Mathematics in India

In the previous part of the series [10], we had discussed Brahmagupta’s composition law bhāvanā. We recall the statement in modern language and notation:

Theorem 1. [Brahmagupta’s Bhāvanā] The solution space of the equation Dx^2+z=y^2 admits the binary operations

\[(x_1,y_1,z_1) \odot (x_2,y_2,z_2) =
(x_1y_2 + x_2y_1, Dx_1x_2 + y_1y_2, z_1z_2),\]and
\[(x_1,y_1,z_1) \star (x_2,y_2,z_2) =
(x_1y_2 – x_2y_1, Dx_1x_2 – y_1y_2, z_1z_2).\]In other words, if (x_1,y_1,z_1) and (x_2,y_2,z_2) are solutions of Dx^2+z=y^2, then so are (x_1y_2+x_2y_1, Dx_1x_2+y_1y_2, z_1z_2) and (x_1y_2-x_2y_1, Dx_1x_2-y_1y_2, z_1z_2).

We had also mentioned that Theorem 1 clearly leads to (and is, in fact, equivalent to) the identity

\[\begin{align}\label{Bidentity}
({y_1}^2-D{x_1}^2)({y_2}^2-D{x_2}^2)
=
(Dx_1x_2 \pm y_1y_2)^2-D(x_1y_2 \pm x_2y_1)^2
\end{align}\]which are now called Brahmagupta’s identity. The additive portion has become a basic result used repeatedly in modern number theory.

We had also stated some of the applications of Theorem 1 made in India. In this part, we mention some of the later developments in Europe around the result (1) after it reappeared in the work of Euler in the eighteenth century.

The Ubiquity of bhāvanā

In a plenary talk at the International Congress of Mathematicians (1978), André Weil¹ asserted ([16], pp. 231–232):

Brahmagupta’s bhāvanā provides a grand illustration of the message conveyed by Weil in the above passage. An analysis of Brahmagupta’s methods reveal in his approach the dynamic anticipation of a certain mathematics culture that was to eventually evolve much later, albeit on a more broad and firm foundation. To achieve an understanding in depth of Brahmagupta’s work on the bhāvanā, one has to recognize in it the implicit occurrence of some of the simple but powerful ideas which characterize modern algebra. We have seen in Part 8 that the concepts embedded in the bhāvanā include the all-pervading modern idea of binary composition on an abstractly defined unknown set and the the multiplicativity of the norm function on a ring extension of \mathbb Z. We will see further examples in this part.

We shall first resume our discussion on the possible genesis of Brahmagupta’s bhāvanā. Next we shall discuss Brahmagupta’s identity (1) in the framework of the modern theory of binary quadratic forms and also use it to work out a special case of Gauss’s composition law. We shall then show the crucial role played by Brahmagupta’s bhāvanā in a celebrated theorem of Legendre. Finally we shall make a few concluding remarks. In the last section, we shall present the original verses of Brahmagupta with detailed word-by-word translations.

Bhāvanā: Possible Genesis

We do not have any record of Brahmagupta’s thought process. But it is always useful, especially for a budding researcher, to try to imagine, in retrospect, how a great mathematical idea could possibly have been arrived at, by looking at closely related problems and discoveries of the era. Even if the actual thought-process had been different, meaningful speculations have the potential of providing fresh insights on the mathematics involved.

In Part 8 ([10], p. 41), we had seen that Brahmagupta’s identity (1) may be obtained by splitting the terms {y_1}^2-D{x_1}^2 and {y_2}^2-D{x_2}^2, observing the identity

\[\begin{align}\label{ring}
(y_1+x_1\sqrt{D})(y_2+x_2\sqrt{D})
= Dx_1x_2 + y_1y_2+(x_1y_2 + x_2y_1)\sqrt{D},
\end{align}\]and multiplying this identity by the conjugate identity

\[\begin{align*}(y_1-x_1\sqrt{D})(y_2-x_2\sqrt{D})= Dx_1x_2 + y_1y_2-(x_1y_2 + x_2y_1)\sqrt{D}.\end{align*}\]This amounts to an implicit handling of the set {\mathbb Z}[\sqrt{D}] as a ring and an implicit imparting of the norm function on it, and implicitly arriving at the multiplicativity of the norm function.

We now recall a few ancient results and questions which are closely related to Brahmagupta’s identity (1), and which suggest an alternative route to their discovery.

Integer triples (x,y,z) satisfying x^2+y^2=z^2 have fascinated mathematicians from very ancient civilizations like the Babylonian, Chinese, Vedic Indian and later, the Greek. Practical construction problems often involved such triples. They are called Pythagorean triples. In Part 5 of the series ([8], p. 55), we had seen that the Śulba-sūtras give examples of such Pythagorean triples, and that there is a possibility that the Śulba mathematicians were aware of the formula

\[\begin{equation}\label{Pythagoras}
(2mn)^2+(m^2-n^2)^2=(m^2+n^2)^2.
\end{equation}\]which yield such triples, since the identity (3) readily follows from the identities implicit in some of the Śulba constructions.

Euclid (300 BCE), Diophantus (250 CE) and Brahmagupta (628 CE) explicitly described the general integer solution of the equation x^2+y^2=z^2. Scholars have wondered whether, while working with the Pythagorean triples, ancients had been led to the more general question:

Question 1. When is a positive integer b (not necesarily a square) expressible as a sum of two squares r^2+s^2 (r<s)?

