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History of Hindu Mathematics: A Source Book v1.c8.s12: 12. The Rule of Three

History of Hindu Mathematics: A Source Book v1.c8.s12

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Contents

12. The Rule of Three

v1.c8.s12.1

**Terminology.** The Hindu name for the Rule of Three terms is *trairāśika* ("three terms," hence "the rule of three terms"). The term *trairāśika* can be traced back to the beginning of the Christian era as it occurs in the Bakhshâlî Manuscript,¹ in the *Âryabhaṭîya* and in all other works on mathematics. About the origin of the name Bhâskara I (*c.* 525) remarks:² "Here three quantities are needed (in the statement and calculation) so the method is called *trairâśika* ("the rule of three terms")."

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¹ The term *râśi* is used in the enumeration of topics of mathematics in the *Sthânâṅga-sûtra* (*c.* 300 B.C.) (*Sûtra* 747). There it probably refers to the Rules of Three, Five, Seven, etc.

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² In his commentary on the *Âryabhaṭîya*.

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A problem on the rule of three has the form:

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If p yields f, what will i yield?

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In the above, the three terms are p, f and i. The Hindus called the term p, *pramâṇa* ("argument"), the term f, *phala* ("fruit") and the term i, *icchâ* ("requisition"). These names are found in all the mathematical treatises. Sometimes they are referred to simply as the first, second and third respectively. Âryabhaṭa II differs from other writers in giving the names *mâna*, *vinimaya* and *icchâ* respectively to the three terms. It has been pointed out by most of the writers that the first and third terms are similar, *i.e.*, of the same denomination.

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**The Method.** Âryabhaṭa I (499) gives the following rule for solving problems on the Rule of Three:

v1.c8.s12.8

"In the Rule of Three, the *phala* ("fruit"), being multiplied by the *icchâ* ("requisition") is divided by the *pramâṇa* ("argument"). The quotient is the fruit corresponding to the *icchâ*. The denominators of one being multiplied with the other give the multiplier (*i.e.*, numerator) and the divisor (*i.e.*, denominator)."³

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³ The above corresponds to *âryâ* 26 and the first half of *âryâ* 27 of the *Gaṇitapâda* of the *Âryabhaṭîya*; compare the working of *Example* I, where the interchange of denominators takes place. See also pp. 195f.

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Brahmagupta gives the rule thus:

v1.c8.s12.11

“In the Rule of Three *pramâṇa* (“argument”), *phala* (“fruit”) and *icchâ* (“requisition”) are the (given) terms; the first and the last terms must be similar. The *icchâ* multiplied by the *phala* and divided by the *pramâṇa* gives the fruit (of the demand).”¹

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¹ *BrSpSi*, p. 178.

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Śrîdhara states:

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“Of the three quantities, the *pramâṇa* (“argument”) and *icchâ* (“requisition”) which are of the same denomination are the first and the last; the *phala* (“fruit”) which is of a different denomination stands in the middle; the product of this and the last is to be divided by the first.”²

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² *Triś*, p. 15.

v1.c8.s12.16

Mahâvîra writes:

v1.c8.s12.17

“In the Rule of Three, the *icchâ* (“requisition”) and the *pramâṇa* (“argument”) being similar, the result is the product of the *phala* and *icchâ* divided by the *pramâṇa*.”³

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³ *GSS*, p. 58(2).

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Âryabhaṭa II introduces a slight variation in the terminology. He says:

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“The first term is called *mâna*, the middle term *vinimaya* and the last one *icchâ*. The first and the last are of the same denomination. The last multiplied by the middle and divided by the first gives the result.”⁴

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⁴ *MSi*, p. 149.

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Bhâskara II, Nârâyaṇa and others give the rule in the same form as Brahmagupta or Śrîdhara.

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The Hindu method of working the rule may be illustrated by the following examples taken from the *Triśatikâ*:

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*Example 1*.¹ "If one *pala* and one *karṣa* of sandal wood are obtained for ten and a half *paṇa*, for how much will be obtained nine *pala* and one *karṣa*?"

