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Rh being the mean of two results found in the manner just described. The fifteen values of the right ascension of Arietis agree wonderfully well inter se, the probable error of the mean being only ±6″, but the twenty-four single results in the twelve groups show rather considerable discordances, the greatest and smallest differing by 16′ 30″. But anyhow the final mean adopted by Tycho is an exceedingly good one, agreeing well with the best modern determinations. He adopts for the end of the year 1585 26° 0′ 30″, the modern value for the same date being 26° 0′ 45″.

From the absolute right ascension of Arietis thus determined, and the directly observed declination, Tycho determined the co-ordinates of other stars by measuring the distance from  Arietis and the declination, after which the spherical triangle between the pole and the two stars (in which the three sides were known) gave the angle at the pole or the difference of right ascension. Proceeding thus from one star to another round the heavens, Tycho determined the right ascensions first for four, then for six, and finally for eight principal standard stars; and as the sums of the differences of right ascension in the three cases only differ a few seconds from 360°, he imagined that he had proved his results to be extremely accurate. It is needless to say that the accuracy cannot be so great as Tycho fondly hoped, as the errors of observation would be increased by neglect of refraction and by his ignorance of the existence of aberration and nutation. But it must be conceded that Tycho's results were an immense improvement on the positions of fixed stars as previously known, as the comparison with the best modern star-places for the nine stars reduced to the end of 1585 gives the probable error of Tycho's standard right ascensions equal to ± 24″.1, and that of his