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

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

RATIONAL SOLUTION 253 get the greater root as 3. Hence the statement for the com- position is m=11 l=1 g=3 i=-2 l=1 g=3 i=-2 Here m=multiplier (guṇaka or prakṛti), l=lesser root (kaniṣṭha-mūla), g=greater root (jyeṣṭha-mūla) and i=interpola- tor (kṣepa). Here we have set down successively the lesser root, greater root and interpolator, and below them again set down the same (See Brahmagupta's Lemmas described by Bhāskara II). Now proceeding as before we obtain the roots for the additive 4 : l=6, g=20, (for) i=4. Then by the rule : "If the interpolator (of a varga-prakṛti or Square-nature) divided by the square of an optional number be the interpolator (of another Square-nature), then the two roots (of the former) divided by that optional number will be the roots (of the other). Or, if the interpolator be multiplied, their roots should be multiplied."¹ are found the roots for the additive unity l=3, g=10 (for) i=1. Whence by the Principle of Composition of Equals, we get the lesser and greater roots : l=60, g=199 (for) i= 1. In this way an infinite number of roots can be deduced. Alternative method:—Bhāskara II has given another method for finding the two roots for the additive unity : Or divide twice an optional number by the difference between the square of that optional number and the prakṛti. This (quotient) will be the lesser root (of a Square-nature) when unity is the additive. From that (follows) the greater root.²

1. इष्टवर्गहृतः क्षेपः क्षेपः स्यादिष्टभाजिते । मूले ते स्तोऽथवा क्षेपः क्षुण्णः क्षुण्णे तदा पदे ॥ Bījagaṇita II. 5.
2. Siddhānta-śekhara, XIV. 32.

254 BRAHMAGUPTA AS AN ALGEBRAIST Let us solve the first example 8x²＋1＝y². We assume the optional number to be 3. Its square is 9; the prakṛti of multiplier is 8, their difference is 9–8＝1. Dividing by this twice the optional number (2×3, i.e. 6), namely 6, we get the lesser root for the addi- tive unity as 6. Whence proceeding as before, we get the greater to be 17. Thus here x＝6 and y＝17. Let us use this method for the equation 11x²＋1＝y². Let the optional number be 3. Its square is 9: multiplier or prakṛti is 11; the difference is 11–9＝2; dividing by this twice the optional number (2×3), namely 6, we get 6/2＝3, which is the lesser root. Consequently the greater root would be 10. Thus for this equation x＝3 and y＝10. Solution in Positive Integers The Indian algebraists usually aimed at obtaining solutions of the varga-prakṛti or Square-nature in positive integers or abhinna. The tentative methods of Brahmagupta and Śrīpati always did not furnish solutions in positive integers. These auth- ors, however, discovered that if the interpolator of auxiliary equa- tion in the tentative method be ±1, ±2 or ±4, an integral solu- tion of the equation Nx²＋1＝y² can always be found. Thus Śrīpati says : If 1, 2 or 4 be the additive or subtractive (of the auxi- liary equation), the lesser and greater roots will be integral (abhinna)¹. (i) If k＝±1, then the auxiliary equation will be Nα²±1＝β where α and β are intergers. Then by Brahmagupta' Corollary we get x＝2αβ and y＝β²＋Nα² as the required first solution in positive integers of the equation Nx²＋1＝y²

1. इष्टवर्गं प्रकृत्योर्यद्विवरं तेन वा भजेत् । द्विघ्नमिष्टं कनिष्ठं तत् पदं स्यादेक संयुतौ । ततो ज्येष्ठमिहानन्त्यं भावनाभिस्तथेष्टतः ।। Bījagaṇita, Varga-Prakṛti, 5-6

Here is the complete line-by-line transcription of the page into clean Markdown, preserving all text, mathematical expressions (without LaTeX math mode), and Devanagari Sanskrit:

