved from

B to E, if left to it self, has been impelled towards the earth from

E to F. And when the time of the moon’s passing from B to F is small,

as here it is only one minute, the distance between E and F scarce

differs from the space, through which the moon would descend in the

same time, if it were to fall directly down from B toward A without any

other motion. A B the distance of the earth and moon is about 60 of the

earth’s semidiameters, and the moon completes her revolution round the

earth in about 27 days 7 hours and 43 minutes: therefore the space E F

will here be found by computation to be about 16⅛ feet. Consequently,

if the power, by which the moon is retained in its orbit, be near the

surface of the earth greater, than at the distance of the moon in the

duplicate proportion of that distance; the number of feet, a body would

descend near the surface of the earth by the action of this power upon

it in one minute of time, would be equal to 16⅛ multiplied twice into

the number 60, that is, equal to 58050. But how fast bodies fall near

the surface of the earth may be known by the pendulum[182]; and by the

exactest experiments they are found to descend the space of 16⅛ feet in

a second of time; and the spaces described by falling bodies being in

the duplicate proportion of the times of their fall[183], the number of

feet, a body would describe in its fall near the surface of the earth

in one minute of time, will be equal to 16⅛ twice multiplied by 60, the

same as would be caused by the power which acts upon the moon.

5. IN this computation the earth is supposed to be at rest, whereas

it would have been more exact to have supposed it to move, as well

as the moon, about their common center of gravity; as will easily be

understood, by what has been said in the preceding chapter, where it

was shewn, that the sun is subjected to the like motion about the

common center of gravity of it self and the planets. The action of

the sun upon the moon, which is to be explain’d in what follows, is

likewise here neglected: and Sir ISAAC NEWTON shews, if you take in

both these considerations, the present computation will best agree

to a somewhat greater distance of the moon and earth, viz. to 60½

semidiameters of the earth, which distance is more conformable to

astronomical observations.

6. THESE computations afford an additional proof, that the action of

the earth observes the same proportion to the distance, which is here

contended for. Before I said, it was reasonable to conclude so by

induction from the planets Jupiter and Saturn; because they act in

that manner. But now the same thing will be evident by drawing no other

consequence from what is seen in those planets, than that the power,

by which the primary planets act on their secondary, is extended from

the primary through the whole interval between, so that it would act in

every part of the intermediate space. In Jupiter and Saturn this power

is so far from being confined to a small extent of distance, that it

not only reaches to several satellites at very different distances, but

also from one planet to the other, nay even through the whole planetary

system[184]. Consequently there is no appearance of reason, why this

power should not act at all distances, even at the very surfaces of

these planets as well as farther off. But from hence it follows, that

the power, which retains the moon in her orbit, is the same, as causes

bodies near the surface of the earth to gravitate. For since the

power, by which the earth acts on the moon, will cause bodies near the

surface of the earth to descend with all the velocity they are found

to do, it is certain no other power can act upon them besides; because

if it did, they must of necessity descend swifter. Now from all this

it is at length very evident, that the power in the earth, which we

call gravity, extends up to the moon, and decreases in the duplicate

proportion of the increase of the distance from the earth.

7. THIS finishes the discoveries made in the action of the primary

planets upon their secondary. The next thing to be shewn is, that the

sun acts upon them likewise: for this purpose it is to be observed,

that if to the motion of the satellite, whereby it would be carried

round its primary at rest, be superadded the same motion both in

regard to velocity and direction, as the primary it self has, it will

describe about the primary the same orbit, with as great regularity,

as if the primary was indeed at rest. The cause of this is that law

of motion, which makes a body near the surface of the earth, when let

fall, to descend perpendicularly, though the earth be in so swift a

motion, that if the falling body did not partake of it, its descent

would be remarkably oblique; and that a body projected describes in

the most regular manner the same parabola, whether projected in the

direction, in which the earth moves, or in the opposite direction, if

the projecting force be the same[185]. From this we learn, that if the

satellite moved about its primary with perfect regularity, besides its

motion about the primary, it would participate of all the motion of its

primary; have the same progressive velocity, with which the primary

is carried about the sun; and be impelled with the same velocity as

the primary towards the sun, in a direction parallel to that impulse

of its primary. And on the contrary, the want of either of these,

in particular of the impulse towards the sun, will occasion great

inequalities in the motion of the secondary planet. The inequalities,

which would arise from the absence of this impulse towards the sun are

so great, that by the regularity, which appears in the motion of the

secondary planets, it is proved, that the sun communicates, the same

velocity to them by its action, as it gives to their primary at the

same distance. For Sir ~ISAAC NEWTON~ informs us, that upon

examination he found, that if any of the satellites of Jupiter were

attracted by the sun more or less, than Jupiter himself at the same

distance, the orbit of that satellite, instead of being concentrical to

Jupiter, must have its center at a greater or less distance, than the

center of Jupiter from the sun, nearly in the subduplicate proportion

of the difference between the sun’s action upon the satellite, and upon

Jupiter; and therefore if any satellite were attracted by the sun but

1/1000 part more or less, than Jupiter is at the same distance, the

center of the orbit of that satellite would be distant from the center

of Jupiter no less than a fifth part of the distance of the outermost

satellite from Jupiter[186]; which is almost the whole distance of the

innermost satellite. By the like argument the satellites of Saturn

gravitate towards the sun, as much as Saturn it self at the same

distance; and the moon as much as the earth.

8. THUS is proved, that the sun acts upon the secondary planets, as

much as upon the primary at the same distance: but it was found in the

last chapter, that the action of the sun upon bodies is reciprocally

in the duplicate proportion of the distance; therefore the secondary

planets being sometimes nearer to the sun than the primary, and

sometimes more remote, they are not alway acted upon in the same degree

with their primary, but when nearer to the sun, are attracted more,

and when farther distant, are attracted less. Hence arise various

inequalities in the motion of the secondary planets[187].

9. SOME of these inequalities would take place, though the moon, if

undisturbed by the sun, would have moved in a circle concentrical

to the earth, and in the plane of the earth’s motion; others depend

on the elliptical figure, and the oblique situation of the moon’s

orbit. One of the first kind is, that the moon is caused so to move,

as not to describe equal spaces in equal times, but is continually

accelerated, as she passes from the quarter to the new or full, and is

retarded again by the like degrees in returning from the new and full

to the next quarter. Here we consider not so much the absolute, as the

apparent motion of the moon in respect to us.

10. THE principles of astronomy teach how to distinguish these two

motions. Let S (in fig. 95.) represent the sun, A the earth moving

in its orbit B C, D E F G the moon’s orbit, the place of the moon H.

Suppose the earth to have moved from A to I. Because it has been shewn,

that the moon partakes of all the progressive motion of the earth;

and likewise that the sun attracts both the earth and moon equally,

when they are at the same distance from it, or that the mean action

of the sun upon the moon is equal to its action upon the earth: we

must therefore consider the earth as carrying about with it the moon’s

orbit; so that when the earth is removed from A to I, the moon’s orbit

shall likewise be removed from its former situation into that denoted

by K L M N. But now the earth being in I, if the moon were found in O,

so that O I should be parallel to H A, though the moon would really

have moved from H to O, yet it would not have appeared to a spectator

upon the earth to have moved at all, because the earth has moved as

much it self; so that the moon would still appear in the same place

with respect to the fixed stars. But if the moon be observed in P, it

will then appear to have moved, its apparent motion being measured

by the angle under O I P. And if the angle under P I S be less than

the angle under H A S, the moon will have approached nearer to its

conjunction with the sun.

11. TO come now to the explication of the mentioned inequality in

the moon’s motion: let S (in fig. 96.) represent the sun, A the

earth, B C D E the moon’s orbit, C the place of the moon, when in

the latter quarter. Here it will be nearly at the same distance from

the sun, as the earth is. In this case therefore they will both be

equally attracted, the earth in the direction A S, and the moon in

the direction C S. Whence as the earth in moving round the sun is

continually descending toward it, so the moon in this situation must in

any equal portion of time descend as much; and therefore the position

of the line A C in respect of A S, and the change, which the moon’s

motion produces in the angle under C A S, will not be altered by the

sun.

12. BUT now as soon as ever the moon is advanced from the quarter

toward the new or conjunction, suppose to G, the action of the sun upon

it will have a different effect. Here, were the sun’s action upon the

moon to be applied in the direction G H parallel to A S, if its action

on the moon were equal to its action on the earth, no change would be

wrought by the sun on the apparent motion of the moon round the earth.

But the moon receiving a greater impulse in G than the earth receives

in A, were the sun to act in the direction G H, yet it would accelerate

the description of the space D A G, and cause the angle under G A D to

decrease faster, than otherwise it would. The sun’s action will have

this effect upon account of the obliquity of its direction to that,

in which the earth attracts the moon. For the moon by this means is

drawn by two forces oblique to each other, one drawing from G toward

A, the other from G toward H, therefore the moon must necessarily be

impelled toward D. Again, because the sun does not act in the direction

G H parallel to S A, but in the direction G S oblique to it, the sun’s

action on the moon will by reason of this obliquity farther contribute

to the moon’s acceleration. Suppose the earth in any short space of

time would have moved from A to I, if not attracted by the sun; the

point I being in the straight line C E, which touches the earth’s orbit

in A. Suppose the moon in the same time would have moved in her orbit

from G to K, and besides have partook of all the progressive motion of

the earth. Then if K L be drawn parallel to A I, and taken equal to it,

the moon, if not attracted by the sun, would be found in L. But the

earth by the sun’s action is removed from I. Suppose it were moved down

to M in the line I M N parallel to S A, and if the moon were attracted

but as much, and in the same direction, as the earth is here supposed

to be attracted, so as to have descended during the same time in the

line L O, parallel also to A S, down as far as P, till L P were equal

to I M; the angle under P M N would be equal to that under L I N, that

is, the moon will appear advanced no farther forward, than if neither

it nor the earth had been subject to the sun’s action. But this is upon

the supposition, that the action of the sun upon the moon and earth

were equal; whereas the moon being acted upon more than the earth,

did the sun’s action draw the moon in the line L O parallel to A S,

it would draw it down so far as to make L P greater than I M; whereby

the angle under P M N will be rendred less, than that under L I N. But

moreover, as the sun draws the earth in a direction oblique to I N, the

earth will be found in its orbit somewhat short of the point M; however

the moon is attracted by the sun still more out of the line L O, than

the earth is out of the line I N; therefore this obliquity of the sun’s

action will yet farther diminish the angle under P M N.

