From whence we get, sin. AE × co-t. AH × (sin. AH)² = (sin. EH)² × co-t. E × (sin. E)². But co-t. AH × sin. AH = co-s. AH × 1 (rad.), and co-t. E × sin. E = co-s. E × 1 (rad.): therefore sin. AE × co-s. AH × sin. AH = (sin. EH)² × co-s. E × sin. E; and, consequently, k × sin. AE × co-s. AH × sin. AH/sin. E = k × co-s. E × sin. EH² (= Ee).
Let, now, the sun's longitude EH be denoted by Z (considered as a flowing quantity); then, (sin. Z)² being = ½-½ co-s. 2 Z, we shall have k × co-s. E × (sin. EH)² = ½k × co-s. E × (1-co-s. 2 Z). But the angle described about the axe of rotation Pp, in the time that the sun's longitude is augmented by the particle Ż, will be = T/t × Ż. Therefore (by the general proposition) we have, as 1∶ ½k × co-s. E × (1-co-s. 2 Z) ∷ T/t × Ż∶ ½k × T/t × co-s. E × Ż-Ż co-s. 2 Z, the true regress of the equinoctial point E, during that time: whose fluent, ½k × T/t × co-s. E × (Z- ½ sin. 2 Z), will consequently be the total regress of the point E, in the time that the sun, by his apparent motion, describes the arch HE or Z; which, on the sun's arrival at the solstice, becomes barely = ½k × T/t × co-s. E × an arch of 90°∶ the quadruple whereof, or ½k × T/t × co-s. E × 360° (= 3t/4T × OA²-OP²/OA² × co-s. E × 360°) is therefore the whole annual precession of the equinox caused by the sun. This, in numbers (taking OP/OA = 229/230 ) comes out 3/4 × 366¼ × 2/230½ × 0.917176 × 360° = 21´´ 6´´´.
The very ingenious M. Silvabelle, in his essay on this subject, inserted in the 48th volume of the Philosophical Transactions, makes the quantity of the annual precession of the equinox, caused by the sun, to be the half, only, of what is here determined. But this gentleman appears to have fallen into a twofold mistake. First, in finding the momenta of rotation of the terrestrial spheroid, and of a very slender ring, at the equator thereof; which momenta he refers to an axis perpendicular to the plane of the sun's declination, instead of the proper axe of rotation, standing at right angles to the plane of the equator. The difference, indeed, arising from thence, with respect to the spheroid (by reason of its near approach to a sphere) will be inconsiderable; but, in the ring, the case will be quite otherwise; the equinoctial points thereof being made to recede just twice as fast as they ought to do. This may seem the more strange, if regard be had to the conclusions, relating to the nodes of a satellite, derived from this very assumption. But, that these conclusions are true, is owing to a second, or subsequent mistake, at Art. 27; where the measure of the sun's force is taken the half, only, of the true value; by means whereof the motion of the equinoctial points of the ring is reduced to its proper quantity, and the motion of the equinoctial points of the terrestrial spheroid, to the half of what it ought to be.
That expert geometrician M. Cha. Walmsley, in his Essay on the Precession of the Equinox, printed in the last volume of the Philosophical Transactions, has judiciously avoided all mistakes of this last kind, respecting the sun's force, by pursuing the method, pointed out by Sir Isaac Newton; but, in determining the effect of that force, has fallen into others, not less considerable than those above adverted to.
In his third Lemma, the momentum of the whole Earth, about its diameter, is computed on a supposition, that the momentum or force of each particle is proportional to its distance from the axis of motion, or barely as the quantity of motion in such particle, considered abstractedly. No regard is, therefore, had to the lengths of the unequal levers, whereby the particles are supposed to receive and communicate their motion: which, without doubt, ought to have been included in the consideration.
In his first proposition, he determines, in a very ingenious and concise manner, the true annual motion of the nodes of a ring (or of a single satellite) at the earth's equator, revolving with the earth itself, about its center, in the time of one siderial day. This motion he finds to be = 3co-s. 23° 29´/4 rad. × 1/366¼ × 360°. Then, in order to infer from thence, the motion of the equinoctial points of the earth itself, he, first, diminishes that quantity, in the ratio of 2 to 5: Because (as is demonstrated by Sir Isaac Newton in his 2d Lemma) the whole force of all the particles situated without the surface of a sphere, inscribed in the spheroid, to turn the body about its center, will be only 2-5ths of the force of an equal number of particles uniformly disposed round the whole circumference of the equator, in the fashion of a ring. The quantity (3co-s. 23° 29´/4 rad. × ⅖ × 1/366¼ × 360°) thus arising, will, therefore, express the true motion of the equinoctial points of a ring, equal in quantity of matter to the excess of the whole earth above the inscribed sphere, when the force whereby the ring tends to turn about its diameter is supposed equal to the force whereby the earth itself tends to turn about the same diameter, in consequence of the sun's attraction. Thus far our author agrees with Sir Isaac Newton; but, in deriving from hence the motion of the equinoctial points of the earth itself, he differs from him; and, in the corollary to his third Lemma, assigns the reasons, why he thinks Sir Isaac Newton, in this particular, has wandered a little from the truth. Instead of diminishing the quantity above exhibited (as Sir Isaac has done) in the ratio of all the motion in the ring to the motion in the whole earth, he diminishes it in the ratio of the motion of all the matter above the surface of the inscribed sphere to the motion of the whole earth: which matter, tho' equal to that of the ring, has nevertheless a different momentum, arising from the different situation of the particles in respect to the axis of motion.
But since the aforesaid quantity, from whence the motion of the earth's equinox is derived, as well by this gentleman, as by Sir Isaac Newton, expresses truly the annual regress of the equinoctial points of the ring (and not of the hollow figure formed by the said matter, which is greater, in the ratio of 5 to 4) it seems, at least, as reasonable to suppose, that the said quantity, to obtain from thence the true regress of the equinoctial points of the earth, ought to be diminished in the former of the two ratios above specified, as that it should be diminished in the latter. But, indeed, both these ways are defective, even supposing the momenta to have been truly computed; the ratio, that ought to be used here, being that of the momenta of the ring and earth about the proper axe of rotation of the two figures, standing at right-angles to the plane of the ring and of the equator. Now this ratio, by a very easy computation, is found to be as 230²-229² to ⅖ of 230²; whence the quantity sought comes out = 3co-s. 23° 29´/4 rad. × 1/366¼ × 230²-229²/230² × 360° = 21´´ 6´´´: which is the same that we before found it to be, and the double of what this author makes it.
What has been said hitherto, relates to that part of the motion only, arising from the force of the sun. It will be but justice to observe here, that the effect of the moon, and the inequalities depending on the position of her nodes, are truly assigned by both the gentlemen above-named; the ratio of the diameters of the earth, and the density of the moon being so assumed, as to give the maxima of those inequalities, such as the observations require: in consequence whereof, and from the law of the increase and decrease (which is rightly determined by theory, tho' the absolute quantity is not) a true solution, in every other circumstance, is obtained.
The freedom, with which I have expressed myself, and the liberty I have here taken, to animadvert on the works of men, who, in many places, have given incontestible proofs of skill and genius, may, I fear, stand in need of some apology. 'Tis possible I may be thought too peremptory. Indeed, I might have delivered my sentiments with more caution and address: but, had not I imagined myself quite clear in what has been advanced, from a multitude of concurrent reasons, I should have thought it too great a presumption to have said any thing at all here, on this subject. The great regard I have for this Society, of which I have the honour to be a member, will, I hope, be considered as the motive for my having attempted to rectify some oversights, that have occurred in the works of this learned body.