पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

IV. 30 IV. THREE PROBLEMS 103 Example 10. Find the right ascensions of the points of the ecliptic ending Aries, Taurus, and Gemini, i.e. longitudes 30°, 60° and 90°. From them find their respective differences. Sin 30° = 60′, sin 60° = 103′55″ and sin 90° = 120′. Sin dec. of the points ending Aries etc. are, respectively, 24′24″, 42′15″ and 48′48″. The respec- tive day-diameters are 235′, 2244′38″, and 219′15″. (a) For the point 30°, sin Rt. asc = √60′² – 24′ 24″² × 240 ÷ 235 = 54′ 49″ × 240 ÷ 235 = 55′ 59″. Its arc = 27° 49′. Multiplying by 10, the vināḍis of Rt. asc. are 27° 49′ × 10 = 278. (b) For the point 60°, sin Rt. asc. = √103′ 55″² – 42′ 15″² × 240 ÷ 224′ 38″ = 94′ 57″ × 240′ ÷ 224′ 38″ = 101′ 26″. Its arc = 57° 42′. The vināḍis of Rt. asc. = 57° 42′ × 10 = 577. (c) For the point 90°, sin Rt. asc. = √120′² – 48′ 48″² × 240′ ÷ 219′ 15″ = 109′ 37″.5 × 240′ ÷ 219′ 15″ = 120′. Its arc = 90°. The vināḍis of Rt. asc. arc. 90° × 10 = 900. The difference for Gemini = Rt. asc. for 90° – Rt. asc for 60° = 900 – 577 = 323 The difference for Taurus = Rt. asc. for 60° – Rt. asc. for 30° = 577 – 278 = 299. As the Rt. asc. of the first point of Aries is zero, the difference for Aries = Rt. asc. for 30° – Rt. asc. for 0° = 278 – 0 = 278. All these are the same as given by the author. This is how the formula is arrived at: The time taken by each sign of the ecliptic, beginning from Aries, to rise above the eastern horizon, for an observer on the equator, is in vināḍis 278, 299, etc., and their total is the time taken by any point to rise, after the rising of the First point of Aries. This is represented by the arc of the celestial equator (called the Rt. asc.) measured from the First point of Aries, and we have to find this arc. In Fig. 11, r is the First point of Aries. P is the point on the ecliptic of which the time of rising is required, and Pd is the declination of the point, equal to the arc of the horizon from the east point to the rising point. dr is the arc on the celestial equator, called the Right-ascension of the point P, which is required to be found. From the fundamental formula iv, Sin Rt. asc. = sin dr = sin Pd × cos Prd × Radius ÷ (Cos Pd × sin Prd) = sin Pr × cos Prd ÷ cos Pd (∵ by the fundamental formula ii, sin Prd = sin Pd × radius ÷ sin Pr.) = sin Pr × √Radius² – Radius². sin²Pd ÷ sin² Pr ÷ cos Pd = sin Pr × Radius √sin² Pr – sin² Pd ÷ (sin Pr × cos Pd) = Radius × √sin² Pr – sin² Pd ÷ cos Pd = 120′ × √sin² long. – sin² dec. ÷ 1/2 day-diameter = 240′ × √sin² long. – sin² dec. ÷ day-diameter. Fig. IV. 11 The arc of this is the Rt. asc. As there are 3600 vināḍis for a Rt. asc. of 360°, for the Rt. asc. got, the time is, Rt. asc. × 3600 ÷ 360 = Rt. asc. × 10. Then by subtracting the vināḍis pertaining to the Rt. asc. of the beginning of the sign from that of the end of the sign, the differences are got.