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

BHĀSKARA I AND KUṬṬAKA OPERATIONS 233 furnished by the above and applying the method of such quadratics, obtain the (simplest) solution by the substitution of 2, 3 etc. successively in the general solu- tion). Then calculate the ahargaṇa and the revolu- tions performed by Saturn and Mars in that time toge- ther with the number of solar years elapsed.¹ Let x and y denote the residues of the revolution of Mars and Saturn respectively. Then we have to find out two numbers x and y such that each of the expressions x+y, x-y and xy+1 may be a perfect square. Let x+y=4P² and x-y=4Q², so that x=2P²+2Q² y=2P²-2Q² and therefore xy+1=(2P²-1)²+4(P²-Q⁴) Hence the condition that xy+1 be a perfect square is that P²=Q⁴. Substituting these values, we have x=2(Q⁴+Q²) y=2(Q⁴-Q²) where Q may possess any of the values 2, 3, 4,......but not 1. (We neglect the case when x or y is zero).

1. भाज्यं न्यसेदुपरि हारमधश्च तस्य । खण्ड्यात्परस्परमथो विनिधाय लब्धम् । केनाऽऽहतोऽ

234 BRAHMAGUPTA AS AN ALGEBRAIST Putting Q=2, we get x=40 and y=24, which is the least solution. Assuming now that the residues of the revolution (maṇḍa- laja-śeṣa) of Saturn and Mars are 24 and 40 respectively, we have to obtain the ahargaṇa (which means the number of mean civil days elapsed since the beginning of Kaliyuga, or, in fact, any epoch). The revolution-number of Saturn is 146564, and the number of civil days in a yuga is 1,577,917,500. In the present problem, these are respectively the dividend and the divisor. Their H.C.F. is 4, so that dividing them out by 4 we get 36641 and 394,479,375 as the abraded dividend and abraded divisor respectively. We have, therefore, to solve the pulveriser 36641x–24 ────────── = y 394479375 where x and y denote the ahargaṇa and the revolutions respecti- vely made by Saturn. Mutually dividing 36641 and 394479375, we get 36641) 394479375 (10766 394477006 ───────── 2369) 36641 (15 35535 ───── 1106) 2369 (2 157) 1106 (7 1099 7) 157 (22 154 3) 7 (2 6 ── 1×27–24=3)3(1 3 ── 0 We have chosen here the number 27 as the optional number (mati). In fact, mati may be chosen at any stage after an even number of quotients are obtained.

BHĀSKARA I AND KUṬṬAKA OPERATIONS 235 Writing down the quotients one below the other as pres- cribed in the rule, we get the chain 10766 15 2 7 22 2 (mati) 27 1 Reducing the chain, we successively get 10766 10766 10766 10766 10766 10766 3108044439 (multiplier) 15 15 15 15 15 288689 288689 (quotient) 2 2 2 2 18665 18665 7 7 7 8714 8714 22 22 1237 1237 2 55 55 (mati) 27 27 1 (it would be seen in this reduction of chain that mati or 27 × 2 plus 1 is 55; 55 × 22 plus 27 is 1237; 1237 × 7 plus 55 is 8714; 8714 × 2 plus 1237 is 18665; 18665 × 15 plus 8714 is 288689; and finally 288689 × 10766 plus 18665 is 3108044439 which is the multiplier). Dividing 3108044439 by 394479375, and 288689 by 36641, we obtain 346688814 and 32202 respectively as remainders, (This division is performed only when the multiplier and quotient are greater than the divisor and dividend respectively). These are the minimum values of x and y satisfying the above equation. Therefore, the required ahargaṇa = 346688814. and the revolutions performed by Saturn = 32202. To obtain the ahargaṇa and the revolutions of Mars, one has to solve the equation : 191402 z — 40 ───────────── = w 131493125

