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

174 BRAHMAGUPTA AND ARITHMETIC and the form p/q + r/s of p/q + t/u of (p/q + r/s of p/q) + ......... is written as ┌───┐ │ p │ │ q │ ├───┤ │ r │ │ s │ ├───┤ │ t │ │ u │ └───┘ (iv) Bhāgāpavāha, i.e., the form (a - b/c) is written as ┌───┐ │ a │ │ .b│ │ c │ └───┘ and the form p/q - r/s of p/q - t/u of (p/q - r/s of p/q) - ........ is written as ┌───┐ │ p │ │ q │ ├───┤ │ .r│ │ s │ ├───┤ │ .t│ │ u │ └───┘ (v) Bhāga-bhāga : The form (a ÷ b/c) or (p/q ÷ r/s) There does not appear to have been any notation for divi- sion, such compounds being written as ┌───┐ ┌───┐ │ a │ │ p │ │ b │ or │ q │ │ c │ ├───┤ └───┘ │ r │ │ s │ └───┘ just as for bhāgānubandha. That division is to be performed was known from the problem, e.g., 1 ÷ 1/6 was written as ṣaḍ-bhāga- bhāga, i.e., "one-sixth bhāga-bhāga" or "one divided by one- sixth". It is only in the Bakhshālī Manuscript that the term bha is sometimes placed before or after the quantity affected. (vi) Bhāga-mātṛ, i.e., combinations of forms enumerated above. Mahāvīra, the author of the Gaṇitasārasaṃgraha (850

FRACTIONS 175 A.D.) gives twenty-six variations of this class. We shall illus- trate it by the following example from Śrīdhara : What is the result when half, one-fourth of one-fourth, one divided by one-third, half-plus half of itself, and one-third diminished by half of itself, are added together ? (Triśatikā, p. 12). A modern writer would have written it as : ½+(¼ of ¼)+(1÷⅓)+(½+½ of ½)(⅓—½ of ⅓) In the old Indian notation, it is written as · | 1 | 1 | 1 | 1 | 1 | 1 | | 2 | 4 | 4 | 1 | 2 | 3 | | | | | 3 | 1 | .1 | | | | | | 2 | 2 | The defect of the notation is obvious: | 1 | 1 | can be read | 4 | 4 | also as ¼+¼ and | 1 | can also be read as 1 ⅓. | 1 | | 3 | And therefore the original meaning is inferred from the context or from the enunciation of the problem. The rules for reduction of the first two classes (bhāga and prabhāga) are those of addition or subtraction and multiplica- tion. The rule for the reduction of the third (bhāgānubandha) and fourth (bhāgāpavāha) classes are given by Brahmagupta in the Brāhmasphuṭa-siddhānta thus : The (upper) denominator is multiplied by the deno- minator and the upper numerator by the same (deno- minator) increased or diminished by its own numera- tor.¹ “Numerator” is known as “aṁśa” and the “denominator” as “cheda.” We give here from Śrīdhara's Pāṭīgaṇita (about 900 A.D., according to K.S. Shukla, 750 A.D. according to Datta and Singh) a rule for reducing a fraction of the bhāgānubandha class (i.e., a whole number increased by a fraction or a fraction incre- ased by a fraction itself) :

1. ऊर्ध्वां शारच्छेदगुखास्तृतीयाजातौ द्वयोः पृथक्परयोः । छेदैश्छेदा गुणिताः स्वांशयुतोनैरुपरीमांशाः ॥ —BrSpSi. XII. 9.

