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20 one, so that they always formed a right angle. The instrument could be used in two ways. Two sights might be fixed at the ends of the shorter rod, and one at the end of the longer rod, and the observer, having placed the latter close before his eye, moved the cross-rod along until he saw through its two sights the two objects of which he wanted to measure the angular distance. Or one of the sights of the shorter arm might be movable, and the observer first arbitrarily placed the shorter arm at any of the graduations on the longer one, and then shifted the movable sight along until he saw the two objects through it, and a sight fixed at the centre of the transversal arm. In either case the graduations and a table of tangents furnished the required angle. Tycho's instrument was of the latter kind, and was made according to the directions of Gemma Frisius. He got his friend Bartholomæus Scultetus to subdivide it by means of transversals, which method of subdividing small intervals was then beginning to be used, and which Tycho ascribes to Homilius. The earliest observations stated to have been made with the radius are from the 1st of May 1564, and Tycho says that he had to use it while his tutor was asleep, from which we see that Vedel had even at that time not given up his resistance to his pupil's scientific labours. The observer soon found that the divisions did not give the angles correctly, and as he could not get money from Vedel for a new instrument, he constructed a table of corrections to be applied to the results of his observations. This is deserving of notice as the first indication of that eminently practical talent which was in the course of years to guide the art of observing into the paths in which modern observers have followed. Kepler, who more than any one else was able to appreciate his great