Two of the smallest such numbers are clearly 5=1^2+2^2 and 13=2^2+3^2. In these two cases, the decomposition is unique; but their product 65 can be expressed as the sum of two squares in two different ways: 65=1^2+8^2=4^2+7^2. A mature algebraic mind would realize (maybe, after observing a few more of such examples) that the non-uniqueness in the decomposition of 65 is no accident – it is a consequence of the algebraic identity

\[\begin{align}\label{identity}
&\hskip -20pt({y_1}^2+{x_1}^2)({y_2}^2+{x_2}^2)\nonumber\\
&=(y_1y_2 – x_1x_2)^2 + (x_1y_2 + x_2y_1)^2\nonumber\\
& =(y_1y_2 + x_1x_2)^2 + (x_1y_2 – x_2y_1)^2.
\end{align}\]Thus,\[\begin{align*}
65&=5 \times 13\\
& = (2^2+1^2)(3^2+2^2)\\
&=(6-2)^2+(3+4)^2\\
&=(6+2)^2+(3-4)^2.
\end{align*}\]The identity (4) is stated explicitly by Fibonacci (1225 CE); Diophantus appears to have been familiar with the identity; for he mentioned that 65 is expressible as sum of two different squares in two different ways since 65 is the product of 13 and 5, each of which is a sum of two squares.

It is surmised that ancient Indians too had examined Question 1 and in the process arrived at the identity (4) and therefore, Brahmagupta, while considering the more intricate equation Dx^2+z=y^2, had been on the lookout for an analogous identity.

Brahmagupta’s identity (1) readily follow from (4) by replacing x_1 by x_1 \sqrt{-D} and x_2 by x_2 \sqrt{-D}. Weil remarks ([17], p. 14) that “this could not have been fully realized until the eighteenth century”. Complex numbers appeared in formal mathematics only in the sixteenth century and its vigorous use can be seen from the eighteenth century.

However, though the validity of complex numbers does not appear to have been recognized in Brahmagupta’s time, it is still possible that Brahmagupta might have heuristically used them as a secret trick – an inspired “rough work” – for guessing the magic identity which could then be verified by recognized algebraic manipulations.²

Before ending the section, we point out that Question 1 above can now be fairly easily solved by analysing it in `the ring of Gaussian integers’ (denoted by {\mathbb Z}[i]). It is presented in several basic textbooks on Algebra (and Number Theory); for instance, in M. Artin’s Algebra [1]. One of the answers to Question 1 is the following result:

Theorem 2. If -1 is a square modulo b (i.e., if there exists m such that b divides m^2+1), then there exist integers r,s such that b=r^2+s^2.

The proof of Theorem 2 is usually presented in books for the case when b is a prime p, but it can be seen that the general case easily follows from the prime case by repeated applications of the identity (4).

Binary Quadratic Form y^2-Dx^2

Fermat’s researches in number theory leads him to a deep study of the binary quadratic form y^2-2x^2 which must have resulted in his realization of the far-reaching importance of the equation y^2-Dx^2=1. As mentioned earlier, Fermat not only investigated the Brahmagupta-Pell equation; he also inspired others to take interest in the problem.

One of the fruitful theories to emerge from the research on the Brahmagupta-Pell equation is the theory of binary quadratic forms (with integer coefficients), systematically developed by Lagrange and Legendre after initial contributions from Fermat and Euler.

For a binary quadratic form Q(x,y):=ax^2+bxy+cy^2, a,b,c \in {\mathbb Z}, the discriminant is defined to be b^2-4ac; in particular, y^2-Dx^2 is a binary quadratic form with discriminant 4D.

In the language of quadratic forms, Brahmagupta’s identity says that two such forms (say y^2-Dx^2, v^2-Du^2) can be composed to yield another such form with the same discriminant 4D in a new pair of variables (xv \pm yu, yv \pm Dxu). The theory of composition of quadratic forms, an important and rich topic initiated by Gauss and Dirichlet, is still an active area of research.

Before mentioning another aspect of Theorem 1 in the context of quadratic forms, we recall a few definitions. Two quadratic forms f(x,y) and g(x,y) over an integral domain K are defined to be equivalent if there exists a homogeneous linear change of variables which takes the form f to the form g; more precisely, if there exists an invertible matrix A with entries in K such that g(x)=f(A x) where x denotes the vector (x,y)^T. Thus f and g are equivalent if there exist a, b, c, d in K such that g(x,y)=f(a x+ b y, c x+ d y) and ad-bc is a unit³ in K.

An element c in K is said to be represented by a binary form f(x,y) if the value c is attained by f, i.e., if there exist a,b \in K such that c=f(a,b). A quadratic form f is said to be strongly multiplicative if f is equivalent to cf for every unit c represented by f.

Theorem 1 of Brahmagupta may be viewed as the result:

Theorem 3. The binary form y^2-Dx^2 is strongly multiplicative (over rationals).

For, if f denotes the form y^2-Dx^2, c a nonzero number represented by f, say c=b^2-Da^2, and g=cf, then

\[
g(x,y)=(b^2-Da^2)(y^2-Dx^2).
\]Now Brahmagupta’s identity (1) prescribes the substitution that has to be made to obtain g(x,y)=f(u,v): namely, the homogeneous linear transformation given by
\[
u=b x + a y, v=D a x+ b y
\]which is invertible (since the determinant of Brahmagupta’s transformation is c \not=0).

