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

212 BRAHMAGUPTA AS AN ALGEBRAIST The total value (of the unknown quantities) plus or minus the individual values (of the unknowns) multi- plied by an optional number deing severally (given), the sum (of the given quantities) divided by the num- ber of unknowns increased or decreased by the multi- plier will be the total value; thence the rest (can be determined).¹ Σx±cx₁=a₁, Σx±cx₂=a₂, Σx±cx₃=a₃,...... Σx±cx =an Therefore Σx = (a₁ + a₂ + a₃ + ...... + an) / (n ± c) Hence x₁ = 1/c ( ±a₁ ∓ (a₁ + a₂ + a₃ + ...... + an) / (n ± c) ); and so on for x₂, x₃ etc. Now we shall give the rule enunciated by Brahmagupta for solving linear equations involving several unknowns : Removing the other unknowns from (the side of) the

QUADRATIC EQUATIONS 213 be performed all operations conformably to the state- ment of the example and thus should be carefully framed two or more sides and also equations. Equi-clear- ance should be made first between two and two of them and so on to the last : from one side one un- known should be cleared, other unknowns reduced to a common denominator and also the absolute numbers should be cleared from the side opposite. The residue of other unknowns being divided by the residual coefficient of the first unknown will give the value of the first unknown. If there be obtained several such values, then with two and two of them, equations should be formed after reduction to common denominators. Proceeding in this way to the end find out the value of one unknown. If that value be (in terms of) another unknown then the coefficients of those two will be reciprocally the values of the two unknowns. If, however, there be present more un- knows in that value, the method of the pulveriser should be employed. Arbitrary values may then be assumed for some of the unknowns. Datta and Singh have said that the above rule of Brahma- gupta, and also the one indicated in the commentary of Pṛthudaka Svāmī, embraces the solution of indeterminate as well as the determinate equations. In fact, all the examples given by Brahmagupta in illustration of the rule are of indeterminate character. So far as the determinate simultaneous equations are concerned, Brahmagupta's method for solving them will be easily recognised to be the same as our present one. Quadratic Equations The geometrical solution of a quadratic equation in this country would take us to the Vedic Śulba period. The Bakha- śālī Manuscript also contains certain problems which need the solving of quadratic equations. I shall quote one out of the numerous available : A certain person travels s yojana on the first day and b yojana more on each successive day. Another who travels at the uniform rate of S yojana per day, has a start of t days. When will the first man overtake the second ?

214 BRAHMAGUPTA AS AN ALGEBRAIST This problem would today be expressed in terms of the following equation : S(t+x)=x {s+((x-1)/2)b}, where x is the number of days after which the first overtakes the second. We may write this equation as bx²-{2(S-s)+b}x=2tS whence the value x would be after solving the quadra- tic : x= (√{2(S-s)+b}²+8bts+{2(S-s)+b}) / 2b The Bakhasālī Manuscript gives this solution as follows : The daily travel (S) diminished by the march of the first day (s) is doubled; this is increased by the common increment (b). That (sum) multiplied by itself is designated (as the kṣepa quantity). The product of the daily travel and the start (t) being multiplied by eight times the common increment, the kṣepa quantity is added. The square-root of this (is increased by the kṣepā quantity; the sum divided by twice the common increment will give the required number of days). (BMS. Folio 5, recto) Āryabhaṭa I (499 A.D.) is regarded as the founder of algebra, since he gives the solutions of a few quandratic problems. For example, to find the number of terms of an arithmetical pro- gression (A.P.), he gives the following rule : The sum of the series multiplied by eight times the common difference is added by the square of the dif- ference between twice the first term and the common difference: the square-root (of the result) is diminished by twice the first term and (then) divided by the com- mon difference : half of this quotient plus unity is the number of terms.¹ In the modern notations of algebra, the solution would be expressed as follows :

1. गच्छोऽष्टोत्तर गुणिताद् द्विगुणाद्युत्तर विशेषवर्गयुतात् । मूलं द्विगुणाद्यूनं स्वोत्तर भजितं सरूपार्थं ॥ —Ārya. II, 20