176 BRAHMAGUPTA AND ARITHMETIC In the bhāgānubandha class, the whole number (rūpa- gaṇa) is multiplied by the denominator (of the frac- tion) should be increased by the numerator (of the fraction) or the upper denominator having been multiplied by the lower denominator, the initial numerator (i.e. the upper numerator) should be multi- plied by the sum of the lower numerator and denomi- nator.¹ (Pātīgaṇita, 39 cf. BrSpSi. XII. 9 (ii); GSS. (iii) 113 This means that (i) a + b/c = (ac + b)/c (ii) a/b + c/d of a/b (which was written by Indians in the style ┌───┐ │ a │ │ b │ │ c │ │ d │ └───┘ is equal to a(d + c)/bd Addition and Subtraction of Fractions In the Brāhmasphuṭa-siddhānta, Brahmagupta gives the rule for the addition and subtraction of fractions : If the denominators (cheda) of fractions are different then reduce these fractions to a common denomina- tor. Now for the additions, unite the numerators and take their difference in case of subtraction.² Brahmagupta and Mahāvīra give the method under Bhāga- jāti. Multiplication Brahmagupta says : The product of the numerators divided by the pro-

1. भागानुबन्धजातौ रूपगणश्छेद सङ्गुणः सांशः । अवरहरन्धोर्ध्वं हरेऽथोंशाद्युतहरघ्न आद्यंशः ॥ —Patīganita 39.
2. विपरीतच्छेदगुणाः राश्योश्छेदांशकाः समच्छेदाः । संकलितेंऽशा योज्या व्यवकलितेऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2

DIVISION OF FRACTIONS 177 duct of the denominators is the (result of) multiplica- tion of two or more fractions.¹ While all other writers give the rule in the same way as Brahmagupta, Mahāvīra in the Gaṇitasārasaṃgraha refers to cross reduction in order to shorten the work : In the multiplication of fractions, the numerators are to be multiplied by the numerators and the denomi- nators by denominators, after carrying out the pro- cess of cross reduction, if that be possible.² Division of Fractions The Āryabhaṭīya does not explicitly give the rule of divi- sion, but under the Rule of Three, we have an indication of this operation. The Rule of Three states the result as (f × i) / p, where f stands for phala i.e. "fruit", i for icchā, i.e., demand or requisition, and p for pramāṇa i.e. argument. When these quantities are fractional, we get an expression of the form (a/b × c/d) / (m/n) for the evaluation of which Āryabhaṭa I states : The multipliers and the divisor are multiplied by the denominators of each other. These quantities are written in the following way ┌───────┐ │ a m │ │ b n │ ├───────┤ │ c │ │ d │ └───────┘ Transferring the denominators we have ┌───────┐ │ a m │ │ n b │ │ c d │ └───────┘ Performing multiplication, the result is anc / mbd. The above interpretation of the obscure line in the Āryabhaṭīya is based

1. रूपाणिच्छेद गुण्यान्यंशयुतानि द्वयोर्बहूनां वा । प्रत्युत्पन्नो भवति च्छेदवधेनोद्धृतोऽशवधः ॥ —BrSpSi. XII. 3
2. GSS. p. 25. (2)

178 BRAHMAGUPTA AND ARITHMETIC on the commentaries of Sūryadēva and Bhāskara I (the commen- tary of Parameśvara on this line is vague and misleading). Sūryadeva in this connection says : Here by the word guṇakāra is meant the multiplier and multiplicand, i.e., the phala and icchā quantities that are multiplied together. By Bhāgahāra is meant the pramāṇa quantity. The denominators of the phala and icchā are taken to the pramāṇa. The denominator of the pramāṇa is taken with the phala and icchā. Then multiplying these, i.e., (the numerators of) the phala and icchā and this denominator, and dividing by (the product of) the numbers standing with the pramāṇa the result is the quotient of the fractions. Brahmagupta gives the method of division as follows : The denominator and numerator of the divisor having been interchanged, the denominator of the dividend is multiplied by the (new) numerator. Thus division of *proper fractions is performed.*¹ Square and Square-Root of Fractions Brahmagupta says as follows in this connection:— The square of the numerator of a proper fraction divi- ded by the square of the denominator gives the square². This rule of Brahmagupta has been followed by other authors also. The rule regarding the square-root as given by Brahmagupta is as follows : The square-root of the numerator of a proper fraction divi- ded by the square-root of the denominator gives the square- root.³ The Rule of Three : The Indian term in Sanskrit for the Rule of Three is Trai- rāśika (literally, "three terms"). The term occurs in the Bakh- śālī Manuscript also, and also in the Āryabhaṭīya, indicating the