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¹ *Triś*, p. 15.

v1.c8.s12.26

Here 1 *pala* and 1 *karṣa* = 1¼ *pala*, and 9 *pala* and 1 *karṣa* = 9¼ *pala* are the similar quantities. The "fruit" 10½ *paṇa* corresponding to the first quantity (1¼ *pala*) is given, so that

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{lcr} pramāṇa (argument) & = & 1 1/4 / phala (fruit) & = & 10 1/2 / icchā (requisition) & = & 9 1/4

v1.c8.s12.28

The above quantities are placed in order as

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{|r|r|r|} \hline 1 & 10 & 9 / 1 & 1 & 1 / 4 & 2 & 4 / \hline

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Converting these into proper fractions we have

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{|r|r|r|} \hline 5 & 21 & 37 / 4 & 2 & 4 / \hline

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Multiplying the second and the last and dividing by the first, we have

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{|r|r|} \hline 21 & 5 / 2 & 4 / \hline 37 & \multicolumn{r}{} / 4 & \multicolumn{r}{} / \cline{1-1} = (21/2 × 37/4)/(5/4).

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Or transferring denominators {|r|r|} \hline 21 & 5 / 4 & 2 / 37 & 4 / \hline = (21 · 4 · 37)/(5 · 2 · 4) pala

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= 4 *purāṇa*, 13 *paṇa*, 2 *kākiṇī* and 16 *varāṭaka*.

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In actual working the intermediate step $(21/2 × 37/4)/(5/4)$ was not written. The denominators of the multipliers were transferred to the side of the divisor and that of the divisor to the multipliers, thus giving at once $21.4.37/5.².4.$

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*Example II.*¹ "Out of twenty necklaces each of which contains eight pearls, how many necklaces, each containing six pearls, can be made?"

v1.c8.s12.38

¹ *Triś*, p. 17.

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Firstly, we have

v1.c8.s12.40

| 1 | 8 | 20 |

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The result (performing the operation of the Rule of Three) is 160 pearls.

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Secondly, perform the operation of the Rule of Three on the following:

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If 6 pearls are contained in one necklace, how many necklaces will contain 160 pearls?

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Placing the numbers, we have

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| 6 | 1 | 160 |

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Result: necklaces 26, part of necklace 2 / 3 /.

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**Inverse Rule of Three.** The Hindu name for the Inverse Rule of Three is *vyasta-trairāśika* (lit. "inverse rule of three terms"). After describing the method of the Rule of Three the Hindu writers remark that the operation should be reversed when the proportion is inverse. Thus Śrīdhara observes:

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"The method is to multiply the middle term by the first and to divide by the last, in case the proportion is different."²

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² *Triś*, p. 18.

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Mahâvîra says:

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“In the case of this (proportion) being inverse, the operation is reversed.”¹

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¹ *GSS*, p. 58(2).

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Bhâskara II writes:

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“In the inverse (proportion), the operation is reversed.”²

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² *L*, p. 17.

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He further observes:

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“Where with increase of the *icchâ* (requisition) the *phala* decreases or with its decrease the *phala* increases, there the experts in calculation know the method to be the Inverse Rule of Three.”³

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³ *L*, p. 17.

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“Where the value of living beings is regulated by their age; and in the case of gold, where the weight and touch are compared; or when heaps are subdivided,⁴ let the Inverse Rule of Three be used.”⁵

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⁴ “When heaps of grain, which have been meted with a small measure, are again meted with a larger one, the number decreases … .” (Com. of Sûryadâsa).

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⁵ *L*, p. 18.

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*Example*: Example II given under the Rule of Three above has been solved also by the application of the Inverse Rule as follows:

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“Statement | 8 | 20 | 6 |. Result: necklaces | 26 |

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| 2 |
| 3 |”

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Here if the *icchâ*, i.e., the number of pearls in a necklace, increases, the *phala*, *i.e.*, the number of necklaces, decreases, so that the Inverse Rule of Three is applied.

Appreciation of the Rule of Three

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The Rule of Three was highly appreciated by the Hindus because of its simplicity and its universal application to ordinary problems. The method as evolved by the Hindus gives a ready rule which can be applied even by the "ignorant person" to solve problems involving proportion, without fear of committing errors. Varâhamihira (505) writes:

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"If the sun performs one complete revolution in a year, how much does he accomplish in a given number of days? Does not even an ignorant person calculate the sun in such problems by simply scribbling with a piece of chalk?"¹

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¹ *PSi*, iv. 37.

