The planetary spheres were much more difficult to arrange. Eudoxus used four spheres for each, and these had in every case to be carefully adjusted to the very different periods and amplitudes of the planetary oscillations. It must be confessed that the scheme failed with the difficult case of Mars, and was not quite satisfactory with Venus, but it represented remarkably well the movement—so far as then known—of Sun and Moon, Saturn, Jupiter, and Mercury. It was certainly a feat for those days, whether we consider it merely as the solution of a mathematical problem, or as an embodiment of astronomical knowledge. The periods of the planets as known to Eudoxus, stated in round numbers only, are given in the following table. They are taken from Simplicius, who describes the system of Eudoxus, but as in the so-called Papyrus of Eudoxus the synodical revolution of Mercury is given as 116 days, the same as the modern value, Eudoxus may have had much more exact data. It will be seen that his synodic period for Mars is the only one which is totally wrong, and the large error is difficult to explain. _Modern Zodiacal Modern_ _Planet. Synodic Period. Value. Period._[34] _Value._ Mercury 110 days 116 days 1 year 1.0 year Venus 19 months 584 ” ” 1.0 ” Mars 8 months, 20 days 780 ” 2 years 1.88 ” Jupiter 13 months 399 ” 12 ” 11.86 ” Saturn 13 months 378 ” 30 ” 29.46 ” ------------------------------------------------------------------------ [34] The period in which a planet is seen to revolve round the zodiac, and return to the same star, varies greatly, because complicated by its retrograde movements; but if the average of a sufficient number of periods be taken, it coincides for Mercury and Venus with the sidereal year; for Mars, Jupiter, and Saturn, with the period in which each is actually revolving round the sun (its “sidereal period”). ------------------------------------------------------------------------