The actual accuracy obtained by Tycho in his observations naturally varied considerably according to the nature of the observation, the care taken, and the period of his career at which it was made. The places which he assigned to nine stars which were fundamental in his star catalogue differ from their positions as deduced from the best modern observations by angles which are in most cases less than 1′, and in only one case as great as 2′ (this error being chiefly due to refraction (chapter II., § 46), Tycho’s knowledge of which was necessarily imperfect). Other star places were presumably less accurate, but it will not be far from the truth if we assume that in most cases the errors in Tycho’s observations did not exceed 1′ or 2′. Kepler in a famous passage speaks of an error of 8′ in a planetary observation by Tycho as impossible. This great increase in accuracy can only be assigned in part to the size and careful construction of the instruments used, the characteristics on which the Arabs and other observers had laid such stress. Tycho certainly used good instruments, but added very much to their efficiency, partly by minor mechanical devices, such as the use of specially constructed “sights” and of a particular method of graduation,[65] and partly by using instruments capable only of restricted motions, and therefore of much greater steadiness than instruments which were able to point to any part of the sky. Another extremely important idea was that of systematically allowing as far as possible for the inevitable mechanical imperfections of even the best constructed instruments, as well as for other permanent causes of error. It had been long known, for example, that the refraction of light through the atmosphere had the effect of slightly raising the apparent places of stars in the sky. Tycho took a series of observations to ascertain the amount of this displacement for different parts of the sky, hence constructed a table of refractions (a very imperfect one, it is true), and in future observations regularly allowed for the effect of refraction. Again, it was known that observations of the sun and planets were liable to be disturbed by the effect of parallax (chapter II., §§ 43, 49), though the amount of this correction was uncertain. In cases where special accuracy was required, Tycho accordingly observed the body in question at least twice, choosing positions in which parallax was known to produce nearly opposite effects, and thus by combining the observations obtained a result nearly free from this particular source of error. He was also one of the first to realise fully the importance of repeating the same observation many times under different conditions, in order that the various accidental sources of error in the separate observations should as far as possible neutralise one another.