This version of Theorem 1 was generalized in 1965 by A. Pfister using, what are now called, “Pfister forms”. Pfister’s discovery opened up new directions in the theory of quadratic forms. In his research monograph [14] in this area, Manuel Ojanguren begins Chapter 5 by quoting Brahmagupta’s original Sanskrit verses describing Theorem 1; the chapter itself is titled

Also sprach Brahmagupta

In English:

Thus Spake Brahmagupta

The first result in the chapter (Lemma 5.1) states ([14], p. 54):

Readers familiar with the relevant terminology can see that Lemma 5.1 is a version of the fact that

y^2-Dx^2 is strongly multiplicative.

Theorem 5.2 in ([14], p 55) states the generalization of Lemma 5.1 by Pfister in 1965 (not a typing error, it is 1965!): “Pfister forms are multiplicative”.

M. Ojanguren refers to Lemma 5.1 as “Brahmagupta’s lemma” ([14], p 55). Just before the statement of Lemma 5.1, Ojanguren writes ([14], p. 54):

With a quiet humour characteristic of Ojanguren, Item 5 is listed in “References” as:

5. Brahmagupta: Brāhmasphuṭasiddhānta, Bhīnmāla, 628 (preprint).

The word “preprint” usually refers to a draft of a paper or a book that is in circulation before getting printed; here it alludes to Brahmagupta’s treatise being composed before printing was even invented and became standard.

A Special Case of the Composition Law of Gauss

The article [7] in the inaugural issue of this magazine made a brief discussion on the composition law of Gauss on general binary quadratic forms of a given discriminant, and Manjul Bhargava’s higher composition laws which enormously simplify and generalize the laws of Gauss.

In this section we show how Brahmagupta’s identity helps one to verify the following special case of Gauss’s composition law.

Theorem 4. One has the composition law

\[\begin{align*}(A_1x^2+Bxy+A_2Cy^2)(A_2z^2+Bzw+A_1Cw^2)
=A_1A_2t^2+Bts+Cs^2,\end{align*}\]where t=xz-Cyw, s=A_1xw+A_2yz+Byw.

Proof. The discriminant N:=B^2-4A_1A_2C, so that \frac{A_2C}{A_1}=\frac{B^2-N}{(2A_1)^2}. Hence

\[\begin{align*}A_1x^2+Bxy+A_2Cy^2&=A_1[x^2+2x\frac{By}{2A_1}+ (B^2-N)\frac{y^2}{(2A_1)^2}]\\
&=A_1[(x+\frac{By}{2A_1})^2-N(\frac{y}{2A_1})^2]=A_1 ({x_1}^2-N{y_1}^2),\end{align*}\]where x_1=x+\frac{By}{2A_1}, y_1=\frac{y}{2A_1}. Similarly, A_2z^2+Bzw+A_2Cw^2 =A_2 ({z_1}^2-N{w_1}^2), where z_1=z+\frac{Bw}{2A_2}, w_1=\frac{w}{2A_2}, and A_1A_2t^2+Bts+Cs^2=A_1A_2 ({t_1}^2-N{s_1}^2), where \displaystyle t_1=t+\frac{Bs}{2A_1A_2} and \displaystyle s_1=\frac{s}{2A_1A_2}.

Therefore, by Brahmagupta’s identity (1),
\[\begin{align*}{\rm LHS}&=A_1A_2({x_1}^2-N{y_1}^2)({z_1}^2-N{w_1}^2)\\
&=A_1A_2 ({t_2}^2-N{s_2}^2),\end{align*}\] where
t_2=x_1z_1+Ny_1w_1, s_2=x_1 w_1 +y_1z_1.
Now
\[\begin{align*}t_2&=x_1z_1+Ny_1w_1\\&=(x+\frac{By}{2A_1})(z+\frac{Bw}{2A_2})+ (B^2-4A_1A_2C) \frac{yw}{4A_1A_2}\\
&=xz-Cyw+\frac{B(A_1xw+A_2yz+Byw)}{2A_1A_2}\\&=t+\frac{Bs}{2A_1A_2}=t_1;\\
s_2&=x_1 w_1 +y_1z_1\\&=(x+\frac{By}{2A_1})\frac{w}{2A_2}+\frac{y}{2A_1}(z+\frac{Bw}{2A_2})\\&=\frac{A_1xw+Byw+A_2yz}{2A_1A_2}\\&=\frac{s}{2A_1A_2}\\&=s_1.\end{align*}\]Therefore, {\rm LHS}=A_1A_2 ({t_1}^2-N{s_1}^2)={\rm RHS}.□

Bhāvanā in a Theorem of Legendre

In this section, we shall show how Brahmagupta’s bhāvanā is a crucial ingredient in one of the main theorems of Adrien-Marie Legendre (1752-1833) in number theory (Theorems 6 and 7). This result, which occurs in his Recherches d’analyse indéterminée (1785), leads to an important local-global theorem in number theory, and is perhaps his most famous contribution in mathematics. We shall sketch the proof that is presented in ([12], pp. 272-275).

In Part 5 of the series ([8], pp. 60-61) we had briefly discussed the idea of Fermat’s descent and its resemblance with the Indian kuṭṭaka algorithm. A descent-like reduction argument is involved in the above proof of Legendre’s theorem. We shall demonstrate (Theorem 5) how Brahmagupta’s bhāvanā (1) is a key step in the reduction. We shall omit those basic algebraic arguments in the proof which are not directly relevant to our central discussions. The reader can find the arguments in ([12], pp. 272-275) and several other texts on number theory. For the convenience of the reader, we shall use the same notation as in ([12], pp. 272-275).

For given integers a,b, the homogeneous equation ax^2+by^2=z^2 always has the “trivial” integer solution (0,0,0). Any integer solution other than (0,0,0) will be called a non-trivial integer solution.

Before stating Legendre’s main theorem, we first note the following observation.

