Prof. Manjul Bhargava on Symmetries of the Khajuraho Magic Square and Generalisations to Higher Dimensions
Mandyam D Srinivas
K S Kannan
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Hon. Prof., Chanakya University, Bangalore.,
Emeritus Prof. and Director, Vishnugupta VishwaVidyapeetham, Gokarna.,
Sant Rajinder Singh Ji Maharaj Chair Professor (Retd.), IIT-Madras.
Member, Expert Committee for Review of Criticism of Indian Knowledge Traditions, Central Sanskrit University (under MoE, GoI), Ganganath Jha Campus, Prayagraj.
Adjunct Faculty, Dept of Heritage Science and Technology, IIT Hyderabad.
Member, Academic Council, Veda Vijnana Shodha Samsthana.
Academic Director, Swadeshi Indology.
Nominated Member, IIAS, Shimla.
Former Director, Karnataka Samskrit
University, Bangalore.
Former Head, Dept. of Sanskrit, The
National Colleges, Bangalore.
Gadi Manoj
Yogananda CS
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K S Kannan
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Rajaram Krishnamurthy
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Ankur Kakkar
Krishna Kumar Pillalamarri
Soumya Kumar
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K S Kannan
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K S Kannan
In case you
Ideas
The Proofs Historians Said Indian Mathematics Lacked
R S Hariharan
Sep 30, 2026 | Updated Sep 29, 2026, 06:25 PM GMT+5:30
- Indian mathematicians wrote their rules in verse and explained them in separate texts. Later Western scholars who studied only the verses assumed the explanations did not exist.
In December 1832, the Royal Asiatic Society in London heard a remarkable paper from a civil servant of the East India Company.
Charles Matthew Whish had been studying mathematical manuscripts from Kerala, and he expected what he had found to astonish European mathematicians. The manuscripts described infinite series for π, the sine, and the cosine, which predated similar discoveries in Europe by centuries.
Here was evidence that Indian mathematicians had ventured into territory long associated with Gregory, Leibniz, and the beginnings of calculus.
Among the results Whish reported was a deceptively simple infinite series.
π/4 = 1 − 1/3 + 1/5 − 1/7 + …
Keep adding the fractions, alternating the signs, and the sum creeps ever closer to a quarter of π. The same series would not appear in Europe until the seventeenth century.
Yet, elegant as the formula is, it is not the most interesting thing in these manuscripts. Whish himself chose to close his paper with a short proof of Pythagoras’ theorem taken from one of them, the Yuktibhāṣā.
Whish promised a second paper on the demonstrations in the Yuktibhāṣā, but he died in April 1833 without publishing it.
Histories of mathematics continued to follow a familiar script, in which Greek mathematics was celebrated as the birthplace of proof and Europe carried the subject forward through the Renaissance and the Scientific Revolution.
India found an honourable place in the narrative for its arithmetic, algebra, and astronomy, but a qualification was usually added. Indian mathematicians, readers were told, possessed extraordinary computational skill yet lacked the rigorous tradition of proof that distinguished Greek mathematics. The assumption became so entrenched that it was seldom questioned.
Even some of the greatest modern mathematicians repeated it. André Weil, one of the most influential figures of the twentieth century, suggested that the cakravāla, a celebrated Indian method for solving equations, could only have been an experimental finding for those who used it, a conclusion drawn from many worked cases rather than a proof. It was Fermat, he wrote, who first understood that a general demonstration was required.
The physicist and historian M D Srinivas, whose essay ‘Proofs in Indian Mathematics’ gathers much of the evidence discussed here, has pointed out the irony in that choice. Fermat is remembered above all as the man who announced proofs he never supplied, most famously his claim that a certain margin was too narrow to hold his argument.
Srinivas is careful to concede that Weil’s particular example holds, since none of the available Indian sources demonstrates that the method always works, and the first such proof came only in 1929–30 from the mathematician A A Krishnaswamy Ayyangar.
Even so, to credit Fermat with first seeing the need for a general proof, while denying the same to a tradition that preserved its demonstrations in abundance, reflects the expectations later historians brought to the manuscripts rather than anything in the mathematics itself.
The more one reads the mathematical works themselves, the harder it becomes to accept the old verdict. The issue is not whether Indian mathematicians valued proof, which they certainly did. The real question is what they meant by a proof and where they chose to present it.
Indian mathematicians drew a sharp line between a rule that happens to work and a rule that has been proved. Kṛṣṇa Daivajña, writing around the year 1600, makes the point with a deliberately false rule.
Four times the product of two numbers equals the square of their sum when the numbers are two and two, or three and three, or four and four, yet the rule fails whenever the two numbers differ. A result seen to hold in a few instances is therefore of no consequence, he argues, because a contrary case may always turn up. Only upapatti, a reasoned demonstration, can settle the matter.
Although upapatti is often translated simply as “proof,” its meaning is broader. It conveys the ideas of justification, rational explanation, and demonstration. A mathematical result was not regarded as complete merely because it produced the correct answer, and it deserved acceptance only when the reasoning behind it had been made clear. This was not an isolated opinion expressed by a single scholar, but one that reflected a wider mathematical culture.
Few figures illustrate this better than Bhāskara II. His Līlāvatī remains one of the most celebrated mathematical works produced in India, while the Bījagaṇita secured his reputation as one of the greatest algebraists of the medieval world.
Yet one of his most revealing observations appears in the Golādhyāya of the Siddhāntaśiromaṇi. Before presenting a series of demonstrations, Bhāskara explains why they are necessary. Without upapatti, he writes, a mathematician will carry no weight in an assembly of scholars and cannot himself be free of doubt.
It is a striking statement because it rejects the authority of the mathematician in favour of the authority of reason. Bhāskara insists that every mathematical result must ultimately stand on its own reasoning.
