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

IV. 22 IV. THREE PROBLEMS 95
[मध्याह्नच्छाया]
अपमोनयुताऽक्षज्यां त्रिज्यात्कृतिविशेषमूलेन ।
छिन्द्याद् द्वादशगुणितां लब्धा माध्याह्निकी छाया ॥ २२ ॥
Sine zenith distance
22. Subtract the Sun’s declination from the latitude (of the place), if the decli-
nation is north, and add if it is south. The midday Sun’s Z.D. is got. Find its
sine and multiply by twelve. Divide this by the root of the difference of the
squares of the radius and sine ZD. The mid-day shadow is obtained in aṅgulas.
The following are the rules:
i. Degrees of zenith distance = Latitude ∓ Sun’s declination (the upper sign being used for
north declination and the lower for the south.)
ii. Mid-day shadow = 12 × sin ZD ÷ √120² − sin² ZD (where 120 is written for radius).
It must be noted that the incompleteness mentioned in connection with the second formula if the
previous work is found here too.
Example 6. The latitude of a place is 25° 9′. The Sun’s declination is 11° 44′ , south, (the Sun being in the
part of the ecliptic beginning from Libra). Find the mid-day shadow.
The declination being south, ZD = 25° 9′ + 11° 44′ = 36° 52′. Sin ZD = 72′.
∴ Midday shadow = 12 × 72 ÷ √120² − 72² = 12 × 72 ÷ 96 = 9 aṅgulas.
The rules are explained thus:
From the previous rule, Latitude = zenith distance ± declination, (+ for north declination, and
− for south declination), we have, zenith distance = latitude ∓ declination, (for north and south
declinations, respectively).
From this sine zenith distance is got. Using this, the shadow is obtained from the previous rule,
sin ZD = shadow × 120 ÷ √shadow² + 144.
Squaring both sides, sin² ZD = shadow² × 120² ÷ (shadow² + 144).
sin² ZD × (shadow² + 144) = shadow² × 120²
sin² ZD × shadow² + 144 sin² ZD = shadow² × 120²
120². shadow² − sin² ZD. shadow² = 144 sin² ZD
shadow² = 144 sin² ZD ÷ (120² − sin² ZD)
shadow = 12 sin ZD ÷ √120² − sin² ZD.
22. Quoted by Utpala on BS 2, p. 61.
22a. A.D. अपनोन. A.C.D. U. ॰क्षज्या c. A.C.D. गुणिता
b. A. तांत्रिकृति; C.D. U. तत्तिर्ज्याकृति. A. मूला d. A. माध्याह्नकी