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

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

194 BRAHMAGUPTA AS AN ALGEBRAIST flourished, seems then, from the concurrence of all these arguments, to be satisfactorily settled as antecedent to the earliest dawn of the culture of sciences among the Arabs; and consequently establishes the fact, that the Hindus were in possession of algebra before it was known to the Arabians. Brahmagupta's treatise, however, is not the earliest work known to have been written on the same subject by an Indian author. The most eminent scholiast of Bhāskara II (Gaṇeśa) quotes a passage of Āryabhaṭa specifying algebra under the designation of Bīja, and making separate mention of Kuṭṭaka, which more par- ticularly intends a problem subservient to the general method of resolution of indeterminate problems of the first degree : he is understood by another of Bhāskara's commentators to be at the head of the elder writers, to whom the text then under consideration adverts, as having designated by the name of Madhyamāharaṇa the resolution of affected quadratic equations by means of the completion of the square. It is to be presumed, therefore, that the treatise of Āryabhaṭa then extant, did extend to quadratic equations in the determinate analysis, and to indeterminate problems of the first degree; if not to those of the second likewise, as most probably it did. This ancient astronomer and algebraist, so says Cole- brooke, was anterior to both Varāhamihira and Brahmagupta, being repeatedly named by the latter; and the determination of the age when he flourished is particularly interesting as his astronomical system, though on some points agreeing, essentially dis- agreed on others, with that which the Hindu astrono- mers still maintain. He, as Colebrooke says, is considered by the commen- tators of the Sūryasiddhānta and Śiromaṇi, as the earliest of uninspired and mere human writers on the science of astronomy, as having introduced requisite corrections into the system of Parāśara, from whom he took the numbers for the planetary mean motions; as

ALGEBRA GOES TO EUROPE FROM INDIA 195 having been followed in the tract of emendation, after a sufficient interval to make further correction requisite, by Durgāsinha and Mihira; who were again succeeded after a further interval by Brahmagupta, son of Jiṣṇu. In short, says Colebrooke, Āryabhaṭa was founder of one of the sets of Indian astronomers, as Puliśa, an author likewise anterior to both Varāhamihira and Brahmagupta, was of another : which were distingui- shed by names derived from the discriminative tenets respecting the commencement of planetary motions at sunrise according to the first, but at midnight accord- ing to the latter, on the meridian of Laṅkā, at the beginning of the great astronomical cycle. A third sect began the astronomical day, as well as the great period, at noon. Āryabhaṭa’s name accompanied the intimation which the Arab astronomers (under the Abbasside Khalifs, as it would appear,) received, that three distinct astro-

• nomical systems were current among the Hindus of those days : and it is but slightly corrupted, certainly not at all disguised, in the Arabic representation of it Arjabahar, or rather Arjabhar, (corrupted form of Āryabhaṭa). The two other systems were, first, Brahmagupta’s Siddhānta which was the one they became best acquainted with, and to which they apply the denomination of the sind-hind; and second, that of Arca, the Sun, which they write Arcand a corruption still prevalent in the vulgar Hindi. Āryabhaṭa appears to have had more correct notions of the true explanation of celestial phenomena than Brahmagupta himself, so says Colebrooke; who in a few instances, correcting errors of his predecessor, but oftener deviating from that predecessor’s juster views, has been followed by the herd of modern Hindu astro- nomers, in a system not improved, but deteriorated, since the time of the more ancient author. Considering the proficiency of Āryabhaṭa in astronomi- cal science, and adverting to the fact of his having