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Bhâskara II has eulogised the method highly at several places in his work. His remarks are:

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"The Rule of Three is indeed, (the essence of) arithmetic."²

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² *L*, p. 15. The same remark occurs in *SiŚi*, *Golâdhyâya*, *Praśnâdhyâya*, verse 3.

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"As Lord Śri Nârâyaṇa, who relieves the sufferings of birth and death, who is the sole primary cause of the creation of the universe, pervades this universe through His own manifestations as worlds, paradises, mountains, rivers, gods, men, demons, etc., so does the Rule of Three pervade the whole of the science of calculation. . . . Whatever is computed whether in algebra or in, this (arithmetic) by means of multiplication and division may be comprehended by the sagacious learned as the Rule of Three. What has been composed by the sages through the multifarious methods and operations such as miscellaneous rules, etc., teaching its easy variations, is simply with the object of increasing the comprehension of the duller intellects like ourselves."³

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³ *L*, p. 76.

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On another occasion Bhâskara II observes:

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“Leaving squaring, square-root, cubing and cube-root, whatever is calculated is certainly variation of the Rule of Three, nothing else. For increasing the comprehension of duller intellects like ours, what has been written in various ways by the learned sages having loving hearts like that of the bird *cakora*, has become arithmetic.”¹

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¹ *SiŚi, Golâdhyâya, Praśnâdhyâya*, verse 4.

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**Proportion in the West.** The history of the Hindu rules of proportion shows how much the West was indebted to India for its mathematics. The Rule of Three occurs in the treatises of the Arabs and mediæval Latin writers, where the Hindu name ‘Rule of Three’ has been adopted. Although the Hindu names of the terms were discarded, the method of placing the terms in a line, and arranging them so that the first and last were similar, was adopted. Thus Digges (1572) remarked,² “Worke by the Rule ensueing ..... Multiplie the last number by the seconde, and diuide the Product by the first number,” ... “In the placing of the three numbers this must be observed, that the first and third be of one Denomination.” The rule, as has been already stated, was perfected in India in the early centuries of the Christian era. It was transmitted to the Arabs probably in the eighth century and thence travelled to Europe, where it was held in very high esteem³ and called the “Golden Rule.”

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² Quoted by Smith, *l.c.* p. 488.

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³ The Arabs, too, held the method in very high esteem as is evidenced by Al-Bîrûnî’s writing a separate treatise, *Fi rasîkat al-hind* (“On the *rasîka* of the Hindus”) dealing with the Hindu Rules of Three or more terms. Compare also *India* (I. 313) where an example of *vyasta-trairâśika*, (“the Inverse Rule of Three”) is given.

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**Compound Proportion.** The Hindu names for compound proportion are the Rule of Five, the Rule of Seven, the Rule of Nine, etc., according to the number of terms involved in the problems. These are sometimes grouped under the general appellation of the “Rule of Odd Terms.” The above technical terms as well as the rules were well-known in the time of Âryabhaṭa I (499), although he mentions the Rule of Three only. That the distinction between the Rule of Three and Compound Proportion is more artificial than real was stressed by Bhâskara I (*c.* 525) in his commentary on the *Âryabhaṭîya*. He says:

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“Here Âcârya Âryabhaṭa has described the Rule of Three only. How the well-known Rules of Five, etc. are to be obtained? I say thus: The Âcârya has described only the fundamentals of *anupâta* (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 Three, in the Rule of Seven, three Rules of Three, and so on. This I shall point out in the examples.”

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Remarks similar to the above concerning the Rules of Five, Seven, etc., have been made by the commentators of the *Lîlâvatî*, especially by Gaṇeśa and Sûryadâsa.¹

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¹ Noted by Colebrooke, *l.c.*, p. 35, note.

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In problems on Compound Proportion, two sets of terms are given. The first set which is complete is called *pramâṇa pakṣa* (argument side) and the second set in which one term is lacking is called the *icchâ pakṣa* (requisition side).

