LECTUBE I. 



HISTORICAL SKETCH. 



[THE Lunar Theory may be said to have had its commencement with 

 Newton. Many irregularities in the Moon's motion were known before his 

 time, but it was he that first explained the cause of those irregularities 

 and calculated their amounts from theory. 



Of the inequalities which are due to the action of the Sun, the first, 

 which is called the Evection, was discovered by Ptolemy, who lived at 

 Alexandria in the first half of the second century of our era, under the 

 reigns of Hadrian and Antoninus Pius. At a very early period the relative 

 distance of the Moon at different times could be told from the angle it 

 subtended, and its orbit could thus be mapped out. By such means Ptolemy 

 found that its form was not the same from month to month, and that 

 the longer axis moved continually though not uniformly in one direction. 

 He represented this change by a motion of the centre of the ellipse, as 

 we would put it, in an epicycle round the focus, obtaining thus a variable 

 motion for the longer axis and a variable eccentricity. 



The representation of position by means of epicycles is intimately related 

 to the modern method of developing the coordinates in harmonic series ; 

 thus if we have 



x = Aj cos (nj + Oj) + A,, cos (n.i + 0^) + ... 



y = A^ sin (n^t + Oj) + A^ sin (n + a 2 ) + . . . 



the motion of the point (x, y) is that on a circle of radius A l with angular 

 velocity n lt around a centre which moves on a circle of radius A, with 

 angular velocity 2 , and so on ; and if, more generally, we have 



x = Aj cos (nf + Oj) + . . . 

 y = B l sin (nf + 0^) + ... 



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