ABSTRACT
Christiaan Huygens faced a challenge. In 1680, the Dutch astronomer was designing a mechanical planetarium for l’Acade´mie Royale des Sciences in France. He calculated that in one year, the Earth goes through an angle of 359◦45′40′′31′′′ of arc, where 31′′′ denotes 31/60 seconds. This is
(359 · 603 + 45 · 602 + 40 · 60 + 31)/60 = 77708431/60
seconds of arc (note that this is slightly less than 360◦, which is why we need leap years). Saturn, which goes slower, goes through 2640858/60 seconds of arc in a year. Therefore, in his planetarium, he needed gears in the ratio of
2640858 ≈ 29.42545
in order to make the relative motions of Earth and Saturn accurate. Since it was impossible to make a gear with 77708431 teeth, he approximated the ratio as
2640858 ≈ 206
7 ≈ 29.42857.
