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

164 BRAHMAGUPTA AND ARITHMETIC bhaṭīya : The continued product of three equals and also the solid having twelve (equal) edges are called ghana.¹ A method of cubing applicable to numbers written in the decimal place-value notation, has been in use in this country from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had the familiarity with this method; he, however, does not give the method of cubing in his treatise, though he describes the inverse process of extracting the cube-root. Brahmagupta gives the method of cubing in the following verse : Set down the cube of the last (antya); then place at the next place from it, thrice the square of the last multi- plied by the succeeding; then place at the next place thrice the square of the succeeding multiplied by the last, and (at the next place) the cube of the succeeding. This gives the cube.² The rule may be illustrated by an example. Example : To cube 1357. The given number has four places, i.e., four portions. First we take the last digit 1 and the succeeding digit 3, i.e. 13 and apply the method of cubing thus : (i) Cube of the last (1³) = 1 (ii) Thrice the square of the last (3.1²) multiplied by the succeeding (3) gives (3.3.1²) = 9 (placing at the next place) (iii) Thrice the square of the succeeding, multiplied by the last gives (3.3².1) = 27 (placing at the next place) (iv) Cube of the succeeding (3³) = 27 (placing at the next place) ———————————————————————————————————————————————— Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रयस्यात् ।। — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसङ्गुणा च तत्प्रथमात् । उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.

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164 BRAHMAGUPTA AND ARITHMETIC bhaṭīya : The continued product of three equals and also the solid having twelve (equal) edges are called ghana.¹ A method of cubing applicable to numbers written in the decimal place-value notation, has been in use in this country from before the 5th century A.D. Āryabhaṭa I (499 A.D.) had the familiarity with this method; he, however, does not give the method of cubing in his treatise, though he describes the inverse process of extracting the cube-root. Brahmagupta gives the method of cubing in the following verse : Set down the cube of the last (antya); then place at the next place from it, thrice the square of the last multi- plied by the succeeding; then place at the next place thrice the square of the succeeding multiplied by the last, and (at the next place) the cube of the succeeding. This gives the cube.² The rule may be illustrated by an example. Example : To cube 1357. The given number has four places, i.e., four portions. First we take the last digit 1 and the succeeding digit 3, i.e. 13 and apply the method of cubing thus : (i) Cube of the last (1³) = 1 (ii) Thrice the square of the last (3.1²) multiplied by the succeeding (3) gives (3.3.1²) = 9 (placing at the next place) (iii) Thrice the square of the succeeding, multiplied by the last gives (3.3².1) = 27 (placing at the next place) (iv) Cube of the succeeding (3³) = 27 (placing at the next place) —————— Thus 13³ is the sum 2197

1. सदृशत्रयसंवर्गो घनस्तथा द्वादशाश्रस्स्यात् ॥ — Ārya, II. 3.
2. स्थाप्योऽन्त्यघनोऽन्त्यकृतिस्त्रिगुणोत्तरसंगुणा च तत्प्रथमात् । उत्तरकृतिरन्त्यगुणा त्रिगुणा चोत्तर घनश्च घनः । — BrSPSi. XII, 6.

C U B E 165 After this we take the next figure, 5, i.e., the number 135, and in this consider 13 as the last and 5 as the succeeding. Then the method proceeds thus : (i) The cube of the last (13³) as already obtained = 2197 (ii) Thrice the square of the last multiplied by the succeeding, i.e. 3.13².5 = 2535 (placing at the next place) (iii) Thrice the square of the succeeding multiplied by the last, i.e. 3.5².13 = 975 (placing at the next place) (iv) Cube of the succeeding, i.e. 5³ = 125 (placing at the next place) Thus 135³ is the sum 2460375 Now the remaining figure 7 is taken, so that the number is 1357, of which 135 is the last and 7 the succeeding. The method proceeds thus : (i) Cube of the last, i.e. (135³) as already obtained = 2460375 (ii) Thrice the square of the last into the succee- ding, i.e. 3. (135)². 7 = 382725 (placing at the next place) (iii) Thrice the square of the succeeding into the last, i.e. 3.7². 135 = 19845 (placing at the next place) (iv) Cube of the succeeding i.e. 7³ = 343 (placing at the next place) ———————— Thus (1357)³ is the sum 2498846293 Evidently these methods of cubing are based on the identity: (a+b)³=a³+3a²b+3ab²+b³ and keeping in mind the place values of numerals in a given