SOLUTION IN POSITIVE INTEGERS 255 (ii) Let k=±2; then the auxiliary equation is Nα²±2=β² By Brahmagupta's Corollary, we have N(2αβ)²+4=(β²+Nα²)² or N (αβ)²+1=( (β²+Nα²)/2 )² Hence the required first solution is x=αβ, y=½(β²+Nα²) Since Na²=β² ∓ 2, we have ½ (β²+Nα²)=β² ∓ 1=a whole number. (iii) Now suppose k=+4: so that Nα²+4=β² With an auxiliary equation like this, the first integral solution of the equation Nx²+1=y² is x=½αβ y=½(β²-2); if α is even; or x=½

256 BRAHMAGUPTA AS AN ALGEBRAIST Substituting the value of N in the right-hand side expres- sion from (i), we have N.( αβ / 2 )² + 1 = ( (β²-2) / 2 )² (iii) Composing (ii) and (iii), N { α/2 (β²-1) }² + 1 = { β/2 (β²-3) }² Hence x= ½ αβ, y= ½ (β²--2); and x= ½α(β²-1), y= ½β(β²-3); are solutions of Nx²+1=y². If β be even, the first values of (x,y) are integral. If β be odd, the second values are integral. (iv) Finally, suppose k=-4; the auxiliary equation is Nα²-4 = β² Then the required first solution in positive integers of Nx²+1=y² is x= ½αβ(β²+3) (β²+1) y=(β²+2) { ½(β²+3) (β²+1)-1 }. Brahmagupta says : In the case of 4 as subtractive, the square of the second is increased by three and by unity; half the product of these sums and that as diminished by unity (are obtained). The latter multiplied by the first sum less unity is the (required) second root; the former multi- plied by the product of the (old) roots will be the first root corresponding to the (new) second root.¹ The rationale of this solution, as given by Datta and Singh is as follows : Nα²-4=β² (i) N(α/2)²-1=(β/2)² Hence by Brahmagupta's Corollary, we get N ( αβ / 2 )² + 1 = ( β²/4 + N α²/4 )²

1. चतुरूनेऽन्यपद कृती त्र्येकयुते वधदलं पृथग्व्येकम् । व्येकाद्वाहतमन्त्यं पदवध गुणमाद्यमान्त्यपदम् ॥ BrSpSi. XVIII. 68

CAKRAVĀLA OR CYCLIC METHOD 257 = {½(β² + 2)}² (ii) Again applying the Corollary, we get N {½αβ(β² + 2)}² + 1 = {½(β⁴ + 4β² + 2)}² (iii) Now by the Lemma we obtain from (ii) and (iii) N {½αβ(β² + 3) (β² + 1)}² + 1 = [(β² + 2){½(β² + 3) (β² + 1) − 1}]² Hence x = ½αβ(β² + 3) (β² + 1), y = (β² + 2){½(β² + 3) (β² + 1) − 1} is a solution of Nx² + 1 = y² This can be proved without difficulty that these values of x and y are integral. Since if β is even, β² + 2 is also even. And hence the above values of x and y are integral. On the other hand, if β is odd, β² is also odd; under these conditions β² + 1 and β² + 3 are even. In this also, therefore, the above values must be integral. Putting p = αβ, q = β² + 2, we can write the above solution in the form x = ½p (q² − 1). y = ½q (q² − 3). This was the form in which the solution was found by Euler. Cakravāla or Cyclic Method We have shown in the preceding articles that the most fundamental step in Brahmagupta's method for the general solution in positive integers of the equation Nx² + 1 = y² where N is a non-square integer, is to form an auxiliary equation of the kind Na² + k = b² where a and b are positive integers and k = ±1, ±2 or ±4. From this auxiliary equation, by the Principle of Composition, applied repeatedly whenever necessary, one can derive, as we have alrea- dy shown above, one positive integral solution of the original Varga-prakṛti or Square-Nature. And thence again, by means of the same principle, an infinite number of other solutions in integers can be obtained. How to form an auxiliary equation of

