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apparent places of the sun, two sides (real and apparent declination) and one angle (180° minus the parallactic angle) were known, from which the third side could be computed, which was the effect of refraction in altitude. This is a most inconvenient and troublesome method, and must have given the computers plenty to do, if the observations were really extensively used in this way for the construction of his refraction tables. For these investigations he assumed the real declination (i.e., corrected for refraction) as equal to the declination as observed on the meridian, as he thought the refraction at the meridian altitude at summer solstice (571/2°) insensible. He assumed, in fact, that refraction disappeared already at 45°, where it in reality amounts to 58″. Unluckily Tycho spoiled the refraction table which he constructed from his solar observations by assuming, with all previous astronomers, from Ptolemy down to Copernicus, that the horizontal parallax of the sun was equal to 3′. It is remarkably strange that Tycho should not have endeavoured to deduce this important constant from new observations which ought to have shown him that it was for his instruments insensible. This was the only astronomical quantity which he borrowed from his predecessors, and it was a wrong one.^[1] The refractions, as given by him, must therefore be diminished by 3′ × cosine of altitude, and it is interesting to see that he was well aware of the fact that the refractions as found by the stars were different from those which he had mixed up with the imaginary solar parallax, as he gives a separate table of stellar refraction, in which the quantities are smaller than those in the solar refraction table by 4′ 30″; so that according to him refraction becomes insensible in the case of stars at 20° altitude (where it is in reality 2′ 37″). Possibly the refraction of stars was not as carefully looked into as that of the sun,

1. Among Tycho's original observations there is at the beginning of the year 1581 a table of solar parallax for every degree (beginning with 2′ 58″ at 1°) up to 60° altitude. Compare Progymn., p. 80.