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altitudes of the two celestial bodies were measured, and occasionally their azimuths, while their declinations were observed by the armillæ, and their meridian altitudes as often as opportunity offered. After sunset the same sextant was employed to measure the distance of Venus from certain conspicuous stars near the zodiac (Aldebaran, Pollux, and some others in the same constellations), while, as before, altitudes and declinations were also observed. In deducing the positions of the stars observed, the motion of Venus and the sun in the interval between the day and night observations was taken into account. By simple trigonometrical operations the difference of right ascension between the sun and a zodiacal star was computed, and as the right ascension of the sun was known from the solar tables, the absolute right ascension of the star was thus found from the observations, while the declination was directly measured. All the stars thus determined were connected by distance measures with the star α Arietis, which he preferred to γ Arietis, which by Copernicus had been adopted as principal standard star, as being nearest to the vernal equinox, but which Tycho found too faint to be conveniently observed by moonlight. Each observation thus gave a value for the right ascension of α Arietis. During the following six years Tycho repeated these observations as often as an opportunity offered, and, in order to eliminate the effect of parallax and refraction, he combined the results in groups of two, so that one was founded on an observation of Venus while east of the sun, the other on an observation of Venus west of the sun; while the observations were selected so that Venus and the sun as far as possible had the same altitude, declination, and distance from the earth in the two cases. From the observations of 1582 Tycho selects three single determinations, and from the years 1582–88 twelve results, each