QUADRATIC EQUATIONS 215 n = 1/2 { [√(8bs + (2a - b)²) - 2a] / b + 1 } There is another certain interest problem¹, the solution of which has been provided in the Āryabhaṭīya as x = [√(Apt + (p/2)²) - p/2] / t which is the solution of the quadratic equation : tx² + px - Ap = 0 Āyabhaṭa I has thus given the solutions of a few quadratic equations, but he nowhere gives the procedure of solving these equations. We give here the Rules of Brahmagupta for the solution of quadratic equations. He undoubtedly is not the discoverer of these rules; but perhaps for the first time in the history of algebra we find the process of solving a quadratic equation so clearly indicated. First Rule : The quadratic : the absolute quantities multiplied by four times the coefficient of the square of the unknown are increased by the square of the coefficient of the middle (i.e. unknown); the square-root of the result being diminished by the coefficient of the middle and divided by twice the coefficient of the square of the unknown, is (the value of) the middle."² This expressed in the modern notations would mean x = [√(4ac + b²) - b] / 2a It would be noted that in this rule, Brahmagupta has emp- loyed the term madhya (middle) to imply the simple unknown as well as its coefficient. The origin of the term is doubtless connected with the mode of writing the quadratic equation in the form ax² + bx + 0 = 0x² + 0x + c so that there are three terms on each side of the equation.

1. मूलफलं सफलं कालमूलं गुणमर्धमूल कृति युक्तम् । मूलं मूलाधोनं कालहृतं स्यात्स्वमूलफलम् ॥ —Ārya. II. 25.
2. वर्गं चतुर्गुणितानां रूपाणां मध्यवर्गसहितानाम् । मूलं मध्येनोनं वर्ग द्विगुणोद्धृतं मध्यः ॥ —BrSpSi. XVIII. 44.

216 BRAHMAGUPTA AS AN ALGEBRAIST Second Rule : The absolute term multiplied by the coefficient of the square of the unknown is increased by the square of half the coefficient of the unknown; the square-root of the result diminished by half the coefficient of the unknown and divided by the coefficient of the square of the unknown is the unknown.² This when expressed in the modern algebraic notations would be x = [√(ac+(b/2)²) - (b/2)] / a Here if the quadratic equation is ax² + bx + c = 0 the 'absolute term' is c (the one without the unknown x), 'the coefficient of the square of unknown' means the coefficient of x², i.e. a, and the 'coefficient of the unknown' means the coefficient of x, i.e. b. The above two methods of Brahmagupta are exactly the same as were suggested by Āryabhaṭa I. The root of the quadratic equation for the number of terms of an arithmetic progression (A.P.) is given by Brahmagupta according to the first rule ² : n = [√(8bs+(2a-b)²) - (2a-b)] / 2b Third Rule : Brahmagupta also suggests a Third Rule which is very much the same as is used commonly now. Though it has not been expressedly suggested as a new rule, we find its application in a few instances. For example this rule has been suggested in connection with the following problem on interest : A certain sum (p) is lent out for a period (t₁); the interest accrued (x) is lent out again at this

1. वर्गाहत रूपाणामव्यक्ताधैंकृति संयुतानां यत् । पदमव्यक्ताधोनं तद्वर्ग विभक्तमव्यक्तः ॥ —BrSpSi. XVIII. 45
2. उत्तरहीनद्विगुणादि शेषवर्गं धनोत्तराष्टवधे । प्रक्षिप्य पदं शेषोनं द्विगुणोत्तरहृतं गच्छः ॥ —BrSpSi. XII. 18

QUADRATIC EQUATIONS 217 rate of interest for another period (t₂) and the total amount is A. Find x. The equation for determining x is (t₂ / pt₁) x² + x = A. The solution of this equation would be : x = √((pt₁ / 2t₂)² + (A / t₂) · pt₁) - pt₁ / 2t₂ Brahmagupta has stated the result in exactly the same form. Pṛthūdaka Svāmī has illustrated it in solving the following pro- blem of interest : Problem : A sum of five hundred paṇas (p) is lent out for a period of 4 months (t₁); the interest accrued (x) is lent out again at this rate of interest for another period of 10 months (t₂) and the total amount is 78 (A). Give the pramāṇa-phala, i.e., the interest accrued x. Here pramāṇa-kāla (t₁) = 4 months pramāṇa-dhana (p) = 500 paṇas para-kāla (t₂), the subsequent period = 10 months miśra dhana or the total interest accrued (A) = 78 paṇas. Brahmagupta states his solution of such quadratics like this : Take the product of the pramāṇa-dhana (p) or the sum originally lent out and pramāṇa-kāla, i.e. the period for which originally lent out (t₁); and divide by the para- kāla or the subsequent time (t₂); place this result at two places. Multiply the one placed at the first place with the miśra-dhana (A), that is with the total inter- est accrued; in this product add the square of half the one placed in the second place; now take the square- root of it, and from it subtract half of the one placed at the second place.¹

