called the moon's evection. Astronomers before the time of Ptolemy being accustomed to observe the longitudes of the moon only at the times of the eclipses, that is, when the sun and moon are in conjunction or opposition; on comparing the longitudes of the moon deduced from the mean motion as above mentioned, with the longitude obtained from the observation of the eclipse, they considered the difference to be (for the distance of the moon at that instant from apogee or perigee) the value of what was called the first inequality, or the equation of the centre. But, on obtaining the longitude of the moon from observations made when she was in quadrature, or at either extremity of a diameter of her orbit at right angles to that which joins the points of conjunction and opposition, it was discovered that, in order to make the longitudes given by the tables agree with those which were observed in such situations of the moon, the first inequality ought to be augmented. The variable quantity by which the equation of the centre should be increased was subsequently called the evection: it was found, after many trials, to be a maximum when the excess of the mean longitude of the moon over that of the sun is 90 degrees, and when at the same time the mean anomaly of the moon is also 90 degrees. It was found to vanish when the mean longitude of the sun is equal to that of the moon, and also when the mean anomaly of the moon is 180 degrees or zero.

Now, when any deviation from a law of the angular motion which one body may have about another is observed to be periodical; that is, to be zero in a certain position, to attain a maximum when at 90 degrees, and again to vanish when at 180 degrees from thence; that deviation may evidently be considered as depending upon the sine of the angular distance of the body from the first position. Therefore, since the sine of an angle goes through all its variations, while the angle increases from zero to 180 degrees, and twice while it increases from zero to 360 degrees; the formula expressing the amount of the deviation or inequality may be represented by P sin. Q, where P is some constant number to be determined by observation and Q is a variable angle: thus, M being the mean longitude of the moon, s that of the sun, A the moon's mean anomaly, and E the evection, the equation may be represented by

E=P sin. {2 (MS)-A}.

From the latest observations the value of P is found to be 1° 20′ 29′′.54, and the period in which the evection goes through all its variations is 31.811939 da.

221. The longitudes of the moon computed from tables which include the evection, on being compared with the longitudes obtained from observations at the times of syzygy and quadrature, were found to agree very nearly; but Tycho Brahe discovered that there was a third inequality which attained its maximum value when the moon was in the octants, or when the difference between the mean longitude of the moon and that of the sun was 45 degrees, and vanished both in the quadratures and syzygies; this inequality is therefore represented by P' sin. 2 (MS). The value of P' is found from observations to be equal to 35'41".64, and the period in which the inequality goes through all its variations is evidently half a synodical revolution of the moon about the earth. This inequality is what is called the moon's variation.

222. A fourth inequality, which is called the moon's annual equation, was discovered on comparing the observed with the computed longitude of the moon when the earth was at its mean distance from the sun. For, after correcting all the above deviations, it was found that then the excess of the computed above the mean longitude was greater than that of the observed longitude above the latter: the excess was found to diminish gradually, and to vanish when the earth is in the aphelion and perihelion points. From these points the increase of the observed above the mean motion recommences, and it again becomes the greatest at the mean distances of the earth from the sun. The form of the equation is p" sin. A', where A' is the sun's mean anomaly, and p" is equal to 11'11".976. It is evident that the period in which the inequality compensates itself is half an anomalistic year. Besides these, there is a small inequality called secular, of the moon's motion in longitude, which with several others have been detected by the aid of physical astronomy.

223. The latitude of the moon is, in like manner, affected by periodical variations, of which the most important was discovered by Tycho Brahe. When its value is the greatest it changes the inclination of the moon's orbit to the ecliptic by 8' 47".15, and it depends on the cosine of twice the distance of the sun from the moon's ascending node. There is also an inequality depending on the sine of the sun's distance from the node; and others have been obtained from theory. The radius vector of the moon's orbit, and consequently the equatorial parallax of the moon, which depends on the radius vector, is also subject to considerable variations: a mean equatorial parallax whose value is 57' 4".165, has been determined both by theory and by observation, and this is called the constant of the parallax.

224. The distance of the moon from the earth being ascertained, it is possible by its aid to find that of the earth from the sun by plane trigonometry: it must be admitted, however, that the result so obtained is inaccurate; yet, as it will be advantageous that an approximative knowledge of that distance should be given previously to an explanation of the phenomena by which, in the present state of astronomy, it is determined, the method may without impropriety be here stated.