236 BRAHMAGUPTA AS AN ALGEBRAIST where z and w denote the ahargaṇa and the revolutions per- formed by Mars respectively. The general solution of this equation is z = 131493125 s + 118076020 w = 191402 s + 171872 where s = 0, 1, 2, 3......When s = 0, we have the least solution. Brahmagupta’s Rules Concerning Indeterminate Analysis of the First Degree For the solution of Āryabhaṭa’s problem, Brahmagupta gives the following rule : What remains when the divisor corresponding to the greater remainder is divided by the divisor correspond- ing to the smaller remainder—that (and the latter divisor) are mutually divided and the quotients are severally set down one below the other. The last residue (of the reciprocal division after an even number of quo- tients has been obtained) is multiplied by such an optio- nal integer that the product being added with the differ- ence of the (given) remainders will be exactly divisible (by the divisor corresponding to that residue). That optional multiplier and then the (new) quotient just obtained should be set down (underneath the listed quotients). Now, proceeding from the lower-most number (in the column), the penultimate is multiplied by the number just above it and then added by the number just below it. The final value thus obtained (by repeating the above process) is divided by the divisor corresponding to the smaller remainder. The residue being multiplied by the divisor corresponding to the greater remainder and added to the greater remainder will be the number in view.¹

1. अधिकाग्रभागहारादूनाग्रच्छेद भाजिताच्छेषम् । यत् तत् परस्परहृतं लब्धमधोधः पृथक् स्थाप्यम् ॥ शेषं तदिष्टगुणितं यथाऽनयोरन्तरेण संयुक्तम् । शुध्यति गुणकः स्थाप्यो लब्धं चान्त्यादुपान्त्यगुणः ॥ स्वोर्ध्वोऽन्त्य युतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् । अधिकाग्रच्छेदहत मग्रिकाप्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII, 3-5

BRAHMAGUPTA'S RULES OF ANALYSIS 237. Brahmagupta further observes : Such is the process when the quotients (of mutual division) are even in number. But if they be odd, what has been stated before as negative should be made as positive, or as positive should be made negative.¹ Regarding the direction for dividing the divisor corres- ponding to the greater number by the divisor corresponding to the smaller remainder, Pṛthūdaka Svāmī (860A.D.) observes that it is not absolute, rather optional; so that the process may be conducted in the same way by starting with the division of the divisor corresponding to the smaller remainder by the divisor corresponding to the greater remainder. But in this case of in- version of the process, he continues, the difference of the remain- ders, must be negative. That is to say, the equation by=ax+c can be solved by transforming it first to the form ax=by-c so that we shall have to start with the division of b by a. For the details of the "Theory of the pulveriser" as applied to the problems in Astronomy, the reader is referred to the writ- ings of Bhaṭṭa Govind, translated by K.S. Shukla, and given as an Appendix to the edition of the Laghu-Bhāskarīya. For the rationale of the rules in relation to kuṭṭaka or the pulveriser operation, one may also refer to the chapters by Datta and Singh in the History of Hindu Mathematics: Algebra. Solution of by=ax ± 1. This simple indeterminate equation has a special use in astronomical calculations and therefore, Indian algebraists have paid special attention to it. In fact, this equation is solved exactly in the same way as the equation by=ax ± c; it is a parti-

1. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् । ऋणधनयोर्व्यस्तत्वं गुण्य प्रक्षेपयोः कार्यम् ॥ —BrSpSi. XVIII. 13.

238- BRAHMAGUPTA AS AN ALGEBRAIST cular case only of the more general latter equation. Of course, there is a little justification also for treating it separately, since both the types of equations represent two different physical conditions of the astronomical problems. In the case of by=ax±c, the conditions are such that the value of either y or x, more particu- larly of the latter, has to be found and the rules for solution formulated with that objective. But in the case of the equation by=ax±1, the physical conditions require the values of both y and x. The equation by=ax±1 is usually known by the name sthira-kuṭṭaka, literally meaning the 'constant pulveriser' Pṛthū- daka Svāmī also names it as dṛḍha-kuṭṭaka meaning firm-pulveri- ser. Later on this term dṛḍha-was confined to another sense, equivlent to nicched (having no divisor) or nirapavarta (irreduci- ble). The origin of the name sthira-kuṭṭaka or constant pulveriser has been explained by Pṛthūdaka Svāmī as being due to the fact that the interpolator (±1) is here invariable. For the solution of this equation, we shall quote Bhāskara I's rule and the rule by Brahmagupta. Bhāskara I writes in this connection as follows : The method of the pulveriser is applied also after subtracting unity. The multiplier and quotient are respectively the numbers above and underneath. Multi- plying those quantities by the desired number divide by the reduced divisor and dividend; the residues are in this case known to be the (elapsed) days and (resid- ues of) revolutions respectively¹. The pulveriser ax - c -------- = y ... (1) b may be written as aX - 1 -------- = Y ... (2) b where x=cX and y=cY. If X=α, Y=β is a solution of (2), then x=cα, y=cβ will be a solution of (1). Hence the above rule.