1. कालप्रमाणघातः परकालहृतो द्विधाऽऽप्तमिश्रवधात् । अन्यार्धकृति युतात् पदमन्यार्धोनं प्रमाण फलम् ॥ —BrSpSi. XII. 15.

218 BRAHMAGUPTA AS AN ALGEBRAIST Thus in the above example the product of pramāṇa-dhana and pramāṇa kāla divided by parakāla is (pt₁/t₂)—is (500 × 4) / 10 = 200. This is first multiplied by the total interest accrued (A); it becomes 200 × 78 = 15600. To this is now added square of half of 200 (which is 10000) ; it becomes 15600 plus 10000 = 25600. Its square-root is taken which is 160. From this is subtracted half of the quantity (i.e. half of 200 which is 100). Thus 160–100 = 60, which is the answer. It was the interest which first accrued (x). Another Quadratic Problem : Brahmagupta refers to an astronomical problem which involves the quadratic equation (72 + a²)x² ∓ 24 apx = 144 ( R² / 2 - p² ), where a = agra (the sine of the amplitude of the Sun), b = palabha (the equinoctial shadow of a gnomon 12 aṅguli long), R = radius, and x = koṇaśaṅku (sine of the altitude of the Sun when his altitude is 45°). Dividing out by (72 + a²) we have x² ∓ 2mx = n, where m = 12 ap / (72 + a²), n = 144(R²/2 - p²) / (72 + a²) Therefore we have x = √(m² + n) ± m, as stated by Brahmagupta. We find the same result in the Sūrya-siddhānta and in the text of Śrīpati. Āryabhaṭa II (1150) also followed the method of Āryabhaṭa I and Brahmagupta in solving a quadratic equation in connection with finding out the number of terms in an arithmetical progression (A.P.) whose first term is (a), common difference is b and the sum is s. The number of terms n is given by¹ n = [√(2bs + (a - b/2)²) - a + b/2] / b Two Roots of a Quadratic Equation and Brahmagupta A quadratic equation has two roots. This must have been known to Indian algebraists even at a very early stage. Bhāskara II in his Bījagaṇita has quoted a rule ascribed to an ancient writer Padmanābha whose works are not available now :

1. Mahāsiddhānta. Bhāskara II, XV. 50

TWO ROOTS OF A QUADRATIC EQUATION 219 If (after extracting roots) the square-root of the absolute side (of the quadratic) be less than the negative abso- lute term on the other side, then taking it negative as well as positive two values (of the unknown) are found¹. The term used here is dvividhotpadyate mitiḥ which means that two values are obtained. The existence of two roots of a quadratic equation appears to have been known also to Brahmagupta (628 A.D.). In illustra- tion of his rules for the solution of a quadratic he has stated two problems involving practically the same equation : Problem I : The square-root of the residue of the revolution of the Sun less 2 is diminished by 1, multi- plied by 10 and added by 2; when will this be equal to the residue of the revolution of the Sun less 1, on Wednesday ?² Problem II : When will the square of one-fourth the residue of the exceeding months less three, be equal to the residue of the exceeding months ? We shall follow Pṛthūdaka Svāmī in solving the Problem I. In this problem the residue of the revolutions of the Sun may be supposed to be x² + 2; then by the question, we have 10 (x - 1) + 2 = x² + 1, or x² - 10x = -9 Again in Problem II, if we put 4x for the residue of the exceeding month, then we have (x - 3)² = 4x or x² - 10x = -9. Now by the second rule of Brahmagupta, retaining both the signs of the radical, we get : x = 5 ± √(25 - 9) = 9 or 1.

