The circle A was fixed perfectly upright in the meridian, so that the greatest altitude of the sun each day could be observed; the circle B was then placed exactly in the plane of the earth’s equator by adjusting the line joining C and D to the part of the heavens between the Bears, about which the stars appear to revolve. This done, the occurrence of the equinox was waited for, at which time the shadow of the part of the circle E must fall upon the part marked F, so as exactly to cover it. [Illustration: FIG. 7.—The First Instrument Graduated into 360° (West Side). ] We now come to the time when the circle began to be divided into 360 divisions or degrees—about the time of Hipparchus (160 B.C.). There are two instruments described by Ptolemy for measuring the altitude of the sun in degrees instead of in fractions of a circle. They, like the gnomon, were used for determining the altitude of the sun. The first, Fig. 7, consisted of two circles of copper, one, C D, larger than the other, having the smaller one, B, so fitted inside it as to turn round while the larger remained fixed. The larger was divided into 360°, and the smaller one carried two pointers. This instrument was placed perfectly upright and in the plane of the meridian, and with a fixed point, C, always at the top by means of a plumb-line hanging from C over a mark, D. On this small circle are two square knobs projecting on the side, E and F. When the sun was on the meridian the small circle was turned so as to bring the shadow of the knob E over the knob F, and then the degree to which the pointer pointed was read off on the larger circle. And of course, as the position of the knobs had to be changed as the sun moved in altitude, the angle through which the sun moved was measured, and the circle being fixed, the sun’s altitude could always be obtained. The other instrument consisted of a block of wood or stone, one side of which was placed in the plane of the meridian; and on the top corner of this side was fixed a stud; and round it as a centre a quarter of a circle was described, divided into 90°. Below this stud was another, and by means of a plumb-line one stud could always be brought over the other; so that the instrument could always be placed in a true position. At midday then, when the sun was shining, the shadow of the upper stud would fall across the scale of degrees, and at once give the altitude of the sun. Ptolemy, who used this instrument, found that the arc included between the tropics was 47⅔°. The result of all these accurate determinations of the solstices and equinoxes was the fixing of the length of the year. We have so far dealt with the methods of observation which depend upon the use of the horizon and of the meridian; we will now turn our attention to extra-meridional observations, or those made in any part of the sky.