166 BRAHMAGUPTA AND ARITHMETIC number (this accounts for keeping the results of each of the four operations at the next place). Square-Root Indian synonyms for square-root are vargamūla or pada of a kṛti. The word mūla means the "root" of a tree, which may also mean the "foot" or the lowest part or bottom of a thing and hence "pada" or foot also became a synonym of root. Brahma- Gupta defines square-root as follows : The pada (root) of a kṛti (square) is that of which it is the square.¹ While the word mūla for root is the oldest in Indian litera- ture (it occurs in Anuyogadvāra-sūtra, c. 100 B.C.), the word pada for root probably for the first time occurs in the writings of Brahmagupta. The term mūla was borrowed by the Arabs who translated it by jadhr, meaning "basis of square". The Latin term radix also is a translation of the term mūla. In the Śulba litera- ture and in the Prākṛta texts, we find a term karaṇī for square- root. In geometry, this term karaṇī means a "side", In later days, the term karaṇī was reserved for surds, i.e. a square-root which cannot be exactly evaluated, but which may be represented by a line. We would like to quote here a rule for determining square- root of numbers from the Āryabhaṭīya : Always divide the even place by twice the square-root (upon the preceding odd place); after having subtracted from the odd place the square (of the quotient), the quotient put down at the next place (in the line of the root) gives the root². As an illustration, we shall proceed to find the square-root of 18225. The odd and even places are marked out by vertical (I) and horizontal (—) lines : The other steps are as follows :

1. पदं कृतिय॑त् तत् BrSpSi. XVIII. 35
2. भागं हरेदवर्गान्नित्यं द्विगुणेन वर्गमूलेन । वर्गाद्वर्गे शुद्धे लब्धं स्थानान्तरे मूलम् ॥ — Ārya, II. 4.

CUBEROOT 167 | - | - | 1 8 2 2 5 Subtract square 1 root = 1 Divide by twice ——— the root 2) 8 (3 placing quotient at the 6 next place, the root=13 ——— 22 Subtract square of quotient 9 Divide by twice 26)132(5 placing quotient at the the root 130 next place, the root=135 Subtract square 25 of the quotient 25 The process ends. The square-root of 18225 is thus 135. It has been stated by Kaye, that Āryabhaṭa’s method of finding out the square-root is algebraic in character, and that it resembles the method given by Theon of Alexandria. Ārya- bhaṭa’s method is purely arithmetic and not algebraic is the view of Datta and Singh who do not agree with Kaye on this point. Cube Root The Sanskrit term for cube-root is ghanamūla or ghanapada. The first mention of the operation of cube-root is found in the Āryabhaṭīya of Āryabhaṭa I (499 A.D.), though the operation is given in only a concise form : Divide the second aghana place by thrice the square of the cube-root; subtract from the first aghana place the square of the quotient multiplied by thrice the preceding (cube-root); and (subtract) the cube (of the quotient) from the ghana place; (the quotient put down) at the next place (in the line of the root) gives (the root).¹ As has been explained by all the commentators on the Āryabhaṭīya, the units place is ghana; the tens place is first aghana, the hundreds place is the second aghana, the thousands place is ghana, the ten thousands place is first aghana, the hun-

1. अघनाद् भजेद् द्वितीयात् त्रिगुणेन घनस्य मूलवर्गेण। वर्गस्त्रिपूर्व गुणितश्शोध्यः प्रथमाद् घनश्च घनात् ॥ —Ārya. II. 5

168 BRAHMAGUPTA AND ARITHMETIC dred-thousands place is second aghana, and so on. Thus to find out the cube-root, one has to mark out the ghana, first aghana and second aghana places, then the process of finding out the cube-root begins with the subtraction of the greatest cube num- ber from the figures up to the last ghana place. Though this has not been explicitly mentioned in the rule, the commentators say that it is implied in the expression ghanasya mūla-vargeṇa etc. ("by the square of the cube-root etc.") We are reproducing here an illustration given by Datta and Singh. Example. Find the cube-root of 1953125. The places are divided into groups of three by marking them as below [ghana ( | ) first aghana (—) and second aghana (--)]: | — — | — — | 1 9 5 3 1 2 5 Subtract cube 1 ... ... ... ... ... (c) Root=1 Divide by thrice —————————————— square of root, i.e. 3.1² 3)9(2 ... (a) Placing quotient Subtract square 6 after the root 1 of quotient mul- —— gives the root 12 tiplied by thrice 35 the previous root, 12 ... (b) i.e. 2².3.1 —— Subtract cube of 233 quotient, i.e. 2³ 8 ... (c) Divide by thrice square of the root, i.e. 3.12² 432)2251(5 ... (a) Placing quotient Subtract square 2160 after the root quotient multiplied ———— 12 gives the by thrice the pre- 912 root 125 vious root, i. e. 5².3.12 900 ———— Subtract cube of 125 ... (b) quotient, i.e. 5³ 125 ... (c) ———— Thus the cube-root=125. From the details given, it would be clear that the present

