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of Mars's, then at equal distances from the Sun they would be urged towards him by equal forces: for such equality of force is an essential condition of Newton's fifteenth Proposition of the third Section.

Now if, at equal distances from the Sun, Mars, Jupiter, Venus, &c. were urged towards him by equal forces, or if they then gravitated equally to him, it was no very improbable supposition that these forces arose from some attraction in the Sun. It, at the least, involved no contradiction, to suppose that the matter or particles of the Sun caused the gravitation of the planets towards him. The mode by which this was effected formed no part of such supposition, nor was, in any sort, implied by it.

But the supposition that the matter of the Sun caused the gravitation of the planets, if it involved no contradiction, would naturally give rise to conjecture and enquiry. If the Earth and Jupiter gravitated to the Sun by virtue of his attracting particles, would not Jupiter's satellites towards Jupiter, and the Moon towards the Earth, gravitate from like particles resident in Jupiter and the Earth? and might not these gravitations, at equal distances, be in proportion, respectively, to the number of particles, or the masses of the central bodies? At this point of Newton's research, if we were permitted to feign its theoretical history, might be supposed to have arisen the momentous question concerning Universal Gravitation: a question of great extent, and not admitting of any summary determination.

Not summarily to be decided on, except its connected theory were false in that case one impugning instance would overthrow the theory but if true, a thousand instances would only tend to establish it. The proof of the truth of Newton's Theory is only the accumulation of individual arguments, derived from various instances, and all conspiring and the first in the series of arguments, to prove that all bodies gravitated, the one towards the other, was derived from the Moon's Gravitation.

The drift of this first argument was to shew that the descent of a heavy body near the Earth's surface, and the deflection of the Moon

was not such The relation But the dif

from the tangent of her orbit, were like effects, or of the same class; or, which would make the analogy closer, that the latter deflection, and the deflection from the tangent of a parabola described by a heavy body projected near the Earth's surface were like effects. The criterion of their being like effects consists in their obeying the Law of Gravity: the two deflections, therefore, ought to bear to each other that numerical relation which subsists between the squares of the Earth's radius, and of the radius of the Moon's orbit. Now these latter quantities were inaccurately known at the beginning of Newton's researches: our great philosopher, therefore, in the first instance, found that the relation between the two deflections, or between the sagitta of an arc of the Moon's orbit and the space described (in the same time as the arc) by a body near the Earth's surface, as it ought to be, were the Law of Gravity true. was nearly, but not exactly, according to that law. ference was quite sufficient to make Newton suspend his decision on the truth of the law. Some years afterwards, however, the dimensions of the Earth being determined by Picart more accurately than they were before, Newton resumed his investigation, (such as we find it in the fourth Proposition of the third Book of the Principia), and found from them that the Moon gravitated. The signification of that expression has been already explained: if it required farther illustration, we might say, that, a heavy body removed to the Moon's orbit and suffered to fall, would, in a second of time, fall through a space equal to the sagitta of an arc of the Moon's orbit described in the same time; or, that the Moon brought down to the vicinity of the Earth, and a body there projected and describing a parabola would (the resistance of the air being supposed to be abstracted) be equally deflected from the tangents of their curves; the deflection being about sixteen feet.

The Proposition of the Principia to which we have referred is an easy instance of the application of the mathematical results, obtained by Newton in the preceding books, to the system of the universe. If PQ be an arc of the Moon's orbit described in 1", then,

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is the value of the sagitta, or of the deflection of the Moon from

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the tangent of her orbit, or (by the second law of motion) of the space through which the Moon, or a heavy body at the Moon's distance, would fall in the same time. Now, according to the Law of Gravity, a space corresponding to RQ at the Earth's surface, would equal

RQ× (CP)3, or (see Astron. p. 95.)

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responding to RQ, at the Earth's surface, is equal

2.'s radius X (3.14159)

(D's parallax)3. (D's period)

which, by computation is nearly equal to the space fallen through by a heavy body at the Earth's surface.

This result most simply and clearly illustrated Newton's Theory. The deflection of the Moon, from the tangent of her

* Nearly equal; for, the process needs several corrections.

First, the space RQ, the deflection of the Moon from the tangent, does not expound the whole effect of the Earth's attraction: for, by reason of the Sun's disturbing force, the Moon's gravity is diminished and by about its th part; consequently, instead of R2, R2×

358

(1+
(1 +353),

expounds the Earth's attraction, and the corresponding space at the Earth's surface (see p. 24. l. 9.) ought to be

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Secondly, the deflection, or the descent of the Moon in a given time, towards the Earth (whether or not we consider the Sun's disturbing force) does not arise solely from the Earth's attraction, but from the joint attractions of the Earth and Moon. For, according to the Principle of Gravity, every particle of matter attracts; the particles of the Moon, therefore, as well as those of the Earth. The approach, therefore, of the Moon to the Earth arises from their mutual action, and, consequently, that part of the approach which is due solely to the Earth is less than the whole in the proportion that the Earth's mass () is less than the sum of the masses of the Earth and Moon (+ D): and, accordingly, the computed descent at the Earth's surface arising solely from the Earth's attraction, is now

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This space, in order that Newton's Principle may be proved to be true, ought to equal the descent of a heavy body (ascertained by means of the pendulum) in one second of time; and it is to be observed, that, in strictness of principle, although they are far too minute to affect the computation, the two last corrections apply to the descent of a heavy body near the Earth's surface, or to its deflection from the tangent to a parabola: for such descent and deflection must be less (the question

orbit, towards the Earth, and the fall of a body near the Earth's surface were shewn to be like effects. The mode of producing those effects formed no part of the enquiry; but, without absurdity, or the obtrusion of a theory, they might be said to proceed from the same cause, namely, the Earth's attraction. 'Vis quâ Luna,' (says Newton in that remarkable Proposition* that has been just quoted) in suo orbe retinetur, illa ipsa est quam nos gravitatem dicere solemus.'

This important point of the Moon's Gravitation being gained, there was opened to Newton an immense field for the farther

is not about the degree) than it would be were the Sun away; and must be, in part, attributable to the attraction of the heavy body.

The third correction applies to the Moon's parallax. The parallax ought to be that which (see Astronomy, p. 315.) is called the constant, and which is the angle at the Moon subtended by that radius of the Earth which is drawn from the centre to a parallel, the square of the sine of which is. In such latitude the centrifugal force = 23

2 gravity
3. 288

cen-.

trifugal force at the equator = The descent (s), therefore, of a heavy body in this latitude does not expound the whole

S 432

does consequently, for the purpose of

effect of gravity: but s + verifying Newton's Theory, or, more correctly, in order to prove that the Moon gravitates, this equality ought to subsist.

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The corrections are almost as curious and important as the theory itself.

* We cannot, even at this distance of time, view without interest and anxiety, the momentous trial and test to which Newton thus subjected his system. Had that equality, which is stated at the end of the last Note, been found not to subsist, the System of Gravitation would have been as baseless as the Vortices of Descartes. We should have had no Celestial Mechanics. The Principia would have been reduced to its second Book: and Newton must then have gone down to posterity as an extraordinary man for his discoveries in Optics and pure Mathematics.

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