Lemma 1. Let a,b be square-free positive integers with GCD d. Suppose that

\[
ax^2+by^2=z^2
\]has a non-trivial integer solution. Then the following conditions hold:

1. a is a square modulo b.
2. b is a square modulo a.
3. -(\frac{ab}{d^2}) is a square modulo d.

Proof. Let (x_0,y_0,z_0) be a non-trivial solution. Dividing by the common factors if necessary, we may assume that x_0,y_0, z_0 do not have any common prime factor. Then it follows that x_0, y_0 will be coprime.

We claim that x_0 and b will also be coprime. For, if p is a prime which divides both x_0 and b, then the relation
\[\begin{equation}\label{L01}
a{x_0}^2+b{y_0}^2={z_0}^2
\end{equation}\]shows that the prime p divides z_0 (since p divides LHS and hence RHS) and hence p^2 divides b{y_0}^2 (={z_0}^2-a{x_0}^2). Since b is square-free, it will follow that p divides y_0, a contradiction. This proves the claim.

Since x_0 and b are coprime, from the kuṭṭaka algorithm of Āryabhaṭa discussed in Part 5 of the series [8], one gets integers u,v such that ux_0-vb=1, i.e., ux_0 \equiv 1 modulo b.

Since d divides b, it also follows that x_0 and d are coprime and that ux_0 \equiv 1 modulo d. Therefore, multiplying both sides of (\ref{L01}) by u^2, one gets a \equiv (uz_0)^2 modulo b, proving (i).

(ii) can be proved similarly.

(iii) Since a and b are square-free, so is d. Hence, by (\ref{L01}), d divides z_0. Therefore, by (\ref{L01}),

\[
(\frac{a}{d}){x_0}^2+(\frac{b}{d}){y_0}^2=d(\frac{z_0}{d})^2
\]Hence, multiplying both sides by -\frac{b}{d}u^2, we have
\[
-(\frac{ab}{d^2}) \equiv ({\frac{b}{d}{y_0}u})^2 {\rm ~~ modulo ~~} d.
\]□

Legendre proved (Theorem 6) that the converse holds, that is, the obvious necessary conditions (i), (ii), (iii) above are also sufficient for the equation ax^2+by^2=z^2 to have a non-trivial integer solution. We separately state below two key steps for the proof that we will present. As in the case of the kuṭṭaka algorithm, the main idea in this proof is to “reduce” the given problem to the problem of solving an equation Ax^2+by^2=z^2 with 0<A<a, i.e., with a smaller coefficient, and arrive at a situation where the equation can be solved easily.

Lemma 2. Let a, A,b be square-free positive integers such that aAm^2+b is a square for some integer m. If the conditions (i), (ii), (iii) in Lemma 1 are satisfied by the pair (a,b), then they will also be satisfied by the pair (A,b), i.e., we will have A is a square modulo b, b is a square modulo A and -(\frac{Ab}{{\delta}^2}) is a square modulo \delta, where \delta is the GCD of A and b.

The proof of the above result involves only elementary modular arithmetic ([12], p. 274). Brahmagupta’s bhāvanā (Theorem 1) is the key to the next result which will enable one to prove Legendre’s theorem by a `descent’ like reduction, using Lemma 2.

Theorem 5. Let a,b, A be square-free positive integers such that aAm^2+b is a square for some integer m. If the equation

\[\begin{equation}\label{L11}
Ax^2+by^2=z^2
\end{equation}\]has a nontrivial integral solution, then so does the equation

\[
ax^2+by^2=z^2.
\]

Proof. Let (X,Y,Z) be a non-trivial solution of the equation (\ref{L11}), i.e., we have

\[\begin{equation}\label{L13}
bY^2+ AX^2= Z^2.
\end{equation}\]By hypothesis, we also have a positive integer c such that

\[\begin{equation}\label{L14}
b1^2+ aAm^2= c^2.
\end{equation}\]Recall the additive version of Brahmagupta’s bhāvanā (Theorem 1): if D{x_1}^2+z_1={y_1}^2 and D{x_2}^2+z_2={y_2}^2, then

\[
D(x_1y_2+x_2y_1)^2+z_1z_2= (Dx_1x_2+y_1y_2)^2.
\]Applying the result with D=b, (x_1,y_1,z_1)=(Y, Z, AX^2) and (x_2,y_2,z_2)=(1, c, aAm^2), we have

\[\begin{equation}\label{L15}
b(Yc+Z)^2+ aA^2m^2X^2= (bY+Zc)^2,
\end{equation}\]that is,
\[\begin{equation}\label{L16}
a(AmX)^2 +b(Yc+Z)^2 = (bY+Zc)^2.
\end{equation}\]As b is square-free, by (7) and (8), we have AmX \ne 0. Thus, (x,y,z)= (AmX, cY+Z, bY+cZ) is a non-trivial integral solution of the equation ax^2+by^2=z^2. □

We now prove a version of Legendre’s main theorem, a converse of Lemma 1.

Theorem 6. Let a,b be square-free positive integers with GCD d. Suppose that the following conditions hold:

1. a is a square modulo b.
2. b is a square modulo a.
3. -(\frac{ab}{d^2}) is a square modulo d.

Then the equation

\[\begin{equation}\label{L17}
ax^2+by^2=z^2
\end{equation}\]has a non-trivial integer solution.

Proof. If a=1, then (1,0,1) is a non-trivial solution of (11).

If a=b, then by (iii), -1 is a square modulo b and hence, by Theorem 2, there exist integers r,s for which a=b=r^2+s^2. Then x=r, y=s give a non-trivial solution of (11).