196 BRAHMAGUPTA AS AN ALGEBRAIST written algebra, as well as to the circumstance of his being named by numerous writers as the founder of a sect, or author of a system in astronomy, and being quoted at the head of algebraists, when the commen- tators of extant treatises have occasion to mention early and original writers on this branch of science, it is not necessary to seek further for a mathematician qualified to have been the great improver of the ana- lytic art, and likely to have been the person by whom it was carried to the pitch to which it is found to have attained among the Hindus, and at which it is observ- ed to be nearly stationary through the long lapse of ages which have since passed : the later additions being few and unessential in the writings of Brahmagupta, of Bhāskara and of Jñānarāja, though they lived at intervals of centuries from each other. Āryabhaṭa, Colebrooke rightly says, then being the earliest author known to have treated of Algebra among the Hindus, and being likely to be, if not the inventor, the improver of that analysis, by whom too it was pushed nearly to the whole degree of excellence which it is found to have attained among them; it becomes in an especial manner interesting to investigate any discoverable trace in the absence of better and more direct evidence, which may tend to fix the date of his labours; or to indicate the time which elapsed between him and Brahmagupta, whose age is more accurately determined. Taking Āryabhaṭa, for reasons given, to have preceded Brahmagupta and Varāhamihira by several centuries; and Brahmagupta to have flourished more than twelve hundred years ago, and Varāhamihira, concerning whose works and age, Colebrooke has given a few notes, and has placed him at the beginning of the sixth century after Christ, it appears probable that this earliest of known Hindu algebraists wrote as far back as the fifth century of the Christian era; and perhaps in an earlier age. Hence it is concluded that he is nearly as ancient as the Grecian algebraist Diophantus, sup-

ALGEBRA GOES TO EUROPE FROM INDIA 197 posed on the authority of Abulfaraj, to have flourished in the time of the Emperor Julian or about A. D. 360. Colebrooke further says : Admitting the Hindu and Alexandrian authors to be nearly equally ancient, it must be conceded in favour of the Indian algebraist, that he was more advanced in the science; since he appears to have been in possession of the resolution of equations involving several unknowns, which it is not clear, nor fairly presumable, that Diophantus, knew; and a general method of indeterminate problems of at least the first degree, to a knowledge of which the Grecian algebraist had certainly not attained ; though he displays infinite sagacity and ingenuity in particu- lar solutions ; and though a certain routine is indiscer- nible in them. Colebrooke appears to be of the view that Greeks. were the first to discover the solution of equations involving one unknown; and this knowledge was passed to ancient Indians by their Greek instructors in impro- ved astronomy. But "by the ingenuity of the Hindu- scholars, the hint was rendered fruitful and the algeb- raic method was soon ripened from that slender beginn- ing to the advanced state of a well arranged science, as it was taught by Āryabhaṭa, and as it is found in treatises compiled by Brahmagupta and Bhāskara." We do not agree with this analysis in entirety. Indian algebra is entirely of Indian roots. It had its beginning in the times of Saṃhitās and Brāhmaṇas. Some of the equations and problems were solved by geometric methods. It must have had its origin in the Śulba period if not before. Āryabhaṭa undoubtedly was the discoverer of many algebraic solutions of equations of the first and higher order with one and more unkno- wns. It is rather too much to trace the influence of Diophantus on Indian algebra which developed in this country independently. Brahmagupta is one of the most brilliant algebraists we ever had in the entire history of mathematics.

198 BRAHMGUPTA AS AN ALGEBRAIST Technical Terms Coefficient— In the ancient Indian algebra, there is no systematic term for the coefficient. Usually, the power of the unknown is men- tioned when the reference is to the coefficient of that power. At one place, for example, we find Pṛthūdaka Svāmī (the commenta- tor of Brahmagupta's Brāhmasphuṭasiddhānta) writing “the num- ber (aṅka) which is the coefficient of the square of the unknown is called the ‘square’ and the number which forms the coefficient of the (simple) unknown is called the ‘unknown quantity’ (avyakta- māna).”¹ However, at many places, we find the use of a technical term also. Brahmagupta once calls the coefficient saṁkhyā² (number) and on several other occasions guṇaka³ or guṇakāra⁴ (multiplier). Pṛtthūdaka Svāmī (860 A.D.) calls it aṅka (number) or prakṛti (multiplier). These terms may also be seen in the works of Śrīpati⁶ (1039) and Bhāskara II⁷ (1150 A.D.). The former also used the word rūpa for the same purpose.⁸ Unknown Quantity The unknown quantity has been termed as yāvat-tāvat (meaning so-much-as or as-many-as) in literature as early as 300 B. C. (vide the Sthānāṅga-sūtra⁹). In the Bakhasālī Manuscript, it has been termed as yadṛcchā, vāñchā or kāmikā (or any desired quantity)¹⁰. Āryabhaṭa I in one of his verses calls the unknown as gulikā¹¹ (literally meaning a shot) From the early seventh century A.D., the word avyakta was used for unknown quantities. Brahmagupta uses this term in his Brāhmasphuṭasiddhānta¹²

