394 On the Variation of the Moon’s Motion. 
(221 x ED)? ' 5 oes 
60xcED? or about 672 : 649. If it be inquired how the ra- 
dius can be longer and yet the moon nearer the earth, we reply 
that owing to the disturbing influences that we have been con- 
sidering, the moon does not revolve around the earth as a centre, 
but round a point more distant at C and D, and less so at A and B. 
As this oval is very nearly an ellipse we may regard it as pos- 
sessing its properties, considering AB as the transverse and CD 
the conjugate axis. Now it is proved in works on the conic sec- 
tions, that the radius of curvature at any point in an ellipse is 
proportional to the cube of the diameter conjugate to that which 
passes through that point: Hence; AB* : CD*: ‘the radius of a 
circle having the same curvature as the oval at C or D : that of 
one having the same curvature as the oval at A or B. But these 
radii we have proved to be to each other in the. ratio 672 : 649 ; 
therefore AB? : CD22:672 : 649, and AB : CD: 72/672 : 7/649 
*:1110117. Sir I. Newton and after him other astronomers 
have resolved the problem with more accuracy than is here at- 
tempted, taking into account some minor circumstances that we 
have disregarded, and make the ratio 1 : 1-0147, or about 68 : 69. 
It has been already shown that the absolute velocity of the moon 
in quadrature : its absolute velocity in syzygy : 221 Xx ED + 223 
xEA. But adopting Sir I. Newton’s result, ED ; EA: :68 1 69; 
therefore the absolute velocity in quadrature ; the absolute velocity 
in syzygy : 1221 x68 . 223 x69: :15028 : 15387. And the dif- 
ference of the apparent or angular velocity is still farther inereas- 
ed in the ratio 68 : 69, by reason of the diminished distance of 
the moon from the earth when in syzygy. Hence, from all causes 
combined, the angular velocity in quadrature : the angular velo- 
city in syzygy::15028x68 : 15387 x69::51 : 53 nearly; and 
the numbers 51, 52, and 53 are proportional to the velocity in 
quadrature, the mean velocity, and the velocity in syzygy- 
The moon’s mean horary motion is 1977”; hence according 
to our calculations it must exceed that in syzygy by 38”, and be 
less in quadrature by the same amount. According to the most 
accurate observations the difference is about 41”. 
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