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**The Method.** The rule relating to the solution of problems in compound proportion has been given by Brahmagupta as follows:

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“In the case of odd terms beginning with three

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terms¹ upto eleven, the result is obtained by transposing the fruits of both sides, from one side to the other, and then dividing the product of the larger set of terms by the product of the smaller set. In all the fractions the transposition of denominators, in like manner, takes place on both sides.”²

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¹ It should be observed that, as stated above, the Rule of Three is a particular case of the above Rule of Odd terms. Brahmagupta is the only Hindu writer to have included the Rule of Three also in the above rule. Some Arab writers have followed him in this respect by not writing the terms of the Rule of Three in a line, but arranging them in compartments, as for the other rules of odd terms.

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² *BrSpSi*, p. 178.

v1.c8.s12.t1.25

Srîdhara says:

v1.c8.s12.t1.26

“Transpose the two fruits from one side to the other, then having transposed the denominators (also in like manner) and multiplied the numbers (so obtained on each side), divide the side with the larger number of terms by the other (side).”³

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³ *Tris*, p. 19.

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Mahâvîra⁴ and Âryabhaṭa II⁵ have given the rule in the same way as Srîdhara. Bhâskara II has given it thus:

v1.c8.s12.t1.29

⁴ *GSS*, p. 62 (32).

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⁵ *MSi*, p. 150, rules 26 and 27 (repeated with a slight variation).

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“In the rules of five, seven, nine or more terms, after having taken the *phala* (fruit) and *chid*⁶ from its own side to the other, the product of the larger set of terms divided by the product of the smaller set, gives the result (or produce sought).”¹

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⁶ The commentators differ as regards the interpretation of this word. Some take it to mean “divisor,” *i.e.*, “denominator,” while others say that it means “the fruit of the other side.” The rule is, however, correct with either interpretation. The first interpretation, however, brings Bhâskara's version in line with those of his predecessors. It may be mentioned here that Âryabhaṭa II repeats the rule twice. At first he does not direct the transposition of denominator, and at the second time he does so.

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¹ L, p. 18.

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**Illustration.** We shall illustrate the Hindu method of working by solving the following example taken from the *Lîlâvatî*:

v1.c8.s12.t1.35

“If the interest of a hundred in one month be five, what will be the interest of 16 in 12 months? Also find the time knowing the interest and principal; and tell the principal knowing the time and interest.”

v1.c8.s12.t1.36

*To find interest.*

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The first set of terms (*pramâṇa pakṣa*) is: 100 *niṣka*, 1 month, 5 *niṣka* (*phala*) The second set (*icchâ pakṣa*) is: 16 *niṣka*, 12 months, x *niṣka* The terms are now written in compartments² as below:

v1.c8.s12.t1.38

² The terms of the same denomination are written in compartments in the same horizontal line.

v1.c8.s12.t1.39

| 100 | 16 |
| 1 | 12 |
| 5 | 0 |

v1.c8.s12.t1.40

³

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³ The figures are written in compartments in order to facilitate the writing of fractions and also to denote the side which contains more terms after transposition of fruits. Sometimes, the compartment corresponding to an absent term is left vacant as we find in a copy of Munîśvara’s *Pâṭîsâra* (in the Government Sanskrit Library at Benares). When the terms are written in compartments, the symbol 0 to denote the unknown or absence of a term is unnecessary. In some commentaries on the *Lîlâvatî* (Asiatic Society of Bengal manuscripts) we find the numbers written without compartments, but in such cases the symbol 0 is used to denote the absence of a term. After transposition, the side on which 0 occurs contains a smaller number of terms than the other.

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In the above 5 (written lowest) is the "fruit" of the first side, and there is no "fruit" on the second side. Interchanging the fruits we get

v1.c8.s12.t1.43

| 100 | 16 |
| 1 | 12 |
| 0 | 5 |

v1.c8.s12.t1.44

The larger set of terms is on the second "side." The product of the numbers is 960. The product of the numbers on the side of the smaller set of terms is 100. Therefore, the required result is 960/100 = 48/5, written as | 48 / 5 | or 9 *niṣka*, fraction | 3 / 5 |

v1.c8.s12.t1.45

*To find Time:*

v1.c8.s12.t1.46

Here the sides are 100 *niṣka*, 1 month, 5 *niṣka* and 16 *niṣka*, *x* months, 48/5 *niṣka*