That concern is visible throughout the Indian mathematical tradition. The relation between the sides of a right-angled triangle, familiar today as Pythagoras’ theorem, was not merely stated but demonstrated.
Bhāskara himself gave two proofs in the Bījagaṇita, one relying upon the similarity of triangles and the other transforming the theorem into a problem of areas whose truth became visually evident through geometric rearrangement. The objective was not simply to verify the theorem but to make its necessity apparent.
The proof Whish chose to end his paper with is a rearrangement of this kind. It begins with the squares on the two shorter sides of the triangle, then marks off a few lengths and joins the points to form a new square standing on the hypotenuse.
The pieces of the two small squares that fall outside this new square turn out to be exactly the same shapes as the gaps they leave inside it, so each piece can be moved into a matching gap with nothing left over.
The two smaller squares, cut up and rearranged, become the larger one, and the theorem is seen rather than calculated.
Whish did not even bother to spell out the final check, remarking only that the mathematician would easily prove it. He thought this was probably the very form in which Pythagoras had first discovered the result.
The same attitude shaped algebra. Instead of presenting rules in isolation, Indian mathematicians frequently connected them with ordinary experience.
Kṛṣṇa Daivajña, for instance, explained positive and negative quantities through ideas such as gain and debt, possession and obligation, or movement in opposite directions. The symbolism remained grounded in intuition, making the resulting operations easier to understand rather than merely to memorise.
These examples become even more significant when we turn to the mathematicians of Kerala. Today, Mādhava of Saṅgamagrāma is rightly celebrated for his work on infinite series. His successors extended his methods, refined his ingenious correction terms, and achieved astonishing levels of numerical accuracy.
Modern discussions often emphasise the resemblance between these discoveries and later developments in Europe. That comparison is historically important, but it can also obscure something equally remarkable.
The Kerala mathematicians were not content merely to record formulas but developed careful arguments that led to them. Earlier propositions supported later ones, and arguments were refined, extended, and corrected. The correction terms in particular reveal not only computational brilliance but an awareness of why approximations succeed, where they fail, and how they may be systematically improved.
This is especially evident in the Yuktibhāṣā, written in Malayalam by Jyeṣṭhadeva around 1530 and one of the most extraordinary mathematical texts produced anywhere in the early-modern world. Its very title is revealing. The word yukti signifies reasoning, and the work patiently unfolds the logic behind mathematical results instead of merely announcing them. Reading it leaves little room for the notion that reasoned argument occupied no place in Indian mathematics.
How, then, did the contrary impression arise?
The answer lies partly in the literary form adopted by Sanskrit scientific writing. Most mathematical treatises were composed in verse, and the verses stated rules with extraordinary economy because they were intended to be memorised.
Their brevity was a pedagogical strength, but it could easily mislead anyone encountering them without the accompanying commentaries. The detailed reasoning, intermediate steps, diagrams, and demonstrations were usually supplied by commentators and teachers rather than incorporated into the root text itself.
Reading only the verses is rather like opening a modern mathematics textbook, memorising every formula printed in bold, and then skipping the explanations that follow. The mathematics appears incomplete only because half of it has been set aside.
The earliest European scholars to study Sanskrit mathematics understood this better than they are often given credit for. Henry Thomas Colebrooke observed that Indian mathematicians employed both algebraic and geometric demonstrations. Whish, too, recognised that the achievements of the Kerala school rested upon sustained mathematical reasoning. Yet these insights struggled to reshape a historical narrative that had already become deeply entrenched.
The difficulty, however, lay not merely in the expectations historians brought to the mathematics but in those they brought to the very form of a mathematical text. Euclid’s Elements had become the standard against which proof itself was judged, and a proposition was expected to be followed immediately by a demonstration. Indian mathematical works were organised differently, with an architecture that reflected the rhythms of oral teaching.
This difference in literary form gradually hardened into a difference in intellectual reputation. Brevity came to be mistaken for incompleteness, and a concise statement was assumed to be an unsupported assertion.
The commentarial tradition, where much of the mathematical reasoning actually resided, remained comparatively neglected. The conclusion followed almost automatically. If the proof did not appear where European readers expected to find it, perhaps it did not exist at all.
It is worth remembering that mathematics has never possessed a single universal language of proof. Indian mathematicians contributed their own methods of argument, shaped by their educational practices and literary culture. To insist that every civilisation must resemble Euclid before it can be said to possess rigour is to mistake one historical tradition for a universal standard.
Seen in this light, the old stereotype begins to dissolve. Indian mathematics was never merely a collection of clever algorithms but a tradition that expected mathematical claims to be justified, even if that justification was expressed in forms unfamiliar to later readers.
Bhāskara understood this, and in the same verse of the Golādhyāya he expressed it in a metaphor whose simplicity has ensured its survival long after the technical details of many demonstrations have faded from memory.
On the sphere, he wrote, a demonstration becomes as plain as a berry resting in the palm of one’s hand.
It is an image of remarkable gentleness. A berry held in the hand is open to inspection. One may turn it over, examine it from every side, and satisfy oneself that nothing has been concealed.
It does not demand faith but invites observation, and the truth becomes convincing because it is plainly visible.
Nearly nine centuries ago, Bhāskara chose an image so ordinary that it is easy to overlook its brilliance. A demonstration, he understood, is meant to leave nothing hidden and to be as plain, as immediate, and as certain as a berry resting in the palm of one’s hand.
R S Hariharan, PhD, is Assistant Professor at CAHC, JAIN University. With a background in science and technology and experience across research, linguistics, and industry, his work explores Sanskrit literature and India’s civilisational and scientific thought.
K S Kannan
Rajaram Krishnamurthy
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