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

RULE OF COMPOUND PROPORTION 181 =4 purāṇa, 13 paṇas, 2 kākiṇīs and 16 varāṭakas. (One purāṇa is equivalent to 16 paṇas; one paṇa is equivalent to 4 kākiṇīs, and one kākiṇī is equivalent to 20 varāṭakas or cowries. Inverse Rule of Three This is known as vyasta-trairāśika (literally meaning "inverse rule of three terms)". After having described the rule of three, Brahmagupta proceeds to give an account of this inverse rule of three : Divide the phala with icchā and multiply by pramāṇa; this gives the vyasta-trairāśika inverse rule of three¹. Here pramāṇa is the argument also known as the first term and, and phala is the fruit also known as the middle term and icchā is known as requisition or the last term. As Bhāskara II clearly states, this rule is applied where with the increase of the icchā, the phala decreases or with its decrease the phala increases (Līlāvatī). Rule of Compound Proportion Brahmagupta and other writers call the rule of compound proportions as pañca-rāśika, sapta-rāśika etc., meaning the rule of five terms, rule of seven terms etc. depending on the number of terms involved the problems. These are sometimes grouped under the general application of the "Rule of Odd Terms". Āryabhaṭa I (499 A.D.) though actually gives the rule of three appears to have been quite familiar with the rule of compound proportion also. In fact the difference between the rule of three and compound proportion is more or less arti- ficial. This view was expressed by Bhāskara I (525 A.D.) in his commentary on the Āryabhaṭīya : Here Ācārya Āryabhaṭa has described the Rule of Three only. How the well-known Rules of Five etc. are to be obtained ? I say thus : The Ācārya has described only the fundamentals of anupāta (proportion). All others such as the Rule of Five etc. follow from that fundamental rule of proportion. How ? The Rule of Five etc. consist of combinations of the Rule of Three. ......In the Rule of Five, there are two Rules of

1. व्यस्त त्रैराशिक फलमिच्छा भक्तः प्रमाण फलघातः । त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥ —BrSpSi XII. 11