1. व्यक्त पक्षस्य चेन्मूलमन्यपक्षर्णरूपतः । अल्पं धनर्णगं कृत्वा द्विविधोत्पद्यते मितिः ॥ —Bhāskara, Bījagaṇita
2. मण्डलशेषाद् द्व्यूनान्मूलं व्येकं दशाहतं द्वियुतम् । मण्डलशेषं व्येकं भानोर्ज्ञादिने कदा भवति ॥ —BrSpSi. XVIII. 49
3. अधिमासशेषपादात् त्र्यूनाद्वर्गोऽधिमासशेषसमः । अवमावशेषतो वावमशेषसमः कदा भवति ॥ —BrSpSi. XVIII: 50.

220 BRAHMAGUPTA AS AN ALGEBRAIST As shown by Pṛthudaka Svāmī, the first value is taken by Brahmagupta for the Problem I and second value for the problem II. Thus it is quite clear that Brahmagupta uses sometimes the positive and at other times the negative sign with the radical. Hence we shall say that Brahmagupta knew that a quadratic equation would have two roots, and according to the requisite- ness of the problem, one value out of the two would be utilised. Simultaneous Quadratic Equations Indian authors usually treated problems involving various forms of simultaneous quadratic equations. (i) x — y = d } (ii) x + y = a } xy = b } xy = b } (iii) x² + y² = c } (iv) x² + y² = c } xy = b } x + y = a } For the solution of the combination (i), Āryabhaṭa I gives the following rule in his Āryabhaṭīya . The square-root of four times the product (of two quan- tities) added with the square of their difference, being added and diminished by their difference and halved gives the two multiplicands.¹ This means that x = ½ √(d² + 4b + d) , y = ½ (√(d² + 4b) - d) For the solution of the same combination, Brahmagupta states as follows : The square-root of the sum of the square of the diffe- rence of the residues and two squared times the product of the residues, being added and subtracted by the difference of the residues, and halved (gives) the desi- red residues severally.² (Here by difference of the residues is mesnt x — y; and by product of the residues is meant xy.) Brahmagupta does not seem to give the solution for simulta- neous equations of the combination (ii). Mahāvīra (850 A.D.)

1. द्विकृति गुणात्संवर्गाद् द्वयन्तरवर्गेण संयुतान्मूलम् । अन्तरयुक्तं हीनं तद्गुणकारद्वयं दलितम् ॥ —Ārya. II. 24
2. शेषवधाद् द्वि कृति गुणात् शेषान्तर वर्ग संयुतान्मूलम् । शेषान्तरेण युक्तं दलितं शेषे पृथगभीष्टे ॥ —Br SpSi. XVIII. 99

SIMULTANEOUS QUADRATIC EQUATION 221 has given the solution : Subtract four times the area (of a rectangle) from the square of the semi-perimeter then by saṅkramaṇa bet- ween the square-root of that (remainder) and the semi- perimeter, the base and the upright are obtained.¹ (GSS. VII. 129½) This expressed in the modern notations would be : x = ½(a + √(a² — 4b)), y = ½(a — √(a² — 4b)) For the combination (iii), Mahāvīra in his Gaṇita-Sāra- Saṃgraha gives the following rule : Add to and subtract twice the area (of a rectangle) from the square of the diagonal and extract the square-roots. By saṅkramaṇa between the greater and lesser of these (roots), the side and upright (are found).² This put in modern notations would be : x = ½ √(c + 2b) + √(c — 2b)). y = ½ (√(c + 2b) — √(c — 2b)). For the combination (iv), Āryabhaṭa I gives the following rule : From the square of the sum (of two quantities) subtract the sum of their squares. Half of the remainder is their product.³ The remaining operations will be similar to those for the equations (ii); so that x = ½ (a + √(2c — a²)), y = ½ (a — √(2c — a²)). Brahmagupta in this connection says : Subtract the square of the sum from twice the sum of squares; the square-root of the remainder being added to and subtracted from the sum and halved, (gives) the desired residues.⁴

1. GSS. VII. 129½
2. GSS. VII. 127½
3. संपर्कस्य हि वर्गाद्विशोधयेदेव वर्गसंपर्कम् । यत्तस्य भवत्यर्धं विद्याद् गुणकारसंवर्गम् ॥ —Ārya. II. 23
4. कृति संयोगाद् द्विगुणाद्युति वर्गं प्रोह्य शेष मूलं यत् । तेन युतोनो योगो दलितः शेषे पृथगभीष्टे ॥ —BrSpSi. XVIII. 98

