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334 numerical data he computed tables for the apparent motion of the sun, which he remarks are worthy of considerable confidence, as they depend on observations made with three or four large instruments made of metal, and capable of determining the position of the sun within 10″, or at most 20″; and by comparing the tables with observations by Regiomontanus, Walther, the Landgrave, and Hainzel, he shows that they represent the observed places within a small fraction of a minute, while the Alphonsine tables and those of Copernicus are often 15′ or 20′ in error.

The solar observations at Uraniborg led to a result which Tycho does not seem to have anticipated. The colatitude, as found by the greatest and smallest altitude of the sun at the solstices, differed from that deduced from observations of the Pole Star by a considerable quantity, which sometimes amounted to 4′. Having ascertained that the discrepancy did not arise from instrumental errors, he was led to attribute it to the effect of refraction. As soon as the great armillæ at Stjerneborg were finished, he instituted systematic observations to prove this, and to determine the amount of refractions at various altitudes. Having first found, by following the sun throughout the day with the armillæ, that the declination apparently varied, as stated by Alhazen in his book on optics, he repeatedly in the years 1585 to 1589 devoted a whole day, generally in June, when the declination of the sun changed very slowly, to investigations on refraction. With an altazimuth quadrant he measured at frequent intervals the altitude and azimuth, and from the latitude of the observatory, the azimuth and the decimation, he computed the altitude, which, deducted from the observed altitude, gave the amount of refraction. Another method was by observing simultaneously with the quadrant and with the armillæ. In the triangle between the pole and the true and