1. रूपमेकमपास्यापि कुट्टाकारः प्रसाध्यते । गुणकारोऽथ लब्धं च राशी स्यातामुपर्यधः ॥ —MBh. I. 45

BRAHMAGUPTA'S RULES OF ANALYSIS 239 Brahmagupta's Rule in this connection is as follows : Solution of by=ax-1 : Divide them (i.e., the abraded coefficient of the multi- plier and the divisor) mutually and set down the quo- tients one below the other. The last residue (or the reciprocal division after an even number of quotients has been obtained) is multiplied by an optional integer such that the product being diminished by unity will be exactly divisible (by the divisor corresponding to that residue). The (optional) multiplier and then this quotient should be set down (underneath the listed quotients). Now proceeding from the lower most term to the uppermost, by the penultimate multiply the term just above it and then add the lowermost number. (The uppermost number thus calculated being divided by the reduced divisor, the residue (is the quan- tity required. This is the method of the constant pul- veriser¹. Solution of by+ax=±c Indian algebraists usually transformed this equation as by=-ax+c, so that it appeared as a particular case of by=ax+c, in which a was negative. Brahmagupta has been the first person to solve this equation, but the rule given by him is obscure : The reversal of the negative and positive should be made of the multiplier and interpolator.² Pṛthudaka Svāmī has tried to explain it, but he too is not very clear. He says :

1. हृतयोः परस्परं यच्छ्रेषं गुणकार भागहारकयोः । तेन हृतौ निश्छेदौ तावेव परस्परं हृतयोः ॥ लब्धमधोऽधः स्थाप्यं तथेष्ट गुणकारसङ्गुणं शेषम् । शुद्धस्यति यथैकहीनं गुणकः स्थाप्यः फलं चान्त्यम् ॥ अश्रान्तमुपान्त्येन स्वोर्ध्वो गुणितोऽन्त्य संयुतो भक्तम् । निःशेषभागहारेणैव स्थिरकुट्टकः शेषम् ॥ —BrSpSi. XVIII. 9-11
2. एवं समेषु विषमेष्वृणं धनं धनमृणं यदुक्तं तत् । ऋणधनयोर्व्यस्तत्वं गुण्यप्रक्षेपयोः कार्यम् ॥ BrSpSi XVIII. 13

240 BRAHMAGUPTA AS AN ALGEBRAIST If the multiplier be negative, it must be made positive; and the additive must be made negative : and then the method of the pulveriser should be employed. Pṛthūdaka Svāmī, however, does not indicate how to derive the solution of the equation. by = -ax+c ...(1) from that of the equation by = ax-c ...(2) The method, however, seems to have been this : Let x=α, y=β be the minimum solution of (2). Then we get bβ = a α-c or b(a-β) = -a(a-b)+c Hence x=a-b, y=a-β is the minimum solution of (1). This rule is very clearly indicated by Bhāskara II and others. We shall give two examples from Bhāskara II (Bījagaṇita) to illustrate the rule : Example I. 13y = -60x + 3 By the method described before, we find that the minimum solution of 13y = 60x+3 is x=11, y=51. Subtracting these values from their respective abraders, namely 13 and 60, we get 2 and 9. Then by the maxim : "In the case of the dividend and divisor being of differ- ent signs, the results from the operation of division should be known to be so", making the quotient negative we get the solu- tion of 13y = -60x+3 as x=2, y=-9. Subtracting these values again from their respec- tive abraders (13, 60), we get the solution of 13y = -60x-3 as x=11, y=-51. Example II. 11y = 18x+10

