operations and laws of gravity to be executed, we cannot but ultimately refer them to his immediate care.

Men, in general, know nothing of this interesting power. They consider the sun as dispensing merely light and heat. They perceive that our earth, without its benign influence, would be a dark lump of matter, barren and desolate. Few know that, besides the effects of light and heat, we derive from the sun another source of preservation. Solar gravity is as essential to our welfare as light and heat. The effects of the latter, indeed, we feel instantaneously, but the former is not perceptible by our senses. It silently and incessantly operates in preserving to us all that gladdens our existence here. Nor would its suspension be immediately observed. It might for weeks cease* to act without being noticed by the mass of mankind. But a continued suspension would inevitably be followed by a complete annihilation of the human race, and that under circumstances the most deplorable that the mind of man can imagine.

Mr. Woodhouse in the first chapter deduces the differential equations of motion caused by the action of accelerating and centripetal forces. Here and in a few of the following chapters he uses the illustrations afforded by geometrical figures; this, although not necessary, must be advantageous to the student; it serves to connect, as it were, the learning he brings with him from the Principia' of Newton with that which he is to attain by studying this and other treatises on physical astronomy. We would wish the Principia' to be always made the ground-work on which this science is to be built.

It has been objected to the modern application of mathematics to physics that the want of seeing every distinct step renders it much inferior to processes by geometrical reasoning and geometrical schemes. This is not just. The art of analysis has been so improved, that we can use it with unerring certainty to deduce conclusions dependent upon so many steps that the mind could not possibly embrace them in the manner it embraces the steps of geometrical reasoning. If we object to this, as well might we object to the machine of the mechanist, that produces with speed and unerring exactness the most difficult work, because each part is not successively subjected to the inspection of the eye and the slow operation of the hand. Newton, than whom no one knew better

It may not be uninteresting to consider the result of calculation respecting the effect of a suspension of solar gravity for a few months. Suppose this suspension to commence in the height of our summer, and to last for three months only; the effect would be to make our year about thirty times as long as at present. The first apparent effect would be to protract the duration of summer. During the actual suspension we might appear rather benefited; but the resumed action of solar gravity would not remedy the mischief in store. The consequences of which need not to be stated.

ог

or extended farther the powers of analysis, preferred, it is true, to exhibit his investigations and conclusions in a geometric rather than in an analytic form: but the former is better adapted to the outlines of physical astronomy than the latter; and it was also better adapted to call the attention of the world to his great discoveries. When it was attempted to extend them, the resources of the analytic method were found absolutely necessary. Lagrange, speaking of his Mécanique Analytique,' says, ' On ne trouvera point de figures dans cet ouvrage. Les méthodes que j'y expose ne demandent ni constructions, ni raisonnemens géométriques ou mécaniques, mais seulement des opérations algébraïques, assujéties à une marche régulière et uniforme.'

In this first chapter Mr. Woodhouse gives the differential equations of motion relative to three rectangular co-ordinates, and also relative to the longitude, latitude, and projected radius vector. After which he remarks,

'If we were immediately to press forward to those most commodious and perfect forms, which the ingenuity and labour of mathematicians have given to the differential equations of motion, we should conduct the student, in the outset of his career, over too extended a field of apparently barren speculation. It is better to stop for a while and endeavour to collect some useful truths.'

In the 2d, 3d, and 4th chapters he investigates the consequences that follow from these equations when applied to two bodies acting on each other, and deduces the principal phenomena of the planetary system. They appear to answer admirably the purpose for which the author doubtless intended them, to familiarise the student with the uses of fluxions in physical astronomy, and thus to prepare him for the more difficult computation of the effects of the perturbations. We have nothing particular to remark except that in page 37, it is said, 'I must now resume the three differential equations which are,' &c.; now we have not been able to find any previous statement of these equations, and therefore imagine that, by some oversight, the author omitted them.

The fifth chapter prepares for the solution of the problem of the three bodies. It principally consists in the valuation of the perturbating force of one body on the motions of the other two. Mr. Woodhouse here deduces, with much perspicuity, the expres sions of the perturbating forces by means of partial differential coefficients of a function of the distance, longitude, and latitude. This may, at first, appear to the reader an unnecessary refinement, and one that occasions difficulty and delay in his progress. But he will afterwards discover its importance.

In the seventh chapter we find the commencement of the solution of the problem of the three bodies. With reference to na

ture, the approximate solution, (and this only is within our reach,) has two cases. One refers to the sun, moon and earth, and the problem to be resolved is the investigation of the motions of the moon, as seen from the earth. The other case refers to the sun and two planets, and the problem to be resolved is the investigation of the effects of the one planet on the motion of the other revolving about the sun. The perturbation of the motion of one planet by another is very small indeed, compared with the perturbation of the motions of the moon by the sun. The former can only be discovered by the exactest observations with the best instruments, or after the lapse of a long interval of time. But the perturbations of the motions of the moon are very considerable, and require to be expressed by many different equations, as they are called. Some of them were discovered in the infancy of astronomy, and long before any explanation of their causes could be given. On these accounts, we think, that Mr. Woodhouse has judiciously made the investigation of the lunar precede that of the planetary irregularities.

It would exceed our limits to remark very diffusely on the steps by which Mr. Woodhouse proceeds. They are of such a nature that they will enable the student to follow him with ease.

The principal point in the investigations of Clairaut and of all subsequent authors is the integration of the equation,

where p (u, v) designates a function of u and v, originating solely from the perturbating force: u is the reciprocal of the moon's distance from the earth, and v is the longitude of the moon, considering for simplicity the orbit of the moon coincident with the ecliptic.

