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

IV. 34 IV. THREE PROBLEMS 109 SS₂D₂, right angled at S₂. SS₂ is the declination. Angle S₂D₂S = ZM₂ = latitude. Therefore, from the fundamental formula IV, sin D₂S₂ = Cos S₂D₂S × sin SS₂ × radius ÷ (sin S₂D₂S.cos SS₂) = Cos lat × sin dec × radius ÷ (sin lat.× cos dec.) = sin colat × sin dec × radius ÷ (sin lat.× day-diameter/2) = sin colat × sin dec × 240′ × (sin lat.× day-diameter) (From sin D₂S₂ arc D₂S₂is found and converted into time at 10 vināḍīs per degree, as mentioned before.) We shall prove the author’s method by showing that his formula is equal to this. The author’s formula is: Sin D₂S₂ = √sin² alt. at prime-vertical − sin² dec × 240 ÷ day-diameter = √sin² dec × 120² ÷ sin² lat − sin² dec × 240 ÷ day-diameter (∵ sin D₂S = sin SS₂ ÷ sin S₂D₂S, by fundamental formula II) = √(sin² dec (120² − sin² lat) ÷ sin² lat × 240 ÷ day-diameter. = √sin² dec. sin² colat ÷ sin² lat × 240 ÷ day-diameter). = sin dec × sin colat × 240 ÷ (sin lat × day-diameter) This is identical with the formula derived by us. (The author himself will be giving this form in the next verse.) We shall now show how the formula for the condensed work is got. The formula for half-cara is: Sin half-cara = sin dec × sin lat × 240′ ÷ (sin colat × day-diameter) Multiplying the numerator and the denominator of the formula arrived at by (sin lat × sin colat), we have, Sin Rt. asc. = sin declination × sin lat × sin² colat × 240 ÷ (sin² lat × day-diameter × sin colat) = sin half-cara × sin² colat ÷ sin²lat, given by us. Again, sin² colat ÷ sin² lat. = 120′ × 12 ÷ equinoctial hypotenuse² ÷ (120′ × equinoctial shadow ÷ equinoctial hypotenuse)² = 12²/equinoctial shadow² = 144 ÷ square of equinoctial shadow. So this can be substituted for sin² colat ÷ sin² lat. It must be noted that if the declination is greater than latitude, i.e. if sin dec > sin lat, then sin colat > day-diameter. Therefore sin rt. asc. > 120′, for which there is no arc, which means that at no altitude, or at no time does the Sun cross the prime vertical. This is what was referred to earlier and here shown mathematically. ‘(ख) जीन’घ्नी क्रान्तिज्या लम्बघ्नी ध्रुवगु(ण)हृदै(र्घ्यहृता) । तच्चाप(स्य) ‘रसां’शः सक[T]लः (स) दि(वस)वृद्ध्यर्धः ॥ ३४ ॥ 34. Multiply sine declination by 240 and again by sin co-latitude and divide by the product of the sine of latitude and day-diameter. Find its arc in degrees and divide by six. (The time in nāḍīs, taken by the Sun to move from