Oa I Oe Op2 Oe

sin X

and sin a cos X

sin fi

From these equations it follows, since |0Pi| = |OP2| = |OS|,that sin a . | EO| sin a tan X =

sin fi and

sin X

To obtain k and |EO | /1 OS| from these equations, Hipparchus needed to know the time between the vernal equinox and the summer solstice and that between the summer solstice and the autumnal equinox, and from his observations he was able to improve the known values for the lengths of the seasons. He measured the time taken for the Sun to travel from P1 to P2 as 94 2 days and from P2 to P3 as 922 days (the times from P3 to P4 and from P4 to P1 being 881 and 908 days, respectively). Hence, since ZP1OP2 = a + p + 90° and ZP2OP3 = a - p + 90°, a + fi + 90° =

0 0

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