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But the proofs, derived from the same sources, are not so strict as we find them in other parts of his Work. Those of the eleventh Section were not even proposed, as their Author himself informs us*, as perfect ones. They do not establish (and this is a point more than once insisted on) the Law of Gravity to be exactly according to the inverse square of the distance. Most of the reasonings of the eleventh Section hold good, if the Law, as Clairaut proposed in a memorable instance (see Mem. Acad. 1745.) instead of being as it is, should only be nearly so, and should be expressed by a formula of two terms.

After explaining for what reasons, founded on the Theory of Gravity, the Moon would deviate from the laws of elliptical motion, Newton proceeded to other proofs of that theory: or rather to the explanations of other phenomena which it afforded. The inequalities which he had traced to the Sun's disturbing force, as their source, were periodical: admitting of alternate increase and decrease, and, at the completions of their periods, returning to those values or magnitudes which they had at the beginnings. But the Moon was subject to other inequalities; such as affected not her position in the orbit, but the dimensions of that orbit and its position in space. To the same source, the Sun's disturbing force (which, as it has been often said, itself springs from the principle of gravity) he traced these latter inequalities. On just principles, and, completely, he explained the motion of the Nodes of the Lunar Orbit, their mean Regression, and the change of the orbit's inclination; but imperfectly, since

‘Quæ ad motus Lunares spectant (imperfecta cum sint) in Corollariis Propositionis LXVI, simul complexus sum, ne singula methodo prolixiore quam pro rei dignitate proponere et sigillatim demonstrare tenerer, et seriem reliquarum propositionum interrumpere.' Auctoris Præfatio.

† It seems remarkable that Newton should have forborn announcing in distinct terms that his Theory afforded an explanation of those principal Lunar inequalities which Astronomers had noted and distinguished by technical denominations. Such an announcing would have drawn the attention of the curious most forcibly to his Theory.

on partial principles, the Progression of the apogee. After having explained the Regression of the Nodes he beautifully applies the result of such explanation to the precession of the equinoxes: and then passes on to shew in what manner the phenomena of the Tides are, on the Theory of Gravity, explicable from their causes, which are the attractive, or disturbing forces, of the Sun and Moon.

All these phenomena, the tides, the precession of the equinoxes, the periodical inequalities of the Moon, the inequalities of the elements of its orbit (for the inclination, the eccentricity, the positions of the node and apogee, are technically so called) are familiarly explained (methodo populari) in the eleventh Section. The succeeding Section is occupied with different investigations.

But, although in kind different, they like those of the Lunar inequalities, date their rise from the same source, the principle of Gravitation. According to that principle not only all the bodies of the system, the Sun and planets, separately attract as masses, but the component parts or particles of each mass. Now in some of the Sections previous to the twelfth, which treat of Kepler's Laws and the deviations from those laws caused by disturbing forces, the central, revolving and disturbing bodies are considered as merely physical points. No account is made of their figure. But such are not the conditions in nature. The Earth, which, as the central body, attracts the Moon, and the Sun which as the third and external body disturbs the Moon (disturbs it by attracting it more or less than it does the Earth) are both endowed with volume and figure; and then comes this question; if each particle of the Earth attracts the Moon, as it must do according to the Principle of Gravitation, to what point in the Earth will the result of the several attractions be directed? If it should be the centre of the Earth, then what had been demonstrated in the preceding Sections relative to Kepler's Laws would still hold good. But if not, then the previous results, in order to be adapted to the real circumstances of nature, would, at the least, require some correction.

To this subject of enquiry Newton directs his attention in the twelfth Section. He therein demonstrates that, the law of attrac

tion being the inverse square of the distance, spheres attract just as if all their matter were condensed into their centres. Accordingly, if the planets were spherical bodies, the results which had been proved to belong to merely central points might be transferred to them; and they may be transferred with very little inaccuracy, since the planets, although not exactly spherical bodies, are nearly so.


In truth, however, spherical planets, like elliptical orbits, belong to an ideal system and have no place in nature. orbits are only nearly elliptical and their figures nearly spherical. If we assume the former to be ellipses and the latter spheres, it is for the purpose of conveniently commencing our processes: of making, by such first steps, approximations towards remoter results.

