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

180 BRAHMAGUPTA AND ARITHMETIC In (solving problems on) the Rule of Three, the argu- ment (pramāṇa) and the requisition (icchā), which are of the same denomination, should be set down in the first and last places; the fruit (phala), which is of a different denomination, should be set down in the middle, (this having been done) that (middle quantity multiplied by the last quantity should be divided by the first quantity.¹ We shall illustrate the Rule of Three by an example from the Pāṭīgaṇita (Example 25) : Example, If 1 pala and 1 karṣa of sandalwood are obtai- ned for ten and a half paṇas, then for how much will nine palas and one karṣa (of sandalwood) be obtai- ned ?² Here in this Example. argument = 1 pala and 1 karṣa = 1¼ or 5/4 palas; fruit = 10½ or 21/2 paṇas; and requisition = 9 palas and 1 karṣa = 9¼ or 37/4 palas. According to the Rule we shall write them as :

1	10	9
1	1	1
4	2	4
Converting these into proper fractions we have
5	21	37
---	----	----
4	2	4
Then applying the rule, (i.e. multiplying the second and the
last and dividing by the first), we have
┌────┬───┐
│ 21 │ 5 │
│ 2 │ 4 │ (21/2 × 37/4)
├────┼───┤ = ───────────────
│ 37 │ │ 5/4
│ 4 │ │
└────┴───┘
Or transferring denominators:
┌────┬───┐
│ 21 │ 5 │ 21 · 4 · 37
│ 4 │ 2 │ = ─────────── pala
├────┼───┤ 5 · 2 · 4
│ 37 │ 4 │
└────┴───┘

1. आद्यन्तयोस्त्रिराशावभिन्नजाती प्रमाणमिच्छा च । फलमथ विजातीयं तदन्त्यगुणमादिना विभजेत् ॥ —Pāṭīgaṇita 43.
2. चन्दनपलं सकर्षं सार्धैर्यदि लभ्यते पणैर्दशभिः । तत्किं नु लभ्यन्ते पलानि नव कर्षयुक्तानि ॥ —Pāṭīgaṇita Ex. 25.