ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

324 BRAHMAGUPTA A GREAT CRITIC Rsine of the Sun's prime vertical altitude.¹ Bhāskara also gives the rule in his Mahābhāskarīya : Multiply the Rsine of the (Sun's) greatest declination by the Rsine of the Sun's true (sāyana) longitude; then divide (the product) by the Rsine of the colati- tude, the result is (the Rsine of) the agrā of the true Sun. When that (agrā) is less than the latitude and when the Sun is also in the northern hemisphere, multiply (the Rsine of the Sun's agrā) by (the Rsine of) the colatitude; the result is the Rsine of the Sun's prime vertical altitude.² The condition laid down in the rule that the "Sun's agrā should be less than the latitude" is incorrect. The error was originally comitted by Āryabhaṭa and Bhāskara followed it. This error was noticed by Brahmagupta³. Bhāskara I, however, corrected the error in the Laghu-Bhāskarīya. There he gives the correct conditions. It is not the agrā that should be less than the latitude, it is the Sun's declination (or rather the Rsine of the Sun's northern declination as we have in the Laghu- Bhāskarīya), which should be less than the latitude (or rather the Rsine of the latitude). This condition is necessary for the existence of the prime vertical shadow of the gnomon. It may be pointed out that the commentators of Āryabhaṭa I have interpreted the rule given by Āryabhaṭa I as conveying the correct meaning; they say that Āryabhaṭa also meant declination when he used the term agrā.

1. परमापक्रमजीवामिष्टज्यार्हतां ततो विभजेत् । ज्यालम्बकेन लब्धार्काग्रा पूर्वापरे क्षित्तिजे ॥ सा विषुवज्ज्योना चेद् विषुवदुदग्लम्बकेन संगुणिता । विषुवज्ज्या विभक्ता लब्धः पूर्वापरे शंकुः ॥ — Ārya. V. 30-31
2. स्फुटरविभुजनिघ्नां यां परां क्रांतिजीवां । हरतु समलम्बज्या कलापेन भूयः ॥ स्फुट दिवस कराग्रा सा यदाऽक्षांशहीना । रविरपि षडगोले चोत्तरे लम्बकृघ्नाम् ॥ अक्षज्यया हरेद् भूयः शंकुः स्यात् सममण्डले । तद् वर्गं व्यास कृत्योर्यद् विश्लेषं तत्पदं प्रभा ॥ — MBh. III. 37-38
3. उत्तरगोलेऽग्रायां विषुवज्ज्यातो यदुक्तमूनायाम् । सममण्डलमस्तदसत् क्रान्तिज्यायां यतो भवति ॥ — BrSpSi. XI. 22