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

IV. 28 IV. THREE PROBLEMS 101 iii. Earth-sine × 120 ÷ √sin² earth-sine + sin² dec = sin lat. From this the latitude is found iv. 90° — latitude = co-latitude. Its sine. co-lat. Example 9. At a certain place on a certain day, the vināḍis of cara are 390 1/3. The day-diameter is 219' 15". Find the latitude of the place, and sine co-latitude. The sine of declination required for the formulae is, by (IV.23), √ 120² — (219' 15"/2)² = 48' 48". i. Degree of half-cara = 390 1/3 ÷ 20 = 1171/(3 × 20) = 19° 31'. From this, sine half-cara = 40' 4". ii. (Earth)-sine = 40' 4" × 219' 15" ÷ 240' = 36' 36". iii. Sin lat.= 36' 36" × 120' ÷ √36' 36"² + 48' 48"² = 120 ÷ √1 +16/9 = 120' × 3/5 = 72'. From this, lat = 36° 52'. iv. Co-latitude = 90° — 36° 52' = 53° 8'. From this sine co-latitude = 96'. The rules are thus derived: a. From the rule, vināḍikās of cara = minutes of half-cara ÷ 3. By transposing, we have: Minutes of half-cara = vināḍikās of cara × 3. Degrees of half-cara = vināḍikās of cara × 3/60 = vināḍikās of cara/20, which is (i). b. From the rule, sine half-cara = sin lat.× 240 × sin dec ÷ (sin co-lat x day-diameter), we get; Sin dec = sin half-cara × sin co-lat.× day-diameter ÷ (240 × sin lat) = sin co-lat × earth-sine ÷ sin lat. Using this in (iii) above, we have: Sin lat = earth-sine × 120' ÷ √earth-sine² + sin² co-lat × earth-sine² ÷ sin²lat. 120' ÷ √sin²lat + sin²co-lat ÷ sin²lat. = earth-sine × 120' ÷ (earth-sine) √1 + sin² co-lat = 120' ÷ √sin² lat + sin² co-lat ÷ sin²lat. sin²lat = 120' ÷ √1/sin²lat = √ sin²lat = sin lat, thus proving (iii). From this the latitude is got. Then, ∵ latitude + co-latitude = 90°, Co-latitude = 90° — latitude. It should be noted that of the sin declination and the day-diameter required in the rules, one is sufficient, because the other can be got from that. As for the word khaṇḍa, meaning 'interval' or 'dif- ference', we have already said that it is piṇḍa ('the whole') we get first, and thence the khaṇḍa. As for the reading, we have corrected, carathaṇakapakṣāṁśa, into carakhaṇḍakhapakṣāṁśa, making ka into kha, because 'twenty' is required here as the divisor. This is the only correction we have made. But TS, followed by NP, have made several corrections, not realising that if Bhaṭṭotpala's reading is adopted no other correction would be required.