vs. 4 ] ASCENSIONAL DIFFERENCES OF SIGNS 43 zenith. RETW the equator, and P the north pole. R is the point where tht equator intersects the local meridian. Then the arc ZR defines the latitude of the place. OY is, the gnomon erected at the local place O perpendicular to the plane of the horizon. Let RD be perpendicular to the plane of the horizon and YX parallel to RO. Then we have two right-angled triangles YOX and RDO, right angled at O and D respectively. These triangles are similar and their corresponding sides are as follows: base upright hypotennse DO(=Rsin^) RD^(RsinC) RO(=R). OX(=s) YO(=g) YX^Jg^Tj Comparing these triangles, we have (1) and (2). A rule for finding the ascensional differences of the tropical signs Aries, Taurus, and Gemini* : 4. From the declinations of the last points of the (first three) signs should be obtained, as before, 1 their ascensional differences in terms of asus. When (each of them is) diminish- ed by the preceding (ascensional difference, if any,) they become (the asus of ascensional difference) for Aries, Taurus, and Gemini respectively. That is, if x,y, and z be the ascensional differences of the last points of the signs Aries, Taurus, and Gemini respectively, then the ascensional differ- ences of the signs Aries, Taurus, and Gemini are #, y— x, z-y respectively. We have already seen that the ascensional difference of the Sun is the difference between the times of rising of the Sun on the local and equatorial horizons. The ascensional difference of the sign Aries is the difference between the times of its rising above the local and equatorial horizons. Since the first point of Aries rises simultaneously at both the horizons, therefore the ascensional difference of Aries is equal to the ascensional difference of the last point of Aries (for which the celestial longitude X is equal to 30 d ). Simi- larly, the ascensional difference of Aries and Taurus (taken together j is equal to the ascensional difference of the last point of Taurus (for which X=60°). 1 Vide supra, Chapter II, stanza 18.