Therefore, interchanging x,y if necessary, we may assume a>b.

Claim. There exists a square-free positive integer A such that aAm^2+b is a square for some integer m and A < a.

By (ii), there exists an integer c such that

\[\begin{equation}\label{L18}
c^2-b=aAm^2
\end{equation}\]for some square-free integer A, some integer m and with |c| \leq a/2.

We have
\[
0 \le c^2 = aAm^2+b \lt a(Am^2+1)
\]
showing that A > -1; and since b is square-free, it follows from (\ref{L18}) that A>0.

Again, by (12), we have aAm^2 \lt c^2 \le a^2/4 so that A \le Am^2 < a/4 < a. This proves the Claim.

Now, by Lemma 2, all the hypotheses (i), (ii), (iii) will still be satisfied if a is replaced by A. Further, by Theorem 5, to show that (11) has a non-trivial integer solution, it is enough to show that the equation Ax^2+by^2=z^2, with 0 < A < a has a non-trivial integer solution.

After a finite number of steps, interchanging A and b in case A < b, we arrive at one of the cases A=1 or A=b, each of which has already been settled at the outset. Hence the result.□

From the above version (Theorem 6), one can easily deduce the following, more symmetric, version of Legendre’s Theorem ([12], p. 273):

Theorem 7. Let a, b,c be nonzero, square-free, pairwise coprime (relatively prime) integers, and neither all positive nor all negative. Suppose that the following conditions are satisfied:

1. -ab is a square modulo |c|.
2. -ac is a square modulo |b|.
3. -bc is a square modulo |a|.

Then the equation

\[\begin{equation}\label{L19}
ax^2+by^2+cz^2=0
\end{equation}\]has a non-trivial integer solution.

Note that the conditions (i), (ii), (iii) are obviously needed for the equation (13) to have a non-trivial integer solution.

An important application of the above theorem is the following local-global result ([12], p. 275), a special case of the “Hasse Principle”:

Theorem 8. Let a, b,c be square-free, pairwise coprime (relatively prime) integers, and neither all positive nor all negative. If, for each prime power p^m, the congruence
\[\begin{equation}\label{L20}
ax^2+by^2+cz^2 \equiv 0 (p^m)
\end{equation}\]has a solution in integers (x,y,z) not all divisible by p, then the equation (\ref{L19}) has a nontrivial integer solution.

Bhāvanā defies simplistic labels

Labels, often with antagonistic nuances, like “applied/practical mathematics” vs “pure mathematics”, “algorithmic mathematics” vs “abstract mathematics”, are often used to characterize mathematics of different cultures (eg. “Indian” vs “Greek”). A pitfall of such labelling is the neglect or denial of great achievements which do not conform to such broad labels.

For instance, Indian mathematics is labelled “utilitarian”, unlike Greek mathematics. A pernicious outcome of the label has been the reluctance of several prominent historians of science to accept that ancient Indians could have worked on the equation Dx^2+1=y^2 on their own and led to speculations, without basis, that the Greeks must have solved the problem earlier and Indians got it from the Greeks! An erudite mathematician like van der Warden asks, “Brahmagupta was mainly an astronomer. […] But why should he be interested in the solution of Pell’s equation?” and surmises that Brahmagupta and other Indians must have “copied” from Greek sources. The issue has been discussed in detail in ([6], pp. 192–196).

Again, Indian mathematics is labelled “algorithmic”. The outcome has been the exclusive emphasis on algorithms like kuṭṭaka and cakravāla and the neglect of their underlying principles. The greatness of a fundamental creation (like the bhāvanā) with momentous consequences gets overlooked; the bhāvanā gets projected as a mere “Lemma” for the cakravāla algorithm.

As discussed in ([11], pp. 535–537), Brahmagupta’s bhāvanā defies simplistic labels. It is an important identity, but it is much more – it is a law of composition. It manifests the fundamental principle of modern Abstract Algebra (principle of composition) but does so in the language and framework of classical algebra. It arises out of algorithmic mathematics – it leads to the discovery of, and takes a crucial part in, an astonishing algorithm for solving an important problem. And yet it stands apart as a result whose significance far exceeds that of the original problem. The computational or algorithmic approach and the conceptual abstract approach are inseparably fused in Brahmagupta’s great work on the varga-prakṛti Dx^2+1=y^2.

Epilogue

Bhāskarācārya (1150 CE) declared Brahmagupta to be Gaṇaka-Cakra-Cūḍāmaṇi (Crest-Jewel among the Mathematicians). We have also seen (in Part 8 [10] and this part), glowing tributes to Brahmagupta’s work in general, and the bhāvanā in particular, from leading modern mathematicians like Michael Atiyah, David Mumford, Manuel Ojanguren and Manjul Bhargava. After Part 6 of the present series on Brahmagupta’s contributions to The Foundations of Algebra in India was published [9], David Mumford had written in an email (on 27.4.2023) to C.S. Aravinda, Chief Editor of Bhāvanā:

The sublime soul-stirring impact that the bhāvanā can have on an algebraist can perhaps be felt from an informal e-mail that the present author had received (on 23 June 2004) from his former teacher at TIFR, Late Professor Amit Roy:

The above correspondence took place after Amit Roy read an article on Brahmagupta’s bhāvanā by the present author. In a follow-up mail, Prof. Roy remarked:

Amit Roy further clarified:

Many of the major landmarks in ancient Indian mathematics, from the decimal notation to the calculus and analysis of the Kerala school, were the outcomes of chains of ideas evolving and crystallizing over long periods of time. Brahmagupta’s composition law bhāvanā may well be regarded as the greatest single original idea in the vast and grand spectrum of ancient Indian mathematics.