222 BRAHMAGUPTA AS AN ALGEBRAIST These equations have also been treated by Mahāvīra, Bhāskara II and Nārāyaṇa. Nārāyaṇa has attempted two other forms of quadratic equations : (v) x² + y² = c } (vi) x² - y² = m } x - y = d } xy = b } For their solutions, see Datta and Singh, Algebra, P. 84. Rule of Dissimilar Operations : Datta and Singh say that the process of solving the follow- ing two particular cases of simultaneous quadratic equations was distinguished by most Indian mathematicians by the special designation viṣama-karma or dissimilar operation : (i) x² - y² = m } (ii) x² - y² = m } x - y = n } x + y = p } These equations have been regarded by these mathematicians as if of fundamental importance. They have given the following solutions (expressed in modern algebraic symbols) : For the combination (i) : x = ½ (m/n + n), y = ½ (m/n - n), For the combination (ii) : x = ½ (p + m/p), y = ½ (p - m/p). We shall express these solutions as follows in the words of Brahmagupta : The difference of the squares (of the unknowns) is divi- ded by the difference of the unknowns and the quotient is increased and diminished by the difference and divided by two ; (the results will be the two unknown quantities) ; (this is) dissimilar operation. The same rule is restated by him on a different occasion in the course of solving a problem. If then the difference of their squares, also the differe- 'nce of them (are given) ; the difference of the squares

1. योगोऽन्तरयुग्हीनो द्विहतः संक्रमणमन्तर विभक्तं वा । वर्गान्तरमन्तर-युतहीनं द्विहतं विषमकर्म । BrSpSi. XVIII. 36

RULE OF DISSIMILAR OPERATIONS 223 is divided by the difference of them, and this (latter) is added to and subtracted from the quotient and then divided by two ; (the results are) the residues whence the number of elapsed days (can be found).¹ This viṣama-karma or dissimilar operation has been descri- bed by other Indian algebraists also, as Āryabhaṭa II (Mahāsi- ddhānta, XVII, 22); Śrīpati (Siddhānta-śekhara, XIV. 13) ; Bhā- skara II (Līlāvatī) and Nārāyaṇa (Gaṇita-kaumudī, I, 32). Indeterminate Equations of the First Degree Āryabhaṭa I should be given the credit of giving for the first time a treatment of the indeterminate equation of the first degree. In his Āryabhaṭīya, we find a method for obtaining the general solution in positive integers of the simple indeterminate equation : by – ax = c for integral values of a,b,c, and further indicated how to extend it to get positive integral solutions of simultaneous indeter- minate equations of the first degree. His disciple Bhāskara I (522) showed that the same method might be applied to solve : by – ax = – c and further that the solution of this equation would follow that of by – ax = – 1. These methods of Āryabhaṭa I and Bhā- skara I have also been adopted by Brahmagupta, and in certain cases, the improvement were suggested by Āryabhaṭa II in the middle of the tenth century A.D. The problems which were treated by ancient Indian algeb- raists and which led them to the investigation of the simple inde- terminate equation of the first degree may be classified under three heads : Class I : To find a number N which being divided by two given numbers (a,b) will leave two given remain- ders (R₁, R₂). Thus we have : N = ax + R₁ = by + R₂

1. तद्वर्गान्तरमाधे तदन्तरं चान्तरोद्धृतयुतोनम् । वर्गान्तरं विभक्तं द्वाभ्यां शेषे ततोद्युगणः । BrSpSi, XVIII, 97