CUBE ROOT 169 method of extracting the cube-root is almost a contraction of the method first given by Āryabhaṭa I (499 A.D.) The method of Āryabhaṭa has been invariably followed by Indian mathematicians. Brahmagupta in his Brāhmasphuṭa- siddhānta repeats the method in the following words : The divisor for the second aghana place is thrice the square of the cube-root; the square of the quotient multiplied by three and the preceeding (root) must be subtracted from the next (aghana place to the right). and the cube (of the quotient) from the ghana place (the procedure repeated gives) the root.¹ Śrīdhara and Āryabhaṭa II have further improved on the method of extracting cube-root proposed by Āryabhaṭa I and followed by Brahmagupta. Rule for finding the cube-root as given by Śrīdhara in his Pāṭīgaṇita is as follows : (Divide the digits beginning with the units' place into periods of) one ghana-pada (one "cube" place) and two aghana-padas (two "non-cube" places). Then subtrac- ting the (greatest possible) cube from the (last) ghana- pada and placing the (cube) root underneath the third place (to the right of the last ghana-pada), divide out the remainder up to one place less (than that occupied by the cube-root) by thrice the square of the cube-root, which, is not destroyed. Setting down the quotient (obtained from division) in the line (of the cube-root), (and designating the quotient as the 'first' (ādima) and the cube-root as the 'last' (antya), subtract the square of that quotient, as multi- plied by thrice the 'last' (antya) from one place less than that occupied by the quotient (uparima-rāśi) as before, and the cube of the 'first' (ādima) from its own place. (The number now standing in the line of cube-root is the cube-root of the given number up to its last-but one ghana-pada (cube place) from the left). Again apply the rule, "(placing cube-root) under the third place" etc. (provided there be more than two ghana-padas (cube places) in the given number; and

1. छेदो घनाद् द्वितीयाद् घनमूलकृतेस्त्रिसंगुणाप्ताप्तकृतिः । शोध्या त्रिपूर्वगुणिता प्रथमाद् घनतो घनो मूलम् ॥ —BrSpSi. XII. 7

170 BRAHMAGUPTA AND ARITHMETIC continue the process till all ghana-padas (cube-places) are exhausted). This will give the (cube) root (of the given number).¹ K.S. Shukla in his translation and commentary of this book has given the illustration of extracting cube - root as follows : Example :- To find the cube root of 277167808. Let us indicate ghana-padas or 'cube' places by "c" and aghana-padas or non-cube places as "n" : n n c n n c n n c 2 7 7 1 6 7 8 0 8 Subtract the greatest possible cube (i.e. 6³ or 216) from the last 'cube' place (i.e. from 277) and place the cube- root (i.e. 6) underneath the third place to the right of the last 'cube' place; thus we have n n c n n c n n c 6 1 1 6 7 8 0 8 (remainder) 6 (line of cube-root) Dividing out by thrice the square of the cube-root (i.e. by 3.6² or 108) the remainder up to one place less than that occupied by the cube-root (i.e. 611) and setting down the quotient in the line of the cube-root (to the right of the cube-root), we have n n c n n c n n c 7 1 6 7 8 0 8 (remainder) 6 5 (line of cube-root) Let now quotient 5 be called the 'first' (ādima) and the cube-root 6 the 'last' (antya). Then subtracting the square of the 'first' (ādima) as multiplied by thrice the 'last' (antya) (i.e. 3×6×5² or 450) from one place less than that occupied by the quotient (i.e. from 716), we get

1. घनपदमघनपदे द्वे घन (पद) तोऽपास्य घनमदो मूलम् । संयोज्य तृतीयपदस्याधस्तदनष्टवर्गेण ॥ २९ ॥ एकस्थानोनतया शेषं त्रिगुणेन (सं) भजेत्तस्मात् । लब्धं निवेश्य पङ्क्त्यां तद्वर्गं त्रिगुणमन्त्यहतम् ॥ ३० ॥ जह्यादूपरिमराशेः प्राग्वद् घनमादिमस्य (च) स्वपदात् । भूयस्तृतीय पदस्याध इत्यादिक विधिर्मूलम् ॥ ३१॥ —Śrīdhara, Pāṭīgaṇita, 29-31

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