1. परिवर्त्य भागहारच्छेदांशौ छेद संगुणच्छेदः । अंशोंशगुणो भाज्यस्य भागहारः सवर्णितयोः ॥ —BrSpSi: XII. 4
2. संवर्गितांशवर्गश्छेदकृतिविभाजितो भवति वर्गः । —BrSpSi. XII. 5 (1)
3. संवर्गितांशमूलं छेदपदेनोद्धृतं मूलम् । —BrSpSi. XII. 5 (2)

THE RULE OF THREE 179 antiquity of the term. Bhāskara in his commentary of the Āryabhaṭīya gives a justification of the use of this term for the Rule of Three thus : Here three quantities are needed (in the statement and calculation) so the method is called trairāśika (meaning thereby the "rule of three terms"). The problem of the Rule of Three has the form : If p (pramāṇa) yields f (phala), what will i (icchā) yield ? Āryabhaṭa II (the author of the Mahāsiddhānta, 950 A.D.) uses the terms māna, vinimaya and icchā, instead of pramāṇa, phala and icchā respectively. It has also been pointed out by several authors that the first and third terms are similar, i.e., of the same denomination. We shall give here the Rule of Three as given by Āryabhaṭa I and Brahmagupta : In the Rule of Three, the phala ("fruit"), being multi- plied by the icchā ("requisition") is divided by the pramāṇa ("argument"). The quotient is the fruit corresponding to the icchā The denominators of one being multiplied with the other give the multiplier (i.e. numerator) and the divisor (i.e. denominator).¹ In the Rule of Three pramāṇa ("argument"), phala ("fruit") and icchā ("requisition") are the (given) terms; the first and the last terms must be similar. The icchā multiplied by the phala and divided by the pramāṇa gives the fruit (of the demand).² Śrīdhara also gives the Rule of Three almost in the same words. Bhāskara II, Nārāyaṇa and others follow Brahmagupta and Śrīdhara in the Trairāśika operation. Śrīdhara in his Pāṭīga- nita says :

1. त्रैराशिकफलराशिं तमथेच्छाराशिनाहतं कृत्वा । लब्धं प्रमाणभजितं तस्मादिच्छाफलमिदं स्यात् ।। छेदाः परस्परं हता भवन्ति गुणकार भागहाराणां । छेदगुणं सच्छेदं परस्परं तत्सवर्णत्वम् ।। —Arya. II 26-27.
2. त्रैराशिके प्रमाणं फलमिच्छाद्यन्तयोः सदृशराशी । इच्छाफलेन गुणिता प्रमाणभक्ता फलं भवति ।। —BrSpSi XII. 10

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

RULE OF COMPOUND PROPORTION 181 =4 purāṇa, 13 paṇas, 2 kākiṇīs and 16 varāṭakas. (One purāṇa is equivalent to 16 paṇas; one paṇa is equivalent to 4 kākiṇīs, and one kākiṇī is equivalent to 20 varāṭakas or cowries. Inverse Rule of Three This is known as vyasta-trairāśika (literally meaning "inverse rule of three terms)". After having described the rule of three, Brahmagupta proceeds to give an account of this inverse rule of three : Divide the phala with icchā and multiply by pramāṇa; this gives the vyasta-trairāśika inverse rule of three¹. Here pramāṇa is the argument also known as the first term and, and phala is the fruit also known as the middle term and icchā is known as requisition or the last term. As Bhāskara II clearly states, this rule is applied where with the increase of the icchā, the phala decreases or with its decrease the phala increases (Līlāvatī). Rule of Compound Proportion Brahmagupta and other writers call the rule of compound proportions as pañca-rāśika, sapta-rāśika etc., meaning the rule of five terms, rule of seven terms etc. depending on the number of terms involved the problems. These are sometimes grouped under the general application of the "Rule of Odd Terms". Āryabhaṭa I (499 A.D.) though actually gives the rule of three appears to have been quite familiar with the rule of compound proportion also. In fact the difference between the rule of three and compound proportion is more or less arti- ficial. This view was expressed by Bhāskara I (525 A.D.) in his commentary on the Āryabhaṭīya : Here Ācārya Āryabhaṭa has described the Rule of Three only. How the well-known Rules of Five etc. are to be obtained ? I say thus : The Ācārya has described only the fundamentals of anupāta (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