Original Verses on Brahmagupta’s bhāvanā

Chapter 18 of Brahmagupta’s magnum opus Brāhma Sphuṭa Siddhānta is titled Kuṭṭakādhyāyaḥ (Chapter on Kuṭṭaka). While the word kuṭṭaka is used in Sanskrit mathematics for the problem of solving linear indeterminate equations discussed in this chapter, Brahmagupta evidently uses the word for the subject Algebra ([5], p. 1) for which he lays down the foundation in this chapter, as discussed in [9]. The Sanskrit word kuṭṭaka (pulverizer) indicates that the subject is being regarded as the science of solving problems by a process of simplification – by breaking quantities (for instance, the coefficients or solutions of a given equation) into smaller pieces.

Theorem 1 is stated by Brahmagupta in Verses 64–65 of the Kuṭṭakādhyāyaḥ in the following form:

Atha Varga Prakṛtiḥ

mūlaṁ dvidheṣṭavargād guṇakaguṇādiṣṭa yutavihīnācca
ādyavadho guṇakaguṇaḥ sahāntyaghātena kṛtamantyam
vajravadhaikyaṁ prathamaṁ prakṣepaḥ kṣepavadhatulyaḥ

In symbols, the first line states:
[{\rm If}]~~~~ {y_i}^2= D{x_i}^2 + z_i, ~~~~~~ i=1,2;
while the second and third line taken together mean:
\[\begin{align*}[{\rm then}]~~~~ y&=Dx_1x_2 \pm y_1y_2, \\
x&=x_1y_2 \pm x_2y_1, \\
z&=z_1z_2
{\rm ~~~~satisfy~~~~} \\
y^2 &=Dx^2+z.\end{align*}\]

(Mention of y=Dx_1x_2+y_1y_2 and x=x_1y_2+x_2y_1 is explicit; mention of y=Dx_1x_2 \sim y_1y_2 and x=x_1y_2 \sim x_2y_1 also seem implicit.)

A word-by-word break-up is given below. We shall try to be faithful to both the letter and spirit of the original – even at the cost of elegance! More details on the varied significances of the Sanskrit words are given in [13].

atha : (to commence) “now”. The word has the nuance of an auspicious inception which is difficult to convey in English.

varga : “square”.

prakṛti : “nature”, “principle”, “origin”. Jayadeva explains ([15], p. 4) that since x^2D \pm z has the nature of a square, hence the name varga-prakṛti [for the equation Dx^2 \pm z = y^2].

Line 1

mūla : “root”. (See Remark 1.) Thus varga-mūla = “square root”; ghana-mūla = “cube root”, etc. In the absence of a specific qualifier like varga or ghana, the term mūla denotes “square root”, just as the plain \sqrt{} denotes \sqrt[2]{}.

mūlaṁ : Suppose that the following have square-roots, i.e., are perfect squares {y_i}^2.

dvidhā : “twofold”, “in two ways or parts”, “two cases”. Here, the main sense is that of the modern “i=1,2”. But the word itself has been used here with “twofold” (double) implication. The second implication will be discussed separately in Remark 2 below.

iṣṭa : “sought”, “desired”; in Sanskrit mathematics, the adjective is used for an unknown quantity x (which is to be sought and realized). Recall that in Indian literature, the adjective is usually associated with iṣṭa-devatā – the Personal Deity whose Realization is the aim or desire of a spiritual seeker.

vargād : “squares”.

guṇaka : “multiplier”, i.e., D.

guṇād : “multiplied thereof”. Thus, here dvidheṣṭavargād guṇakaguṇād : =D{x_1}^2, D{x_2}^2.

iṣṭa : Here, the word is used as a noun referring to the “desired” interpolators c_1, c_2.

yuta : “added”.

vihīna : “subtracted”.

ca : “or”.⁵ Thus yuta vihīnācca conveys the modern notation \pm.

A literal translation of Line 1:

In symbols, it is given that {y_i}^2=D{x_i}^2 \pm c_i, ~~ i=1,2, where c_i are positive integers; i.e.,
{y_i}^2=D{x_i}^2 + z_i, ~~ i=1,2, where z_i are integers.

Line 2

ādya : “initial” or “first” [root], i.e., x_i.

vadho : “product of the two”. Thus ādyavadho guṇakaguṇaḥ =Dx_1x_2.

saha : “[combined] with”.

antya : “final” [root], i.e., y_i.

ghātena : “product of”. Thus antyaghātena =y_1y_2.

kṛtam : “attained”.

antyam : [new] “final” [root], i.e., y.

A literal translation of Line 2:

In symbols, y=Dx_1x_2 + y_1y_2 (or y=Dx_1x_2 \pm y_1y_2).

Line 3

vajravadha : “cross-multiplication” x_1y_2, x_2y_1.

aikyaṁ : “united”; “added”.

prathamaṁ : “first” [root], i.e., x. Thus vajravadhaikyaṁ prathamaṁ : x=x_1y_2 + x_2y_1 (or x=x_1y_2 \pm x_2y_1).

prakṣepaḥ : [new] “interpolator”, i.e., z.

kṣepa : [previous] “interpolators”, i.e., z_1, z_2. Thus kṣepavadha =z_1z_2.

tulya : “equal”. Thus prakṣepaḥ kṣepavadhatulyaḥ: z=z_1z_2.