But it must not be supposed, from what has just been said, that the inequalities of motion of the revolving body caused by the oblateness of figure in the central are at all comparable, either for their magnitude or number, fo those that are caused by an external disturbing body. The Lunar inequality caused by the small ellipticity of the terrestrial spheroid may be corrected by one equation, the maximum of which does not exceed seven seconds: whereas there are nearly thirty equations which it is necessary to make account of, in correcting the deviation from elliptical motion caused by the Sun's disturbing force.

The result, therefore, of the seventy-first Proposition (Sect. 12. Book I.) is applicable, with very little error, to the Lunar theory. The inequality that would, virtually, be neglected by such application is less, as we have seen, than seven seconds; a quantity less than the error of those observations which were made in Newton's time.

By theory the oblateness of the Earth causes an inequality in the Moon's motion: and, consequently, which is very curious, we may from the Moon's motion determine the ellipticity or the degree of the Earth's oblateness.

We have now enumerated and slightly described the series of explanations which Newton, by means of his Theory of Gravity, has given of the planetary phenomena; which explanations, under another point of view, are so many proofs of the truths of that theory. They are all to be found in the Principia. But their matter, although by far its most important constituent, occupies less than the half of that extraordinary Work. If it were permitted not to approve of every thing that Newton has done, we might regret that the great argument, by which the Principle and Law of Gravitation are established, should be so mixed up and interrupted, as it is, with foreign matter; more foreign now, indeed, than it then seemed to be, when other systems, hardly out of vogue, were to be put aside or asunder in order to make room for a new one. Still the business of refutation and other matters interrupted the main course of argument. The progress of Physical Astronomy was impeded by the very abundance of the riches of the Principia. Had its contents not much exceeded or not much varied from what Newton originally communicated to the Royal Society: or if it had contained part of the second Section, the third, seventh, ninth, eleventh, twelfth Sections of the first Book and the third Book, it would probably have had more numerous readers.

But be this as it may. Still the Principia is the most prodigious Work in mathematical Philosophy that was ever produced t. To estimate its merit we must view Science as its Author found and as he left it. He did not merely add to or beautify a system. Newton's merit was more than that of having left marble what he found brick. For he laid the very foundations of Physical Astronomy, and furnished the materials and the means of putting them together.

Quippe cum demonstratam a me figuram orbium cœlestium impetraverat, rogare non destitit ut eandem cum Societate Regali communicarem.' Auctoris Præfatio.

+ And it may be justly said, that so many and so valuable phi losophical truths, as are herein discovered and put past dispute, were never yet owing to the capacity and industry of any one man.' Halley's notice of the publication of the Principia: see Phil. Trans. 1687, No. 186.

The Author of the Principia, although he lived in an age rich in men of genius, far outstepped his contemporaries: and from this circumstance, or the abstruse matter of his Work, or its manner, it happened that nearly sixty years elapsed from the first publication of the Principia before any material researches, on its principles, were made in Physical Astronomy. Clairaut began to make such about the year 1743. And then so limited seems to have been the study of Physical Astronomy, that the Author of the Histoire* prefixed to that volume of the Paris Acts which contains Clairaut's Memoir, thinks it expedient to illustrate its subject, and to explain, by instances, the most easy and familiar, after what manner Newton conceived a planet to describe an elliptical orbit round the Sun.

The researches of Clairaut, of which we have just spoken, were given in the Memoirs of the Academy of Sciences at Paris for the year 1743. 1743. And the Memoir was entitled ' De l'Orbite de la Lune dans le Systeme Newtonien. We may discern in it the ground-work of that method which the Author and other foreign mathematicians afterwards used in their researches in Physical Astronomy. The method is widely different from Newton's; and the object of research is shortly stated to consist in the solution of the Problem of the Three Bodies.

By means of the opposite diagram we may easily explain the conditions and object of that problem.

When L revolves round T and is solely acted on by a centripetal force, directed from L towards T as a centre, the curve described by L and the laws of its motion are the objects of solution in the Problem of Two Bodies; and, as we have seen, Kepler's Laws are strictly observed in such system.

A third body placed at S, but not so distant as to attract T and L equally and in parallel directions, prevents the above laws from being observed: it disturbs what would take place were it

*The Histoire contains the abstract and brief explanation of the subjects treated of in the Memoirs and body of the Work.

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