118 TRUB LONGITUDE OF A PLANET Similarly, MT, MT, ,... are the next successive approximations 13, r₁, ... to the radius of the Sun's true epicycle, and ET3, ET4, ... are the next successive approximations H₂, H₁, ... to the Sun's true distance. As before, it can be easily shown that In < 'n+1 and H <H Moreover, from the method of construction 1₁, 1₂, 13, ... are each n less than MT, which is the upper bound of the sequence { }, and H₁, H₂, Hą, ... are each less than ET, which is the upper bound of the sequence {H}. Hence it follows that n+1. 1₁ < 1₂ < 1₂ <. <<... < MT. H₁ <H₂ <H₂ < ... <H₂ < ... < ET. and The sequences {n} and {H} are each monotonic and therefore convergent. The first converges to MT, the radius of the Sun's true epicycle, and the second to ET, the Sun's true distance. It may be seen that the sequences {n} and {H} converge rapid- ly so that the third or fourth approximation will give the result correct to the minute. In actual practice, however, the process of successive approxi- mations is carried on until two successive approximations are the same to minutes of arc. For this reason, this process is sometimes called aviseşakarma ("the process of reducing the difference to zero").¹ The method explained above is applicable to the Sun as also to the Moon. If in the above figure DM (which is equal to ST₂) be assumed to be equal to SS₂, i. e., H₁-R, we shall have ED=R-(H₁-R) =2R-H₁, and likewise, from the similar triangles SDE and TME, we shall get. ET = ESX EM ED R² 2R-H₂' 1 In the above discussion, we have assumed the Sun to be in the first anomalistic quadrant as shown in the figure. When the Sun is in any other quadrant, the procedure is similar.
