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388 From the third and fifth columns we find the probable errors of Tycho's standard right ascensions = ±24″.1, and of his standard declinations = ±25″.9. Of course the accuracy of most of his star-places must be much less, as they were neither as often nor as carefully determined as those of the standard stars, and were vitiated not only by refraction, but also by aberration and nutation. See above, pp. 351 and 353.

In order to remove any doubt which the reader might have as to the correctness of the opinion set forth above on p. 360, that the azimuth error of 15′ detected in some of the instruments in November 1586 does not prove all Tycho's instruments to have been erroneously placed during all the years previous to 1586, I have tested the matter by calculation. Azimuths were never very extensively observed at Hveen, and the only series of observations sufficient for my purpose was that made in the beginning of 1582, when a number of altitudes and azimuths of bright stars were measured. Many of these were taken too near the meridian to be of any use in this case, but there are many observations made in the prime vertical, by which a great error in the assumed zero of the azimuth circle would easily be detected. I have made use of all the observations taken in or near the prime vertical during March 1582 (except of a few which were vitiated by some great error of copying, observing, or in the identification of the star), and of one observation of a Tauri in azimuth 69º on February 25. The following mean declinations for 1582 were computed from Auwers' Bradley the reductions to apparent declinations for 1st March being appended:—

Assuming the observed altitudes to be correct, the azimuth corresponding to each altitude was computed by the formula

cos A = $sin φ sin h - sin δ⁄cos φ cos h$