1. व्यस्त त्रैराशिक फलमिच्छा भक्तः प्रमाण फलघातः । त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥ —BrSpSi XII. 11

182 BRAHMAGUPTA AND ARITHMETIC Three, in the Rule of Seven three Rules of Three, and so on. This I shall point out in the examples. Brahmagupta gives the following rule relating to the solu- tion of problems in compound proportion : In the case of odd terms beginning with three terms up to 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.¹ This may be illustrated by taking an example from the commentary of Pṛthūdaka Svāmī on the Brāhmasphuṭasid- dhānta : Example —If there is an increase of 10 in 3 months on 100 (niṣkas), what would be the increase on 60 (niṣkas) in 5 months. Here the Pramāṇa pakṣa (the first set of terms) is 100 niṣkas, 3 months, 10 niṣkas (phala) The second set or the icchā pakṣa is 60 niṣkas, 5 months, x niṣkas The terms are written in compartments as below : | 100 | 60 | | 3 | 5 | | 10 | 0 | In the above 10 (written lowest) is the fruit of the first side (pramāṇa pakṣa), and there is no fruit on the second side or the iccā pakṣa. Interchanging the fruits we get | 100 | 60 | | 3 | 5 | | 0 | 10 | The larger set of terms is on the second side (icchā pakṣa). The product of the numbers is 3,000. The product of the

1. व्यस्तं त्रैराशिक फलमिच्छा भक्तः प्रमाणफलघातः । त्रैराशिकादिषु फलं विषमेष्वेकादशान्तेषु ॥ फलसंक्रममुभयतो बहुराशि वधोऽल्पवधहृतो ज्ञेयम् । सकलेष्वेवं भिन्नेष्वथवैतच्छेदसंक्रमकम् ॥ —BrSpSi: XII. 11-12.

RULE OF THREE AS A PARTICULAR CASE 183 number on the side of the smaller set of terms is 300. Therefore the required result is 3000 / 300 = 10. Rule of Three as a Particular Case According to Brahmagupta, the above method of "com- pound proportion" may be applied to the Rule of Three. Taking the example solved under the Rule of Three : If one pala and one karṣa of sandal wood are obtained for ten and a half paṇas, for how much will be obtain- ed nine palas and one karṣa ? (4 karṣas = 1 pala). We shall represent them according to the Rule of Com- pound Proportion as Pramāṇa pakṣa : 1 pala, 1 karṣa, 10½ paṇa or 5/4 pala , 21/2 paṇa, Icchā pakṣa : 9 pala, 1 karṣa, x paṇa or 37/4 pala , x paṇa This we shall represent as ┌────┬────┐ │ 5 │ 37 │ │ 4 │ 4 │ ├────┼────┤ │ 21 │ 0 │ │ 2 │ │ └────┴────┘ Transposing the fruits, we have ┌────┬────┐ │ 5 │ 37 │ │ 4 │ 4 │ ├────┼────┤ │ 0 │ 21 │ │ │ 2 │ └────┴────┘ Transposing denominators ┌────┬────┐ │ 5 │ 37 │ │ 4 │ 4 │ ├────┼────┤ │ 0 │ 21 │ │ 2 │ │ └────┴────┘ The product of 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. Hence the result is : (37 . 4 . 21) / (5 . 4 . 2) paṇas

