OF ASPECT, ROTATION, LIBRATION, ETC. 141

sidereal or than the tropical month. And it is very-easy to calculate the exact length of the lunation, or,as it is called, the synodical month. In 27'322 days,the moon has not completed the whole cycle of herphases, but only the portion M/ Mg'wq out of thewhole cycle,-—that is, she has completed the wholecycle, less the portion m l M/. Now, the anglerrii E' M/ is obviously the same as the angle E'SE;hence the part wanting from the complete cycle bearsto the whole cycle the same ratio that E E' bears tothe complete orbit of the earth, or that 27 - 322 dayshears to 365-242 days. The moon, then, in 27 - 322days, completes only HHjths of a lunation (thenumerator being obtained by taking 2 7'322 from365-242). So that a mean synodical month exceedsa mean sidereal month (or 27‘322 days) in the sameproportion that 365'242 exceeds 337-920. Increasing27-322 in this proportion (a mere rule-of-three sum),we obtain 29'531,* which is the length of a lunation.

The phases of the moon are explained in text-booksof astronomy. But a few remarks on the subject mayhe useful.

Let M l M 2 M g . . . . fig. 47, Plate XII., representthe moon’s orbit, the sun being at S, only many timesfarther away than in the figure. The earth and moonare relatively much exaggerated in dimensions ; andthe moon is shown in eight equidistant positions, asthough she performed a complete circuit, while the

* More exact values are given in the tables.