But this is a subject which I propose to press upon the world; -to ask the institutions of my own, and of foreign countries, to reinvestigate; and that I shall, eventually, have supporters, who will dare to risk their reputation by venturing to think for themselves, I have no doubt.

But to refer to some of the difficulties under which the mathematicians and astronomers of Europe have labored, in respect to the law of gravity, as promulgated by Newton, I quote from Mr. Vince, who though a very honest recorder of scientific discoveries, is nevertheless, quite partial to Sir Isaac Newton.

Mr. Vince, in the introductory remarks to his treatise upon the Theory of the Moon, says, "To enter into a computation of all the effects of the disturbing force of the sun upon the moon, and their nature, would require the investigation of the nature of a curve described by a body attracted to two points, called the Problem of three Bodies;" which problem, he suggests, has been solved by but very few mathematicians, amongst whom are Clairaut and some few others. And it appears, that for more than half a century from and after the time that Sir Isaac Newton promulgated his theory and law of gravity, the learned world essayed to apply them to the motions of the moon. Mathematicians were exceedingly perplexed with Sir Isaac Newton's law of gravity; for Mr. Vince proceeds thus: "In the year 1747, M. Clairaut, in a memoir read before the Academy of Sciences, in Paris, made an objection to this law, upon this ground; that it would not account for the motion of the moon's apogee, it giving, according to his calculations, that motion only one half of what it was found to be by observation; and he concluded, that it was necessary to change this law, by adding something to correct it. He however soon after discovered his mistake, and was the first who gave a complete theory of the moon, in which he showed that Sir Isaac Newton's law of gravity would not only account for the motion of the moon's apogee, but also for all the irregularities of the moon," and Mr. Vince proceeds to say, that, "M. Euler has done great justice to M. Clairaut upon his solution of this important problem, in a letter to Rev. Casper Witstien; in which he observes, that, 'This question is of the last importance; and I must own, that till now, I always believed that this theory did not agree with the motion of the apogee of the moon.' M. Clairaut was of the same opinion; but he has publicly retracted it, by declaring that the motion of the apogee is not contrary to the Newtonian theory. Upon this occasion, I have renewed my inquiries on this affair; and after most tedious calculations, I have

at length found to my satisfaction, that M. Clairaut was in the right, and that this theory is entirely sufficient to explain the motion of the apogee of the moon. As this inquiry is of the greatest difficulty, and those who hitherto pretended to have proved this nice agreement of the theory with the truth, have been much deceived, it is to M. Clairaut that we are obliged for this important discovery, which gives quite a new lustre to the theory of the great Newton; and it is but now, that we can expect good astronomical tables of the moon."

And in accordance with the foregoing suggestions of M. Euler, we find from Mr. Vince that various had been the attempts to reconcile the Newtonian law of gravity with fact and observation in respect to the motion of the moon's apogee; for Mr. Vince says that " M. Walmsley, in his theory of the Motion of the Apsides, has computed the mean motion of the moon's apogee, and his conclusion agrees very well with observation; but his principles are altogether wrong; for he has entirely omitted that part of the force which acts in a direction perpendicular to the radius, which, as we have shown, produces just one half of the motion; he also assumes the mean disturbing force in the direction of the radius as acting constantly, instead of the real disturbing force; and he also wrongly computed the periodic time of the moon; it was by accident therefore that he obtained the mean motion; in respect to the true motion his conclusions are erroneous; he says Mr. Machin has not given us his process; we cannot therefore say how far it was just. He says also, in the Phil. Trans. of 1751, that Mr. P. Murdock has given a method of computing the mean motion of the moon's apogee, by first considering only that part of the disturbing force which acts in the direction of the radius; and then instead of supposing the earth to be at rest, by conceiving the earth and moon to revolve about their common centre of gravity, he imputes about one half of the motion to that cause, and thence deduced a conclusion agreeing with observation.

"What we have already observed (Art. 861) is sufficient to show, that no part of the effect can arise from the latter circumstance; and he has also (as we have already shown) omitted a cause which produces about one half the motion; by two mistakes he has therefore fallen upon a true conclusion." And Mr. Vince, in conclusion, makes these remarks: "but in whatever manner this subject is treated, some corrections are applied from observations, in order to render the equations more perfect; not that the principle of attraction is insufficient to furnish conclusions which shall agree with observation," &c. And again: "Thus we have given the reader all the satisfaction we

are able upon this difficult subject, without entering into a direct solution of the problem, which requires the integration of a fluxional equation of the second order, and this can be done only by an approximation of a very intricate nature, and of great labor."

