corresponded with ours only at the equinox; these hours, if we may use the expression, were longer in summer and shorter in winter than our equinoctial hours. This explains why the time of the chief meal, which was usually taken at about five or six in the afternoon, was indicated sometimes by a 7-foot, sometimes by a 10-or 12-foot, or even a 20-foot shadow; for though at midsummer the shadow would be quite small at this time, it would have a considerable length at the equinox, and at the time of the winter solstice it is probable that they did not dine until after sunset. Unfortunately, we have not sufficient information to determine exactly the length of this shadow-pointer, which was doubtless always the same, in order to prevent confusion. The assumption that the pointer was about the average height of a human being, and that people even used their own shadows for measuring time, is very improbable. Such shadow-pointers probably stood in public places, where everyone could make use of them with help of the lines drawn on the ground; they could only be set up in private dwellings when these had large open spaces (which was not often the case) to which the sun could have access all day long. In later times inventions were made which supplied what was wanting in this mode of reckoning time; lines were graven on the stone floor on which the shadow-pointer stood, which gave, at any rate, some indication of the change in the length of the hours according to the months; a network of lines of this description belonged to the obelisk which Augustus set up on the Campus Martius, and also used as a shadow-pointer. The sun-dials, invented later than the shadow-pointers, probably by Aristarchus, about 270 B.C., were different; here the shadow of a stick placed in a semicircle, on which the hours were marked by lines, indicated the time of day. There were three kinds: first, those that were calculated at the place on which they were set up, and could not be moved, and which indicated the hours of the day according as they changed in the course of the year; second, those which were arranged for moving, and could be set up at different places; and, third, those used by mathematicians, which showed the equinoctial hours such as we use to-day. It is impossible, however, to determine whether the Greeks were acquainted with all the three kinds which we find in use in the Roman period.