LINEAR EQUATIONS 241 Proceeding as before, we find the minimum solution of 11y=18x+10 to be x=8, y=14. These will also be the values of x and y in the case of the negative divisor but the quotient for the reasons stated before should be made negative. So the solution of -11y=18x+10 is x=8, y=-14. Subtracting these (i.e., their numerical values) from their respective abraders, we get the solution of -11y=18x-10 as x=3, y=-4, "When the divisor is positive or negative the numeri- cal values of the quotient and multiplier remain the same : when either the divisor or the dividend is negative, the quotient must always be known to be negative"¹. One Linear Equation in More Than Two Unknowns Whenever a linear equation involves more than two unkno- wn's the Indian algebraists used to assume arbitrary values for all the unknowns except two and then to apply the method of kuṭ- ṭaka or "pulveriser". In this connection, Brahmagupta says : The method of the pulveriser (should be employed if there be present many unknowns (in any equation)²,

1. Bhāskara II gives the following rule : "Those(the multiplier and quotient)cbtained for a positive divi- dend being treated in the same manner give the results corres- ponding to a negative dividend." The treatment alluded to in this rule is that of subtraction from the respective abraders. He has further elaborated it thus : The multiplier and quotient should be determined by taking the dividend, divisor and interpolator as positive. They will be the quantities for the additive interpolator. Subtracting them from their respective abraders, the quantities for a negative interpolator are found. If the dividend or divi- sor, be negative, the quotient should be stated as negative, the quotient should be stated as negative. —Bījagaṇita
2. आद्याद्वर्णादन्यान् वर्णान् प्रोह्याद्यमानमाद्यहृतम् । सदृशच्छेदावसकृद् द्वौ व्यस्तौ कुट्टको बहुषु ॥ — BrSpSi. XVIII. 51

242 BRAHMAGUPTA AS AN ALGEBRAIST We shall take up one of the problems posed by Brahmagupta concerning astronomy and leading to the equation :¹ 197x - 1644 y - z = 6302. Hence 1644 y + z + 6302 x = ------------------- 197 The commentator assumes z = 131. Then 1644 y + 6433 x = --------------- ; 197 hence by the usual method of the pulveriser x = 41; y = 1. General Problem of Remainders A certain type of simultaneous indeterminate equations of the first degree arise out of the general problem of remainders which may thus be stated : To find a number N which being severally divided by a₁, a₂, a₃.......aₙ , leaves as remainders r₁, r₂, r₃.........rₙ respectively. While dealing with such a case, we shall have the following series of equations : N = a₁x₁ + r₁ = a₂x₂ + r₂ = a₃x₃ + r₃ = ...... = aₙ xₙ + r . We have reasons to believe that the method of solution of these equations was known to Āryabhaṭa I. In the translation of the verse in the Āryabhaṭīya, II. 32-33 (the translation of which we have already given), the term dvicchedāgram should be translated as "the result will be the remainder corresponding to the product of the two divisors", instead of "the result will be the number corresponding to the two divisors." (the last line of the translation). This explanation is in fact given by Bhāskara I, the direct disciple and earliest commentator of Āryabhaṭa I. Such a rule is clearly stated by Brahmagupta².

1: अंशकशेषेण युतात् लिप्ताशेषात्तदन्तरादथवा । भानोर्भ'दिने द्युगणं यः कथयति कुट्टकज्ञः सः ॥ —BrSpSi. XVIII. 55 2: स्वोच्छेऽन्ययुतोऽग्रान्तो हीनाग्रच्छेदभाजितः शेषम् । अधिकाग्रच्छेदहतमधिकाग्रयुतं भवत्यग्रम् ॥ —BrSpSi. XVIII. 5

