Note 157 — Chapter XXXIX · LEGES AND EPINOMIS. (continued 2)

Plato, Legg. v. p. 738 C-D. ὅπως ἂν ξύλλογοι ἑκάστων τῶν μερῶν κατὰ χρόνους γιγνόμενοι τοὺς προσταχθέντας … μετὰ θυσιῶν.

That such “ordained seasons” for meetings and sacrifices should be punctually attended to — was a matter of great moment, on religious no less than on civil grounds. It was with a view to that object principally that each Grecian city arranged its calendar and its system of intercalation. Plato himself states this (vii. p. 809 D).

Sir George Lewis, in his Historical Survey of the Astronomy of the Ancients, adverts to the passage of Plato here cited, and gives a very instructive picture of the state of the Hellenic world as to Calendar and computation of time (see p. 19; also the greater part of chapter i. of his valuable work). The object of all the cities was to adjust lunar time with solar time by convenient intercalations, but hardly any two cities agreed in the method of doing so. Different schemes of intercalation and periods (trietêric, octaetêric, enneadekaetêric) were either adopted by civic authority or suggested by private astronomers, such as Kleostratus and Meton. The practical dissonance and confusion was great, and the theoretical dissatisfaction also.

Now in this dialogue De Legibus, Plato recognises both the importance of the object and the problem to be solved, yet he suggests no means of his own for solving it. He makes no arrangement for the calendar of his new Magnêtic city. I confess that this is to me a matter of some surprise. To combine an exertion of authority with an effort of arithmetical calculation, is in his vein; and the exactness of observances as respects the Gods, in harmony with the religious tone of the treatise, depended on some tolerable solution of the problem.

We may perhaps presume that Plato refused to deal with the problem because he considered it as mathematically insoluble. Days, months, and years are not exactly commensurable with each other. In the Timæus (p. 36 C) Plato declares that the rotation of the Circle of the Same, or the outermost sidereal sphere, upon which the succession of day and night depends, is according to the side of a parallelogram (κατὰ πλευράν) — while the rotations of the Moon and Sun (two of the seven branches composing the Circle of the Different) are according to the diagonal thereof (κατὰ διάμετρον): now the side and the diagonal represented the type of incommensurable magnitudes among the ancient reasoners. It would appear also that he considers the rotations of the Moon and Sun to be incommensurable with each other, both of them being members included in the Circle of the Different.

Since an exact mathematical solution was thus unattainable, Plato may probably have despised a merely approximative solution, sufficient for practical convenience — to which last object he generally pays little attention. He might also fancy that even the attempt to meddle with the problem betokened that confusion of the incommensurable with the commensurable, which he denounces in this very treatise (vii. pp. 819-820).