Pythian, I have shewn reasons for thinking that they were originally celebrated at intervals of eight instead of four years; certainly this is attested for the Pythian games,(279) and the mode of calculating the Olympiads by alternate periods of fifty and forty-nine lunar months,(280) which added together make up eight solar years, seems to prove that the Olympic cycle of four years was really based on a cycle of eight years, from which it is natural to infer that in the beginning the Olympic, like the Pythian, games may have been octennial instead of quadriennial.(281) Now we know from the testimony of the ancients themselves that the Greeks instituted the eight-years’ cycle for the purpose of harmonising solar and lunar time.(282) They regulated their calendar primarily by observation of the moon rather than of the sun; their months were lunar, and their ordinary year consisted of twelve lunar months. But the solar year of three hundred and sixty-five and a quarter days exceeds the lunar year of twelve lunar months or three hundred and fifty-four days by eleven and a quarter days, so that in eight solar years the excess amounts to ninety days or roughly three lunar months. Accordingly the Greeks equated eight solar years to eight lunar years of twelve months each by intercalating three lunar months of thirty days each in the octennial cycle; they intercalated one lunar month in the third year of the cycle, a second lunar month in the fifth year, and a third lunar month in the eighth year.(283) In this way they, so to say, made the sun and moon keep time together by reckoning ninety-nine lunar months as equivalent to eight solar years; so that if, for example, the full moon coincided with the summer solstice in one year, it coincided with it again after the revolution of the eight years’ cycle, but not before. The equation was indeed not quite exact, and in order to render it so the Greeks afterwards found themselves obliged, first, to intercalate three days every sixteen years, and, next, to omit one intercalary month in every period of one hundred and sixty years.(284) But these corrections were doubtless refinements of a later age; they may have been due to the astronomer Eudoxus of Cnidus, or to Cleostratus of Tenedos, who were variously, but incorrectly, supposed to have instituted the octennial cycle.(285) There are strong grounds for holding that in its simplest form the octennial cycle of ninety-nine lunar months dates from an extremely remote antiquity in Greece; that it was in fact, as a well-informed Greek writer tell us,(286) the first systematic attempt to bring solar and the lunar time into harmony. Indeed, if the Olympiads were calculated, as they appear to have been, on the eight years’ cycle, this of itself suffices to place the origin of the cycle not later than 776 B.C., the year with which the reckoning by Olympiads begins. And when we bear in mind the very remote