v1.c8.s12.t1.47

The terms are written as

v1.c8.s12.t1.48

| 100 | 16 |
| 1 | 0 |
| 5 | 48 5 |

v1.c8.s12.t1.49

Transposing the fruits, *i.e.*, transposing the numbers in the bottom compartment, we get

v1.c8.s12.t1.50

| 100 | 16 |
| 1 | 0 |
| 48 5 | 5 |

v1.c8.s12.t1.51

Transposing the denominators we have

v1.c8.s12.t1.52

| 100 | 16 |
| 1 | 0 |
| 48 | 5 5 |

v1.c8.s12.t1.53

Here, the larger set of terms is on the first side and their product is 4800. The product of the numbers on the side of the smaller set is 400. Therefore, the result is $4800 / 400 / 4800/400 = 12 months.$

v1.c8.s12.t1.54

*To know the principal:*

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The first side is

v1.c8.s12.t1.56

100 *niṣka*, 1 month, 5 *niṣka*

v1.c8.s12.t1.57

The second side is

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x *niṣka*, 12 months, 48/5 *niṣka*

v1.c8.s12.t1.59

This is written as

v1.c8.s12.t1.60

| 100 | 0 |
| 1 | 12 |
| 5 | 48 5 |

v1.c8.s12.t1.61

After transposition of fruits (*i.e.*, the terms in the bottom cells) we have

v1.c8.s12.t1.62

| 100 | 0 |
| 1 | 12 |
| 48 5 | 5 |

v1.c8.s12.t1.63

Transposing denominators we get

v1.c8.s12.t1.64

| 100 | 0 |
| 1 | 12 |
| 48 | 5 5 |

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The product of the numbers in the larger set divided by the product of the numbers in the smaller set, gives $4800 / 300 = 16 niska.$

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**Rule of Three as a Particular Case.** According to Brahmagupta, the above method may be applied to the Rule of Three. Taking the first example solved under the Rule of Three, above, and placing the terms we have¹

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¹ Here, we consider 5/4 *pala* of sandal wood as the "fruit" of 21/2 *paṇa* (money). The previous method forces us to consider 21/2 *paṇa* as the "fruit" or the middle term, because the "first" and "third" are directed to be alike. It will be observed that any of the terms may be considered to be the fruit in the alternative method given here.

v1.c8.s12.t1.68

| 21 2 | 0 |
| 5 4 | 37 4 |

v1.c8.s12.t1.69

Transposing the fruits, we have

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| 21 2 | 0 |
| 37 4 | 5 4 |

v1.c8.s12.t1.71

Transposing denominators, we get

v1.c8.s12.t1.72

| 21 | 0 2 |
| 37 4 | 5 4 |

v1.c8.s12.t1.73

Therefore, the result is 21.37.4/2.5.4 as before.

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If we consider the term corresponding to the unknown as the fruit, the terms should be set as below:

v1.c8.s12.t1.75

| 5 | 37 |
| 4 | 4 |
| 21 | |
| 2 | 0 |

v1.c8.s12.t1.76

Hence, as before,¹ the result is (37·4·21)/(5·4·2)

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¹ The product of the numbers on the side of the larger set is divided by the product of the numbers on the side of the smaller set. 0 in this case is not a number. It is the symbol for the unknown or absence.

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The above method of working the Rule of Three is found among the Arabs,² although it does not seem to have been used in India after Brahmagupta. This points to the indebtedness of the Arabs to Brahmagupta especially, for their knowledge of Hindu arithmetic.

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² Thus Rabbi ben Ezra wrote 47 / 7 63 / 0 for 47:7=63:x. See Smith, *l.c.*, p. 489f.

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Written as above the method of working the Rule of Three appears to be the same as the method of proportion. In the same way the rule of other odd terms, when properly translated into modern symbolism, is nothing but the method of proportion. It has been stated by Smith³ that the Hindu methods of solution "fail to recognize the relation between the Rule of Three and proportion." This statement appears to have been made without sufficient justification, for the solutions have been evidently obtained by the use of the ideas of proportionality and variation. The aim of the Hindu works is to give a method which can be readily used by common people. For this very reason, the cases in which the variation is inverse have been enumerated. Considered as a method which stimulated the student to think for himself, the method is certainly defective, but for practical purposes, it is, in our opinion, the best that could be devised.

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³ *l.c.*, p. 488.

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