13. THUS the moon at the point G receives an impulse from the sun,

whereby her motion is accelerated. And the sun producing this effect in

every place between the quarter and the conjunction, the moon will move

from the quarter with a motion continually more and more accelerated;

and therefore by acquiring from time to time additional degrees of

velocity in its orbit, the spaces, which are described in equal times

by the line drawn from the earth to the moon, will not be every where

equal, but those toward the conjunction will be greater, than those

toward the quarter. But now in the moon’s passage from the conjunction

D to the next quarter the sun’s action will again retard the moon, till

at the next quarter in E it be restored to the first velocity, which it

had in C.

14. AGAIN as the moon moves from E to the full or opposition to the

sun in B, it is again accelerated, the deficiency of the sun’s action

upon the moon, from what it has upon the earth, producing here the same

effect as before the excess of its action. Consider the moon in Q,

moving from E towards B. Here if the moon were attracted by the sun in

a direction parallel to A S, yet being acted on less than the earth,

as the earth descends toward the sun, the moon will in some measure be

left behind. Therefore Q F being drawn parallel to S B, a spectator

on the earth would see the moon move, as if attracted from the point

Q in the direction Q F with a degree of force equal to that, whereby

the sun’s action on the moon falls short of its action on the earth.

But the obliquity of the sun’s action has also here an effect. In the

time the earth would have moved from A to I without the influence of

the sun, let the moon have moved in its orbit from Q to R. Drawing

therefore R T parallel to A I, and equal to the same, for the like

reason as before, the moon by the motion of its orbit, if not at all

attracted by the sun, must be found in T; and therefore, if attracted

in a direction parallel to S A, would be in the line T V parallel to

A S; suppose in W. But the moon in Q being farther off the sun than

the earth, it will be less attracted, that is, T W will be less than

I M, and if the line S M be prolonged toward X, the angle under X M

W will be less than that under X I T. Thus by the sun’s action the

moon’s passage from the quarter to the full would be accelerated, if

the sun were to act on the earth and moon in a direction parallel to A

S: and the obliquity of the sun’s action will still more increase this

acceleration. For the action of the sun on the moon is oblique to the

line S A the whole time of the moon’s passage from Q to T, and will

carry the moon out of the line T V toward the earth. Here I suppose the

time of the moon’s passage from Q to T so short, that it shall not pass

beyond the line S A. The earth also will come a little short of the

line I N, as was said before. From these causes the angle under X M W

will be still farther lessened.

15. THE moon in passing from the opposition B to the next quarter will

be retarded again by the same degrees, as it is accelerated before

its appulse to the opposition. Because this action of the sun, which

in the moon’s passage from the quarter to the opposition causes it

to be extraordinarily accelerated, and diminishes the angle, which

measures its distance from the opposition; will make the moon slacken

its pace afterwards, and retard the augmentation of the same angle in

its passage from the opposition to the following quarter; that is, will

prevent that angle from increasing so fast, as otherwise it would. And

thus the moon, by the sun’s action upon it, is twice accelerated and

twice restored to its first velocity, every circuit it makes round the

earth. This inequality of the moon’s motion about the earth is called

by astronomers its variation.

16. THE next effect of the sun upon the moon is, that it gives the

orbit of the moon in the quarters a greater degree of curvature,

than it would receive from the action of the earth alone; and on the

contrary in the conjunction and opposition the orbit is less inflected.

17. WHEN the moon is in conjunction with the sun in the point D, the

sun attracting the moon more forcibly than it does the earth, the

moon by that means is impelled less toward the earth, than otherwise

it would be, and so the orbit is less incurvated; for the power, by

which the moon is impelled toward the earth, being that, by which it is

inflected from a rectilinear course, the less that power is, the less

it will be inflected. Again, when the moon is in the opposition in B,

farther removed from the sun than the earth is; it follows then, though

the earth and moon are both continually descending to the sun, that

is, are drawn by the sun toward it self out of the place they would

otherwise move into, yet the moon descends with less velocity than

the earth; insomuch that the moon in any given space of time from its

passing the point of opposition will have less approached the earth,

than otherwise it would have done, that is, its orbit in respect of

the earth will approach nearer to a straight line. In the last place,

when the moon is in the quarter in F, and equally distant from the

sun as the earth, we observed before, that the earth and moon would

descend with equal pace toward the sun, so as to make no change by

that descent in the angle under F A S; but the length of the line F

A must of necessity be shortned. Therefore the moon in moving from F

toward the conjunction with the sun will be impelled more toward the

earth by the sun’s action, than it would have been by the earth alone,

if neither the earth nor moon had been acted on by the sun; so that

by this additional impulse the orbit is rendred more curve, than it

would otherwise be. The same effect will also be produced in the other

quarter.

18. ANOTHER effect of the sun’s action, consequent upon this we have

now explained, is, that though the moon undisturbed by the sun might

move in a circle having the earth for its center; by the sun’s action,

if the earth were to be in the very middle or center of the moon’s

orbit, yet the moon would be nearer the earth at the new and full, than

in the quarters. In this probably will at first appear some difficulty,

that the moon should come nearest to the earth, where it is least

attracted to it, and be farthest off when most attracted. Which yet

will appear evidently to follow from that very cause, by considering

what was last shewn, that the orbit of the moon in the conjunction

and opposition is rendred less curve; for the less curve the orbit of

the moon is, the less will the moon have descended from the place it

would move into, without the action of the earth. Now if the moon were

to move from any place without farther disturbance from that action,

since it would proceed in the line, which would touch its orbit in that

place, it would recede continually from the earth; and therefore if

the power of the earth upon the moon, be sufficient to retain it at the

same distance, this diminution of that power will cause the distance

to increase, though in a less degree. But on the other hand in the

quarters, the moon, being pressed more towards the earth than by the

earth’s single action, will be made to approach it; so that in passing

from the conjunction or opposition to the quarters the moon ascends

from the earth, and in passing from the quarters to the conjunction and

opposition it descends again, becoming nearer in these last mentioned

places than in the other.

19. ALL these forementioned inequalities are of different degrees,

according as the sun is more or less distant from the earth; greater

when the earth is nearest the sun, and less when it is farthest off.

For in the quarters, the nearer the moon is to the sun, the greater is

the addition to the earth’s action upon it by the power of the sun; and

in the conjunction and opposition, the difference between the sun’s

action upon the earth and upon the moon is likewise so much the greater.

20. This difference in the distance between the earth and the sun

produces a farther effect upon the moon’s motion; causing the orbit to

dilate when less remote from the sun, and become greater, than when at

a farther distance. For it is proved by Sir ~ISAAC NEWTON~, that the

action of the sun, by which it diminishes the earth’s power over the

moon, in the conjunction or opposition, is about twice as great, as

the addition to the earth’s action by the sun in the quarters[188]; so

that upon the whole, the power of the earth upon the moon is diminished

by the sun, and therefore is most diminished, when the action of the

sun is strongest: but as the earth by its approach to the sun has its

influence lessened, the moon being less attracted will gradually recede

from the earth; and as the earth in its recess from the sun recovers by

degrees its former power, the orbit of the moon must again contract.

Two consequences follow from hence: the moon will be most remote from

the earth, when the earth is nearest the sun; and also will take up a

longer time in performing its revolution through the dilated orbit,

than through the more contracted.

21. THESE irregularities the sun would produce in the moon, if the

moon, without being acted on unequally by the sun, would describe a

perfect circle about the earth, and in the plane of the earth’s motion;

but though neither of these suppositions obtain in the motion of the

moon, yet the forementioned inequalities will take place, only with

some difference in respect to the degree of them; but the moon by not

moving in this manner is subject to some other inequalities also.

For as the moon describes, instead of a circle concentrical to the

earth, an ellipsis, with the earth in one focus, that ellipsis will be

subjected to various changes. It can neither preserve constantly the

same position, nor yet the same figure; and because the plane of this

ellipsis is not the same with that of the earth’s orbit, the situation

of the plane, wherein the moon moves, will continually change; neither

the line in which it intersects the plane of the earth’s orbit, nor the

inclination of the planes to each other, will remain for any time the

same. All these alterations offer themselves now to be explained.

22. I SHALL first consider the changes which are made in the plane

of the moon’s orbit. The moon not moving in the same plane with the

earth, the sun is seldom in the plane of the moon’s orbit, viz. only

when the line made by the common intersection of the two planes, if

produced, will pass through the sun, as is represented in fig. 97.

where S denotes the sun; T the earth; A T B the earth’s orbit described

upon the plane of this scheme; C D E F the moon’s orbit, the part C

D E being raised above, and the part C F E depressed under the plane

of this scheme. Here the line C E, in which the plane of this scheme,

that is, the plane of the earth’s orbit and the plane of the moon’s

orbit intersect each other, being continued passes through the sun in

S. When this happens, the action of the sun is directed in the plane of

the moon’s orbit, and cannot draw the moon out of this plane, as will

evidently appear to any one that shall consider the present scheme: for

suppose the moon in G, and let a straight line be drawn from G to S,

the sun draws the moon in the direction of this line from G toward S:

but this line lies in the plane of the orbit; and if it be prolonged

from S beyond G, the continuation of it will lie on the plane C D E;

for the plane itself, if sufficiently extended, will pass through the

sun. But in other cases the obliquity of the sun’s action to the plane

of the orbit will cause this plane continually to change.

23. SUPPOSE in the first place, the line, in which the two planes

intersect each other, to be perpendicular to the line which joins the

earth and sun. Let T (in fig. 98, 99, 100, 101.) represent the earth; S

the sun; the plane of this scheme the plane of the earth’s motion, in

which both the sun and earth are placed. Let A C be perpendicular to

S T, which joins the earth and sun; and let the line A C be that, in

which the plane of the moon’s orbit intersects the plane of the earth’s

motion. To the center T describe in the plane of the earth’s motion

the circle A B C D. And in the plane of the moon’s orbit describe the

circle A E C F, one half of which A E C will be elevated above the

plane of this scheme, the other half A F C as much depressed below it.

24. NOW suppose the moon to set forth from the point A (in fig. 98.) in

the direction of the plane A E C. Here she will be continually drawn

out of this plane by the action of the sun: for this plane A E C, if

extended, will not pass through the sun, but above it; so that the sun,

by drawing the moon directly toward it self, will force it continually

more and more from that plane towards the plane of the earth’s motion,

in which it self is; causing it to describe the line A K G H I, which

will be convex to the plane A E C, and concave to the plane of the

earth’s motion. But here this power of the sun, which is said to draw

the moon toward the plane of the earth’s motion, must be understood

principally of so much only of the sun’s action upon the moon, as

it exceeds the action of the same upon the earth. For suppose the

preceding figure to be viewed by the eye, placed in the plane of that

scheme, and in the line C T A on the side of A, the plane A B C D will

appear as the straight line D T B, (in fig. 102.) and the plane A E C

F as another straight line F E; and the curve line A K G H I under the

form of the line T K G H I.