258 BRAHMAGUPTA AS AN ALGEBRAIST this type was a problem, write Datta and Singh, which could not be solved completely nor satisfactorily by Brahmagupta. In fact, Brahmagupta had to depend on trial. Success in this direc- tion was, however, remarkably attained by Bhāskara II. He evol- ved a simple and elegant method which assisted in deriving an auxiliary equation having the required interpolator ±1, ±2, or ± 4, simultaneously with its two integral roots, from another auxi- liary equation empirically formed with any simple integral value of the interpolator, positive or negative. This method has been technically known as Cakravāla or the cyclic method. This is so called because it proceeds as in a circle, the same set of opera- tions being applied again and again in a continuous round. For the details of this method, our reader is requested to consult the Algebra of Bhāskara II and the narrative on this method as given by Datta and Singh under the title "Cyclic Method" in their History of Hindu Mathematics: Algebra, 1962 Edition, pp. 161-72. Solution of Indeterminate Quadratic Equation It is remarkable to see that Brahmagupta was the first algebraist in the history of mathematics to find a general solu- tion of the indeterminate quadratic equation Nx²±c=y² in positive integers. We have the following verse in the Brāhmasphuṭasiddhānta in this connection: From two roots (of a Square-nature or varga-prakṛti) with any given additive or subtractive, by making (combination) with the roots for the additive unity other first and second roots (of the equation having) the given additive or subtractive (can be found).¹ Let us take the following two equations: a₁k=an+b; and b₁k=bn+Na From them we get : by eliminating n a₁b−ab₁=1

1. रूप प्रक्षेपपदे पृथगिष्टक्षेपशोध्यमूलाभ्याम् । कृत्वाऽऽन्त्याथपदे ये प्रक्षेपे तेने ॥ —BrSpSi. XVIII. 66

SOLUTION OF INDETERMINATE 259 Hence b₁ = (a₁b - 1) / a = a whole number. Now n² - N = ((a₁k - b)² - Na²) / a² = (a₁²k² - 2bka₁ + k) / a² = k(a₁²k - 2ba₁ + 1) / a² Therefore (k / a²)(a₁²k - 2ba₁ + 1) is a whole number. Since a, k have no common factor, it follows that (a₁²k - 2ba₁ + 1) / a² = (n² - N) / k = k₁ = an integer. Also k₁ = (n² - N) / k = (a₁²k - 2ba₁ + 1) / a² = (a₁²(b² - Na²) - 2ba₁ + 1) / a² = ((a₁b - 1) / a)² - Na₁². Thus having known a single solution in positive integers of the equation Nx² ± c = y², says, Brahmagupta, an infinite number of other integral solutions can be obtained by making use of the integral solutions of Nx² + 1 = y². If (p, q) be a solution of the former equation found empirically and if (α, β) be an integral solution of the latter, then by the principle of Composition x = pβ ± qα; y = qβ ± Npα will be a solution of the former. Repeating the operations, we can easily deduce as many solutions as we like. FORM Mn²x² ± c = y² : In this connection, Brahmagupta says : If the remainder is that divided by a square, the first root is that divided by its root¹. This seems to mean that if we have the equation Mn²x² ± c = y² (i) such that the multiplier (i.e. the coefficient of x²) is divisible