GENERAL PROBLEM OF REMAINDERS 243 The rationale of this method is not difficult. I shall quote it from the book of Datta and Singh: Starting with the considera- tion of the first two divisors, we have N = a₁x₁ + r₁ = a₂x₂ + r₂. By the method described before, we can find the minimum value α of x₁ satisfying this equation. Then the minimum value of N will be a₁α + r₁. Hence the general value of N will be given by N = a₁ (a₂t + α) + r₁ = a₁a₂ t + a₁α + r₁ where t is an integer. Thus a₁α + r₁ is the remainder left on dividing N by a₁ a₂ as stated by Āryabhaṭa I and Brahmagupta. Now taking into consideration the third condition, we have N = a₁a₂t + a₁αr₁ = a₃x₃ + r₃ which can be solved in the same way as before. Proceeding in this way successively, we shall ultimately arrive at a value of N satisfying all the conditions ; Pṛthūdaka Svāmī remarks : Wherever the reduction of two divisors by a common measure is possible, there 'the product of the divisors' should be understood as equivalent to the product of the divisor corresponding to the greater remainder and quotient of the divisor corresponding to the smaller remainder as reduced (i.e. divided) by the common mea- sure.¹ When one divisor is exactly divisible by the other, then the greater remainder is the (required) remainder and the divisor corresponding to the greater remainder is taken as 'the product of the divisors'. (The truth of) this may be investigated by an intelligent mathematician by taking several symbols. As an illustration we shall take up a problem quoted by Bhāskara II in his Bījagaṇita, and which in its solution follows the method of Āryabhaṭa 1. Pṛthūdaka Svāmī while commenting on serveral verses from Brahmagupta (BrSpSi. XVIII. 3-6)

1. i.e., if p be the L.C.M. of a, and a2, the general value of N satisfying the above two conditions will be N = pt + a₁a + r, instead of N = a₁a₂t + a₁a + r₁.

244 BRAHMAGUPTA AS AN ALGEBRAIST observes that such problems were very popular amongst the an- cient Indian mathematicians. Problem : To find a number N which leaves remainders 5, 4,3,2 when divided by 6,5,4,3 respectively. That is to solve the equations : N = 6x + 5 = 5y + 4 = 4z + 3 = 3w + 2. We have since N = 6x + 5 = 5y + 4, x = (5y - 1) / 6 But x must be integral, so y = 6t + 5, x = 5t + 4 Hence N = 30t + 29 Again N = 30t + 29 = 4z + 3 Therefore, t = (2z - 13) / 15 Since t must be integral, we must have z = 15s + 14; hence t = 2s + 1. Therefore N = 60s + 59. The last condition is identically satisfied. The method given here is the one followed by Pṛthūdaka Svāmī. Thus when N = 60s + 59 = 6x + 5 x = (60s + 54) / 6 = 10s + 9 ...(1) Again, when N = 60s + 59 = 5y + 4, y = (60s + 55) / 5 = 12s + 11 Again when N = 60s + 59 = 4z + 3 z = (60s + 56) / 4 = 15s + 14 Lastly, when N = 60s + 59 = 3w + 2. w = (60s + 57) / 3 = 20s + 19, Varga Prakṛti or Kṛti Prakṛti or Square-Nature The word varga-prakṛti (literally meaning 'square-nature') has been given by Indian algebraists to the indeterminate quadra- tic equation Nx² ± c = y²

VARGA PRAKṚTI 245 Here in this equation the absolute number c should be rūpa (or unity). which means the equation Nx²±1=y² or it may be any absolute number. The most fundamental equation of this class has been regarded as Nx²+1=y² where N is a non-square integer. This branch of mathematics has originated from the number which is the prakṛti of the square of yāvat, etc. (the unknown x etc.), and therefore, it is called varga-prakṛti. The quantity N of the above equation is known as Prakṛti. Brahmagupta uses the term GUṆAKA (multiplier) for the same purpose¹. This term guṇaka together with its variation guṇa appears occasionally also in the writings of later authors. For example, Śrīpati (Siddhānta-śekhara. XIV. 32) employs the term guṇaka where as Bhāskara II and Nārāyaṇa use the term guṇa in their Bījagaṇitas. In this connection, we would now like to quote from Pṛthūdaka Svāmī (863 A.D.) from his commentary on the Brāhmasphuṭasiddhānta : Here are stated for ordinary use the terms which are well known to people.. The number whose square, multiplied by an optional multiplier and then increased or decreased by another optional number, becomes capable of yielding a square-root, is designated by the term the "lesser root" kaniṣṭha pada or the "first root" ādya-mūla). The root which results, after those opera- tions have been performed is called by the name the "greater root" (jyeṣṭha pada) or the "second root" (anya-mūla). If there be a number multiplying both these roots, it is called the "augmenter" (udvartaka); and on the contrary, if there be a number dividing the roots, it is called the "abridger" (apavartaka), Thus in the equation Nx²±c=y²,

1. मूलं द्विधेष्ट वर्गाद् गुणक गुणादिष्टयुत विहीनाच्च । आद्यवधो गुणकगुणः सहान्त्यघातेन कृतमन्त्यम् ॥ —BrSpSi. XVIII. 64
2. BrSpSi. XVIII. 64 (Com.)

