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

96 PAÑCASIDDHĀNTIKĀ IV. 23 [लम्बज्या दिनव्यासश्च] विषुवज्ज्याऽऽयामार्थवर्गविश्लेषमूलमवलम्बकः | क्रान्तित्रिज्याकृत्योरन्तरपदं द्विगुणं दिनव्यासः || २३ || Sine Co-latitude and Day-diameter 23. Square the sine of latitude and deduct from the square of the radius. Its square root is the ‘sine of co-latitude’, (its arc being the ‘co-latitude’). Square the sine of declination, deduct from the square of the radius and find its root. Twice the result is the ‘day diameter’. Now, we have (i) sine co-latitude = √(radius² − sin² latitude) (ii) Day-diameter = 2 × √(radius² − sin² declination) Example 7 (a). sin lat. = 72. Find sin co-lat, and its arc, viz. the co-lat. sin co-lat. = √(120² − 72²) = 96'. Arc 96' = 53° 8' = co-latitude. Example 7 (b). The Sun is at the end of the sign Aries. Find the day-diameter. The Sun’s longitude = rāśi. 1-0-0. Sine rāśi. 1-0-0 = 60'. ∴ sin declination = 60' (60+1)/150 = 24' 24". The day-diameter = 2 × √(120² − 24' 24"²) = 2 × 117' 30" = 235'. In the right angled triangle having the radius as the hypotenuse and the sine of latitude as the base, the sine of the co-latitude stands as the perpendicular or lamba. Therefore it is called lambajyā. By the analogy with co-sine for sine, co-tangent for tangent, and co-secant for secant, the term co- latitude for latitude, has been invented for 90'−latitude, for convenience of expression. Therefore: (since base² + perpendicular² = hypotenuse²), sin²lat + sin² co-lat = radius². From this, sin² co-lat = radius² − sin² lat. ∴ sin co-lat = √(radius² − sin² lat. As for the day-diameter, by the diurnal rotation of the earth on its axis, the Sun apparently moves round the earth every day in a circular path, at a distance from the celestial equator equal to the latitude, with the axis of the earth perpendicular to the plane of the circle. This circle is called the diurnal circle or day-circle and its diameter, the day- diameter. (See this shown in Fig.6.) The diameter can be measured thus: see Fig.9. [Fig. IV. 9: Circle with vertical diameter NP-SP, horizontal diameter CQ, center E; parallel chord SB above CQ with perpendicular SD to CQ, meeting NP-SP at A] 23. Quoted by Utpala on BS 2, p. 60. 23a. A. विवच्छायामात्यार्द्ध b. A. मूलवले लबः; C. मूलभवो लब्धः c-d. A. ॰क्रान्तिज्यात्रिज्याक्रांत्यन्तरपदं; C-D. ॰कृत्यन्तरात् पदाद् दिनव्यासः (D. पदद्द्विदिनव्यासः) d. A. द्विदिन