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II.11 II. VĀSIṢṬHA-SIDDHĀNTA 37 [छायातो लग्नं लग्नतः छाया च] द्वादशभिः सच्छायैर्मध्याह्नोनेर्भजें'द्रसहुताशम्' | अपराह्ने चक्रार्धाद्विशोध्य सार्कं भवति लग्नम् || ११ || Lagna from shadow and vice versa 11. Add 12 to the shadow (of the twelve-digit-gnomon, measured in digits) at any time of the day, and deduct from it the mid-day shadow for the day. Divide 36 by this and take the result. This result taken as rāśis, plus the Sun in rāśis is the lagna at the moment, if it is forenoon. If afternoon, deduct this from the Sun plus six rāśis and the lagna is got. The formulae (a) for the forenoon and (b) afternoon respectively can be expressed thus: (a) Lagna = Sun + 36/(12 + shadow – noon shadow). (b) Lagna = Sun + 6 – 36/(12 + shadow – noon shadow). What is called lagna is the Orient Ecliptic Point, i.e. the point of the ecliptic rising on the eastern horizon. Example 10. (a) On a certain day, the Sun is 9 rāśis and the mid-day shadow 12. At a time in the morning the gnomonic shadow is 36. Find the lagna for the moment. (b) On a certain day, the longitude of the Sun is 6 rāśis and the noon-shadow 6 digits. At a time in the evening the gnomonic shadow is 24 digits, find the lagna for that moment. (a) From formula (a), Lagna = 9 + 36/(12 + 36 – 12) = 9 + 1 = 10, rāśis. Hence the first point of Kumbha is rising in the east. (b) From formula (b), Lagna = 6 + 6 – 36/(12 + 24 – 6) = 12 – 1 6/30 = 10 24/30 rāśis. Hence the 25th degree of Kumbha is rising in the east. These rules are rough and there is no question of strictly proving them. But we can explain them thus. From sunrise to noon, as the altitude of the Sun increases, the lagna goes on increasing and the shadow decreasing, till it is shortest at noon. Therefore the increase in lagna can be roughly expressed as, a/(shadow + b), where a and b are constants to be determined. Now, if the place is supposed to be situated on the equator, and the ecliptic on which the Sun moves is supposed to coincide with the celestial equator, then at noon the shadow will be zero. At that time the increase in lagna (after sunrise) would be 3 rāśis, as the Sun has reached an altitude of 90°. Therefore 3 = a/(0 + b). Again, seven and a half nāḍīs after sunrise, the Sun would have risen to an altitude of 45°. So the increase in lagna now is 1 1/2 rāśis and the shadow is 12 tan 45° = 12 digits. Therefore, 1 1/2 = a/(12 + b). Solving these two equations for a and b we get a = 36, b = 12. Therefore the increase in lagna is 36/(shadow + 12), of course, on the given two assumptions. But the place may not be on the equator and the ecliptic does not coincide with the celestial equator and the Sun moving on it has a varying declination, with the result that generally the noon-shadow is not zero. According to the length of the noon-shadow at other times also there will be an increase in the shadow over what 11a. A1. द्वादशभिः; A2. द्वादभिः. A. सछायै A2. द्रसंजताशं b. B. मध्याह्नाने. B. हृतांशः. A1. ०द्रसजताशं; c. A.B. चक्रार्द्धाद्