M

b

m 'n

E

Let E be the centre of the earth, M that of the moon, and s that of the sun; all these points being supposed to be in one plane, which is that of the ecliptic; then a plane passing through amb perpendicular to the ecliptic and to the line SM will separate the enlightened from the dark hemisphere of the moon. Let a plane pass through мn perpendicular to the ecliptic and to the line EM; then, the semidiameter Mn of the moon being very small compared with her distance from the earth, the line Ebm may be considered as perpendicular to Mn, and mn which represents the visible breadth of the enlightened part

a

S

of the moon's disk to an observer on the earth will have to Mn the ratio that the versed sine of the angle bмn has to radius. Now, by instruments, the angles subtended at the earth by Mn and mn can be measured: and by trigonometry,

therefore the angle bмn may be found. Then, since sмb is a right angle, the angle SMn is the complement of bмn, and EMn being a right angle, the angle EMS is found. Consequently, if at the time of measuring the breadth mn, the angular distance MES, of the moon from the sun be observed (both sun and moon being above the horizon), there will be given, in the plane triangle EMS, the side EM and all the angles; to find Es, which is the required distance of the earth from the sun, (about 95,000,000 miles).

The difficulty of the problem consists in obtaining the measurement of Mn and mn by the micrometer with sufficient accuracy; and, if the angular distance of the sun from the moon be found immediately by observation, in rightly determining the effects of parallax.

CHAP. X.

APPARENT DISPLACEMENTS OF THE CELESTIAL BODIES ARISING FROM THE FIGURE AND MOTION OF THE EARTH.

THE EFFECTS OF PRECESSION, ABERRATION, AND NUTATION.

225. THE observations of the ancients on the places of the fixed stars were not sufficiently precise to allow the law of their apparent motions to be accurately determined by comparisons of those places with such as have been obtained in a later age. In endeavouring to ascertain the value and law of the motions, it has therefore been found proper to compare the observations made in recent times with those of a date not more remote than the middle of the eighteenth century, when celestial observations first acquired the correctness necessary to render a comparison advantageous.

On comparing the longitudes and latitudes computed from the right ascensions and declinations of many hundred stars observed at different epochs since the year 1750, it has been found that the longitude of each has increased at a mean rate of 50".2 annually, while the latitudes undergo in a year changes not exceeding half a second: and, as it is scarcely probable that all the stars have equal angular movements about the axis of the ecliptic, the circumstance suggests the hypothesis that the equinoctial point, from which the longitudes are reckoned, has a retrograde movement on that great circle of the sphere at the mean rate above mentioned, as stated in describing the solar orbit (art. 174.). This hypothesis being adopted, it should follow that the annual variations in right ascension and declination, when computed on that hypothesis, by the rules of trigonometry, agree those which are found by observation to take place.

with

226. In order to find the laws of the variations in declination and right ascension: let YCL be part of a great circle of the sphere representing the trace of the ecliptic, which may be considered as fixed, and let p be its pole.

Let Yq be part of the equator, and P its pole: so that the angle Cq, the obliquity of the ecliptic, is measured by Pp. Let s be the place of a star, and draw the great circle psT; then TT will be the star's longitude, and Ts its latitude. Draw also the declination circle Psm; then, at a given epoch,

suppose at the commencement of any year (

being then the place of the equinoctial point), rm will be the right ascension, and ms the declination of the star.

E=50′′.2; and

P

Now, at the commencement of the next year (for example) let the equinoctial point be at E, so that let the position of the equator be Eq, having P' for its pole. Draw the declination circle P'sn; then, at the second epoch, En will be the right ascension, and sn the declination of the star. Again, PP' or its equal the angle CEq will be the obliquity of the ecliptic, which, without sensible error, may be considered the same as at the former epoch: also, since r

ΕΥ·

E

P

S

L

C

m

T

is the pole of the circle pPC, and E is the pole of a circle passing through p, P'; the angle PpP' will be measured by Then, in the spherical triangle PPP', which, since the angle at p is very small, may be considered as having a right angle at P or P', we have (art. 60. (e)) r sin. PP′ = sin. p sin. Pp, or substituting small arcs and angles, in seconds, for their sines, PP' (in seconds) = p sin. pp.

Again, since is the pole of pc, the equinoctial colure PV may be supposed to pass through P'; and PP' being very small, the angle YPS (the right ascension of s) may be considered as equal to YP's: then producing SP' till it meets a perpendicular Pt let fall upon it from P; in the triangle P P't, which may be considered as plane, we have PP', the angle PP't (P's) and the right angle at t; to find r't: thus Pt PP'cos. Y P's, and by substitution,

P't (in seconds) = p sin. Pp cos. Y P's; or ( being the obliquity, and P's being considered as the given right ascension (a) of the star),

P't (in seconds) =50".2 sin. cos. a.

But P't may be considered as the difference between the polar distances PS and P's, or the variation of the declination between the two epochs. Thus the annual precession in declination is equal to 50".2 sin. cos. a: but 50".2 sin. is nearly equal to 20".06; therefore the precession in declination may be represented by 20".06 cos. a.

This variation is negative, or the effect of precession on a star whose right ascension is less than 6 hours, is to diminish the polar distance, and when the right ascension is zero, the variation has the greatest negative value, being then about