A literal translation of Line 3:

Remark 1. In Sanskrit, the word mūla has various meanings of allied significances including “root” (of a plant or tree; but also figuratively the foot or lower part or bottom of anything), “basis”, “foundation”,“cause”, “origin”.

The word mūla has been introduced in Sanskrit mathematics for the operation of taking roots. For instance, the word varga-mūla is introduced for the concept of “square-root” evidently as the square-root can be considered the mūla (i.e., basis, cause, origin) of the varga (square).

Mūla is also used in the more general sense of “root of an equation”; e.g., ādya-mūla = “initial (i.e., first) root”; antya-mūla = “final (i.e., second) root”.

In commercial problems on Interest, the word mūla has been used for the “Principal”, i.e., the “capital”, the “foundation” or “basis” on which interest is computed.

The use of the word mūla for the mathematical concept of “root” can be seen in the Anuyogadvāra-sūtra (c.100 BCE) and in all Sanskrit mathematical works. Another word pada has also been used in Sanskrit mathematics for the “root”, at least from the seventh century – it occurs in the work of Brahmagupta.

This mathematical use of mūla for the concept of “root” has subsequently been emulated in Arabic and Latin mathematics, through the corresponding words jadhr and radix respectively.

More details are given in the articles “On Mūla, the Hindu Term for `Root”' and “On the Origin of the Hindu Terms for `Root”' by B. Datta in the American Math Monthly Vol 34 pp. 420–423 (1927) and Vol 38 pp. 371–376 (1931); and the book History of Hindu Mathematics Vol 1 by B. Datta and A.N. Singh, pp. 169–170.

Remark 2. As pointed out by Datta-Singh ([5], p 146), the word dvidhā in Line 1 has two-fold implications:

(i) Consider two triples satisfying Dx^2+z=y^2.

(ii) Consider two copies of a triple satisfying Dx^2+z=y^2.

Thus (ii) is a special case of (i) and, together with the next two lines, indicate Corollary 1:

\[\begin{align}\label{corollary1}
\mbox{ If~~} Dp^2+m=q^2, \mbox{ ~~then~~} D(2pq)^2+m^2=(Dp^2+q^2)^2.
\end{align}\]That the audience is also expected to read the implication (ii) in Line 1, and therefore Corollary 1 in Lines 1–3, is confirmed by the very next Line 4:

prakṣepaśodhakahṛte mūle prakṣepake rūpe

prakṣepa: additive, interpolator.

śodhaka: subtractive.

hṛte: divided.

mule: roots of.

rūpe: 1.

A literal translation of Line 4:

Thus, we have, Corollary 2:

Line 4 is in harmony with the previous lines if the implication (ii) is also considered; otherwise the language used would appear abrupt.

Remark 3. Both the additive and subtractive versions of Theorem 1 seem to find simultaneous mention in Brahmagupta's verses through the expressions saha and aikyaṁ with the ambiguous sense of “combining” Dx_1x_2 with y_1y_2 and x_1y_2 with x_2y_1. The word “combining” certainly includes the implication “adding”. Thus the reference to the more important additive principle (producing new roots Dx_1x_2+y_1y_2 and x_1y_2 +x_2y_1) is obvious. Presumably, “combining” also includes “taking difference”; the subtractive principle (generating Dx_1x_2 \sim y_1y_2 and x_1y_2 \sim x_2y_1) is unlikely to have been overlooked by Brahmagupta. (Also see Remark 4 below.) In any case, Jayadeva, Bhāskara II and subsequent writers clearly describe the two principles successively in separate verses.

Remark 4. During our literal translation, we had to insert several brackets [ ] to include expressions which are implicit in Brahmagupta's verses. The three lines, conveying Theorem 1, could appear obscure for a person not already familiar with the essence of the result.

As highlighted earlier in this series ([8], p. 57), brevity is a general feature of the original treatises of earlier greats like Āryabhaṭa (499 CE) and Brahmagupta (628 CE) who indicate broad hints and not complete details; there is an emphasis on the use of a learner's own intellect for filling up the details. Detailed expositions on the works of the Masters are meant to be orally transmitted through the guru-śiṣya (mentor-disciple) link. The commentaries too try to clarify the implications of the passages of the original treatises. Brahmagupta's verses are explained by Pṛthudaka Svāmi (c. 860 CE); Bhāskara II's verses by Kṛṣṇa (1548 CE) and Sūryadāsa (1541 CE).

Besides, for ancient Indians, conciseness had been a practical necessity. Their typical writing materials like palm-leaf manuscripts do not last long. A treatise could be preserved only through memorization or through repeated copying. In either case, the more concise the work, the better its chance of survival through faithful memorization or copying. Brevity would have been particularly indispensable for an ancient author, like Brahmagupta, with enormous output. One has to note that, in spite of its oppressive terseness, the number of verses in Brāhma Sphuṭa Siddhānta exceeds a thousand.

With passage of time and improvement of writing materials, the later treatises, even by stalwarts, present increasingly elaborate versions of the results of their predecessors. Jayadeva's description of the bhāvanā ([15], pp. 4–8) has greater clarity than Brahmagupta's; Bhāskarācārya's version (quoted below) is still more lucid.

Note that while Brahmagupta calls the roots x and y of the equation Dx^2+z=y^2 as ādya (initial/first) and antya (final) respectively, Bhāskarācārya calls them hrasva (lesser) and jyeṣṭha (greater) respectively; since clearly x<y. The multiplier D which is called guṇaka by Brahmagupta is called prakṛti by Bhāskarācārya.

