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sine, the terms involving cose will disappear in the summation of each set; and following the process of the method will give

B+C cosa M

suppose.

Dividing again into two sets corresponding to the positive and negative values of cose, the terms in sin✪ will be cancelled, and the same process will give

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Treating the observations in the same way with respect to the angle, we get two results,

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M, N, M', N' are connected by the equation of condition,

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When the periods of 0 and are nearly, but not exactly, the same, this equation of condition will not hold, and the preceding values of B and C would not be exactly correct, but yet they would be very approximate, especially if the mean between the two values of B be taken.

106. We may also, after having taken one of these slightly erroneous values for B, make a further correction by establishing as it were a counterbalancing error in the value of C. Let B' be the value so found for B; then, from the of each of the observations subtract the value B' sin ê,

the result U will be very nearly equal to A+ C sino + &c., and from the n equations

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a value C' of C will be obtained, by the rule of Art. (62), which will be very approximate, and, at the same time, agree better with B' in satisfying the equations than C itself would do.

107. When two terms whose periods are nearly equal do occur, it is plain, by examining the values of M and M', that the errors which would be committed by following the rule, without taking account of this peculiarity, would be the taking B+C cosa and C+ B cosa for B and C respectively.

CHAPTER VIII.

HISTORY OF THE LUNAR PROBLEM BEFORE NEWTON.

108. The idea which most probably suggested itself to the minds of those men who first considered the motion of the moon among the stars, was that this motion is uniform and circular about the earth as a centre.

This first result is represented in our value of the longitude by neglecting all small terms and writing 0=pt.

109. It must, however, have been very soon perceived that the actual motion is far from being so simple, and that the moon moves with very different velocities at different times.

The earliest recorded attempts to take into account the irregularities of the moon's motion were made by Hipparchus, (140 B.C.) He imagined the moon to move with uniform velocity in a circle, of which the earth occupied, not the centre, but a point nearer to one side. By a similar hypothesis he had accounted for the irregularities in the sun's motion, and his success in this led him to apply it also to the moon.

It is clear that, on this supposition, the moon would seem to move faster when nearest the earth or in perigee, and slower when in apogee, than at any other points of her orbit, and thus an apparent unequal motion would be produced.

Let BAM be a circle, CA a radius, E a point in AC near C; CB, ED two parallel lines making an angle a with CA.

Suppose a body M to describe this circle uniformly with an angular velocity p, the time being reckoned from the instant when the body was at B, and the longitude as seen from E being reckoned from the line ED;

therefore

EC

DEM=0,

B
D

BCM=pt,

AEM=0-α, ACM=pt-a.

Now is a small fraction, and if we represent it by e,

CM

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this would give M, and then 0 by the formula 0=pt+ M. This was called an eccentric, and the value of e was called the eccentricity, which, for the moon, Hipparchus fixed at sin 5° 1'.

110. Another method of considering the motion was by means of an epicycle, which led to the same result.

A small circle PM, with a radius equal to EC of previous figure, has its centre in the circumference of the circle RPD (which has the same radius as that of the eccentric), and moves round E with the uniform angular velocity p, the the body M being carried in the circumference of the smaller

circle, the radius PM remaining parallel to itself, or, which

is the same thing, revolving from the radius PE with the same angular velocity p, so that the angle EPM equals PEA.

Now, when the angle AEP equals the angle ACM of the former figure, it is easily seen that the two triangles EPM, ECM are equal, and there

E

M

fore the distance EM and the angle AEM will be the same in both, that is, the two motions are identical.

111. The value of e being small, we find, rejecting e3, &c., Me sin (pt - a),

therefore

0=pte sin (pt — a).

If we reject terms of the second order in our expression for the longitude, and make c=1, we get, Art. (51),

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which will be identical with the above if we suppose the eccentricity of the eccentric to be double that of the elliptic orbit.

Ptolemy (A. D. 140) calculated the eccentricity of the moon's orbit, and found for it the same value as Hipparchus, viz.

sin5° 1′, nearly.

=

The eccentricity in the elliptic orbit is, we know, about. These values will pretty nearly reconcile the two values of given above, and this shews us, that for a few revolutions the moon may be considered as moving in an eccentric, and her positions in longitude calculated on this supposition will be correct to the first order.

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