246 BRAHMAGUPTA AS AN ALGEBRAIST x is known as the lesser root, y is the greater root, N is the multi- plier (guṇaka) and c is interpolator or kṣepaka. Bhāskara II has used the word "hrasvamūla" for kaniṣṭha pada or ādya-mūla lite- rally meaning "lesser root". The earlier terms, the "first root" (ādyamūla) for the value of x and the "second root" or the "last root" antya-mūla for the value of y are quite free from ambiguity Their use is found in the algebra of Brahmagupta. The later terms appears in the works of his commentator Pṛthūdaka Svāmī. Brahmagupta uses the term kṣepa, prakṣepa or prakṣepaka in the sense of "interpolator." Again, when negative, the inter- polator is sometimes distinguished as the "subtractive" or śodhaka and the positive interpolator is then called "the addi- tive." Lemmas of Brahmagupta Prior to our giving the general solution of the Square-nature or Varga-Prakṛti, it would be better to give two Lemmas esta- blished by Brahmagupta. We have the following in the Brāhma- sphuṭasiddhānta: Of the square of the optional number multiplied by the guṇaka and increased or decreased by an other optional number, iṣṭa, (extract) the square root. (Proceed) twice. The product of the first roots multiplied by the guṇaka together with the product of the second roots will give a (fresh) second root; the sum of their cross-pro- ducts will be a (fresh) first root. The (corresponding) interpolator will be equal to the product of the (previ- ous) interpolators.¹ There is a little difficulty in ascertaining the real sense of the rule given in these lines since the word dvidhā (twice) has two implications. Firstly, it may mean that the earlier operations of finding roots are made on two optional numbers with two optio- nal interpolators, and with the results thus obtained the subse-

1. मूलं द्विधेष्टवर्गाद् गुणक गुणादिष्टयुत विहीनाच्च । आद्यवधो गुणकगुणः सहान्त्यघातेन कृतमन्त्यम् ॥ वज्रवधैक्यं प्रथमं प्रक्षेपः क्षेपवध तुल्यः । प्रक्षेपशोधकहृते मूले प्रक्षेपके रूपे ॥ —BrSpSi. XVIII 64-65

LEMMAS OF BRAHMAGUPTA 247 quent operations of their composition are performed. Secondly, it may also mean that the earlier operations are made with one optionally chosen number and one interpolator, and the subsequent ones are carried out after the repeated statement of those roots for the second time. It is also implied that in the composition of the quadratic roots, their products may be added together or subtracted from each other. In other words, if x=a, y=β be a solution of the equation : Nx² + k = y², and x=a', y=β' be a solution of Nx² + k' = y², then according to the above x = aβ' ± a'β, y = ββ' ± Naa' is a solution of the equation Nx² + kk' = y². In other words, if Na² + k = β² Na'² + k' = β'² then · N(aβ' ± a'β)² + kk' = (ββ' ± Na')² (I) In particular, taking a=a', β=β' and k=k', Brahmagupta finds from a solution x=a, y=β of the equation Nx² + k² = y² a solution x=2aβ, y=β² + Na² of the equation Nx² + k = y² That is, if Na² + k = β² then N(2aβ)² + k² = (β² + Na²)² (II) This result will be hereafter called Brahmagupta's Corol- lary. Thus Brahmagupta's First Lemma says that if two solutions of the equation (of the Square-nature) Nx² + 1 = y² are known, then any number of other solutions can be found. For example if two solutions of the Square-nature are (a, b) and also (a', b'), then two other solutions will be : x = ab' ± a'b, y = bb' ± Naa'.

