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318 from the first one by the equator being movable and attached to a revolving (but not graduated) declination circle, while a smaller and graduated declination circle carried sights. The undivided circle might very well have been left out and the graduated one fixed to the equator. The outer circle (meridian) was nearly five feet in diameter.

The third and most important instrument of this kind was mounted in the largest crypt of Stjerneborg and was far more extensively used by Tycho, who considered it one of his most accurate instruments. It consisted merely of a declination circle 9$1⁄2$ feet in diameter, and a semicircle, which represented the part of the equator below the horizon, and rested on eight stone piers. The former has two pointers turning round a small cylinder in the centre of the polar axis, and perpendicular to the plane of the circle, and each furnished with an eye-piece sight, while a third sight slides along the equator. The polar axis (of iron, but hollow) could be adjusted in inclination and azimuth by screws, which acted on a square plate in a hole in which the lower pointed end of the axis fitted. By reversing the circle double observations of declination might be taken, using first one sight and then the other, and Tycho remarks that this instrument had the further advantage over the two others that stars near the equator were as easily observed as those more distant from it, as the equatorial arc was at some distance behind the declination circle.

Circles and semicircles had naturally been in use from a very early date. We need only refer to the astrolabium of Ptolemy, which consisted of a graduated circle inside which another circle could slide, carrying two small cylinders diametrically opposite to each other, while the instrument was kept vertical by a plumb-line. This astrolabium was imitated by many successive astronomers; among others, by Abul Wefa,