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

CUBEROOT 171 n n c n n c n n c 2 6 6 7 8 0 8 (remainder) 6 5 (line of cube-root) And subtracting the cube of the 'first' (ādima) (i.e. 5³ or 125) from its own place (i.e. from 2667), we get n n c n n c n n c 2 5 4 2 8 0 8 (remainder) 6 5 (line of cube-root) One round of the operation is now over; and the num- ber 65 standing in the line of the cube-root is the cube- root of the given number (277167808) up to its last-but- one 'cube' place ( ghana pada ) from the left (i.e. of 277167), As there is one more 'cube' place (ghana-pada) on the right, the process is repeated. Thus placing the cube- root (i.e. 65) under the third place beginning with the last-but-one 'cube' place (ghana-pada), we have n n c n n c n n c 2 5 4 2 8 0 8 (remainder) 6 5 (line of cube-root) Dividing out 25428 by 3.65² (=12675) as before, and placing the quotient in the line of the cube-root, we have n n c n n c n n c 7 8 0 8 (remainder) 6 5 2 (line of cube-root) Subtracting 3 × 65 × 2² (=780) we get. n n c n n c n n c 8 (remainder) 6 5 2 (line of cube-root) Finally subtracting 2³=8 from 8, we get n n c n n c n n c 0 (remainder) 6 5 2 (line of cube-root) The second round of operation is now over. There being no more of ghana-pada ('cube' place) on the right, the process ends. The quantity in the line of cube root, viz., 652, is the cube-root of the given

172 BRAHMAGUPTA AND ARITHMETIC number. The remainder being zero, the cube-root is exact. Fractions The concept of fractions in India can be traced to very early times. In the Ṛgveda,¹ we find such terms as one-half (ardha) and three-fourths (tri-pāda). In a passage of the Maitrāyaṇī Saṁhitā² are mentioned the fractions one-sixteenth (kalā), one-twelfth (kuṣṭha), one-eighth (śapha) and one-fourth (pāda). In the Śulba Sūtras³ we have not only a mention of fractions, but they have been used in the statement and solution of problems of geometric nature. Here in the Śulba, unit frac- tions are denoted by the use of cardinal number with the term bhāga or aṁśa; thus pañca-daśa-bhāga (literally “fifteen parts”) is equivalent to one-fifteenth, sapta-bhāga (literally, “seven parts”) is equivalent to one-seventh, and so on... The use of ordinal numbers with the term bhāga or aṁśa is also quite common : thus pañcama bhāga stands for one-fifth. The composite fractions like tri-aṣṭama stands for three-eighths and dvi-saptama for two- sevenths. In the Bakhshālī Manuscript, the term tryaṣṭa occurs for 3/8 and 3⅜ is called trayastrayasta (three-three-eighths). The Sanskrit term for fraction is bhinna (literally meaning ‘broken’). Obviously the European terms as fractio, fraction, roupt, rotto or rocto are translations of the same term; they are derived from the Latin fractus (frangere) or ruptus meaning ‘broken’. The Indian term bhinna has a few more connotations; it stands for such numbers of the form : (a/b ± c/d), (a/b of c/d), (a/b ± c/d of a/b) or (a ± b/c). These forms were termed jāti’ i.e., ‘classes’, and the Indian treatises contain special rules for their reduction to proper frac- tions. Śrīdhara and Mahāvīra each enumerate six jātis, while our author, Brahmagupta, gives only five (Bhāskara II gives only four). The need for division of fractions in ‘classes’ arose out of the lack of proper symbolism to indicate mathematical opera- tions. (Datta and Singh Arithmetic, p. 188). The only operational symbol in use was a dot, standing for the negative sign.

1. Ṛv. X, 90,4
2. Mait S, III, 7,7.
3. B. Datta, Śulba, pp. 212ff.

FRACTIONS 173 Reduction to lowest terms.—A non-mathematical work, Tattvārthādhigama-Sūtra-Bhāṣya by Umāsvāti (c.150A.D.) casua- lly mentions as follows in the context of a philosophic discourse: Or, as when the expert mathematician, for the purpose of simplifying operations, removes common factors from the numerator and denominator of a fraction, there is no change in the value of the fraction, so....¹ Reduction to common denominator. Whenever we have to add or subtract fractions, we follow this reduction operation to a common denominator. Brahmagupta gives the reduction along with the similar processes : By the multiplication of the numerator and denomi- nator of each of the (fractional) quantities by other denominators, the quantities are reduced to a common denominator. In addition, the numerators are united, In subtraction their difference is taken.² Fractions in combination :—Since there was no proper symbolism available to these early Indian mathematicians, they divided combination of fractions into four classes : Bhāga, prabhāga, bhāgapavāha and bhāga-bhāga. (i) Bhāga has been mentioned by Brahmagupta (BrSpSi. XII, 8) thus : ( a/b ± c/d ± e/f ± ........ ) usually written as ┌───┬───┬───┐ ┌───┬────┬────┐ │ a │ c │ e │ or │ a │ .c │ .e │ ├───┼───┼───┤ ├───┼────┼────┤ │ b │ d │ f │ │ b │ d │ f │ └───┴───┴───┘ └───┴────┴────┘ where the dots denote subtraction. (ii) Prabhāga : The form ( a/b of c/d of e/f ......... ) This is written as ┌───┬───┬───┐ │ a │ c │ e │ ├───┼───┼───┤ │ b │ d │ f │ └───┴───┴───┘ (iii) Bhāgānubandha : The form ( a + b/c ) is written as ┌───┐ │ a │ ├───┤ │ b │ ├───┤ │ c │ └───┘

1. II, 52.
2. विपरीतच्छेदगुणा : राश्योश्छेदांशाः समच्छेदाः । संकलितेंऽशा योज्या व्यवकलितेंऽशान्तरं कार्यम् ॥ —BrSpSi. XII. 2.

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).

The uploaded image is completely blank (solid white) and does not contain any visible text, diagrams, or manuscript content to transcribe.

Please provide a valid image or page of the manuscript.

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.