[Illustration]

Now it is plain, that the earth and moon being both attracted by the

sun, if the sun’s action upon both was equally strong, the earth T,

and with it the plane A E C F or line F T E in this scheme, would be

carried toward the sun with as great a pace as the moon, and therefore

the moon not drawn out of it by the sun’s action, excepting only from

the small obliquity of the direction of this action upon the moon

to that of the sun’s action upon the earth, which arises from the

moon’s being out of the plane of the earth’s motion, and is not very

considerable; but the action of the sun upon the moon being greater

than upon the earth, all the time the moon is nearer to the sun than

the earth is, it will be drawn from the plane A E C or the line T E by

that excess, and made to describe the curve line A G I or T G I. But it

is the custom of astronomers, instead of considering the moon as moving

in such a curve line, to refer its motion continually to the plane,

which touches the true line wherein it moves, at the point where at

any time the moon is. Thus when the moon is in the point A, its motion

is considered as being in the plane A E C, in whose direction it then

essaies to move; and when in the point K (in fig. 99.) its motion is

referred to the plane, which passes through the earth, and touches the

line A K G H I in the point K. Thus the moon in passing from A to I

will continually change the plane of her motion. In what manner this

change proceeds, I shall now particularly explain.

25. LET the plane, which touches the line A K I in the point K (in fig.

99.) intersect the plane of the earth’s orbit in the line L T M. Then,

because the line A K I is concave to the plane A B C, it falls wholly

between that plane, and the plane which touches it in K; so that the

plane M K L will cut the plane A E C, before it meets with the plane of

the earth’s motion; suppose in the line Y T, and the point A will fall

between K and L. With a semidiameter equal to T Y or T L describe the

semicircle L Y M. Now to a spectator on the earth the moon, when in A,

will appear to move in the circle A E C F, and, when in K, will appear

to be moving in the semicircle L Y M. The earth’s motion is performed

in the plane of this scheme, and to a spectator on the earth the sun

will appear always moving in that plane. We may therefore refer the

apparent motion of the sun to the circle A B C D, described in this

plane about the earth. But the points where this circle, in which the

sun seems to move, intersects the circle in which the moon is seen at

any time to move, are called the nodes of the moon’s orbit at that

time. When the moon is seen moving in the circle A E C D, the points A

and C are the nodes of the orbit; when she appears in the semicircle

L Y M, then L and M are the nodes. Now here it appears, from what has

been said, that while the moon has moved from A to K, one of the nodes

has been carried from A to L, and the other as much from C to M. But

the motion from A to L, and from C to M, is backward in regard to the

motion of the moon, which is the other way from A to K, and from thence

toward C.

26. FARTHER the angle, which the plane, wherein the moon at any time

appears, makes with the plane of the earth’s motion, is called the

inclination of the moon’s orbit at that time. And I shall now proceed

to shew, that this inclination of the orbit, when the moon is in K, is

less than when she was in A; or, that the plane L Y M, which touches

the line of the moon’s motion in K, makes a less angle with the plane

of the earth’s motion or with the circle A B C D, than the plane A E

C makes with the same. The semicircle L Y M intersects the semicircle

A E C in Y; and the arch A Y is less than L Y, and both together less

than half a circle. But it is demonstrated by the writers on that part

of astronomy, which is called the doctrine of the sphere, that when a

triangle is made, as here, by three arches of circles A L, A Y, and Y

L, the angle under Y A B without the triangle is greater than the angle

under Y L A within, if the two arches A Y, Y L taken together do not

amount to a semicircle; if the two arches make a complete semicircle,

the two angles will be equal; but if the two arches taken together

exceed a semicircle, the inner angle under Y L A is greater than the

other[189]. Here therefore the two arches A Y and L Y together being

less than a semicircle, the angle under A L Y is less, than the angle

under B A E. But from the doctrine of the sphere it is also evident,

that the angle under A L Y is equal to that, in which the plane of the

circle L Y K M, that is, the plane which touches the line A K G H I in

K, is inclined to the plane of the earth’s motion A B C; and the angle

under B A E is equal to that, in which the plane A E C is inclined to

the same plane. Therefore the inclination of the former plane is less

than the inclination of the latter.

27. SUPPOSE now the moon to be advanced to the point G (in fig. 100.)

and in this point to be distant from its node a quarter part of the

whole circle; or in other words, to be in the midway between its two

nodes. And in this case the nodes will have receded yet more, and the

inclination of the orbit be still more diminished: for suppose the

line A K G H I to be touched in the point G by a plane passing through

the earth T: let the intersection of this plane with the plane of the

earth’s motion be the line W T O, and the line T P its intersection

with the plane L K M. In this plane let the circle N G O be described

with the semidiameter T P or N T cutting the other circle L K M in P.

Now the line A K G I is convex to the plane L K M, which touches it in

K; and therefore the plane N G O, which touches it in G, will intersect

the other touching plane between G and K; that is, the point P will

fall between those two points, and the plane continued to the plane of

the earth’s motion will pass beyond L; so that the points N and O, or

the places of the nodes, when the moon is in G, will be farther from A

and C than L and M, that is, will have moved farther backward. Besides,

the inclination of the plane N G O to the plane of the earth’s motion A

B C is less, than the inclination of the plane L K M to the same; for

here also the two arches L P and N P taken together are less than a

semicircle, each of these arches being less than a quarter of a circle;

as appears, because G N, the distance of the moon in G from its node N,

is here supposed to be a quarter part of a circle.

28. AFTER the moon is passed beyond G, the case is altered; for then

these arches will be greater than quarters of the circle, by which

means the inclination will be again increased, tho’ the nodes still go

on to move the same way. Suppose the moon in H, (in fig. 101.) and that

the plane, which touches the line A K G I in H, intersects the plane of

the earth’s motion in the line Q T R, and the plane N G O in the line T

V, and besides that the circle Q H R be described in that plane; then,

for the same reason as before, the point V will fall between H and G,

and the plane R V Q will pass beyond the last plane O V N, causing

the points Q and R to fall farther from A and C than N and O. But the

arches N V, V Q are each greater than a quarter of a circle, N V the

least of them being greater than G N, which is a quarter of a circle;

and therefore the two arches N V and V Q together exceed a semicircle;

consequently the angle under B Q V will be greater, than that under B N

V.

29. IN the last place, when the moon is by this attraction of the sun,

drawn at length into the plane of the earth’s motion, the node will

have receded yet more, and the inclination be so much increased, as

to become somewhat more than at first: for the line A K G H I being

convex to all the planes, which touch it, the part H I will wholly fall

between the plane Q V R and the plane A B C; so that the point I will

fall between B and R; and drawing I T W, the point W will be farther

remov’d from A than Q. But it is evident, that the plane, which passes

through the earth T, and touches the line A G I in the point I, will

cut the plane of the earth’s motion A B C D in the line I T W, and

be inclined to the same in the angle under H I B; so that the node,

which was first in A, after having passed into L, N and Q, comes at

last into the point W; as the node which was at first in C has passed

successively from thence through the points M, O and R to I: but the

angle under H I B, which is now the inclination of the orbit to the

plane of the ecliptic, is manifestly not less than the angle under E C

B or E A B, but rather something greater.

30. THUS the moon in the case before us, while it passes from the

plane of the earth’s motion in the quarter, till it comes again into

the same plane, has the nodes of its orbit continually moved backward,

and the inclination of its orbit is at first diminished, viz. till it

comes to G in fig. 100, which is near to its conjunction with the sun,

but afterwards is increased again almost by the same degrees, till

upon the moon’s arrival again to the plane of the earth’s motion, the

inclination of the orbit is restored to something more than its first

magnitude, though the difference is not very great, because the points

I and C are not far distant from each other[190].

31. AFTER the same manner, if the moon had departed from the quarter in

C, it should have described the curve line C X W (in fig. 98.) between

the planes A F C and A D C, which would be convex to the former of

those planes, and concave to the latter; so that, here also, the nodes

should continually recede, and the inclination of the orbit gradually

diminish more and more, till the moon arrived near its opposition to

the sun in X; but from that time the inclination should again increase,

till it became a little greater than at first. This will easily

appear, by considering, that as the action of the sun upon the moon,

by exceeding its action upon the earth, drew it out of the plane A E

C towards the sun, while the moon passed from A to I; so, during its

passage from C to W, the moon being all that time farther from the sun

than the earth, it will be attracted less; and the earth, together with

the plane A E C F, will as it were be drawn from the moon, in such

sort, that the path the moon describes shall appear from the earth, as

it did in the former case by the moon’s being drawn away.

32. THESE are the changes, which the nodes and the inclination of the

moon’s orbit undergo, when the nodes are in the quarters; but when the

nodes by their motion, and the motion of the sun together, come to

be situated between the quarter and conjunction or opposition, their

motion and the change made in the inclination of the orbit are somewhat

different.

33. LET A G C H (in fig. 103.) be a circle described in the plane of

the earth’s motion, having the earth in T for its center. Let the

point opposite to the sun be A, and the point G a fourth part of the

circle distant from A. Let the nodes of the moon’s orbit be situated

in the line B T D, and B the node, falling between A, the place where

the moon would be in the full, and G the place where the moon would

be in the quarter. Suppose B E D F to be the plane, in which the moon

essays to move, when it proceeds from the point B. Because the moon

in B is more distant from the sun than the earth, it shall be less

attracted by the sun, and shall not descend towards the sun so fast

as the earth: consequently it shall quit the plane B E D F, which we

suppose to accompany the earth, and describe the line B I K convex

thereto, till such time as it comes to the point K, where it will be in

the quarter: but from thenceforth being more attracted than the earth,

the moon shall change its course, and the following part of the path

it describes shall be concave to the plane B E D or B G D, and shall

continue concave to the plane B G D, till it crosses that plane in L,

just as in the preceding case. Now I say, while the moon is passing

from B to K, the nodes, contrary to what was found in the foregoing

case, will proceed forward, or move the same way with the moon[191];

and at the same time the inclination of the orbit will increase[192].

