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

IV. 16 IV. THREE PROBLEMS 87 Greeks solved them by an extension of Manelau's Theorem to figures on the sphere.) We shall con- tent ourselves with giving the formulae for solution and refer to them whenever necessary by way of proof. Let ABC (fig. 5) be a spherical triangle, right angled at C, and R the radius of the sphere. I. Cos AB = Cos AC × Cos BC ÷ R II. sin BC = sin AB × sin A ÷ R III. Cos A = R × Cos AB. sin AC/sin AB. Cos AC IV. Cos A = Cos BC × sin B ÷ R V. Cos AB = R. Cos A × Cos B/(sin A × sin B) These, together with the general identities, cosθ = sin (90° – θ), sin² θ + Cos² θ = 1, will suffice for explaining the formulae occurring in the text. [Fig. IV. 5] [रविचन्द्रयोः क्रान्तिः] जीवाऽध्यर्धशतां' (शघ्नै) काषष्टि (र्दि) नेशकाष्ठा (ज्या) । चन्द्रस्य सविक्षेपस्तदपक्र(मो) राशिपादे(भ्यः) ॥ १६ ॥ Declination of the Sun and the Moon 16. The sine of Sun's declination is found by multiplying the sine of its longitude by 61 and dividing by 150. Its arc is the declination. The declination of the Moon found thus is the mean declination. Its true declination is the mean declination plus latitude. The intervals of declinations for intervals of quarter signs are given (in the next two verses, 17-18). This is the formula: sin dec. = sin sāyana long. × 61/150. From this the arc forming the declina- tion is found by using the tables, and then the true declination of the Moon, using this. It must be noted that when the longitude reckoned from r (i.e. sāyana long.) is within 6 signs, the declination is North, and when more than 6 signs it is South. In the case of the Moon, if the declination and the latitude are of the same direction they should be added to get the true declination. If they are of different directions, their difference is the true declination, its direction being that of the greater. The author has not mentioned this because it is obvious. Example 3 (a). Find the maximum declination of the Sun. Obviously, the maximum sine of the Sun's longitude (i.e. when the Sun is 3 signs or 9 signs) will give the maximum declination. It is therefore given by sin dec. = 120' × 61 ÷ 150 = 48' 48". Its arc, 24° is the maximum declination. 16a. A. जीवाध्यार्द्ध०; B. जीवा व्या-र्द्ध; C. जीवाव्यर्द्ध; D. [साङ्कुलिप्ता] . A.B. काष्टान्तः; C. काष्ठान्तः; D. जीवाव्यध्यर्ध. B. सिताशा; A.C.D. शतांशाः D. काष्टान्तः b. A. सैकाःषष्टि; B. सैका षष्ठि; C. सैका षष्टि; d. A.B.C.D. तदपक्रम. A.C. ०पादेन्यः