be in quadratures (fig. 1), the angle mEM' will still be zero, or there will be no error in the longitude. But, if the apse line is in syzygy (fig. 2), the angle mEM' attains its greatest value.*

Ptolemy, as we have said, found this greatest value to be 2° 39′, the angle mEr being then 5° 1'.

117. Copernicus (A. D. 1543), having seen that Ptolemy's hypothesis gave distances totally at variance with the observations on the changes of apparent diameter,† made another and a simpler one which accounted equally well for the inequality in longitude, and was at the same time more correct in its representation of the distances.

Let E be the earth, OD an epicycle whose centre C describes the circle C'CC" about E with the moon's mean angular velocity.

Let CO, a radius of this epicycle, be parallel to the

apse

* If Ptolemy had used the hypothesis of an eccentric instead of an epicycle for the first inequality of the moon, an epicycle would have represented the second inequality more simply than his method did. Dr. Whewell's History of the Inductive Sciences, vol. 1. p. 230.

+ See Delambre, Ast. Moderne, vol. 1. p. 116. Whewell's History of Inductive Sciences, vol. I. p. 395.

line EA, and about O as centre let a second small epicycle be described, the radii CO and OM being so taken that

CO-OM
CE

=

sin 5° 1′, and

CO+OM
CE

= = sin 7° 40'.

The radius OM must now be made to revolve from the radius OC twice as rapidly as EC moves from ES, so that the angle COM may be always double of the angle CES.

From this construction, it follows that in syzygies the angle CES being 0° or 180°, the angle COM is 0° or 360°; and therefore C and M are at their nearest distances, as in the positions C' and C"" in the figure. Then CM=CO−OM, and the angle CEM will range between 0° and 5° 1′, the greatest value being attained when the apse line is in quad

rature.

When the moon is in quadrature CES=90° or 270°, and therefore, COM=180° or 540° and C and M are at their greatest distance apart, as in the position C"; then, CM=CO+ OM, and the angle CEM will range between 0° and 7° 40′, the former value when the apse line is itself in quadrature, and the latter when it is in syzygy.

118. Thus the results attained by Ptolemy's construction are, as far as the longitudes at syzygies and quadratures are concerned, as well represented by that of Copernicus; and the variations in the distances of the moon will be far more exact, the least apparent diameter being 28′ 45′′ and the greatest 37′ 33′′; whereas, Ptolemy's would make the greatest diameter 1°.*

The values which modern observations give vary between 28' 48" and 33′ 32".

* Delambre, Ast. Modernes

119. It will not now be difficult to shew that the introduction of this small epicycle corresponds with that of the term me sin{(2 − 2m − c) pt − 2B+a} in our value of 0. For, referring to the preceding figure, we have

=

OEM sin OEM

=

sin (2.SEC-AEM)

sin {2 (moon's mean long. - sun's long.)

- (moon's true long. - long. of apse)},

and OEM being a small angle whose maximum is 1° 19′, we may write moon's mean longitude instead of the true in the argument, and also EC for OE; therefore,

OM

OEM= sin {2 (moon's mean longitude - sun's longitude) EC

=

— (moon's mean longitude – longitude of apse)}

= 794′ sin [2 { pt — (mpt + B)} — { pt − (1 − c) pt+a}]

= 4770′′ sin {(2 – 2m−c) pt−2B+a}.

The value of the coefficient is from modern observations found to be 4589.61".

120. In Art. (70), we have considered the effect of this second inequality in another light, not simply as a small quantity additional to the first or elliptic inequality, but as forming a part of this first; and therefore, modifying and constantly altering the eccentricity and the uniform progression of the apse line.

Boulliaud (A.D. 1645), by whom the term Evection was first applied to the second inequality, seems to hint at some

thing of this kind in the rather obscure explanations of his lunar hypothesis, which, never having been accepted, it would be useless to give an account of.*

In Ptolemy's theory, Art. (115), the evection was the result of an apparent increase of the first lunar epicycle caused by its approaching the earth at quadratures; but, in this second method, it is the result of an actual change in the elements of the elliptic orbit.

D'Arzachel, an Arabian astronomer, who observed in Spain about the year 1080, seems to have discovered the unequal motion of the apsides, but his discovery must have been lost sight of, for Horrocks, about 1640, re-discovered it in consequence of his attentive observations of the lunar 'diameter: he found that when the distance of the sun from 'the moon's apogee was about 45° or 225°, the apogee was 'more advanced by 25° than when that distance was about '135° or 315°. The apsides, therefore, of the moon's orbit 'were sometimes progressive and sometimes regressive, and ' required an equation of 12° 30', sometimes additive to their mean place and sometimes subtractive from it.'†

Horrocks also made the eccentricity variable between the limits 06686 and 04362.

The combination of these two suppositions was a means of avoiding the introduction of Ptolemy's eccentric or the second epicycle of Copernicus: their joint effect constitutes the evection.

Après avoir établi les mouvemens et les époques de la lune, Boulliaud revient à l'explication de l'évection ou de la seconde inégalité. Si sa théorie n'a pas fait fortune, le nom du moins est resté. En même temps que la 'lune avance sur son cône autour de la terre, tout le système de la lune est déplacé; la terre emportant la lune, rejette loin d'elle l'apogée, et rap'proche d'autant le périgée; mais cette évection à des bornes fixées.'

Delambre, Hist. de l'Ast. Mod., tom. II. p. 157.

+ Small's Astronomical Discoveries of Kepler, p. 307.

Variation.

121. After the discovery of the evection by Ptolemy, a period of fourteen centuries elapsed before any further addition was made to our knowledge of the moon's motions. Hipparchus's hypothesis was found sufficient for eclipses, and when corrected by Ptolemy's discovery, the agreement between the calculated and observed places was found to extend also to quadratures; any slight discrepancy being attributed to errors of observation or to the imperfection of instruments.

But when Tycho Brahé (A.D. 1580) with superior instruments extended the range of his observations to all intermediate points, he found that another inequality manifested itself. Having computed the places of the moon for different parts of her orbit and compared them with observation, he perceived that she was always in advance of her computed place from syzygy to quadrature, and behind it from quadrature to syzygy; the maximum of this variation taking place in the octants, that is, in the points equally distant from syzygy and quadrature. The moon's velocity therefore, so far as this inequality was concerned, was greatest at new and full moon, and least at the first and third quarter.*

*It appears that Mohammed-Aboul-Wefa-al-Bouzdjani, an Arabian 'astronomer of the tenth century, who resided at Cairo, and observed at 'Bagdad in 975, discovered a third inequality of the moon, in addition to 'the two expounded by Ptolemy, the equation of the centre and the evec'tion. This third inequality, the variation, is usually supposed to have 'been discovered by Tycho Brahé, six centuries later...... In an almagest ' of Aboul-Wefa, a part of which exists in the Royal Library at Paris, ' after describing the two inequalities of the moon, he has a Section IX., "Of the third anomaly of the moon called Muhazal or Prosneusis"...... 'But this discovery of Aboul-Wefa appears to have excited no notice among his contemporaries and followers; at least it had been long quite 'forgotten, when Tycho Brahé, re-discovered the same lunar inequality.' Whewell's Hist. of Inductive Sciences, vol. 1. p. 243.