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.) × (⅟366¼) × 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 × (⅟366¼) × 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.