1. वर्गेच्छिन्ने गुणके प्रथमं तन्मूल भाजितं भवति । —BrSpSi. XVIII.70

260 BRAHMAGUPTA AS AN ALGEBRAIST by n², then we are justified in saying that if we put nx=u, the equation (i) becomes Mu²±c=y² (ii), and clearly the first root of (i) is equal to the first root of (ii) divided by n. The corresponding second root will be the same for both the equations. FORM a²x²±c=y² : We find Brahmagupta giving the following rule in this connection : This is a solution of a particular form of a varga-prakṛti or Square-nature. If the multiplier be a square, the interpolator divided by an optional number and then increased and decreased by it, is halved. The former (of these results) is the second root; and the other divided by the square-root of the multiplier is the first root.¹ Thus the solutions of the equation a²x²±c=y² are : x = 1/(2a) (±c/m - m) y = 1/2 (±c/m + m) where m is an arbitrary number. Bhāskara II and Nārāyaṇa have also given the same solutions as proposed by Brahmagupta. Rational Geometrical Figures In the days of the Taittirīya Saṁhitā and the Śatapatha Brāhmaṇa, Indian mathematicians got familiarity with the solution of such equations x²+y²=z² and the results were arrived geometrically on the basis of the law of rectangle as propounded by Baudhāyana in the Śulba Sūtras and which goes by his name. The reader is referred to the Chapter on Baudhāyana, the first Geometer in the author's "Founders of Sciences in Ancient India". Baudhāyana (c. 800 B.C.) gave a

1. वर्गे गुणके क्षेपः केनचिदुद्धृतयुतोनितो दलितः । प्रथमोऽन्यन्मूलमन्त्यो गुणकारपदोद्धृतः प्रथमः ॥ —BrSpSi. XVIII. 69

, is there a space or a matra? Let's compare त in कोटिः or तद्दलं: त has a left loop. In आयतचतुरस्रस्य: आ य त च तु र स्र स्य. No, it's आयतचतुरस्रस्य. Wait, let me re-examine: आयताचतुरस्रस्य? Wait, look at त: the right side of त goes down, then there is a च. It's आयतचतुरस्रस्य. Wait, what about चेष्टस्येष्टाधिका? Let's look at चे ष्टस्येष्टाधिका: चे has a space after it? Wait, look at the footnote line 1: 1. इष्टस्य भुजस्य कृतेर्भक्तो नेष्टेन तद्दलं कोटिः । Line 2: आयतचतुरस्रस्य चे ष्टस्येष्टाधिका कर्णः ॥ —BrSpSi. XII. 35 Wait, look at चे ष्टस्येष्टाधिका: There is चे, then a space, then ष्टस्येष्टाधिका. Wait, could it be चेष्टस्येष्टाधिका with just a broken space? Yes, in De

262 BRAHMAGUPTA AS AN ALGEBRAIST The sum of the squares of two unequal numbers is the side; their product multiplied by two is the altitude, and twice the difference of the squares of those two unequal numbers is the base of an isosceles triangle.¹ Thus if m,n be two integers such that m is not equal to n, the sides of all rational isosceles triangles with integral sides are given by m² + n², m² + n², 2(m² - n²) and the altitude of the triangle is 2mn. This method was also followed by Mahāvīra and other Indian mathematicians. In fact, their solutions are based on the juxtaposition of two rational right triangles, equal so that they have a common leg. It is remarkably a powerful device, for every rational triangle or quadrilateral may be formed by the juxtaposition of two or four rational right triangles. Isosceles Triangles with a Given Altitude Here we have a rule given by Brahmagupta for finding out all rational isosceles triangles possessing the same altitude : The (given) altitude is the producer (karaṇī). Its square divided by an optional number is increased and diminished by that optional number. The smaller is the base and half the greater is the side.² Thus if m be any rational number then for a given definite altitude a, the sides of the rational isosceles triangles are ½ (a²/m + m) each and the base is (a² - m²)/m . We shall illustrate it by an example taken from the commentary of Pṛthūdaka Svāmī The given altitude is 8; let us take any rational number m = 4 then the two equal sides of the isosceles are given by ½ ((8² + 4²)/4) = 10 each and the base is (8² - 4²)/4 = 12. Thus the three sides of the

1. कृति युतिर सदृशराश्योर्बाहुर्घातो द्विसंगुणो लम्बः । कृत्यन्तरमसदृशयोर्द्विगुणं द्विसमत्रिभुज भूमिः ॥ —BrSpSi. XII. 33
2. करणी लम्बस्तत्कृतिरिष्टहतेष्टोन संयुताऽल्पा भूः । अधिको द्विहृतो बाहुः संक्षेप्यो यद्वधो वर्गः ॥ —BrSpSi. XII. 37.