Trairāśika

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 $\tfrac=\tfrac$ then $ad=bc$ or $a=\tfrac$. 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.

The ''trairāśika'' rule

Basically ''trairāśika'' is a rule which helps to solve the following problem:
:"If $p$ produces $h$ what would $i$ produce?"
Here $p$ is referred to as ''pramāṇa'' ("argument"), $h$ as ''phala'' ("fruit") and $i$ as ''ichcā'' ("requisition"). 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. ''Phala'' can be a of a different denomination. 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:
:"In ''trairāśika'', the ''phala'' is multiplied by ''ichcā'' and then divided by ''pramāṇa''. The result is ''icchā-phala''."
In modern mathematical notations, $\text=\tfrac.$

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."

Illustrative examples

1. This example is taken from '' Bījagaṇita'', a treatise on algebra by the Indian mathematician Bhāskara II (c. 1114–1185).

: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'' = $\tfrac$ ''nishca'', ''phala'' = $2\tfrac$ ''pala''-s of saffron, ''icchā'' = $9$ ''nishca''-s and we have to find the ''icchā-phala''. $\text=\tfrac=\tfrac = 52\tfrac$ ''pala''-s of safron.

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.

: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''. $\text=\tfrac=\tfrac = \tfrac$ measures of rice.

''Vyasta-trairāśika'': Inverse rule of three

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". 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, $\text = \tfrac.$

Illustrative example

This example is from ''Bījagaṇita'':

: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 $\text = \tfrac =\tfrac=25\tfrac$ ''nishca''-s.

Compound proportion

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:
:
Indian mathematicians have generalized this problem to the case where there are more than one ''pramāṇa''. Let there be ''n'' ''pramāṇa''-s ''pramāṇa''-1, ''pramāṇa''-2, . . ., ''pramāṇa''-''n'' and the corresponding ''phala''. Let the ''iccha''-s corresponding to the ''pramāṇa''-s be ''iccha''-1, ''iccha''-2, . . ., ''iccha''-''n''. The problem is to find the ''phala'' corresponding to these ''iccha''-s. This may be represented in the following tabular form:
:
This is the problem of compound proportion. The ''ichcā-phala'' is given by $\text = \tfrac.$

Since there are $2n+1$ quantities, the method for solving the problem may be called the "rule of $2n+1$". In his ''Bǐjagaṇita'' Bhāskara II has discussed some special cases of this general principle, like, "rule of five" (''pañjarāśika''), "rule of seven" (''saptarāśika''), "rule of nine" ("navarāśika") and "rule of eleven" (''ekādaśarāśika'').

Illustrative example

This example for rule of nine is taken from ''Bǐjagaṇita'':
:Problem: If thirty benches, twelve fingers thick, square of four wide, and fourteen cubits long, cost a hundred ishcas 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:
:
:''iccha-phala = $\tfrac=\tfrac=16\tfrac$.

Importance of the ''trairāśika''

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."

Additional reading

* For advanced applications of ''trairāśika'' in astronomy, see: .
* For a complete discussion on ''trairāśika'', see:
* For applications of ''trairāśika'' in Indian architecture, see: (Chapter V ''Trairāśika'' (Rule of Three) in Traditional Architecture)

References

{{Indian mathematics

Fractions (mathematics)
Arithmetic
Indian mathematics
Elementary algebra