The reader or inquirer who will take the trouble to examine the moon's motions, and especially the motion of its apogee, as contained in Mr. Vince's Astronomy, will find that in all cases, the little motions and inequalities of motion, are, in whole or in part, assigned to the disturbing force of the sun upon the moon; and after much laboring of the author on the subject, we find this allegation, namely: "If the force perpendicular to the radius vector be neglected, the motion of the apsides comes out one half of what is here determined, and this we know from other principles. This, therefore, tends to confirm the legality of the method here employed," which allegation, after all, I think, must be taken for granted, rather than as being proved; and in respect to the whole matter, I have come to the following doubts:

I think it doubtful, whether the student or inquirer, on a careful examination of the whole subject, will readily agree with the modus operandi by which the observed phenomena of nature have been supposed to be reconciled with the Newtonian law.

Whether he will find the ratio between the force applied to the moon, and the moon's motion in its orbit, clearly and distinctly assigned or stated.

Whether he will clearly discover how they have obtained the force which acts perpendicular to the radius vector, which is said to give just half the motion of the apogee.

Whether he will be able to discover by what law of gravity, the distance may be assumed at unity, and its mean force of gravity at less than unity; or in other words, whether according to Mr. Vince, the mean force of the moon towards the earth is equal to the natural gravity of the moon towards the earth, diminished by the disturbing force of the sun upon the moon in the line of the radius vector,-together with many other matters equally dubious, which the inquirer is required to pass through, in order that he may be brought to render assent to their results, or rather to their conclusions.

And finally, I am doubtful whether the inquirer will readily discover, that even Sir Isaac Newton understood the subject in the same way and manner in which it is now supposed to be understood, notwithstanding Mr. Vince's supposition of what Newton must have meant by some very obcure passages in his Principia; and more especially, when it is acknowledged by

Mr. Vince, that Sir Isaac Newton left out the force which is now said to act perpendicularly to the radius vector; and which is said to give just one half of the motion to the apogee.,

Nevertheless, Walmsley, Murdock, and others, have been somewhat cavalierly treated, for their attempts to make the Newtonian law of gravity, account for the phenomena of nature, which, manifestly, are the result of some kind of law of gravity; notwithstanding, they made their conclusions agree very well with observation. And I must say, that I think the treatment they received from the inquisition-from those who would settle the controversy by establishing articles of implicit faith, however my terious. was somewhat harsh and arbitrary.

Walmsley and others, made use of all the force that was allowed them by the Newtonian law of gravity; and as to the remainder that was found necessary for the purposes intended, each and every one, (not even excepting Clairaut himself,) picked it up from such sources as were in his view most available; and I have yet to learn why others, (as well as Clairaut,) had not the right to call to their aid, for the purpose of triumphing over great difficulties, such auxiliary forces as they could best command.

But wisdom would have been profitable to direct; and had those great mathematicians, in lieu of confiding in their high-priest, just spent an hour or two in examining, to see from whence he derived his authority, they would not have been thrown into all these inexplicable dilemmas. They would have found just force enough to account for all the observed phenomena, without resorting to anything factitious or unnatural.

Nevertheless, however gloomy the stubborn fact may be, if Sir Isaac Newton's law of gravity be erroneous, it wholly destroys his whole system of philosophy; and in such case, his theory of universal gravity, (as by him promulgated,) ceases to exist; and the universe will again be governed by immutable and controlling laws, in lieu of depending on fortuity or chance.

CHAPTER III.

Of the Proper Elements from which to determine the Laws of Force and Motion, incident to the Heavenly Bodies in their Eternal Rounds.

SECTION FIRST.

HAVING attempted to show that what astronomers have called the deflection of a planet, or the deflection of a planet from a tangent to its orbit, and which they have made use of as an element, in their endeavors to account for the agreement between the force and motion by which a planet revolves, is, at least, but a factitious element, and not applicable to the laws of force and motion, as operating in the curve of an orbit, nor identical in its mathematical construction with that of the law of the rectilinear fall of a body through space, I will now proceed to treat of the proper elements of the planets from which to deduce the modus operandi by which they are retained and controlled in their orbits while performing their eternal rounds, which elements I will call by the appellations of Time, Distance, Motion, Force, and Convergency; the qualities or principles of which I will define as follows: Time always has direct reference to the Period, namely the periodic time, or entire revolution of the planet from any given point in its orbit, to the same point again; which period or periodic time, may be divided and subdivided, as occasion may require, when comparing other elements with that of time; and hence the period or whole periodic time of a planet, as also the mean distance of a planet from the centre of gravity may, in contradistinction to the elements of Motion, Force, and Convergency, properly be called positive quantities, by which, what may be called the rates (ratios or proportions) of the elements of motion, force and convergency may be compared.

Distance, or the mean distance of a planet, may always be abstractly expressed by some numerical quantity, by which the mean distance of any given planet of a system is proportioned to the mean distance of that planet of the system which