184 BRAHMAGUPTA AND ARITHMETIC The above method of working Rule of Three is found among Arabs, although it does not seem to have been used in India after Brahmagupta. Problem Containing Quadratic Equation Perhaps Āryabhaṭa I is the first man in the history of mathematics to give a solution of a quadratic equation (499 A.D.). In his Āryabhaṭīya, he gives a rule for the solution of the following problem (I am reproducing it as described by Datta and Singh) : The principal sum p (=100) is lent for one month (interest unknown = x). This unknown interest is then lent out for t(=six) months. After this period, the original interest (x) plus the interest on this interest amounts to A(=16). The rate-interest (x) on the principal (p) is required. This problem requires the solution of the quadratic equation :— tx² + px - AP = 0 which gives x = (-p/2 ± √((p/2)² + Apt)) / t The negative value of the radical does not give a solution of the problem; so that the result is x = (√(Apt + (p/2)²) - p/2) / t This solution is stated by Āryabhaṭa I in the following words : Multiply the sum of the interest on the principal and the interest (A) by the time (t) and by the principal (p). Add to this result the square of half the principal (p/2)². Take square-root this. Subtract half the principal (p/2) and divide the remainder by the time (t). The result will be the (unknown) interest (x) on the principal.¹ Here the Sanskrit terms are mūla for principal and phala for interest.

1. मूलफलं सफलं कालमूलगुणमर्धमूलकृतियुक्तम् । मूलं मूलार्धोनं कालहृतं स्यात्स्वमूलफलम् ॥ Ārya. II. 25,

A PROBLEM ON INTEREST 185 Brahmagupta (628 A.D.) gives a more general rule : He enunciates his problem thus : The principal (p) is lent out for t₁ months and the unknown interest on this (=x) is lent out for t₂ months at the same rate and becomes A. To find x. This evidently gives the quadratic : x² + (pt₁ / t₂) x - (Apt₁ / t₂) = 0 whose solution is x = ± √[ Apt₁/t₂ + (pt₁ / 2t₂)² ] - pt₁ / 2t₂ The negative value of the radical does not give a solution of the problem, so it is discarded. Brahmagupta states the formula thus : Multiply the principal (p) by its time (t₁) and divide by the other time (t₂) (placing the result) at two places. Multiply the first of these by the mixture (A). Add to this the square of half the other. Take the square-root of this (sum). From the result subtract half the other. This will be the interest (x) on the principal.¹ A Problem on Interest Brahmagupta gives a solution of a problem on interest : In what time will a given sum s, the interest on which for t months is r, become k times itself ? The rule for the solution of this problem as given by Brah- magupta is : The given sum multiplied by its time and divided by the interest (phala), being multiplied by the factor (guṇa) less one, is the time (required).² Miscellaneous Problems Brahmagupta in his Gaṇitādhyāya of the Brāhmasphuṭa- siddhānta gives numerous solutions in relation to miscellaneous problems. Here I shall be quoting a few of the problems which

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽद्यमिश्रवधात् । अन्यार्धकृतियुतात् पदमन्यार्धोनं प्रमाणफलम् ॥ —BrSpSi. XII. 15.
2. कालगुणितं प्रमाणं फलभक्तं व्येकगुणहतं कालः । स्वफलयुतरूपभक्तं मूलफलैक्यं भवति मूलम् ॥ —BrSpSi. XII. 14.