[Illustration]

34. WHEN the moon is in the point I, let the plane M I N pass through

the earth T, and touch the path of the moon in I, cutting the plane of

the earth’s motion, in the line M T N, and the plane B E D in the line

T O. Because the line B I K is convex to the plane B E D, which touches

it in B, the plane N I M must cross the plane D E B, before it meets

the plane C G B; and therefore the point M will fall from B towards G,

and the node of the moon’s orbit being translated from B to M is moved

forward.

35. I SAY farther, the angle under O M G, which the plane M O N makes

with the plane B G C, is greater than the angle under O B G, which

the plane B O D makes with the same. This appears from what has been

already explained; because the arches B O, O M are each less than the

quarter of a circle, and therefore taken both together are less than a

semicircle.

36. AGAIN, when the moon is come to the point K in its quarter, the

nodes will be advanced yet farther forward, and the inclination of the

orbit also more augmented. Hitherto the moon’s motion has been referred

to the plane, which passing through the earth touches the path of the

moon in the point, where the moon is, according to what was asserted at

the beginning of this discourse upon the nodes, that it is the custom

of astronomers so to do. But here in the point K no such plane can be

found; on the contrary, seeing the line of the moon’s motion on one

side the point K is convex to the plane B E D, and on the other side

concave to the same, no plane can pass through the points T and K but

will cut the line B K L in that point. Therefore instead of such a

touching plane, we must here make use of what is equivalent, the plane

P K Q, with which the line B K L shall make a less angle than with any

other plane; for this plane does as it were touch the line B K in the

point K, since it so cuts it, that no other plane can be drawn so,

as to pass between the line B K and the plane P K Q. But now it is

evident, that the point P, or the node, is removed from M towards G,

that is, has moved yet farther forward; and it is likewise as manifest,

that the angle under K P G, or the inclination of the moon’s orbit in

the point K, is greater than the angle under I M G, for the reason so

often assigned.

37. AFTER the moon has passed the quarter, the path of the moon being

concave to the plane A G C H, the nodes, as in the preceding case,

shall recede, till the moon arrives at the point L; which shews, that

considering the whole time of the moon’s passing from B to L, at the

end of that time the nodes shall be found to have receded, or to be

placed backwarder, when the moon is in L, than when it was in B. For

the moon takes a longer time in passing from K to L, than in passing

from B to K; and therefore the nodes continue to recede a longer time,

than they moved forwards; so that their recess must surmount their

advance.

38. IN the same manner, while the moon is in its passage from K to L,

the inclination of the orbit shall diminish, till the moon comes to

the point, in which it is one quarter part of a circle distant from

its node; suppose in the point R; and from that time the inclination

shall again increase. Since therefore the inclination of the orbit

increases, while the moon is passing from B to K, and diminishes itself

again only, while the moon is passing from K to R, and then augments

again, till the moon arrive in L; while the moon is passing from B to

L, the inclination of the orbit is much more increased than diminished,

and will be distinguishably greater, when the moon is come to L, than

when it set out from B.

39. IN like manner, while the moon is passing from L on the other side

the plane A G C H, the node shall advance forward, as long as the moon

is between the point L and the next quarter; but afterwards it shall

recede, till the moon come to pass the plane A G C H again in the point

V, between B and A: and because the time between the moon’s passing

from L to the next quarter is less, than the time between that quarter

and the moon’s coming to the point V, the node shall have more receded

than advanced; so that the point V will be nearer to A, than L is to C.

So also the inclination of the orbit, when the moon is in V, will be

greater, than when the moon was at L; for this inclination increases

all the time the moon is between L and the next quarter; it decreases

only while the moon is passing from this quarter to the mid way between

the two nodes, and from thence increases again during the whole passage

through the other half of the way to the next node.

40. THUS we have traced the moon from her node in the quarter, and

shewn, that at every period of the moon the nodes will have receded,

and thereby will have approached toward a conjunction with the sun.

But this conjunction will be much forwarded by the visible motion of

the sun itself. In the last scheme the sun will appear to move from

S toward W. Suppose it appeared to have moved from S to W, while the

moon’s node has receded from B to V, then drawing the line W T X,

the arch V X will represent the distance of the line drawn between

the nodes from the sun, when the moon is in V; whereas the arch B A

represented that distance, when the moon was in B. This visible motion

of the sun is much greater, than that of the node; for the sun appears

to revolve quite round each year, and the node is near 19 years in

making one revolution. We have also seen, that when the node was in

the quadrature, the inclination of the moon’s orbit decreased, till

the moon came to the conjunction, or opposition, according to which

node it set out from; but that afterwards it again increased, till it

became at the next node rather greater than at the former. When the

node is once removed from the quarter nearer to a conjunction with the

sun, the inclination of the moon’s orbit, when the moon comes into the

node, is more sensibly greater, than it was in the node preceding; the

inclination of the orbit by this means more and more increasing till

the node comes into conjunction with the sun; at which time it has

been shewn above, that the sun has no power to change the plane of the

moon’s motion; and consequently has no effect either on the nodes, or

on the inclination of the orbit.

41. AS soon as the nodes, by the action of the sun, are got out of

conjunction toward the other quarters, they begin again to recede as

before; but the inclination of the orbit in the appulse of the moon

to each succeeding node is less than at the preceding, till the nodes

come again into the quarters. This will appear as follows. Let A (in

fig. 104.) represent one of the moon’s nodes placed between the point

of opposition B and the quarter C. Let the plane A D E pass through

the earth T, and touch the path of the moon in A. Let the line A F G H

be the path of the moon in her passage from A to H, where she crosses

again the plane of the earth’s motion. This line will be convex toward

the plane A D E, till the moon comes to G, where she is in the quarter;

and after this, between G and H, the same line will be concave toward

this plane. All the time this line is convex toward the plane A D

E, the nodes will recede; and on the contrary proceed, while it is

concave to that plane. All this will easily be conceived from what has

been before so largely explained. But the moon is longer in passing

from A to G, than from G to H; therefore the nodes recede a longer

time, than they proceed; consequently upon the whole, when the moon is

arrived at H, the nodes will have receded, that is, the point H will

fall between B and E. The inclination of the orbit will decrease, till

the moon is arrived to the point F, in the middle between A and H.

Through the passage between F and G the inclination will increase, but

decrease again in the remaining part of the passage from G to H, and

consequently at H must be less than at A. The like effects, both in

respect to the nodes and inclination of the orbit, will take place in

the following passage of the moon on the other side of the plane A B E

C, from H, till it comes over that plane again in I.

42. THUS the inclination of the orbit is greatest, when the line drawn

between the moon’s nodes will pass through the sun; and least, when

this line lies in the quarters, especially if the moon at the same time

be in conjunction with the sun, or in the opposition. In the first of

these cases the nodes have no motion, in all others, the nodes will

each month have receded: and this regressive motion will be greatest,

when the nodes are in the quarters; for in that case the nodes have

no progressive motion during the whole month, but in all other cases

the nodes do at some times proceed forward, viz. whenever the moon is

between either quarter, and the node which is less distant from that

quarter than a fourth part of a circle.

43. IT now remains only to explain the irregularities in the moon’s

motion, which follow from the elliptical figure of the orbit. By

what has been said at the beginning of this chapter it appears, that

the power of the earth on the moon acts in the reciprocal duplicate

proportion of the distance: therefore the moon, if undisturbed by

the sun, would move round the earth in a true ellipsis, and the line

drawn from the earth to the moon would pass over equal spaces in equal

portions of time. That this description of the spaces is altered by

the sun, has been already declared. It has also been shown, that the

figure of the orbit is changed each month; that the moon is nearer the

earth at the new and full, and more remote in the quarters, than it

would be without the sun. Now we must pass by these monthly changes,

and consider the effect, which the sun will have in the different

situations of the axis of the orbit in respect of that luminary.

44. THE action of the sun varies the force, wherewith the moon is drawn

toward the earth; in the quarters the force of the earth is directly

increased by the sun; at the new and full the same is diminished; and

in the intermediate places the influence of the earth is sometimes

aided, and sometimes lessened by the sun. In these intermediate

places between the quarters and the conjunction or opposition, the

sun’s action is so oblique to the action of the earth on the moon, as

to produce that alternate acceleration and retardment of the moon’s

motion, which I observed above to be stiled the variation. But besides

this effect, the power, by which the earth attracts the moon toward

itself, will not be at full liberty to act with the same force, as if

the sun acted not at all on the moon. And this effect of the sun’s

action, whereby it corroborates or weakens the action of the earth, is

here only to be considered. And by this influence of the sun it comes

to pass, that the power, by which the moon is impelled toward the

earth, is not perfectly in the reciprocal duplicate proportion of the

distance. Consequently the moon will not describe a perfect ellipsis.

One particular, wherein the moon’s orbit will differ from an ellipsis,

consists in the places, where the motion of the moon is perpendicular

to the line drawn from itself to the earth. In an ellipsis, after the

moon should have set out in the direction perpendicular to this line

drawn from itself to the earth, and at its greatest distance from the

earth, its motion would again become perpendicular to this line drawn

between itself and the earth, and the moon be at its nearest distance

from the earth, when it should have performed half its period; after

performing the other half of its period its motion would again become

perpendicular to the forementioned line, and the moon return into

the place whence it set out, and have recovered again its greatest

distance. But the moon in its real motion, after setting out as before,

sometimes makes more than half a revolution, before its motion comes

again to be perpendicular to the line drawn from itself to the earth,

and the moon is at its nearest distance; and then performs more than

another half of an intire revolution before its motion can a second

time recover its perpendicular direction to the line drawn from the

moon to the earth, and the moon arrive again to its greatest distance

from the earth. At other times the moon will descend to its nearest

distance, before it has made half a revolution, and recover again

its greatest distance, before it has made an intire revolution. The

place, where the moon is at its greatest distance from the earth, is

called the moon’s apogeon, and the place of the least distance the

perigeon. This change of the place, where the moon successively comes

to its greatest distance from the earth, is called the motion of the

apogeon. In what manner the sun causes the apogeon to move, I shall now

endeavour to explain.

45. OUR author shews, that if the moon were attracted toward the

earth by a composition of two powers, one of which were reciprocally

in the duplicate proportion of the distance from the earth, and the

other reciprocally in the triplicate proportion of the same distance;

then, though the line described by the moon would not be in reality

an ellipsis, yet the moon’s motion might be perfectly explained by

an ellipsis, whose axis should be made to move round the earth; this

motion being in consequence, as astronomers express themselves, that

is, the same way as the moon itself moves, if the moon be attracted by

the sum of the two powers; but the axis must move in antecedence, or

the contrary way, if the moon be acted on by the difference of these

powers. What is meant by duplicate proportion has been often explained;

namely, that if three magnitudes, as A, B, and C, are so related, that

the second B bears the same proportion to the third C, as the first A

bears to the second B, then the proportion of the first A to the third

C, is the duplicate of the proportion of the first A to the second B.