The version of Bhāskarācārya (Bhāskara II)

Bhāskarācārya's version ([3], p. 22) of Brahmagupta's bhāvanā occurs in the section “Varga Prakṛti” of his text Bījagaṇita (Verses 70–71). We present below the transliteration followed by a translation.

iṣtaṁ hrasvaṁ tasya vargaḥ prakṛtyā kṣuṇṇau yukto varjitā vā sa yena।
mūlaṁ dadyāt kṣepakaṁ taṁ dhanarṇaṁ mūlaṁ tacca jyeṣṭhamūlaṁ vadanti॥

hrasvajyeṣṭha kṣepakān nyasya teṣāṁ tānanyānvā'dho niveśya krameṇa।
sādhyānyebhyo bhāvanābhirbahūni mūlānyeṣāṁ bhāvanā procyate'taḥ॥

vajrābhyāsau jyeṣṭhalaghvostadaikyaṁ hrasvaṁ laghvorāhatiśca prakṛtyā।
kṣuṇṇā jyeṣṭhābhyāsa yug jyeṣṭhamūlaṁ tatrābhyāsaḥ kṣepayoḥ kṣepakaḥ syāt॥

hrasvaṁ vajrābhyāsayorantaraṁ vā laghvorghāto yaḥ prakṛtyā vinighnaḥ।
ghāto yaśca jyeṣṭhayostadviyogo jyeṣṭhaṁ kṣepo'trāpi ca kṣepaghātaḥ॥

References

• [1] M. Artin, Algebra, Prentice-Hall.
• [2] Brahmagupta, Brāhma Sphuṭa Siddhānta (628). The Sanskrit text, edited by R.S. Sharma, has been published by the Indian Institute of Astronomical Research (1966).
• [3] Bhāskarācārya, Bījagaṇita (1150). The Sanskrit text, edited and translated by S.K. Abhyankar, has been published by the Bhaskaracharya Pratishthana, Pune (1980).
• [4] H.T. Colebrooke, Algebra, with Arithmetic and Mensuration, from the Sanscrit of Brahmegupta and Bhascara, John Murray, London (1817).
• [5] B. Datta and A.N. Singh, History of Hindu Mathematics: A Source Book Part II (Algebra), Motilal Banarasidass, Lahore (1935); Asia Publishing House, Bombay (1962).
• [6] A.K. Dutta, Kuṭṭaka, Bhāvanā and Cakravāla, Studies in the History of Indian Mathematics (ed C.S. Seshadri), CHOM 5, Hindustan Book Agency (2010), pp. 145–199.
• [7] A.K. Dutta, The bhāvanā in Mathematics, Bhāvanā, Vol. 1(1), 13–19 (January 2017).
• [8] A.K. Dutta, Mathematics in Ancient India Part 5: The Kuṭṭaka Algorithm, Bhāvanā, Vol. 7(1), 52–67 (January 2023).
• [9] A.K. Dutta, Mathematics in Ancient India Part 6: The Foundations of Algebra – Glimpses, Bhāvanā, Vol. 7(2), 52–68 (April 2023).
• [10] A.K. Dutta, Mathematics in Ancient India Part 8: Brahmagupta's Composition Law, bhāvanā - I Bhāvanā, Vol. 7(4), 31–44 (October 2023).
• [11] A.K. Dutta, Algebraic Insights in Indic Algorithms, The Computational Meme: Computational Thinking in the Indic Tradition (eds. K. Gopinath and S.D. Sharma), IISc Press (2022), pp. 467–543.
• [12] K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer Second Ed. (Indian reprint 2014).
• [13] M. Monier-Williams, Sanskṛit-English Dictionary, Clarendon Press, Oxford (1899); reprinted Munshiram Manoharlal, New Delhi (2002).
• [14] Manuel Ojanguren, The Witt Group and The Problem of Luroth, Dottorato di ricerca in Mathematica, ETS EDITRICE PISA, University of Pisa (1990).
• [15] K.S. Shukla, Ācārya Jayadeva, The Mathematician, Gaṇita 5 (1954), pp. 1–20.
• [16] André Weil, History of Mathematics: Why and How, Proceedings of the International Congress of Mathematicians, Helsinki 1978 Vol 1 (ed. O. Lehto), Helsinki (1980), pp. 227–236.
• [17] André Weil, Number Theory: An approach through history from Hammurapi to Legendre, Birkhauser (1984).

Footnotes

1. André Weil (1906–98), one of the giants of 20th century mathematics, noted for his outstanding contributions in number theory and algebraic geometry, had a lifelong interest in the history of mathematics. ↩
2. To give an analogy, recall the non-commutative ring-theoretic result: “If 1-ab is invertible, then so is (1-ba)^{-1}.” and consider the effective trick of guessing the inverse of 1-ba through power-series expansion of (1-ba) even when the latter does not make formal mathematical sense. ↩
3. Meaning, ad-bc has an inverse in K. That is, there exists \lambda in K such that \lambda(ad-bc)=1. ↩
4. Looking for a picture of the mathematician Legendre, we came across this very interesting article titled `Changing Faces: The Mistaken Portrait of Legendre' by Peter Duren in the December 2009 edition of Notices of AMS which explains why, in order not to cause any further confusion, we are using this authentic image of the letter instead of a photo. ↩
5. ca usually means “and”, “both”, “also”, “as well as”, etc, but is sometimes used in the sense of “or”. ↩
6. The last sentence comes from the phrase bhāvanā procyate'taḥ which can also be translated as “Therefore, it is called bhāvanā.” Both meanings might have been intended. After all, Bhāskara II began the text with a deliberate pun on sāṁkhyāḥ, satpuruṣa, bīja, vyakta, etc!. ↩