186 BRAHMAGUPTA AND ARITHMETIC have been quoted by his commentator Pṛthūdaka Svāmī in connection with one of his karaṇa-sūtra.¹

1. A horse was purchased by (nine) dealers in partner- ship, whose contributions were one, etc. up to nine; and was sold by them for five less than five hundred. Tell me what was each man's share of the sale proceed²
2. Four colleges (mathas), containing an equal num- ber of pupils, were invited to partake of a sacrificial feast. A fifth, a half, a third and a quarter (of the total number of pupils in the college) came from the respective colleges to the feast; and added to one, two, three and four, they were found to amount to eighty- seven; or, with those deducted, they were sixty seven. Find the actual number of the pupils that came from each college.³
3. Three (unequal) jars of liquid butter, of water and of honey, contained thirty-two, sixty and twenty-four Pala respectively; the whole was mixed together and the jars filled again. Tell me the quantity of butter, of water and of honey in each jar⁴.
4. प्रक्षेपयोगहृत्या लब्ध्वा प्रक्षेपका गुणा लाभाः । ऊनाधिकोत्तरस्तुतोनया स्वफलमूनयुत ॥ BrSpSi. XII. 16.
5. एकाद्यैर्नव पर्यन्तैर्वणिग्भिर्मूलराशिभिः । क्रीतो हयोऽसौ विक्रीतः पञ्चोनैः पञ्चभिः शतैः । किमैकैकस्य तत्रासीद् ब्रूहि त्वं मिश्रकान् मम ॥
6. मठस्थानानि चत्वारि छात्राणां समसंख्यया । भोक्तुं संमन्त्रितान्यासन् दीक्षायां किल यज्वना ॥ पञ्चार्धत्रिचतुर्थांशास्तेभ्यो भोक्तुं समागताः । एकद्वित्रिचतुर्युक्ता दृष्टाशीतिः ससप्तका ॥ एवोत्तरैरथवा हीना सप्तषष्टिश्चतैऽशकाः । मठेभ्यश्छात्रसंख्यां मे ब्रूहि ये चागता यतः
7. घृतोदक मधूनां ये त्रयः कलसकाः पलैः । रदषष्टिजिनैः पूर्णा एकीभूतास्ततः पुनः ॥ मिश्रेण पूरिता यावत् तावत् संख्यां न वेद्म्यहम् । घृतोदकमधूनां तामेकैकत्र गतां वद ॥ —: ० :—

MISCELLANEOUS PROBLEMS 187 Reference B. Datta and A.N. Singh : History of Hindu Mathematics, Part I (1962). K.S. Shukla : The Pāṭīgaṇita of Śrīdharācārya (1959). Kern : The Āryabhaṭīya. Bhaṭadīpikā of Parameśvara (1875).

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CHAPTER IX Brahmagupta as an Algebraist Ancient Indian name for algebra is Bījagaṇita where bīja means element or analysis and gaṇita stands for the science of calculation. As early as 860 A.D., Pṛthūdaka Svāmī used this epithet for algebra in his commentary. Brahmagupta calls algebra as Kuṭṭakagaṇita or merely kuṭṭaka, a term which was later on used for "pulveriser" which deals with that special sec- tion of algebra which is connected with indeterminate equations of the first degree. Algebra is often also known as avyakta- gaṇita or the calculations with unknowns, in contrast to arith- metic which was known as vyakta-gaṇita or the calculations with knowns. Algebra goes to Europe from India In the history of mathematical sciences, as Colebrooke rightly remarks, it has long been a question to whom the inven- tion of algebraic analysis is due. There is no doubt that Europe got algebra from Arabs mediately or immediately. But the Arabs themselves scarcely pretend to the discovery of algebra. Colebrooke says that they were not in general inventors but scholars during the short period of their successful culture of the sciences; and the germ at least of the algebraic analysis is to be found among the Greeks in an age not precisely determined, but more than probably anterior to the earliest dawn of civilisation among the Arabs; and this science in a more advanced state subsisted among the Hindus prior to the earliest disclosure of it by the Arabians to modern Europe. (Colebrooke: Disserta- tion on the Algebra of the Hindus)¹. Colebrooke based his observations on the texts he could procure for his studies. These were: Bhāskara II's Bījagaṇita or Vījagaṇita (1150 A.D.) and Līlāvatī (1150 A.D.), the Gaṇitādhyāya and Kuṭṭakādhyāya of Brahmagupta in his famous treatise the Brahma Siddhānta or rather the Brāhmasphuṭasiddhānta (628