Now if a fourth magnitude, as D, be assumed, to which C shall bear the

same proportion as A bears to B, and B to C, then the proportion of A

to D is the triplicate of the proportion of A to B.

46. THE way of representing the moon’s motion in this case is thus. T

denoting the earth (in fig. 105, 106.) suppose the moon in the point

A, its apogeon, or greatest distance from the earth, moving in the

direction A F perpendicular to A B, and acted upon from the earth by

two such forces as have been named. By that power alone, which is

reciprocally in the duplicate proportion of the distance, if the moon

let out from the point A with a proper degree of velocity, the ellipsis

A M B may be described. But if the moon be acted upon by the sum of

the forementioned powers, and the velocity of the moon in the point

A be augmented in a certain proportion[193]; or if that velocity be

diminished in a certain proportion, and the moon be acted upon by the

difference of those powers; in both these cases the line A E, which

shall be described by the moon, is thus to be determined. Let the point

M be that, into which the moon would have arrived in any given space of

time, had it moved in the ellipsis A M B. Draw M T, and likewise C T D

in such sort, that the angle under A T M shall bear the same proportion

to the angle under A T C, as the velocity, with which the ellipsis A

M B must have been described, bears to the difference between this

velocity, and the velocity, with which the moon must set out from the

point A in order to describe the path A E. Let the angle A T C be taken

toward the moon (as in fig. 105.) if the moon be attracted by the sum

of the powers; but the contrary way (as in fig. 106.) if by their

difference. Then let the line A B be moved into the position C D, and

the ellipsis A M B into the situation C N D, so that the point M be

translated to L: then the point L shall fall upon the path of the moon

A E.

47. THE angular motion of the line A T, wereby it is removed into the

situation C T, represents the motion of the apogeon; by the means of

which the motion of the moon might be fully explicated by the ellipsis

A M B, if the action of the sun upon it was directed to the center

of the earth, and reciprocally in the triplicate proportion of the

moon’s distance from it. But that not being so, the apogeon will not

move in the regular manner now described. However, it is to be observed

here, that in the first of the two preceding cases, where the apogeon

moves forward, the whole centripetal power increases faster, with the

decrease of distance, than if the intire power were reciprocally in

the duplicate proportion of the distance; because one part only is in

that proportion, and the other part, which is added to this to make up

the whole power, increases faster with the decrease of distance. On

the other hand, when the centripetal power is the difference between

these two, it increases less with the decrease of the distance, than

if it were simply in the reciprocal duplicate proportion of the

distance. Therefore if we chuse to explain the moon’s motion by an

ellipsis (as is most convenient for astronomical uses to be done, and

by reason of the small effect of the sun’s power, the doing so will

not be attended with any sensible error;) we may collect in general,

that when the power, by which the moon is attracted to the earth, by

varying the distance, increases in a greater than in the duplicate

proportion of the distance diminished, a motion in consequence must

be ascribed to the apogeon; but that when the attraction increases

in a less proportion than that named, the apogeon must have given

to it a motion in antecedence[194]. It is then observed by Sir IS.

NEWTON, that the first of these cases obtains, when the moon is in the

conjunction and opposition; and the latter, when the moon is in the

quarters: so that in the first the apogeon moves according to the order

of the signs; in the other, the contrary way[195]. But, as was said

before, the disturbance given to the action of the earth by the sun

in the conjunction and opposition being near twice as great as in the

quarters[196], the apogeon will advance with a greater velocity than

recede, and in the compass of a whole revolution of the moon will be

carried in consequence[197].

48. IT is shewn in the next place by our author, that when the line A B

coincides with that, which joins the earth and the sun, the progressive

motion of the apogeon, when the moon is in the conjunction or

opposition, exceeds the regressive in the quadratures more than in any

other situation of the line A B[198]. On the contrary, when the line

A B makes right angles with that, which joins the earth and sun, the

retrograde motion will be more considerable[199], nay is found so great

as to exceed the progressive; so that in this case the apogeon in the

compass of an intire revolution of the moon is carried in antecedence.

Yet from the considerations in the last paragraph the progressive

motion exceeds the other; so that in the whole the mean motion of the

apogeon is in consequence, according as astronomers find. Moreover, the

line A B changes its situation with that, which joins the earth and

sun, by such slow degrees, that the inequalities in the motion of the

apogeon arising from this last consideration, are much greater than

what arises from the other[200].

49. FARTHER, this unsteady motion in the apogeon is attended with

another inequality in the motion of the moon, that it cannot be

explained at all times by the same ellipsis. The ellipsis in general

is called by astronomers an eccentric orbit. The point, in which the

two axis’s cross, is called the center of the figure; because all

lines drawn through this point within the ellipsis, from side to side,

are divided in the middle by this point. But the center, about which

the heavenly bodies revolve, lying out of this center of the figure

in one focus, these orbits are said to be eccentric; and where the

distance of the focus from this center bears the greatest proportion

to the whole axis, that orbit is called the most eccentric: and in

such an orbit the distance from the focus to the remoter extremity of

the axis bears the greatest proportion to the distance of the nearer

extremity. Now whenever the apogeon of the moon moves in consequence,

the moon’s motion must be referred to an orbit more eccentric, than

what the moon would describe, if the whole power, by which the moon

was acted on in its passing from the apogeon, changed according to the

reciprocal duplicate proportion of the distance from the earth, and by

that means the moon did describe an immoveable ellipsis; and when the

apogeon moves in antecedence, the moon’s motion must be referred to an

orbit less eccentric. In the first of the two figures last referred

to, the true place of the moon L falls without the orbit A M B, to

which its motion is referred: whence the orbit A L E, truly described

by the moon, is less incurvated in the point A, than is the orbit A M

B; therefore the orbit A M B is more oblong, and differs farther from

a circle, than the ellipsis would, whose curvature in A were equal to

that of the line A L B, that is, the proportion of the distance of the

earth T from the center of the ellipsis to its axis will be greater in

the ellipsis A M B, than in the other; but that other is the ellipsis,

which the moon would describe, if the power acting upon it in the point

A were altered in the reciprocal duplicate proportion of the distance.

In the second figure, when the apogeon recedes, the place of the moon

L falls within the orbit A M B, and therefore that orbit is less

eccentric, than the immoveable orbit which the moon should describe.

The truth of this is evident; for, when the apogeon moves forward, the

power, by which the moon is influenced in its descent from the apogeon,

increases faster with the decrease of distance, than in the duplicate

proportion of the distance; and consequently the moon being drawn more

forcibly toward the earth, it will descend nearer to it. On the other

hand, when the apogeon recedes, the power acting on the moon increases

with the decrease of distance in less than the duplicate proportion of

the distance; and therefore the moon is less impelled toward the earth,

and will not descend so low.

50. NOW suppose in the first of these figures, that the apogeon A is

in the situation, where it is approaching toward the conjunction or

opposition of the sun. In this case the progressive motion of the

apogeon is more and more accelerated. Here suppose that the moon, after

having descended from A through the orbit A E as far as F, where it

is come to its nearest distance from the earth, ascends again up the

line F G. Because the motion of the apogeon is here continually more

and more accelerating, the cause of its motion is constantly upon the

increase; that is, the power, whereby the moon is drawn to the earth,

will decrease with the increase of distance, in the moon’s ascent

from F, in a greater proportion than that wherewith it increased with

the decrease of distance in the moon’s descent to F. Consequently

the moon will ascend higher than to the distance A T, from whence it

descended; therefore the proportion of the greatest distance of the

moon to the least is increased. And when the moon descends again, the

power will yet more increase with the decrease of distance, than in the

last ascent it decreased with the augmentation of distance; the moon

therefore must descend nearer to the earth than it did before, and the

proportion of the greatest distance to the least yet be more increased.

Thus as long as the apogeon is advancing toward the conjunction or

opposition, the proportion of the greatest distance of the moon from

the earth to the least will continually increase; and the elliptical

orbit, to which the moon’s motion is referred, will be rendered more

and more eccentric.

51. AS soon as the apogeon is passed the conjunction with the sun or

the opposition, the progressive motion thereof abates, and with it

the proportion of the greatest distance of the moon from the earth to

the least distance will also diminish; and when the apogeon becomes

regressive, the diminution of this proportion will be still farther

continued on, till the apogeon comes into the quarter; from thence this

proportion, and the eccentricity of the orbit will increase again.

Thus the orbit of the moon is most eccentric, when the apogeon is in

conjunction with the sun, or in opposition to it, and least of all when

the apogeon is in the quarters.

52. THESE changes in the nodes, in the inclination of the orbit to the

plane of the earth’s motion, in the apogeon, and in the eccentricity,

are varied like the other inequalities in the motion of the moon, by

the different distance of the earth from the sun; being greatest, when

their cause is greatest, that is, when the earth is nearest to the sun.

53. I SAID at the beginning of this chapter, that Sir ISAAC NEWTON

has computed the very quantity of many of the moon’s inequalities.

That acceleration of the moon’s motion, which is called the variation,

when greatest, removes the moon out of the place, in which it would

otherwise be found, something more than half a degree[201]. In the

phrase of astronomers, a degree is 1/360 part of the whole circuit of

the moon or any planet. If the moon, without disturbance from the sun,

would have described a circle concentrical to the earth, the sun will

cause the moon to approach nearer to the earth in the conjunction and

opposition, than in the quarters, nearly in the proportion of 69 to

70[202]. We had occasion to mention above, that the nodes perform their

period in almost 19 years. This the astronomers found by observation;

and our author’s computations assign to them the same period[203].

The inclination of the moon’s orbit when least, is an angle about

1/18 part of that angle, which constitutes a perpendicular; and the

difference between the greatest and least inclination of the orbit is

determined by our author’s computation to be about 1/18 of the least

inclination[204]. And this also is agreeable to the observations

of astronomers. The motion of the apogeon, and the changes in the

eccentricity, Sir ~ISAAC NEWTON~ has not computed. The apogeon performs

its revolution in about eight years and ten months. When the moon’s

orbit is most eccentric, the greatest distance of the moon from the

earth bears to the least distance nearly the proportion of 8 to 7; when

the orbit is least eccentric, this proportion is hardly so great as

that of 12 to 11.

