But to show for many ages the obliquity of the ecliptic has always decreased. this particularly the secular motion of Jupiter's node, by prop. 2, is 10′ 221⁄2′′; and the annual motion of the equinox being 50", its motion in the same time is 1° 23′ 20′′; therefore the difference of the motions of the node and equinox, is to the motion of the node, as 7.0331 to 1; therefore the time of the node's transit from the vernal equinox to the autumnal, which gives the termination of the diminution of the ecliptic's obliquity, will be in 14803 years, independent of the small acceleration due to the force of Saturn. Jupiter's node then being at present in 81° of Cancer, it appears that for 8000 years (if such be supposed the age of the world) the obliquity of the ecliptic has decreased, and that it will continue to decrease for more than 6000 years longer, and will not recover its first situation till after a period of 29606 years. But the total diminution, which in the aforesaid time can be generated in the obliquity of the ecliptic by the force of Jupiter, produces by the theorem in this prop. 22′ 30′′; and therefore this is the maximum of the variation. If there be required the decrease of the ecliptic's obliquity in the space of the last 1000 years, it will be thus easily computed. The motion of Jupiter's node, by prop. 2, in 1000 years is 1° 43′ 44"; and the precession of the equinoxes in the same time is 13° 53′ 20′′, the difference of which motions is 12° 9′ 36′′; hence, supposing the place of the node in the beginning of the year 1755 to be in 8° 20' of Cancer, according to Dr. Halley's astronomical tables, the distances of the node from the equinox at the beginning and end of the given time would be 93° 49′ 36′′ and 81° 40′; and hence, by the foregoing theorem is obtained 2′ 22′′ 56" for the required decrement by the force of Jupiter. In like manner, the motion of Saturn's node, by prop. 2, in 1000 years, is 5′ 56′′; hence the difference between the motion of the node and that of the equinox, is to the motion of the node, as 139.265 to 1; but the distances of the node from the equinox, at the beginning and end of the given time, according to this ratio would be 68° 38′ 24′′ and 82° 25′ 48′′, supposing the node according to the same tables to be in 21° 21′ 36" of Cancer, in the beginning of the year 1755; and hence, the inclination of Saturn's orbit to the ecliptic being 2° 30′ 10", by the same theorem the decrement produced by the force of Saturn, comes out 15" 2". Therefore the whole decrement of the ecliptic's obliquity in the last 1000 years, by the united forces of Jupiter and Saturn, becomes 2′ 38′′. So that from the time of Hipparchus the diminution of the obliquity of the ecliptic is about 5'. Thus also, if Jupiter's ascending node in the beginning of the year 1750, be in 8° 15′ 50′′ of Cancer, and Saturn's node in 21° 20′ 6′′ of Cancer, as in Halley's tables, the following little table will come out: Collation of the Theory with Phenomena. For the proper comparing of the theory with phenomena, the observations of the ancients should be consulted and compared with the moderns; but the former are more imperfect than what can serve for minute considerations of this kind. We can therefore only make use of those of the latter, though less fit for the purpose. 1. M. Le Monnier relates, in the Memoirs of the Paris Academy for 1738, that Picart observed there the altitude of the sun's centre at the summer solstice, and found it in the year 1669 to be 64° 39′, and in 1670 to be 64° 38′ 58′′; we shall take the medium 64° 38' 59". Le Monnier himself, in the same Memoirs for 1743, found the altitude of the sun's centre there at the solstice to be 64° 38' 45". Also the mean place of the moon's ascending node, by the corresponding observations, was about 27° of Y at the former time, and 16° of 8 in 1743; hence in the former case the nutation of the earth's axis was 8", the whole being 18", and in the latter 6" 15""; which quantities being respectively deducted, the altitude of the sun's centre at the former becomes 64° 38′ 51′′, and at the latter 64° 38′ 38′′ 45′′", the difference 12′′ 15'" is the decreasing in the mean obliquity of the ecliptic in the interval of 734 years. By the proposition the decrease from the force of Jupiter in the same interval of time was 10′′ 27′′′′; and by that of Saturn 1" 5"". Therefore the whole decrease in the obliquity according to theory was 11" 32". 