1. Colebrooke, H. T., Miscellaneous Essays, Vol. II, 1872, p. 418.

190 BRAHMAGUPTA AS AN ALGEBRAIST A.D.). There can be no doubt regarding the age of these two authors. Bhāskara II completed his great work on the Siddhānta- śiromaṇi in 1072 Śaka, and Karaṇa-kutūhala a practical astro- nomical treatise in 1105 Śaka; these dates are based on the passages given by Bhāskara himself in his works. The Bīja- gaṇita and the Līlāvatī form parts of the great treatise, the Siddhānta-śiromaṇi. The genuineness of the text is established, as Colebrooke says, with no less certainty by numerous commen- tators in Sanskrit, besides a Persian version of it. Those com- mentaries comprise a perpetual gloss, in which every passage of the original is noticed and interpreted : and every word of it is repeated and explained. From comparison and collation of various texts, it appears then that the work of Bhāskara, exhibit- ing the same uniform text which the modern transcripts of it do, was in the hands of both Muhammedans and Hindus, between two or three centuries ago : and numerous copies of it having been diffused throughout India, at an earlier period, as of a performance held in high estimation, It was the subject of study and habitual reference in countries and places so remote from each other as the north and west of India and the Southern Peninsula. This though not marking any extraordinary antiquity, nor approaching to that of the author himself, was a material point to be determined : as there will be in the sequel, so says Cole- brooke, occasion to show, that modes of analysis, and in parti- cular, general methods for the solution of indeterminate prob- lems both of the first and second degrees, are taught in the Bīja-gaṇita, and those for the first degrees repeated in the Līlāvatī, which were unknown to the mathematicians of the West, until invented anew in the last two centuries by algebraists of France and England.¹ Bhāskara who himself flourished more than six hundred and fifty years ago, was in this respect a compiler and took those methods from Indian authors as much more ancient than himself. Regarding the age of the precursors of Bhāskara II, Cole- brooke says : The age of his precursors cannot be determined with equal precision. He then proceeds to examine the evidence as follows :

1. Colebrooke, H. T., Miscellaneous Essays, p. 421.

ALGEBRA GOES TO EUROPE FROM INDIA 191 Towards the close of his treatise on Algebra, Bhāskara II informs us, that it is compiled and abridged from the more diffuse works on the same subject, bearing the names Brāhme (meaning no doubt Brahmagupta), Śrīdhara and Padmanābha; and in the body of his treatise, he has cited a passage of Śrīdhara's algebra and another of Padmanābha. He repeatedly adverts to preceding writers and refers to them in general terms, where his commentators understand him to allude to Āryabhaṭa, to Brahmagupta to the latter's scholiast Caturveda Pṛthūdaka Svāmī and to the other writers above mentioned. Most, if not all, of the treatises, to which he thus alludes, must have been extant, and in the hands of his commentators, when they wrote; as appears from their quotations of them; more especially those of Brahmagupta and Āryabhaṭa, who are cited, and particularly the first mentioned, in several instances. A long and diligent research in various parts of India, has, however, failed of recovering any part of the Padmanābha Bīja (or the algebra of Padmanābha) and of the algebraic and other works of Āryabhaṭa. But the translator has been more fortunate in regard to the works of Śrīdhara and Brahmagupta, having in his collection Śrīdhara's compendium of arithmetic, and a copy incomplete however, of the text and scholia of Brahmagupta's Brahmasiddhānta comprising among other no less interesting matter, a chapter treating of arithmetic and mensuration; and another, the subject of which is algebra : both of them fortu- nately complete. The commentary is a perpetual one; successively quoting in length each verse of the text; proceeding to the interpretation of it, word by word; and subjoining elucidations and remarks; and its colo- phon, at the close of each chapter, gives the title of the work and the name of the author. Now the name which is there given, Caturveda Pṛthudaka Svāmī, is that of a celebrated scholiast of Brahmagupta, frequently cited as such by the commentaries of