Mr. Woodhouse follows Clairaut's method, which is certainly best adapted to a student. For the interesting circumstances connected with Clairaut's first attempts, we must refer to the volume itself. We shall only add a few brief observations.

The integration of the above equation deprived of its last term,

Authors in general do not seem aware, that Euler had arrived at and integrated a similar equation, long before Clairant commenced his investigations on the lunar theory. It occurs in Euler's Essay on the Tides, section 77. The mode in which he integrates it is deserving of notice, as containing the germ of the method of making the constant arbitrary quantities vary. Perhaps a richer page cannot be found in the works of mathematicians than that of Euler here referred to. In it he first denotes the sine of an arc by sin. z. It also seems worthy of notice, that Euler avails himself of the integration of the equation dp+pZd: S dz, in which Z and S are functions of x; which equation had been integrated by James Bernoulli not long after the publication of the Principia. Thus the progress of physical astronomy was not impeded by the state of analysis.

or

or the simple elliptic theory, gives, as is well known, u=(1+e cos. v), e being the eccentricity. By substituting this value of u, in (u, v) of equation (1), we find that (u, v) may be expressed by the form A cos. mv + B cos. pv + &c. This substitution being made, the integration of equation (1) is easily performed, excepting the case of m or p=1. In that case the integration introduces arcs of circles, and the orbit resulting would be entirely different from the orbit of the moon. This difficulty must have soon occurred to Clairaut, and he obviated it by assuming u=(1+ecos. cv) a form either suggested to him by the ninth section of the Principia,' or by the actual motion of the lunar apogee.

In consequence of this substitution, almost the next step furnished, as he conceived, a test of the Newtonian theory of gravity. The integrations of the resulting equation furnished a new value of u, which compared with the substituted value afforded an equation for determining c, and 1-c would express the mean motion of the apogee, that of the moon being unity. To determine c, the coefficients of cos. cv are equated. Now it will easily appear by a reference to the coefficient of cos. co in the value of u deduced from integration, that the tangential force does not enter into it, and that therefore it is precisely the same as if Clairaut bad considered only the perturbating force in the direction of the radius vector, and consequently the result ought to be the same as Newton had before found from the consideration of a perturbating force in the direction of the radius vector only. The mean motion of the apsid appeared to be only half of that shewn by observation. Mr. Woodhouse, after his account of the failure of Clairaut, remarks—

This is a brief notice and description of that notorious error, which, on its first appearance, caused (if we may so express ourselves) so great a sensation in the mathematical world. In one of the most remarkable of the heavenly phenomena, the progressions of the aphelia of the planetary orbits, theory and calculation were erroneous to the amount of half the real quantity. So erroneous a defalcation seemed to portend to Newton's system, that fate which, not long before, Descartes's had experienced.'-p. 147.

But it appears to us that Clairaut rather precipitately called in question the accuracy of the Newtonian law of gravity. He, as we have observed above, could, from the nature of his process, find the same result only as Newton had found. Newton himself had not from thence deduced an argument against the accuracy of the law of gravity, although the difficulty must have appeared in full force to him.

Clairaut's method of treating the subject, however, had greatly the advantage over that of Newton, in affording an easy solution of the difficulty, one simple in the outline of the process, although tedious in the detail. The first approximate value of u, from the integration of equation (1), afforded among its terms one of the form A e cos. (v—2 mv), or rather of the form A e cos. (2 v—2 mo -cv), in which the coefficient A was much greater than in any of other new terms, depending on the perturbating force. He might have remarked, that on this term principally depended the variation of the eccentricity of the lunar orbit. Newton had shewn, Cor. 8. 9. Prop. 66. 1 Lib. Princip. how much the variation of the eccentricity and motion of the apogee were connected. Consequently Clairaut might, à priori, have concluded, that the substitution of the value of u, containing this new term, in (u, v), would sensibly affect the resulting motion of the apogee. He was fortunate, after having excited considerable notice, by the annunciation of the failure of the Newtonian law of gravity, to be the first to correct his own error.

Mr. Woodhouse refers also to the methods used by D'Alembert, Simpson, and Laplace.

'Laplace, in his Mécanique Céleste, (tom. iii. pp. 191, &c.) although in the main he follows D'Alembert's suggested method, yet follows it not so closely as Simpson has done. He first, on the assumption of the elliptical value of u, deduces the values of the coefficients of the terms of the differential equation, and expresses them by means of the quantities, m, e, é, c, &c.'

It is not, we conceive, strictly correct to say that Simpson followed D'Alembert's' suggested method.' He states* himself that what he did was done in 1750, and he refers to an intimation at the conclusion of his Fluxions published in that year. D'Alembert's suggested method' appeared in 1754. We would willingly give the priority to Simpson, but we are not certain that we should be justified in so doing. With respect to Laplace, however, the passage above is certainly incorrect; and we think Laplace himself, by the inaccuracy of his expression, contributed to lead our author into the mistake. It is not the elliptical value of u that Laplace first assumes. He says indeed,+Supposons que du soit la partie de u, due à la force perturbatrice,' &c. &c.

1

The value of u then becomes h2(1+y3) { 1 + ecos. (cv—π)+&c } +ô u.

The former part of this value depends upon the perturbating force, as well as the latter, &u. The quantities h, e, &c. are modified by the perturbating force, and c differs from unity on account of the perturbating force.

Preface to his Tracts, 1757.

† Mécanique Céleste, tom. iii. p. 200.

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