2. From the observations of Walther, compared among themselves, La Caille (in the Paris Memoirs for 1749) collected that the obliquity of the ecliptic about the year 1496 was 23° 29′ 32", which at present is stated at 23° 28' 30"; and therefore in the 260 years the obliquity has decreased about 1'. Now by our theorem that decrease by Jupiter's force would be 37" 2", and by Saturn's 3" 50"; hence the whole decrease in the same time becomes 40" 52", or about 41′′. If instead of Cassini's table of refractions that of Newton be used, the obliquity of the ecliptic deduced from Walther's observations, will come out a few seconds less, and so come nearer to the determination by theory. But because of the uncertainty of the refractions and of the latitudes of places, it seems that the variation of the obliquity can be most safely determined by observations at the summer solstices made at the same place. If the variation at length accurately derived from experiments, should, as in the example above, exceed the variation assigned by this theory, that excess will be due to the actions of the other planets Mars and Venus, which, since the ascending nodes of both are within the first six signs, conspire also to diminish the obliquity of the ecliptic. For which reason, if at any time by observations there can be accurately known both this variation and the progress of the earth's aphelion, then we may also come to know the forces of the planets Mars and Venus, and to weigh their masses. PROP. 4. To determine the Motion of the Equinoxes due to the preceding Causes. Here the motion of the equinoctial point is not investigated, so far as to the earth's equator, because of the redundant matter there, may change its situation in respect of the ecliptic by the force of Jupiter and Saturn, like as it can be done by the forces of the sun and moon; for this kind of mutation arising from the actions of Jupiter and Saturn must be insensible: but we inquire after that motion of the equinox, arising from the variation, which we have showed above can be made in the situation of the plane of the ecliptic. The same things remaining therefore as in the preceding proposition, from the point m, where the equator cuts the circle dɛ, demit mn perpendicular on DE; then because Dg: mn:: 1: cos. DL or CK, and Dd: Dg:: 1: s, it will be Dd: mn:: 1: CK X s; or drawing the radius cs perpendicular to CL, and on cs the perpendiculars DR, er, HG, it will be Dd: mn :: De: Rr X s; therefore the sum of all the mn, while by the difference of the motions of the equinox and node the arc DH is described, will be to the sum of all the Dd, as the sum of all the Rr X s, is to the sum of as many arcs De, that is, as RG XS to the arc DH. Therefore the sum of all the mn, that is, the latitude of the equinoctial point, or its distance from the plane DCE considered as immovable, is = LG X SX 2, the let RG DH ter n denoting the motion of the node while the arc DH is described. Now, as RG is equal to the difference or sum of the sines of the arcs DL, HL, according as the points R, G lie on the same or on contrary sides of the centre c, the circle ID representing the orbit of either Jupiter or Saturn, hence will result the following theorem: " Radius is to the sine of the inclination of the orbit of Jupiter or Saturn to the ecliptic, as the difference or sum of the sines of the distances of the node from the equinox at the beginning and end of a given time, to a certain sine: then, as the difference of the motions of the node and equinox, is to the motion of the node, so is the sine just found, to the sine of the variation in the latitude of the equinoctial point." Or again, since the variation in the obliquity of the ecliptic is, from the preceding prop. = X, and DH KM XNX S RG XnX is the variation in the latitude of the equinoctial point, we thence have this other theorem: "The variation in the latitude of the equinox, is to the variation in the obliquity of the ecliptic, as the sum or difference of the sines of the distances of the node from the equinox at the beginning and end of a given time, is to the sum or difference of the cosines of these distances." And, because it is always, as Ln is to mn, so is the cosine of the obliquity of the ecliptic, to the sine of the same obliquity, or as radius to the tangent of the same obliquity, then will the sum of all the Ln in a given time, i. e. the variation of the equinoctial point according to longitude measured from a fixed point in the plane DCE, be to the variation of the same according to latitude, in the same ratio, and therefore is given. a. E. I. COROL. Hence it follows that the variation of the vernal equinox according to latitude, computed from an immovable plane, is always made towards the north, in the passage of the ascending node of Jupiter or Saturn from the summer solstice to the winter; and towards the south when the same node transits from the winter solstice to the summer. The contrary must be said of the autumnal equinox. But the variation of the equinox according to longitude, counted from a given place in that immovable plane, is