192 BRAHMAGUPTA AS AN ALGEBRAIST . Bhāskara and by other astronomical writers; and the title of the work, Brāhmasiddhānta or sometimes Brāh- masphuṭasiddhānta, corresponds, in the shorter form, to the known title of Brahmagupta's treatise in the usual references to it by Bhāskara's commentators, and answers, in the longer form, to the designation of it, as indicated in an introductory couplet which is quoted from Brahmagupta by Lakṣmīdāsa, a scholiast of Bhās- kara II. Remarking this coincidence, the translator proceeded to collate, with the text and commentary, numerous quotations from both, which he found in Bhāskara's writings or in those of his expositors. The result confirmed the indication and established the identity of both text and scholia as Brahmagupta's treatise, and the gloss of Pṛthūdaka. The authenticity of the Brāhmasiddhānta is further confirmed by numer- ous quotations in the commentary of Bhaṭṭotpala on the Saṁhitā of Varāhamihira : as the quotations from the Brāhmasiddhānta, in that commentary, (which is the work of an author who flourished eight hundred and fifty years ago) are verified in the copy under consideration. A few instances of both will suffice, and cannot fail to produce conviction. It is confidently concluded, that the chapters on arith- metic and algebra, fortunately entire in a copy in many parts imperfect, of Brahmagupta's celebrated work as here described, are genuine and authentic. It remains to investigate the age of the author. Mr. Davis, who first opined to the public a correct view of the astronomical computations of the Hindus, is of opinion, that Brahmagupta lived in the seventh century of the Christian era. Dr. William Hunter, who resided for some time with a British Embassy at Ujjay- inī, and made diligent researches into the remains of Indian science at that ancient seat of Hindu astrono- mical knowledge, was there furnished, by the learned astronomers whom he consulted, with the ages of the principal ancient authorities. They assigned to Brah- magupta the date of 550 Saka; which answers to A.D.

ALGEBRA GOES TO EUROPE FROM INDIA 193 628. The grounds on which they proceeded are unfor- tunately not specified : but as they gave Bhāskara's age correctly, as well as several other dates right, which admit of being verified; it is presumed that they had grounds, though unexplained, for the information which they communicated.. Mr. Bentley, who is little disposed to favour the anti- quity of an Indian astronomer, has given his reasons for considering the astronomical system which Brah- magupta teaches, to be between twelve and thirteen hundred years old (1263 years in A.D. 1799). Now as the system taught by this author is professedly one corrected and adapted by him to conform with the observed positions of the celestial objects when he wrote, the age, when their positions would be conform- able with the results of computations made as by him directed, is precisely the age of the author himself : and so far as Mr. Bentley's calculations may be consi- dered to approximate the truth, the date of Brahma- gupta's performance is determined with like approach . to exactness, within a certain latitude however of uncertainty for allowance to be made on account of the inaccuracy of Hindu observations. The translator has assigned on former occasions the grounds upon which he sees reason to place the author's age, soon after the period when the vernal equinox coincided with the beginning of the lunar mansion and zodiacal asterism Aśvini, where the Hindu ecliptic now commences. He is supported in it by the senti- ments of Bhāskara and other Indian astronomers, who infer from Brahmagupta's doctrine concerning the solistitial points, of which he does not admit a periodi- cal motion, that he lived when the equinoxes did not, sensibly to him, deviate from the beginning of Aśvini and middle of citrā on the Hindu sphere. On these grounds it is maintained, that Brahmagupta is rightly placed in the sixth or beginning of the seventh century of the Christian era, as the subjoined calculations will more particularly show. The age when Brahmagupta