Vateshvarasiddhanta 3.480_44

3.480_44

patropapattiḥ .
citra dvitīyaṃ draṣṭavyam. natra cara rana tribhujāgrajyā' tribhuja-
jyā = dvitīyabhujāgrajyā = koṭijyā'
mūlena
=
√ tribhujāgrajyā (tribhujāgrajyā) (tri-sujāgrajyā)
= √ (tribhujyā) (tribhujyā)
dvitīya bhujāgrajyā koṭijyā .
cara = ravakramajyā. mata vyāsa. mara utkramajyāḥ atha rekhāgaṇita
tṛtīyādhyāyena mara rata cara rakha ujyā (vyāsa-ujyā) ujyā vyāsa
- ujyā = kamajyā
spaṣṭādhikāraḥ
mūlena
√ ujyā (vyāsa-ujyā) = √ ujyā × vyāsa-ujyā = kramajyā
ata upapannamācāryoktam .. 53 ..
hi. bhā- aba bhujajyā aura koṭijyā meṃ se eka dūsare ke jñāna aura kramajyā ke
jñāna kahate haiṃ. trijyā aura bhujāgrajyā ke vargāntaramūla dvitīya bhujāgrajyā hotī hai arthāt
vijyā aura bhujajyā ke vargāntara mūla koṭijyā tathā trijyā, aura koṭijyā ke vargāntaramūla
bhujajyā hotī hai. yā trijyā aura bhujāgrajyā ke yogāntara ghāta mūla dvitīya bhujāgrajyā yā
koṭijyā hotī hai. vyāsa meṃ utkramajyā ko ghaṭākara aura utkramajyā se guṇakara mūla lene se
kamajyā hotī hai. vyāsaguṇita utkramajyā meṃ utkrama jyā varga ghaṭākara mūla lene se kramajyā
hotī hai ..53..

Here is the demonstration: / Figure 2 is to be seen. Here cara² = rana² = radius² - base-perpendicular sine² = radius² - base-sine² = the second base-perpendicular sine² = cosine². / By taking the root: / = √(radius + base-perpendicular sine)(radius - base-perpendicular sine) / = √(radius + base-sine)(radius - base-sine) / = the second base-perpendicular sine, [that is,] the cosine. / cara = rava = sine. mata = diameter. mara = versed sine. Now, by the third chapter of Euclid's Elements, mara × rata = cara × rakha = versed sine × (diameter - versed sine) = versed sine × diameter - versed sine² = sine². / Chapter on True Motion. / By taking the root: / √(versed sine × (diameter - versed sine)) = √(versed sine × diameter - versed sine²) = sine. / Thus what was stated by the Teacher is demonstrated. || 53 || / Hindi commentary: Now he states the determination of one from the other between the base-sine and cosine, and the determination of the sine. The root of the difference of the squares of the radius and base-perpendicular sine is the second base-perpendicular sine; that is, the root of the difference of the squares of the radius and base-sine is the cosine, and the root of the difference of the squares of the radius and cosine is the base-sine. Or, the root of the product of the sum and difference of the radius and base-perpendicular sine is the second base-perpendicular sine or cosine. Subtracting the versed sine from the diameter and multiplying by the versed sine, taking the root yields the sine. From the diameter multiplied by the versed sine, subtracting the square of the versed sine and taking the root yields the sine. || 53