1. BrSpSi. XVIII. 44 (Com.)
2. वर्ण्यप्रमाण भावितघातो भवत्तीष्ट वर्णं संख्येवम् ! —BrSpSi. XVIII 63
3. मूलं द्विषेष्ट वर्गाद् गुणक गुणादिष्ट युत विहीनाच्च । —BrSpSi. XVIII 64 वर्गच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । —BrSpSi. XVIII 70
4. प्रथमोऽन्यमूलमन्यो गुणकार पदोद्धृतः प्रथमः । —BrSpSi. XVIII 69
5. BrSpSi XVIII. 44 (Com.)
6. SiSe XIV. 33-5.
7. Bījagaṇita
8. SiSe XIV. 33-5.
9. Sūtra 747.
10. BMs. Folio 22, verso; 23, recto and verso.
11. गुलिकान्तरेण विभजेद् द्वयोः ।
12. अव्यक्तवर्गं धनवर्गं वर्गपञ्चगत षड्गतादीनाम् ।

TECHNICAL TERMS 199 Power Since long, the word varga has been used for the second power; the word also stands for square (Uttarādhyayana Sūtra¹, B. C. c. 300). The third power is similarly known as ghana, the fourth power as varga-varga (square-square), the sixth power as ghana-varga (cube-square) and the twelfth power as ghana-varga- varga (cube-square-square). In later days, the fifth power was called vargaghana ghāta (here the word ghāta means product; the term means product of cube and square). The former system was multiplicative, rather than additive; wheras the latter was on the additive system. The seventh power on the additive system was known as varga-varga-ghana-ghāta (product of square-square and cube). Brahmagupta, however, uses a more scientific system for expressing the powers more than four. He calls the fifth power as Pañca gata (literally meaning, raised to the fifth), the sixth power as ṣaḍ-gat (raised to the sixth) and so on, thus adding the suffix gata to the name of the number indicating that power.² Bhāskara II has followed the system of Brahmagupta almost consi- stently for powers one and upwards. Equation Perhaps Brahmagupta has for the first time used the term samakaraṇa or samīkaraṇa (literally meaning making equal) or simply sama (equal or equation)³. Pṛthūdaka Svāmī (860) employs the term sāmya (equality or equation) for equation⁴. The equation is said to possess two Pakṣas⁵ (sides) Itara-Pakṣa and apara-pakṣa. Absolute Term Brahmagupta uses the term rūpa (literally meaning appear- ance) for an absolute term. It represents the visible or known

1. Chapter XXX, 10, 11.
2. अव्यक्तवर्गे घनवर्ग-वर्ग-पंचगत-षड्गतादीनाम् । सदृश द्विवधो वर्गस्त्यादि वधस्तद्गतोऽन्य जातिवधः ॥
3. वर्णं प्रमाण भावित-घातो भवतीष्टवर्ण संख्यैवम् । सिध्यति विनाऽपि भावित-समकरणात् किं कृतं तदतः ॥ अव्यक्तान्तर भक्तं व्यस्तं रूपान्तरं समेऽव्यक्तः ।
4. Siśe. XIV, 19.
5. Bījagaṇita.