54. SIR ~ISAAC NEWTON~ shews farther, how, by comparing the

periods of the motion of the satellites, which revolve round Jupiter

and Saturn, with the period of our moon round the earth, and the

periods of those planets round the sun with the period of our earth’s

motion, the inequalities in the motion of those satellites may be

derived from the inequalities in the moon’s motion; excepting only in

regard to that motion of the axis of the orbit, which in the moon makes

the motion of the apogeon; for the orbits of those satellites, as far

as can be discerned by us at this distance, appearing little or nothing

eccentric, this motion, as deduced from the moon, must be diminished.

~CHAP. IV.~

Of ~Comets~.

IN the former of the two preceding chapters the powers have been

explained, which keep in motion those celestial bodies, whose courses

had been well determined by the astronomers. In the last chapter we

have shewn, how those powers have been applied by our author to the

making a more perfect discovery of the motion of those bodies, the

courses of which were but imperfectly understood; for some of the

inequalities, which we have been describing in the moon’s motion,

were unknown to the astronomers. In this chapter we are to treat of

a third species of the heavenly bodies, the true motion of which was

not at all apprehended before our author writ; in so much, that here

Sir ~ISAAC NEWTON~ has not only explained the causes of the motion of

these bodies, but has performed also the part of an astronomer, by

discovering what their motions are.

2. THAT these bodies are not meteors in our air, is manifest; because

they rise and set in the same manner, as the sun and stars. The

astronomers had gone so far in their inquiries concerning them, as to

prove by their observations, that they moved in the etherial spaces far

beyond the moon; but they had no true notion at all of the path, which

they described. The most prevailing opinion before our author was,

that they moved in straight lines; but in what part of the heavens was

not determined. DESCARTES[205] removed them far beyond the sphere of

Saturn, as finding the straight motion attributed to them, inconsistent

with the vortical fluid, by which he explains the motions of the

planets, as we have above related[206]. But Sir ISAAC NEWTON distinctly

proves from astronomical observation, that the comets pass through the

region of the planets, and are mostly invisible at a less distance,

than that of Jupiter[207].

3. AND from hence finding the comets to be evidently within the sphere

of the sun’s action, he concludes they must, necessarily move about the

sun, as the planets do[208]. The planets move in ellipsis’s; but it is

not necessary that every body, which is influenced by the sun, should

move in that particular kind of line. However our author proves, that

the power of the sun being reciprocally in the duplicate proportion of

the distance, every body acted on by the sun must either fall directly

down, or move in some conic section; of which lines I have above

observed, that there are three species, the ellipsis, parabola, and

hyperbola[209]. If a body, which descends toward the sun as low as the

orbit of any planet, move with a swifter motion than the planet does,

that body will describe an orbit of a more oblong figure, than that

of the planet, and have a longer axis at least. The velocity of the

body may be so great, that it shall move in a parabola, and having

once passed about the sun, shall ascend for ever without returning any

more: but the sun will be placed in the focus of this parabola. With a

velocity still greater the body will move in an hyperbola. But it is

most probable, that the comets move in elliptical orbits, though of a

very oblong, or in the phrase of astronomers, of a very eccentric form,

such as is represented in fig. 107, where S is the sun, C the comet,

and A B D E its orbit, wherein the distance of S and D far exceeds that

of S and A. Whence it is, that they sometimes are found at a moderate

distance from the sun, and appear within the planetary regions; at

other times they ascend to vast distances, far beyond the very orbit of

Saturn, and so become invisible. That the comets do move in this manner

is proved by our author, from computations built upon the observations,

which astronomers had made on many comets. These computations were

performed by Sir ~ISAAC NEWTON~ himself upon the comet, which appeared

toward the latter end of the year 1680, and at the beginning of the

year following[210]; but the learned Dr. HALLEY prosecuted the like

computations more at large in this, and also in many other comets[211].

Which computations are made upon propositions highly worthy of our

author’s unparallel’d genius, such as could scarce have been discovered

by any one not possessed of the utmost force of invention;

4. THOSE computations depend upon this principle, that the eccentricity

of the orbits of the comets is so great, that if they are really

elliptical, yet they approach so near to parabolas in that part of

them, where they come under our view, that they may be taken for such

without sensible error[212]: as in the preceding figure the parabola

F A G differs in the lower part of it about A very little from the

ellipsis D E A B. Upon which ground our great author teaches a method

of finding by three observations made upon any comet the parabola,

which nearest agrees with its orbit[213].

5. NOW what confirms this whole theory beyond the least room for

doubt is, that the places of the comets computed in the orbits, which

the method here mentioned assigns them, agree to the observations of

astronomers with the same degree of exactness, as the computations

of the primary planets places usually do; and this in comets, whose

motions are very extraordinary[214].

6. OUR author afterwards shews how to make use of any small deviation

from the parabola, that shall be observed, to determine whether the

orbits of the comets are elliptical or not, and so to discover if the

same comet returns at certain periods[215]. And upon examining the

comet in 1680, by the rule laid down for this purpose, he finds its

orbit to agree more exactly to an ellipsis than to a parabola, though

the ellipsis be so very eccentric, that the comet cannot perform its

period through it in the space of 500 years[216]. Upon this Dr. HALLEY

observed, that mention is made in history of a comet, with the like

eminent tail as this, having appeared three several times before; the

first of which appearances was at the death of JULIUS CESAR, and each

appearance was at the distance of 575 years from the next preceding.

He therefore computed the motion of this comet in such an elliptic

orbit, as would require this number of years for the body to revolve

through it; and these computations agree yet more perfectly with the

observations made on this comet, than any parabolical orbit will

do[217].

7. THE comparing together different appearances of the same comet, is

the only way to discover certainly the true form of the orbit: for

it is impossible to determine with exactness the figure of an orbit

so exceedingly eccentric, from single observations taken in one part

of it; and therefore Sir ~ISAAC NEWTON~[218] proposes to compare the

orbits, upon the supposition that they are parabolical, of such comets

as appear at different times; for if the same orbit be found to be

described by a comet at different times, in all probability it will

be the same comet which describes it. And here he remarks from Dr.

HALLEY, that the same orbit very nearly agrees to two appearances of

a comet about the space of 75 years distance[219]; so that if those

two appearances were really of the same comet, the transverse axis of

the orbit of the comet would be near 18 times the axis of the earth’s

orbit; and the comet, when at its greatest distance from the sun, will

be removed not less than 35 times as far as the middle distance of the

earth.

8. AND this seems to be the shortest period of any of the comets.

But it will be farther confirmed, if the same comet should return a

third time after another period of 75 years. However it is not to be

expected, that comets should preserve the same regularity in their

periods, as the planets; because the great eccentricity of their orbits

makes them liable to suffer very considerable alterations from the

action of the planets, and other comets, upon them.

9. IT is therefore to prevent too great disturbances in their motions

from these causes, as our author observes, that while the planets

revolve all of them nearly in the same plane, the comets are disposed

in very different ones; and distributed over all parts of the heavens;

that, when in their greatest distance from the sun, and moving slowest,

they might be removed as far as possible out of the reach of each

other’s action[220]. The same end is likewise farther answered in those

comets, which by moving slowest in the aphelion, or remotest distance

from the sun, descend nearest to it, by placing the aphelion of these

at the greatest height from the sun[221].

10. OUR philosopher being led by his principles to explain the motions

of the comets, in the manner now related, takes occasion from thence to

give us his thoughts upon their nature and use. For which end he proves

in the first place, that they must necessarily be solid and compact

bodies, and by no means any sort of vapour or light substance exhaled

from the planets or stars: because at the near distance, to which some

comets approach the sun, it could not be, but the immense heat, to

which they are exposed, should instantaneously disperse and scatter any

such light volatile substance[222]. In particular the forementioned

comet of 1680 descended so near the sun, as to come within a sixth

part of the sun’s diameter from the surface of it. In which situation

it must have been exposed, as appears by computation, to a degree of

heat exceeding the heat of the sun upon our earth no less than 28000

times; and therefore might have contracted a degree of heat 2000 times

greater, than that of red hot iron[223]. Now a substance, which could

endure so intense a heat, without being dispersed in vapor, must needs

be firm and solid.

11. IT is shewn likewise, that the comets are opake substances, shining

by a reflected light, borrowed from the sun[224]. This is proved from

the observation, that comets, though they are approaching the earth,

yet diminish in lustre, if at the same time they recede from the sun;

and on the contrary, are found to encrease daily in brightness, when

they advance towards the sun, though at the same time they move from

the earth[225].

12. THE comets therefore in these respects resemble the planets; that

both are durable opake bodies, and both revolve about the sun in conic

sections. But farther the comets, like our earth, are surrounded by

an atmosphere. The air we breath is called the earth’s atmosphere;

and it is most probable, that all the other planets are invested with

the like fluid. Indeed here a difference is found between the planets

and comets. The atmospheres of the planets are of so fine and subtile

a substance, as hardly to be discerned at any distance, by reason of

the small quantity of light which they reflect, except only in the

planet Mars. In him there is some little appearance of such a substance

surrounding him, as stars which have been covered by him are said to

look somewhat dim a small space before his body comes under them, as if

their light, when he is near, were obstructed by his atmosphere. But

the atmospheres which surround the comets are so gross and thick, as to

reflect light very copiously. They are also much greater in proportion

to the body they surround, than those of the planets, if we may judge

of the rest from our air; for it has been observed of comets, that the

bright light appearing in the middle of them, which is reflected from

the solid body, is scarce a ninth or tenth part of the whole comet,

13. I SPEAK only of the heads of the comets, the most lucid part of

which is surrounded by a fainter light, the most lucid part being

usually not above a ninth or tenth part of the whole in breadth[226].

Their tails are an appearance very peculiar, nothing of the same

nature appertaining in the least degree to any other of the celestial

bodies. Of that appearance there are several opinions; our author

reduces them to three[227]. The two first, which he proposes, are

rejected by him; but the third he approves. The first is, that they

arise from a beam of light transmitted through the head of the comet,

in like manner as a stream of light is discerned, when the sun shines

into a darkened room through a small hole. This opinion, as Sir ~ISAAC

NEWTON~ observes, implies the authors of it wholly unskilled in the

principles of optics; for that stream of light, seen in a darkened

room, arises from the reflection of the sun beams by the dust and motes

floating in the air: for the rays of light themselves are not seen,

but by their being reflected to the eye from some substance, upon

which they fall[228]. The next opinion examined by our author is that

of the celebrated DESCARTES, who imagins these tails to be the light

of the comet refracted in its passage to us, and thence affording an

oblong representation; as the light of the sun does, when refracted

by the prism in that noted experiment, which will have a great share

in the third book of this discourse[229]. But this opinion is at once

overturned from this consideration only, that the planets could be

no more free from this refraction than the comets; nay ought to have

larger or brighter tails, than they, because the light of the planets

is strongest. However our author has thought proper to add some farther

objections against this opinion: for instance, that these tails are not

variegated with colours, as is the image produced by the prism, and

which is inseparable from that unequal refraction, which produces that

disproportioned length of the image. And besides, when the light in

its passage from different comets to the earth describes the same path

through the heavens, the refraction of it should of necessity be in all

respects the same. But this is contrary to observation; for the comet

in 1680, the 28th day of December, and a former comet in the year 1577,

the 29th day of December, appear’d in the same place of the heavens,

that is, were seen adjacent to the same fixed stars, the earth likewise

being in the same place at both times; yet the tail of the latter comet

deviated from the opposition to the sun a little to the northward, and

the tail of the former comet declined from the opposition of the sun

five times as much southward[230].