248 BRAHMAGUPTA AS AN ALGEBRAIST We can compose this solution with the previous ones, and get another solution, and thus proceed on to innumerable solu- tions. From Brahmagupta's Corollary to First Lemma we get another set of solutions. If (a, b) be solution of the Square- nature, then another solution of it is x=2ab, and y=b²+Na² Thus even if we have only one solution, we can get the other solution also (since N is known), and thus we can get any number of solutions one after the other by this Principle of Composition. Brahmagupta's Lemmas have been described by Bhāskara II (1150 A.D.) in the following words : Set down successively the lesser root (hrasva), greater root (jyeṣṭha) and interpolator (kṣepaka); and below them should be set down in order the same or an another (set of similar quantities). From them by the Principle of Composition (Bhāvanā) can be obtained numerous roots. Therefore the Principle of Composi- tion will be explained here. (Find), the two cross-pro- ducts (vajrābhyāsa) of the two lesser and the two greater roots; their sum is a lesser root. Add the product of the two lesser roots multiplied by the prakṛti to the product of the two greater roots, the sum will be a greater root. In that (equation) the interpolator will be the product of the two previous interpolators. Again the difference of the two cross-products is a lesser root. Subtract the product of the two lesser roots multiplied by the prakṛti from the product of the two greater roots; (the difference) will be greater root. Here also the interpolator is the product of the two (previous) interpolators.¹

1. ह्रस्वज्येष्ठ क्षेपकान् न्यस्य तेषां तानन्यान् वाऽधो निवेश्य क्रमेण । साध्यान्येभ्यो भावनाभिर्बहूनि मूलान्येषां भावना प्रोच्यतेऽतः ।। वज्राभ्यासौ ज्येष्ठलघ्वोस्तदैक्यं ह्रस्वं लघ्वोराहतिश्च प्रकृत्या । क्षुण्णा ज्येष्ठाभ्यास युग् ज्येष्ठमूलं तत्राभ्यासः क्षेपयोः क्षेपकः स्यात् ।। ह्रस्वं वज्राभ्यासयोरन्तरं वा लघ्वोर्घातो यः प्रकृत्या विनिघ्नः । घातो यश्च ज्येष्ठयोस्तद्वियोगो ज्येष्ठं क्षेपोत्रापि च क्षेपघातः ।। Bhāskara II, Bījaganita, VargaPrakṛti, 2-4

PRINCIPLE OF COMPOSITION 249 Principle of Composition The above results have been technically known amongst Indian algebraists as Bhāvanā (demonstrated or proved, hence theorem or lemma). The word bhāvanā also means "composition or combination" in algebra. Bhāvanā may be of two types : Samāsa Bhāvanā (or addition Lemma, or additive composition) and Antara Bhāvanā (or subtraction Lemma or subtractive composition). Whenever, again, the bhāvanā is made with two equal sets of roots and interpolators, it is technically named as Tulya Bhāvanā (or composition of equals), and when with two unequal sets of values then it is known as Atulya Bhāvanā (or composition of unequals). Proof of Brahmagupta's Lemmas It is significant to be indicated that Brahmagupta's Lemmas were rediscovered by Euler in 1764 and by Lagrange in 1768, and a considerable importance was attached to them. Kṛṣṇa, (1580 A.D.) the commentator on the Bījagaṇita of Bhāskara II gives the following proof of Brahmagupta's Lemmas : Let (α,β) and (α',β') be the two solutions of the equation nx² + k = y². we have Nα² + k = β² Nα'² + k' = β'² Multiplying the first equation by β'², we get Nα²β'² + kβ'² = β²β'² Now, substituting the value of factor β²' of the interpolator from the second equation, we get Nα² β'² + k (Nα'² + k') = β²β'² or N(α²β'² + Nkα'² + kk' = β²β'² Again, substituting the value of k from the first equation in the second term of the left-hand side expression, we have Nα²β'² + Nα'²(β² - Nα²) + kk' = β²β'² or N(α²β'² + α'²β²) + kk' = β²β'² + N²α²α'² Adding ±2Nαβα'β' to both sides, we get N(αβ' ± α'β)² + kk' = (ββ' ± Nαα')²