in the former case made contrary to, but in the latter according to the series of the signs; that is, in the former case the equinox recedes backward, but in the latter precedes forward. If the prints D and и should be situated on different sides of the solstice, i. e. if in the time proposed the node should transit through the sign cancer or capri corn, the theorems in the prop. will give the difference of the contrary variations of the equinoctial point; and h the sum of these may be had is easily seen. SCHOLIUM. Since in the course of the last 1000 years Jupiter's ascending node has possessed the sign Cancer; and because the aforesaid variations have not been made sensible in the whole of that time, we may inquire what they have been in the 500 years counted backward, from the beginning of the year 1755 in which case the difference in the motions of Jupiter's node and the equinox, is by the scholium to the preceding prop. 6° 4′ 48′′; hence proceeding as there for the rest, either theorem in that prop. produces, for the variation of the vernal equinox in north latitude, 6" 37"", and hence the variation in longitude is 15" 14"", by Jupiter's force. By adding, in the former case, 2′′ 26′′ for the force of Saturn, and 5′′ 36′′ in the latter, then the total variation of the equinoctial point, according to latitude, in the last elapsed 500 years, is = 9 3′′′′, and the regression of the saine point 20′′ 50′′. So that the variations of this kind are not sensible unless in a long interval of time. PROP. V. To Investigate the Equations of the Terrestrial Errors. The greatest equations of the angular errors are directly as the forces and the square of the times, and inversely as the diameters of the orbits; and therefore they are as those errors or motions, of which they are the equations, generated in those times but the periodic times are very nearly as the equations. Hence, because of the given errors of the lunar and terrestrial motions, and the pcriods of the equations of the lunar errors, by analogy those of the terrestrial errors will be deduced. Thus, the period of the equation of the lunar apogee, and of the variation of the equation of the moon's centre, since it is proportional to the sun's revolution at the moon's apogee; and because of the similitude of forces similarly applied, the period of the equation of the earth's aphelion, and of the variation of the equation of the centre, must be proportional to the revolution of Jupiter at the earth's aphelion, therefore those lunar equations will be to these like equations of the earth, as the motion of the lunar apogee, in the time of the sun's revolution at the moon's apogee, is to the motion of the earth's aphelion in the time of Jupiter's revolution at the earth's aphelion; that is, the mean annual motion of the moon's apogee being 40° 40′ 43′′, and the annual motion of the earth's aphelion above found 13′′ 2′′ 28′′", as 45° 51′ 40′′ to 2′ 34′ 42′′. Therefore, putting the total variation of the greatest equation of the moon's centre = 2o 41, as it is nearly in astronomical tables, the variation of the greatest equation of the centre of the earth or sun will be 9" 4"". Now, let e denote the greatest mean equation of the sun's centre, then will e+ 4′′ 32′′′′ be the greatest equation, and e—4′′ 32′′′′ the least equation; and by these equations there will also be given the corresponding excentricities. Then, like as the variation of the greatest equation of the moon's centre increases in the duplicate ratio of the sine of the distance of the moon's apogee from its quadrature with the sun, so the variation of the greatest equation of the sun's centre, that is, the increment of the greatest equation, is augmented in the duplicate ratio of the sine of the distance of the earth's aphelion from its quadrature with Jupiter; or, the variation of the mean equation is to half the total variation, viz. to 4" 32"", as the cosine of double the distance of Jupiter from the earth's aphelion is to radius; then adding this to the mean equation when the line of the apses of the orbis magnus passes from its octant with Jupiter to the syzygies, or from the syzygics to the octants; in the other parts it is subducted. In like manner, if the greatest equation of the moon's apogee be stated at 12° 18', then 45° 51' 40' will be to 2′ 34" 42", as 12° 18′ is to the greatest equation of the motion of the earth's aphelion, or of the sun's apogee, which therefore will be 41" 30', viz. where the apses of the earth's orbit is in its octant with Jupiter. In other positions the equation of the aphelion will be to the greatest equation, as the sine of double the distance of Jupiter from the earth's aphelion, is to radius, adding it to the mean motion in the transit of the apses of the |