14. THERE are some other false opinions, though less regarded than

these, which have been advanced upon this argument. These our

excellent author passes over, hastening to explain, what he takes to

be the true cause of this appearance. He thinks it is certainly owing

to steams and vapours exhaled from the body, and gross atmosphere of

the comets, by the heat of the sun; because all the appearances agree

perfectly to this sentiment. The tails are but small, while the comet

is descending to the sun, but enlarge themselves to an immense degree,

as soon as ever the comet has passed its perihelion; which shews the

tail to depend upon the degree of heat, which the comet receives from

the sun. And that the intense heat to which comets, when nearest the

sun, are exposed, should exhale from them a very copious vapour, is a

most reasonable supposition; especially if we consider, that in those

free and empty regions steams will more easily ascend, than here upon

the surface of the earth, where they are suppressed and hindered from

rising by the weight of the incumbent air: as we find by experiments

made in vessels exhausted of the air, where upon removal of the air

several substances will fume and discharge steams plentifully, which

emit none in the open air. The tails of comets, like such a vapour,

are always in the plane of the comet’s orbit, and opposite to the

sun, except that the upper part thereof inclines towards the parts,

which the comet has left by its motion; resembling perfectly the smoak

of a burning coal, which, if the coal remain fixed, ascends from it

perpendicularly; but, if the coal be in motion, ascends obliquely,

inclining from the motion of the coal. And besides, the tails of

comets may be compared to this smoak in another respect, that both

of them are denser and more compact on the convex side, than on the

concave. The different appearance of the head of the comet, after it

has past its perihelion, from what it had before, confirms greatly this

opinion of their tails: for smoke raised by a strong heat is blacker

and grosser, than when raised by a less; and accordingly the heads of

comets, at the same distance from the sun, are observed less bright and

shining after the perihelion, than before, as if obscured by such a

gross smoke.

15. THE observations of HEVELIUS upon the atmospheres of comets still

farther illustrate the same; who relates, that the atmospheres,

especially that part of them next the sun, are remarkably contracted

when near the sun, and dilated again afterwards.

16. TO give a more full idea of these tails, a rule is laid down by

our author, whereby to determine at any time, when the vapour in the

extremity of the tail first rose from the head of the comet. By this

rule it is found, that the tail does not consist of a fleeting vapour,

dissipated soon after it is raised, but is of long continuance; that

almost all the vapour, which rose about the time of the perihelion from

the comet of 1680, continued to accompany it, ascending by degrees,

being succeeded constantly by fresh matter, which rendered the tail

contiguous to the comet. From this computation the tails are found to

participate of another property of ascending vapours, that, when they

ascend with the greatest velocity, they are least incurvated.

17. THE only objection that can be made against this opinion is the

difficulty of explaining, how a sufficient quantity of vapour can

be raised from the atmosphere of a comet to fill those vast spaces,

through which their tails are sometimes extended. This our author

removes by the following computation: our air being an elastic fluid,

as has been said before[231], is more dense here near the surface of

the earth, where it is pressed upon by the whole air above; than it is

at a distance from the earth, where it has a less weight incumbent. I

have observed, that the density of the air is reciprocally proportional

to the compressing weight. From hence our author computes to what

degree of rarity the air must be expanded, according to this rule, at

an height equal to a semidiameter of the earth: and he finds, that

a globe of such air, as we breath here on the surface of the earth,

which shall be one inch only in diameter, if it were expanded to the

degree of rarity, which the air must have at the height now mentioned,

would fill all the planetary regions even to the very sphere of Saturn,

and far beyond. Now since the air at a greater height will be still

immensly more rarified, and the surface of the atmospheres of comets is

usually about ten times the distance from the center of the comet, as

the surface of the comet it self, and the tails are yet vastly farther

removed from the center of the comet; the vapour, which composes those

tails, may very well be allowed to be so expanded, as that a moderate

quantity of matter may fill all that space, they are seen to take up.

Though indeed the atmospheres of comets being very gross, they will

hardly be rarified in their tails to so great a degree, as our air

under the same circumstances; especially since they may be something

condensed, as well by their gravitation to the sun, as that the parts

will gravitate to one another; which will hereafter be shewn to be the

universal property of all matter[232]. The only scruple left is, how so

much light can be reflected from a vapour so rare, as this computation

implies. For the removal of which our author observes, that the most

refulgent of these tails hardly appear brighter, than a beam of the

sun’s light transmitted into a darkened room through a hole of a single

inch diameter; and that the smallest fixed stars are visible through

them without any sensible diminution of their lustre.

18. ALL these considerations put it beyond doubt, what is the true

nature of the tails of comets. There has indeed nothing been said,

which will account for the irregular figures, in which those tails

are sometimes reported to have appeared; but since none of those

appearances have ever been recorded by astronomers, who on the contrary

ascribe the same likeness to the tails of all comets, our author

with great judgment refers all those to accidental refractions by

intervening clouds, or to parts of the milky way contiguous to the

comets[233].

19. THE discussion of this appearance in comets has led Sir ~ISAAC

NEWTON~ into some speculations relating to their use, which I cannot

but extreamly admire, as representing in the strongest light

imaginable the extensive providence of the great author of nature,

who, besides the furnishing this globe of earth, and without doubt

the rest of the planets, so abundantly with every thing necessary

for the support and continuance of the numerous races of plants and

animals, they are stocked with, has over and above provided a numerous

train of comets, far exceeding the number of the planets, to rectify

continually, and restore their gradual decay, which is our author’s

opinion concerning them[234]. For since the comets are subject to such

unequal degrees of heat, being sometimes burnt with the most intense

degree of it, at other times scarce receiving any sensible influence

from the sun; it can hardly be supposed, they are designed for any

such constant use, as the planets. Now the tails, which they emit,

like all other kinds of vapour, dilate themselves as they ascend, and

by consequence are gradually dispersed and scattered through all the

planetary regions, and thence cannot but be gathered up by the planets,

as they pass through their orbs: for the planets having a power to

cause all bodies to gravitate towards them, as will in the sequel of

this discourse be shewn[235]; these vapours will be drawn in process of

time into this or the other planet, which happens to act strongest upon

them. And by entering the atmospheres of the earth and other planets,

they may well be supposed to contribute to the renovation of the face

of things, in particular to supply the diminution caused in the humid

parts by vegetation and putrefaction. For vegetables are nourished by

moisture, and by putrefaction are turned in great part into dry earth;

and an earthy substance always subsides in fermenting liquors; by which

means the dry parts of the planets must continually increase, and the

fluids diminish, nay in a sufficient length of time be exhausted, if

not supplied by some such means. It is farther our great author’s

opinion, that the most subtile and active parts of our air, upon which

the life of things chiefly depends, is derived to us, and supplied

by the comets. So far are they from portending any hurt or mischief

to us, which the natural fears of men are so apt to suggest from the

appearance of any thing uncommon and astonishing.

20. THAT the tails of comets have some such important use seems

reasonable, if we consider, that those bodies do not send out those

fumes merely by their near approach to the sun; but are framed of a

texture, which disposes them in a particular manner to fume in that

sort: for the earth, without emitting any such steam, is more than half

the year at a less distance from the sun, than the comet of 1664 and

1665 approached it, when nearest; likewise the comets of 1682 and 1683

never approached the sun much above a seventh part nearer than Venus,

and were more than half as far again from the sun as Mercury; yet all

these emitted tails.

21. FROM the very near approach of the comet of 1680 our author draws

another speculation; for if the sun have an atmosphere about it, the

comet mentioned seems to have descended near enough to the sun to

enter within it. If so, it must have been something retarded by the

resistance it would meet with, and consequently in its next descent to

the sun will fall nearer than now; by which means it will meet with a

greater resistance, and be again more retarded. The event of which must

be, that at length it will impinge upon the sun’s surface, and thereby

supply any decrease, which may have happened by so long an emission of

light, or otherwise. And something like this our author conjectures

may be the case of those fixed stars which by an additional increase

of their lustre have for a certain time become visible to us, though

usually they are out of sight. There is indeed a kind of fixed stars,

which appear and disappear at regular and equal intervals: here some

more steady cause must be sought for; perhaps these stars turn round

their own axis’s, as our sun does[236], and have some part of their

body more luminous than the other, whereby they are seen, when the most

lucid part is next to us, and when the darker part is turned toward us,

they vanish out of sight.

22. WHETHER the sun does really diminish, as has been here suggested,

is difficult to prove; yet that it either does so, or that the earth

increases, if not both, is rendered probable from Dr. HALLEY’s

observation[237], that by comparing the proportion, which the

periodical time of the moon bore to that of the sun in former times,

with the proportion between them at present, the moon is found to be

something accelerated in respect of the sun. But if the sun diminish,

the periods of the primary planets will be lengthened; and if the earth

be encreased, the period of the moon will be shortened: as will appear

by the next chapter, wherein it shall be shewn, that the power of the

sun and earth is the result of the same power being lodg’d in all their

parts, and that this principle of producing gravitation in other bodies

is proportional to the solid matter in each body.

~CHAP~. V.

Of the BODIES of the SUN and PLANETS.

OUR author, after having discovered that the celestial motions are

performed by a force extended from the sun and primary planets, follows

this power into the deepest recesses of those bodies themselves, and

proves the same to accompany the smallest particle, of which they are

composed.

2. PREPARATIVE hereto he shews first, that each of the heavenly bodies

attracts the rest, and all bodies, with such different degrees of

force, as that the force of the same attracting body is exerted on

others exactly in proportion to the quantity of matter in the body

attracted[238].

