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
The antiquity of the term trairāśika is attested by its presence in the Bakhshali manuscript, a document believed to have been composed in the early centuries of the Common Era.
[1] Basically trairāśika is a rule which helps to solve the following problem: Here
The pramāṇa and icchā must be of the same denomination, that is, of the same kind or type like weights, money, time, or numbers of the same objects.
It is also assumed that phala increases in proportion to pramāṇa.
The unknown quantity is called icchā-phala, that is, the phala corresponding to the icchā.
Āryabhaṭa gives the following solution to the problem:[1] In modern mathematical notations,
The four quantities can be presented in a row like this: Then the rule to get icchā-phala can be stated thus: "Multiply the middle two and divide by the first."
This example is taken from Bījagaṇita, a treatise on algebra by the Indian mathematician Bhāskara II (c.
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.
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.
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: Indian mathematicians have generalized this problem to the case where there are more than one pramāṇa.
This may be represented in the following tabular form: This is the problem of compound proportion.
quantities, the method for solving the problem may be called the "rule of
This example for rule of nine is taken from Bǐjagaṇita:[2] 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!