The first method of measuring the angle was to measure the length of the sun’s shadow at each solstice, and so, by comparison of the length of the shadow with the height of the gnomon, calculate the difference in altitude, the half of which was the angle sought. And this was probably the method of the Chinese, who obtained a result of 23° 38´ 11˝ in the time of Yao; and also of Anaximander in his early days, who obtained a result of 24°. But before trigonometrical tables, the first of which seem to have been constructed by Hipparchus and Ptolemy, were known, in order to find this angle it was constructed geometrically, and then what _aliquot part_ of the circumference it was, or _how much of the circumference_ it contained was determined; for the division of the circle into 360° is subsequent to the first beginning of astronomy—and hence it was that Eratosthenes said that the distance from the tropics was 11/83 of the circumference, and not that it was 47° 46´ 26˝. The gnomon is, without exception, of all instruments the one with which the ancients were able to make the best observations of the sun’s altitude. But they did not give sufficient attention to it to enable it to be used with accuracy. The shadow projected by a point when the sun is shining is not well defined, so that they could not be quite certain of its extremity, and it would seem that the ancient observations of the height of the sun made in this manner ought to be corrected by about half the apparent diameter of the sun; for it is probable that the ancients took the strong shadow for the true shadow; and so they had only the height of the upper part of the sun and not that of the centre. There is no proof that they did not make this correction, at least in the later observations. In order to obviate this inconvenience, they subsequently terminated the gnomon by a bowl or disc, the centre of which answered to the summit; so that, taking the centre of the shadow of this bowl, they had the height of the centre of the sun. Such was the form of the one that Manlius the mathematician erected at Rome under the auspices of Augustus. But in comparatively modern times astronomers have remedied this defect in a still more happy manner, by using a vertical or horizontal plate pierced with a circular hole which allows the rays of the sun to enter into a dark place, and in fact to form a true image of the sun on a floor or other convenient receptacle, as we find is the case in many continental churches.