224 BRAHMAGUPTA AS AN ALGEBRAIST Hence ' by—ax = R₁—R₂ Putting c = R₁ ∽ R₂ we get by—ax = ± c the upper or lower sign being taken according as R₁ is grea- ter than or less than R₂. Class II : To find a number (x) such that its product with a given number (a) being increased or decreased by another given number (δ) and then divided by a third given number (β) will leave no remainder. This means that in other words, we shall have to get the solution of : ax ± r —————— = y β in positive integers. Class III : Here we have to deal with an equation of the form : by ÷ ax = ± c Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms which Brahmagupta has used in regards to the subject of indeter- minate analysis of the first degree. Āryabhaṭa I has also descri- bed this method in brief, but he does not use the word kuṭṭaka. In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā- kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have been translated into English as pulveriser or grinder. According to Datta and Singh, the Hindu method of solving the equation by-ax= ± c is essentially based on a process of deriving from it successively other similar equations in which the values of the coefficients (a,b) become smaller and smaller. Thus the process is indeed the same as that of breaking a whole thing into smaller pieces, and this accounts for its name kuṭṭaka or ‘pulveriser’. In the problems of the Class I, the quantities (a and b) are called’ divisors’ bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂ as ‘remainders’ (agra or śeṣa etc.). while in a problem of the Class II ; β is ordinarily called the ‘divisor’ (bhāgahāra or bhā- jaka) and γ the ‘interpolator’ kṣepa, kṣepaka etc.) ; here a is called the ‘dividend’ (bhājya), the unkown quantity to be found (x) is called the ‘multiplier’ or (guṇaka or guṇakāra etc) and y the

INDETERMINATE EQUATIONS 225 quotient or phala. In later years, Mahāvīra has called the unknown number (x) as rāśi. Preliminary Operations in Kuṭṭaka-Karma Usually it has been suggested that in order that an equation of the form by − ax = ± c or by + ax = ± c may be amenable to solution, the two numbers a and b must not have a common divisor; for otherwise, the equation would be absurd, unless the number c had the same common divisor. So before the rules which we shall give hereafter, could be applied, the numbers a, b, c must be made prime (dṛḍha or firm; niccheda or having no divisor, or nirapavarta, meaning irreducible to each other. In this connection Bhāskara I writes : The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.¹ Similarly we find in the writings of Brahmagupta : Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other.² Āryabhaṭa's Rule : Āryabhaṭa I is probably the first Indian writer on this subject, but the operation given by him is rather obscure. His disciple Bhāskara I has given the solution of inde- terminate equations of the first degree in more satisfactory langu- age. We shall give here the translation of Āryabhaṭa's verse from the Āryabhaṭīya, as rendered by Bibhutibhusan Datta, because other translations of this verse do very often confuse the sense : Divide the divisor corresponding to the greater remain- der by the divisor corresponding to the smaller remain-

1. भूदिनेष्टगणा-न्योन्य भक्तशेषेण भाजितौ । हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोर्विदुः —MBh, I. 41
2. हतयोः परस्परं यच्छेषं गुणकारभागहारकयोः । तेन हृतौ निश्छेदौ तावेव परस्परं हतयोः । —BrSpSi. XVIII. 9.

224 BRAHMAGUPTA AS AN ALGEBRAIST Hence ' by—ax=R₁—R₂ Putting c=R₁∽R₂ we get by-ax=±c the upper or lower sign being taken according as R₁ is grea- ter than or less than R₂. Class II : To find a number (x) such that its product with a given number (a) being increased or decreased by another given number (8) and then divided by a third given number (β) will leave no remainder. This means that in other words, we shall have to get the solution of : ax±r ———— = y β in positive integers. Class III : Here we have to deal with an equation of the form : by ÷ ax= ± c Kuṭṭaka, Kuṭṭākāra and Kuṭṭa : These are the three terms which Brahmagupta has used in regards to the subject of indeter- minate analysis of the first degree. Āryabhaṭa I has also descri- bed this method in brief, but he does not use the word kuṭṭaka. In the Mahābhāskarīya of Bhāskara I we have the terms kuṭṭā- kāra and kuṭṭa (522 A.D.) MBh. I. 41,49). These words have been translated into English as pulveriser or grinder. According to Datta and Singh, the Hindu method of solving the equation by-ax= ± c is essentially based on a process of deriving from it successively other similar equations in which the values of the coefficients (a,b) become smaller and smaller. Thus the process is indeed the same as that of breaking a whole thing into smaller pieces, and this accounts for its name kuṭṭaka or 'pulveriser'. In the problems of the Class I, the quantities (a and b) are called' divisors' bhāgahāra, bhajaka, cheda etc.) and R₁ and R₂ as 'remainders' (agra or seṣa etc.). while in a problem of the Class II ; β is ordinarily called the 'divisor' (bhāgahāra or bhā- jaka) and Y the 'interpolator' kṣepa, kṣepaka etc.) ; here a is called the 'dividend' (bhāj