3. OF this the first proof he brings is from experiments made here

upon the earth. The power by which the moon is influenced was above

shewn to be the same, with that power here on the surface of the earth,

which we call gravity[239]. Now one of the effects of the principle

of gravity is, that all bodies descend by this force from the same

height in equal times. Which has been long taken notice of; particular

methods having been invented to shew that the only cause, why some

bodies were observed to fall from the same height sooner than others,

was the resistance of the air. This we have above related[240]; and

proved from hence, that since bodies resist to any change of their

state from rest to motion, or from motion to rest, in proportion to the

quantity of matter contained in them; the power that can move different

quantities of matter equally, must be proportional to the quantity. The

only objection here is, that it can hardly be made certain, whether

this proportion in the effect of gravity on different bodies holds

perfectly exact or not from these experiments; by reason that the

great swiftness, with which bodies fall, prevents our being able to

determine the times of their descent with all the exactness requisite.

Therefore to remedy this inconvenience, our author substitutes another

more certain experiment in the room of these made upon falling bodies.

Pendulums are caused to vibrate by the same principle, as makes

bodies descend; the power of gravity putting them in motion, as well

as the other. But if the ball of any pendulum, of the same length with

another, were more or less attracted in proportion to the quantity of

solid matter in the ball, that pendulum must accordingly move faster or

slower than the other. Now the vibrations of pendulums continue for a

great length of time, and the number of vibrations they make may easily

be determined without suspicion of error; so that this experiment may

be extended to what exactness one pleases: and our author assures us,

that he examined in this way several substances, as gold, silver, lead,

glass, sand, common salt, wood, water, and wheat; in all which he found

not the least deviation from the proportion mentioned, though he made

the experiment in such a manner, that in bodies of the same weight a

difference in the quantity of their matter less than a thousandth part

of the whole would have discovered it self[241]. It appears therefore,

that all bodies are made to descend by the power of gravity here, near

the surface of the earth, with the same degree of swiftness. We have

above observed this descent to be after the rate of 16⅛ feet in the

first second of time from the beginning of their fall. Moreover it

was also observed, that if any body, which fell here at the surface

of the earth after this rate, were to be conveyed up to the height of

the moon, it would descend from thence just with the same degree of

velocity, as that with which the moon is attracted toward the earth;

and therefore the power of the earth upon the moon bears the same

proportion to the power it would have upon those bodies at the same

distance, as the quantity of matter in the moon bears to the quantity

in those bodies.

4. THUS the assertion laid down is proved in the earth, that the power

of the earth on every body it attracts is, at the same distance from

the earth, proportional to the quantity of solid matter in the body

acted on. As to the sun, it has been shewn, that the power of the sun’s

action upon the same primary planet is reciprocally in the duplicate

proportion of the distance; and that the power of the sun decreases

throughout in the same proportion, the motion of comets traversing the

whole planetary region testifies. This proves, that if any planet were

removed from the sun to any other distance whatever, the degree of

its acceleration toward the sun would yet remain reciprocally in the

duplicate proportion of its distance. But it has likewise been shewn,

that the degree of acceleration, which the sun gives to every one of

the planets, is reciprocally in the duplicate proportion of their

respective distances. All which compared together puts it out of doubt,

that the power of the sun upon any planet, removed into the place of

any ether, would give it the same velocity of descent, as it gives that

other; and consequently, that the sun’s action upon different planets

at the same distance would be proportional to the quantity of matter

in each. It has farther been shewn, that the sun attracts the primary

planets, and their respective secondary, when at the same distance, so

as to communicate to both the same degree of velocity; and therefore

the force, wherewith the sun acts on the secondary planet, bears

the same proportion to the force, wherewith at the same distance it

attracts the primary, as the quantity of solid matter in the secondary

planet bears to the quantity of matter in the primary.

5. THIS property therefore is proved of both kinds of planets, in

respect of the sun. Therefore the sun possesses the quality found in

the earth, of acting on bodies with a degree of force proportional to

the quantity of matter in the body, which receives the influence.

6. THAT the power of attraction, with which the other planets are

endued, should differ from that of the earth, can hardly be supposed,

if we consider the similitude between those bodies; and that it does

not in this respect, is farther proved from the satellites of Saturn

and Jupiter, which are attracted by their respective primary according

to the same law, that is, in the same proportion to their distances, as

the primary are attracted by the sun: so that what has been concluded

of the sun in relation to the primary planets, may be justly concluded

of these primary in respect of their secondary, and in consequence

of that, in regard likewise to all other bodies, viz. that they will

attract every body in proportion to the quantity of solid matter it

contains.

7. HENCE it follows, that this attraction extends itself to every

particle of matter in the attracted body: and that no portion of matter

whatever is exempted from the influence of those bodies, to which we

have proved this attractive power to belong.

8. BEFORE we proceed farther, we may here remark, that this attractive

power both of the sun and planets now appears to be quite of the same

nature in all; for it acts in each in the same proportion to the

distance, and in the same manner acts alike upon every particle of

matter. This power therefore in the sun and other planets is not of a

different nature from this power in the earth; which has been already

shewn to be the same with that, which we call gravity[242].

9. AND this lays open the way to prove, that the attracting power

lodged in the sun and planets, belongs likewise to every part of them:

and that their respective powers upon the same body are proportional to

the quantity of matter, of which they are composed; for instance, that

the force with which the earth attracts the moon, is to the force, with

which the sun would attract it at the same distance, as the quantity of

solid matter contained in the earth, to the quantity contained in the

sun[243].

10. THE first of these assertions is a very evident consequence from

the latter. And before we proceed to the proof, it must first be

shewn, that the third law of motion, which makes action and reaction

equal, holds in these attractive powers. The most remarkable attractive

force, next to the power of gravity, is that, by which the loadstone

attracts iron. Now if a loadstone were laid upon water, and supported

by some proper substance, as wood or cork, so that it might swim;

and if a piece of iron were caused to swim upon the water in like

manner: as soon as the loadstone begins to attract the iron, the iron

shall move toward the stone, and the stone shall also move toward

the iron; when they meet, they shall stop each other, and remain

fixed together without any motion. This shews, that the velocities,

wherewith they meet, are reciprocally proportional to the quantities

of solid matter in each; and that by the stone’s attracting the iron,

the stone itself receives as much motion, in the strict philosophic

sense of that word[244], as it communicates to the iron: for it has

been declared above to be an effect of the percussion of two bodies,

that if they meet with velocities reciprocally proportional to the

respective bodies, they shall be stopped by the concourse, unless their

elasticity put them into fresh motion; but if they meet with any other

velocities, they shall retain some motion after meeting[245]. Amber,

glass, sealing-wax, and many other substances acquire by rubbing a

power, which from its having been remarkable, particularly in amber,

is called electrical. By this power they will for some time after

rubbing attract light bodies, that shall be brought within the sphere

of their activity. On the other hand Mr. BOYLE found, that if a piece

of amber be hung in a perpendicular position by a string, it shall be

drawn itself toward the body whereon it was rubbed, if that body be

brought near it. Both in the loadstone and in electrical bodies we

usually ascribe the power to the particular body, whose presence we

find necessary for producing the effect. The loadstone and any piece of

iron will draw each other, but in two pieces of iron no such effect is

ordinarily observed; therefore we call this attractive power the power

of the loadstone: though near a loadstone two pieces of iron will also

draw each other. In like manner the rubbing of amber, glass, or any

such body, till it is grown warm, being necessary to cause any action

between those bodies and other substances, we ascribe the electrical

power to those bodies. But in all these cases if we would speak more

correctly, and not extend the sense of our expressions beyond what

we see; we can only say that the neighbourhood of a loadstone and a

piece of iron is attended with a power, whereby the loadstone and

the iron are drawn toward each other; and the rubbing of electrical

bodies gives rise to a power, whereby those bodies and other substances

are mutually attracted. Thus we must also understand in the power of

gravity, that the two bodies are mutually made to approach by the

action of that power. When the sun draws any planet, that planet also

draws the sun; and the motion, which the planet receives from the

sun, bears the same proportion to the motion, which the sun it self

receives, as the quantity of solid matter in the sun bears to the

quantity of solid matter in the planet. Hitherto, for brevity sake

in speaking of these forces, we have generally ascribed them to the

body, which is least moved; as when we called the power, which exerts

itself between the sun and any planet, the attractive power of the sun;

but to speak more correctly, we should rather call this power in any

case the force, which acts between the sun and earth, between the sun

and Jupiter, between the earth and moon, &c. for both the bodies are

moved by the power acting between them, in the same manner, as when

two bodies are tied together by a rope, if that rope shrink by being

wet, or otherwise, and thereby cause the bodies to approach, by drawing

both, it will communicate to both the same degree of motion, and cause

them to approach with velocities reciprocally proportional to the

respective bodies. From this mutual action between the sun and planet

it follows, as has been observed above[246], that the sun and planet do

each move about their common center of gravity. Let A (in fig. 108.)

represent the sun, B a planet, C their common center of gravity. If

these bodies were once at rest, by their mutual attraction they would

directly approach each other with such velocities, that their common

center of gravity would remain at rest, and the two bodies would at

length meet in that point. If the planet B were to receive an impulse,

as in the direction of the line D E, this would prevent the two bodies

from falling together; but their common center of gravity would be

put into motion in the direction of the line C F equidistant from B E.

In this case Sir ~ISAAC NEWTON~ proves[247], that the sun and planet

would describe round their common center of gravity similar orbits,

while that center would proceed with an uniform motion in the line C

F; and so the system of the two bodies would move on with the center

of gravity without end. In order to keep the system in the same place,

it is necessary, that when the planet received its impulse in the

direction B E, the sun should also receive such an impulse the contrary

way, as might keep the center of gravity C without motion; for if these

began once to move without giving any motion to their common center of

gravity, that center would always remain fixed.

11. BY this may be understood in what manner the action between the sun

and planets is mutual. But farther, we have shewn above[248], that the

power, which acts between the sun and primary planets, is altogether of

the same nature with that, which acts between the earth and the bodies

at its surface, or between the earth and its parts, and with that which

acts between the primary planets and their secondary; therefore all

these actions must be ascribed to the same cause[249]. Again, it has

been already proved, that in different planets the force of the sun’s

action upon each at the same distance would be proportional to the

quantity of solid matter in the planet[250]; therefore the reaction

of each planet on the sun at the same distance, or the motion, which

the sun would receive from each planet, would also be proportional

to the quantity of matter in the planet; that is, these planets at

the same distance would act on the same body with degrees of strength

proportional to the quantity of solid matter in each.

[Illustration]