250 BRAHMAGUPTA AS AN ALGEBRAIST Brahmagupta's Corollary also follows at once from the above by putting α'=α, β'=β and k'=k. N (2αβ)²+k²=(β²±Nα²)² Thus the roots are x=2αβ and y=β²±Nα² which is the Corollary. It would be seen that modern historians of mathematics are incorrect when they say that Fermat (1657) was the first to state that the equation Nx²+1=y², where N is a non-square integer has an unlimited number of solutions in integers. For this assertion, history takes us to the early Seventh Century A.D. when Brahmagupta wrote his classical treatise, the Brāhmasphu- tasiddhānta, and gave the well known two Lemmas and the Corollary to the first Lemma. Second Lemma of Brahmagupta In the Brāhmasphuṭa siddhānta, we find another important Lemma by Brahmagupta stated as follows : On dividing the two roots (of a square- Nature) by the square-root of its additive or subtractive, the roots for interpolator unity (will be found).¹ This Lemma when expressed in the modern language of algebra would mean that if x=α, y=β be a solution of the equation. Nx²+k²=y² then x=α/k, y=β/k is a solution of the equation Nx²+1=y². This rule, at another place, has been re-enunciated as follows : If the interpolator is that divided by a square then the roots will be those multiplied by its square- root.²

1. प्रक्षेपशोधक हृते मूले प्रक्षेपके रूपे । —BrSpSi. XVIII. 65
2. वर्गाच्छिन्ने क्षेपे तत्पदगुणिते तदा मूले । —BrSpSi. XVIII. 70

SECOND LEMMA OF BRAHMAGUPTA 251 This rule may be expressed in terms of symbols as follows. Suppose the Varga-prakṛti (Square-nature) to be Nx² ± p²d = y², so that its interpolator (kṣepa) p²d is exactly divisible by the square p². Then, putting therein u = x/p, v = y/p, we derive the equation Nu² ± d = v² whose interpolator is equal to that of the original Square-nature divided by p². It is clear that the roots of the original equation are p times those of the derived equation. Rational Solution Indian algebraists have usually suggested the following method to obtain a first solution of Nx² + 1 = y² : Take an arbitrary small rational number, α, such that its square multiplied by the guṇaka N and increased or diminished by a suitably chosen rational number k will be an exact square. In other words, we shall have to obtain empirically a rela- tion of the form Nα² ± k = β² where α, k, and β are rational numbers. Let us call this relation as the Auxiliary Equation. Then by Brahmagupta's Coro- llary, we get from it the relation N(2αβ)² + k² = (β² + Nα²)², or N(2α β / k)² + 1 = ((β² + Nα²) / k)² Hence, one rational solution of the equation Nx² + 1 = y² is given by x = 2αβ / k , y = (β² + Nα²) / k Work on the rational solution of the Square-nature has been also done by Śrīpati. In fact, his solution, given in 1039 A.D. is of historical significance. He derives the rational solution without the aid of the "auxiliary equation." He gives the follo- wing rule :

252 BRAHMAGUPTA AS AN ALGEBRAIST Unity is the lesser root. Its square multiplied by the prakṛti is increased or decreased by the prakṛti com- bined with an (optional) number whose square-root will be the greater root. From them will be obtained two roots by the Principle of Composition¹ Thus if m² be the rational number optionally chosen, one shall have the identity : N.1²+(m²—N)=m², or N.1²—(N—m²)=m² Then by applying Brahmagupta's Corollary we get N(2m)²+(m² ∼ N)²=(m²+N)² ∴ N (2m / (m² ∼ N))² + 1 = ((m² + N) / (m² ∼ N))² Hence x = 2m / (m² ∼ N), y = (m² + N) / (m² ∼ N) where m is any rational number, is a solution of the equation Nx² + 1 = y². This rational solution of the varga-prakṛti which was used by Śrīpati in 1039 A.D. was rediscovered in Europe by Broun- cker in 1657. We shall close this discussion by taking an illustration from Bhāskara II : Problem : Tell me, O mathematician, what is that square which multiplied by 8 becomes, together with unity, a square; and what square multiplied by 11 and increased by unity, becomes a square. This means that we have to solve the equations : 8x² + 1 = y² ......(i) 11x² + 1 = y² ......(ii) In the second example, let us assume 1 as the lesser root. Following the method of Śrīpati, let us multiply its square by the prakṛti (here in eq. ii, prakṛti is 11), then let us subtract 2 (an optional number) and then extracting the square-roots we

1. Śrīpati, Siddhānta-śekhara XIV. 33