INDETERMINATE EQUATIONS 225 quotient or phala. In later years, Mahāvīra has called the unknown number (x) as rāśi. Preliminary Operations in Kuṭṭaka-Karma Usually it has been suggested that in order that an equation of the form by−ax= ± c or by+ax= ± c may be amenable to solution, the two numbers a and b must not have a common divisor; for otherwise, the equation would be absurd, unless the number c had the same common divisor. So before the rules which we shall give hereafter, could be applied, the numbers a, b, c must be made prime (dṛḍha or firm; niccheda or having no divisor, or nirapavarta, meaning irreducible to each other. In this connection Bhāskara I writes : The dividend and divisor will become prime to each other on being divided by the residue of their mutual division. The operation of the pulveriser should be considered in relation to them.¹ Similarly we find in the writings of Brahmagupta : Divide the multiplier and the divisor mutually and find the last residue; those quantities being divided by the residue will be prime to each other.² Āryabhaṭa's Rule : Āryabhaṭa I is probably the first Indian writer on this subject, but the operation given by him is rather obscure. His disciple Bhāskara I has given the solution of inde- terminate equations of the first degree in more satisfactory langu- age. We shall give here the translation of Āryabhaṭa's verse from the Āryabhaṭīya, as rendered by Bibhutibhusan Datta, because other translations of this verse do very often confuse the sense : Divide the divisor corresponding to the greater remain- der by the divisor corresponding to the smaller remain-

1. भूदिनेष्टगणा-न्योन्य भक्तशेषेण भाजितौ । हारभाज्यौ दृढौ स्यातां कुट्टाकारं तयोर्विदुः —MBh, I. 41
2. हत्योः परस्परं यच्छेषं गुणकारभागहारकयोः । तेन हृतौ निश्छेदौ तावेव परस्परं हत्योः । —BrSpSi. XVIII. 9.

226 BRAHMAGUPTA AS AN ALGEBRAIST der. The residue (and the divisor corresponding to the smaller remainder) being mutually divided, the last resi- due should be multiplied by such an optional integer that the product being added(in case the number of quo- tients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders (will be exactly divisible by the last but one remainder. Place the quotients of the mutual division successively one below the other in a column; below them the optional multiplier and underneath it the quotient just obtained). Any number below . : . the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corresponding to the smaller remainder; then multiply the residue by the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors.¹ There is an alternative rendering of this passage also as follows : . Divide the divisor corresponding to the greater remain- der by the divisor corresponding to the smaller remain- der. The residue (and the divisor corresponding to the smaller remainder) being mutually divided (until the remainder becomes zero), the last quotient should be multiplied by an optional integer and then added (in case the number of quotients of the mutual division is even) or subtracted (in case the number of quotients is odd) by the difference of the remainders. (Place the other quotients of mutual division successively one below the other in a column; below them the result just obtained and underneath it the optional integer). Any

1. अधिकाग्रभागहारं छिन्यादूनाग्रभागहारेण । शेषपरस्परभक्तं मतिगुणमग्रान्तरे क्षिप्तम् ॥ अध उपरि गुणितमन्त्ययुगूनाच्छेद भाजिते शेषम् । अधिकाग्रच्छेदगुणं द्विच्छेदाग्रमधिकाग्रयुतम् । —Ārya. II. 32-33

PRELIMINARY OPERATIONS 227 number below (i.e. the penultimate) is multiplied by the one just above it and then added by that just below it. Divide the last number (obtained by doing so repeatedly) by the divisor corres- ponding to the smaller remainder; then multiply the residue ty the divisor corresponding to the greater remainder and add the greater remainder. (The result will be) the number corresponding to the two divisors. Āryabhaṭa’s problem may be enunciated thus : To find a number (N) which being divided by two given numbers (a, b) will leave two given remainders (R₁, R₂). This gives : N=ax+R₁=by+R₂ (where R₁ is a greater remainder and R₂ lesser remainder, and a is the divisor corresponding to greater remainder and b the divisor corresponding to the lesser remainder.) Denoting as before by c the difference between R₁, and R₂, we get (i) by=ax+c, if R₁>R₂ (ii) ax=by+c, if R₂>R₁ the equation being so written as to keep c always positive. Hence the problem now reduces to making either (ax+c)/b or (by+c)/a according as R₁>R₂ or R₂>R₁, a positive integer. So Āryabhaṭa says : Divide the divisor corresponding to the greater remainder etc.” Now we shall proceed with the details of the operation as proposed by Datta and Singh in his History of Hindu Mathema- tics, Part II. Algebra : Suppose R₁>R₂; then the equation to be solved will be ax+c=by ...(i) a, b being prime to each other.

