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

IV. 54 IV. THREE PROBLEMS 129 (iii) Agrā = 48′ 48″ × 60 ÷ 96′ = 30′ 30″. (iv) The Sun being in the six signs from Libra, ‘Perpendicular’ = (36′ + 30′ 30″) × 30 ÷ 120′ = aṅg. 16-37.5 north. (v) ‘Base’ = √(27½² − 16⅝²) = aṅg. 21-54. Or by the simplified formula, Perpendicular = (12 × 72′ + 30 × 24′ 24″)/96 = aṅg. 16-37.5. (+ is taken, as the declination is south). From this the ‘Base’ is calculated to be aṅg. 21-54 as before. The direction is graphically represented thus: [Figure IV. 17: Right-angled triangle ABC with right angle at A. Base AB labeled "Base" and "21-54", perpendicular AC labeled "Perp." and "16-37.5", hypotenuse BC labeled "Shadow" and "27-30". Angle B labeled "Shadow angle". Line CB extends backwards past B towards the southwest, labeled with an arrow "To Sun".] Fig. IV. 17 Here too, the angle of shadow is ABC, and the direction of the Sun is opposite to the shadow, making the same angle. We shall now prove the steps, taking them one by one: (i) Shadow-Hypotenuse: In the right angled triangle having the shadow as base and the twelve digit gnomon as perpendicular, the shadow-hypotenuse is the hypotenuse. Hence by the well-known formula, base² + perpendicular² = hypotenuse², √(shadow² + gnomon²) = shadow- hypotenuse. As the gnomon is 12 aṅgulas and the shadow too is measured in aṅgulas, the shadow- hypotenuse measured in aṅgulas = √(shadow² + 12²). (ii) Sūryāgrā: This is the distance between the line joining the rising and setting points and the diurnal circle (see Fig. 18). This is called śaṅkvagra by the earlier Bhāskara I and his followers and śaṅkutalam by the later Bhāskara II and others. It has been mentioned that, as seen from places on the earth other than the equator, since the circles on the stellar sphere are bent southwards (this is from the point of view of people in the northern hemisphere) the diurnal circles following these are also bent southwards. Therefore by the intersection of the arcs on the stellar sphere and the celestial sphere several right angled triangles are formed by their sine lengths, which triangles are called ‘Latitude-caused triangles’ (Akṣakṣetras). From the similarity of these triangles, when the length elements of one are known the corresponding length elements of another may be calculated by the rule of proportion. Among these, two similar triangles answer to our need, in one, which is well-known, sin lat is the base, sin colat is the perpendicular, and