Trairāśika is the Sanskrit term used by Indian astronomers and mathematicians of the pre-modern era to denote what is known as the "rule of three" in elementary mathematics and algebra. In the contemporary mathematical literature, the term "rule of three" refers to the principle of cross-multiplication which states that if
\tfrac{a}{b}=\tfrac{c}{d}
ad=bc
a=\tfrac{bc}{d}
Basically trairāśika is a rule which helps to solve the following problem:
"If
p
h
i
p
h
i
"In trairāśika, the phala is multiplied by ichcā and then divided by pramāṇa. The result is icchā-phala." In modern mathematical notations,
icchā-phala=\tfrac{phala x icchā
The four quantities can be presented in a row like this:
pramāṇa | phala | ichcā | icchā-phala (unknown)Then the rule to get icchā-phala can be stated thus: "Multiply the middle two and divide by the first."
1. This example is taken from Bījagaṇita, a treatise on algebra by the Indian mathematician Bhāskara II (c. 1114–1185).[2]
Problem: "If two and a half pala-s (a unit of weight) of saffron be obtained for three-sevenths of a nishca (a unit of money); say instantly, best of merchants, how much is got for nine nishca-s?"
Solution: pramāṇa =
\tfrac{3}{7}
2\tfrac{1}{2}
9
icchā-phala=\tfrac{phala x icchā
2. This example is taken from Yuktibhāṣā, a work on mathematics and astronomy, composed by Jyesthadeva of the Kerala school of astronomy and mathematics around 1530.[3]
Problem: "When 5 measures of paddy is known to yield 2 measures of rice how many measures of rice will be obtained from 12 measures of paddy?"
Solution: pramāṇa = 5 measures of paddy, phala = 2 measures of rice, icchā = 12 measures of rice and we have to find the icchā-phala.
icchā-phala=\tfrac{phala x icchā
The four quantities associated with trairāśika are presented in a row as follows:
pramāṇa | phala | ichcā | icchā-phala (unknown)In trairāśika it was assumed that the phala increases with pramāṇa. If it is assumed that phala decreases with increases in pramāṇa, the rule for finding icchā-phala is called vyasta-trairāśika (or, viloma-trairāśika) or "inverse rule of three".[4] In vyasta-trairāśika the rule for finding the icchā-phala may be stated as follows assuming that the relevant quantities are written in a row as indicated above.
"In the three known quantities, multiply the middle term by the first and divide by the last."In modern mathematical notations we have,
icchā-phala=\tfrac{phala x pramāṇa
This example is from Bījagaṇita:[2]
Problem: "If a female slave sixteen years of age, bring thirty-two nishca-s, what will one aged twenty cost?"
Solution: pramāṇa = 16 years, phala 32 = nishca-s, ichcā = 20 years. It is assumed that phala decreases with pramāṇa. Hence
icchā-phala=\tfrac{phala x pramāṇa
In trairāśika there is only one pramāṇa and the corresponding phala. We are required to find the phala corresponding to a given value of ichcā for the pramāṇa. The relevant quantities may also be represented in the following form:
| pramāṇa | ichcā | |
| phala | ichcā-phala |
| pramāṇa-1 | ichcā-1 | ||||
| pramāṇa-2 | ichcā-2 | ||||
| . . . | . . . | ||||
| pramāṇa-n | ichcāa-n | - | phala | ichcā-phala |
ichcā-phala=\tfrac{(ichcā-1 x ichcā-2 x … x ichcā-n) x phala
Since there are
2n+1
2n+1
This example for rule of nine is taken from Bǐjagaṇita:[2]
Problem: If thirty benches, twelve fingers thick, square of four wide, and fourteen cubits long, cost a hundred [nishcas]; tell me, my friend, what price will fourteen benches fetch, which are four less in every dimension?
Solution: The data is presented in the following tabular form:
| 30 | 14 | |
| 12 | 8 | |
| 16 | 12 | |
| 14 | 10 | |
| 100 | iccha-phala |
iccha-phala =
\tfrac{(14 x 8 x 12 x 10) x 100}{30 x 12 x 16 x 14}=\tfrac{100}{6}=16\tfrac{2}{3}
All Indian astronomers and mathematicians have placed the trairāśika principle on a high pedestal. For example, Bhaskara II in his Līlāvatī even compares the trairāśika to God himself!
"As the being, who relieves the minds of his worshipers from suffering, and who is the sole cause of the production of this universe, pervades the whole, and does so with his various manifestations, as worlds, paradises, mountains, rivers, gods, demons, men, trees," and cities; so is all this collection of instructions for computations pervaded by the rule of three terms."[5]