228 BRAHMAGUPTA AS AN ALGEBRAIST Let b) a (q bq ─── r₁) b (q₁ r₁q₁ ─── r₂) r₁ (q₂ r₂q₂ ─── r₃ ... ────── rₘ₋₁) rₘ₋₂ (qₘ₋₁ rₘ₋₁ qₘ₋₁ ─────── rₘ) rₘ₋₁ (qₘ rₘqₘ ───── rₘ₊₁ Then we get (when a < b, we shall have q = 0, r₁ = a) a = bq + r₁ b = r₁q₁ + r₂ r₁ = r₂q₂ + r₃ r₂ = r₃q₃ + r₄ ... ... ... rₘ₋₂ = rₘ₋₁ qₘ₋₁ + rₘ rₘ₋₁ = rₘqₘ + rₘ₊₁ Now, substituting the value of a in the given equation (1), we get by = (bq + r₁)x + c Therefore y = qx + y₁ where by₁ = r₁x + c In other words, since a = bq + r₁, on putting y = qx + y₁ (ii) the given equation (i) reduces to by₁ = r₁x + c (iii) Again, since b = r₁q₁ + r₂

PRELIMINARY OPERATIONS 229 putting similarly x=q₁y₁+x₁ the equation (iii) can be further reduced to r₁x₁=r₂y₁—c (iv) and so on. Writing down the successive values and reduced equations in columns, we have (1) y=qx+y₁ (I.1) by₁=r₁x+c (2) x=q₁y₁+x₁ (I.2) r₁x₁=r₂y₁—c (3) y₁=q₂x₁+y₂ (I.3) r₂y₂=r₃x₁+c (4) x₁=q₃y₂+x₂ (I.4) r₃x₂=r₄y₂—c (5) y₂=q₄x₂+y₃ (I.5) r₄y₃=r₅x₂+c (6) x₂=q₅y₃+x₃ (I.6) r₅x₃=r₆y₃—c ......... ......... (2n-1) yₙ₋₁=q₂ₙ₋₂ xₙ₋₁+yₙ (I. 2n-1) r₂ₙ₋₂ yₙ=r₂ₙ₋₁ xₙ₋₁+c (2n) xₙ₋₁=q₂ₙ₋₁ yₙ+xₙ (I. 2n) r₂ₙ₋₁ xₙ=r₂ₙ yₙ—c (2n+1) yₙ=q₂ₙ xₙ+yₙ₊₁ (I. 2n+1) r₂ₙ yₙ₊₁=r₂ₙ₊₁ xₙ+c Now the mutual division can be continued either (i) to the finish or (ii) so as to get a certain number of quotients and then stopped. In either csse the number of quotients found, negle- cting the first one (q), as is usual with Āryabhaṭa, may be even or odd. Case (i) First suppose that the mutual division is continued until the zero remainder is obtained. Since a, b are prime to each other, the last one remainder is unity. Subcase (i.1.). Let the number of quotients be even. We then have r₂ₙ=1, r₂ₙ₋₁=0, q₂ₙ=r₂ₙ₋₁ The equations (1,2n) and (I.2n+1), therefore become yₙ=q₂ₙ xₙ+c and yₙ₊₁=c respectively. Giving an arbitrary integral value (t) to xₙ we get an integral value of yₙ. From that we can find the value of xₙ₋₁ by the equation (2n). Procceding backwards step by step we ultimately find the values of x and y in positive integers. So that the equation (I) is solved. Subcase (i. 2) : If the number of quotients be odd, we shall have r₂ₙ₋₁=1, r₂ₙ=0, q₂ₙ₋₁=r₂ₙ₋₂.