124 The method It is based on the explained as follows: TRUB LONGITUDE OF A PLANET stated in the text relates to the determination of ET. process of successive approximations and may be With centre E and radius ET, draw an arc of a circle cutting ET at the point S₁, and through S, draw a line parallel to EM meeting MT at T₂. Again with centre E and radius ET, draw an arc of a circle cutting ET at S₂, and through Są draw a line parallel to EM to meet MT at Tg. Continue this process repeatedly. Also let SD, S₁D₁, S₂D₂,... be parallel to EU. The method begins with assuming MT, as the first approximation r₂ to MT and likewise ET, is taken as the first approximation H, to ET. Now from the similar triangles S₁D₂E and SDE, we have SDXES₁_₁XH₁. S₂D₁=: But S₂D₁=MT₂. Therefore, MT, =XH₂ R Therefore, from the triangle T,BE, right-angled at B, we have ET₂ = MB+MT₂)³+BE², where MT₂ is given by (1), MT₂ is taken as the second approximation r₂ to MT, and ET, as the second approximation H₂ to ET. H₂, Similarly, MT3, MT4, ... to MT, and ET3, ET4, to ET. Obviously, R Since H, R, therefore r₂ > 1₁; and consequently, H₂> H₁. and R are the next successive approximations r3, are the next successive approximations H₂, rn <¹+1 and H₁ <Hn+1. n ... are each Moreover, from the construction it is clear that r₁, T₂, T3, less than MT, which is the upper bound of the sequence { }; and H₁, H₂, H3, are each less than ET, which is the upper bound of the sequ- ence {H}. n n Hence it follows that (1) 1₁ < 1g < 1₂ << H₂ <H₂ <H₂ < <H < < ... < MT <ET.
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