Chapter 61: What Modern Science and Mathematics Actually Inherited from India

1. The question must be asked as a transmission history, not a victory claim

There is a legitimate historical question behind modern arguments about “Indian origins” of mathematics and science: which ideas are independently attested in Indian sources, which later entered Arabic and European traditions, and which claims are simply modern nationalistic stories? The answer is neither “everything came from India” nor “the Greeks and Arabs invented it all.” The surviving record shows a network of transmission in which Babylonian, Greek, Indian, Persian, Arabic, Chinese and European scholars sometimes developed ideas independently and sometimes borrowed, translated, adapted or recombined them.

The strongest cases are those in which a dated Indian text survives, the mathematical or astronomical procedure can be identified, and a later transmission route is independently documented. The weakest cases are claims based only on resemblance, an attractive modern name, or a statement that an ancient Sanskrit word “sounds like” a modern scientific concept. This chapter therefore uses a deliberately conservative standard.

2. The clearest case: the decimal place-value numeral system

The modern decimal place-value system is conventionally called “Sanatan Dharma-Arabic” because its mathematical forms travelled from India through the Islamic world and then into Europe. The historical evidence strongly supports an Indian origin for the mature positional decimal system used with the nine numerals and zero, although the visual shapes of individual digits evolved substantially over time. It is therefore misleading to call the system simply “Arabic” if that is meant to imply Arabic invention.

One important early external witness is Severus Sebokht, a seventh-century Syriac Christian scholar, who wrote in 662 CE that Indian methods of calculation used nine signs and praised Indian astronomical and computational knowledge. A later report records an Indian scholar bringing a siddhanta-related astronomical work to the court of the Abbasid caliph al-Mansur around 776–777 CE; the work was ordered translated into Arabic. These are unusually valuable pieces of evidence because they document transmission rather than merely similarity.

Brahmagupta's Brāhmasphuṭasiddhānta (628 CE) is especially important because its arithmetic chapters give systematic operations involving positive and negative quantities and zero. Chapter 12 is devoted to arithmetic; chapter 18 treats algebra (kuṭṭaka and related material). Brahmagupta's treatment of division by zero is not modern and must not be misrepresented as such, but his explicit arithmetic rules for zero and negative numbers are historically significant.

The later European transmission is equally clear. Fibonacci's Liber abaci (1202; revised later) explicitly describes the modus Indorum, “method of the Indians,” and says he considered it superior for numerical practice. Its opening chapter is devoted to the “nine figures of the Indians” and the zero. Thus the chain is not “Europe invented Arabic numerals”: an Indian positional system travelled through Arabic-language scholarship and then entered European mathematical practice.

3. Zero: what India contributed and what must not be exaggerated

Zero has a longer and more complicated prehistory than a single invention story. Ancient civilizations had placeholders or concepts of absence, and the Maya independently developed a positional system with zero. What is distinctive in the Indian history is the integration of zero into a decimal place-value system and the formulation of arithmetic rules for zero as a number.

Brahmagupta's Brāhmasphuṭasiddhānta 18.30–18.35 is a primary textual witness for arithmetic involving zero, positive and negative quantities. His rule for zero divided by zero is not correct by modern standards, and division by zero was not solved in the modern sense. The historically defensible statement is therefore: Brahmagupta gave an early systematic arithmetic treatment of zero within Indian algebraic mathematics. It is not defensible to say he possessed the modern theory of division by zero.

4. Algebra: neither “Greek invention” nor simply “Indian invention”

Algebra has multiple historical ancestors. Babylonian mathematicians solved numerical problems equivalent to quadratic equations long before the Common Era; Greek mathematicians developed powerful geometrical methods; Indian mathematicians developed their own arithmetic-algebraic procedures; and Islamic mathematicians synthesized and extended several traditions.

India nevertheless deserves substantial credit for advances in algebraic calculation. Aryabhata's Āryabhaṭīya (499 CE) contains procedures for indeterminate problems, while Brahmagupta's 628 treatise gives rules for equations and operations with signed quantities. Later Indian authors, including Bhāskara II, developed the treatment of indeterminate equations further. Modern historians Victor Katz and Karen Hunger Parshall explicitly treat medieval Indian algebra as a major chapter in the history of algebra, including the work of Aryabhata, Brahmagupta, later Indian mathematicians and the Kerala school.

At the same time, it would be false to claim that al-Khwarizmi merely copied Brahmagupta. His Kitāb al-jabr wa'l-muqābala created a systematic Arabic exposition of equation-solving whose terminology produced the word “algebra.” Islamic mathematics also drew upon Greek, Persian and other traditions and developed substantially beyond its sources. The accurate description is transmission plus transformation, not simple copying.

5. The “Pell equation” and Indian indeterminate equations

The equation now called Pell's equation, generally written x² − Dy² = 1, has a long history. The name “Pell's equation” is not evidence that John Pell discovered it. Brahmagupta gave important methods for solving indeterminate quadratic equations of this family, and later Indian mathematicians developed them further. The modern label is therefore historically convenient but attributionally imperfect.

However, the chapter must not claim that Brahmagupta invented every method now associated with Pell equations. Earlier mathematical cultures had related problems, and European mathematicians later developed their own theory. The defensible claim is that Indian mathematicians made major, early and technically sophisticated contributions to the solution of indeterminate quadratic equations, centuries before the modern naming convention.

6. Trigonometry: “sine” is a particularly clear transmission story

Greek astronomy used the chord of a circle. Indian astronomers developed the half-chord or jya/ardhajyā approach that became the conceptual ancestor of the modern sine function. Aryabhata's Āryabhaṭīya 1.12 provides a table of 24 sine differences at regular angular intervals. Later Indian astronomers refined trigonometric computation.

When Indian astronomical material entered Arabic scholarship, the Sanskrit jya was transmitted through transliteration and linguistic transformation. The Arabic astronomical tradition used sine tables extensively, while also retaining Greek chord traditions. The modern English word “sine” is ultimately connected to this transmission history. A scholarly history therefore should not describe the sine function as simply an Arabic invention or simply a Greek invention. The Greek chord and Indian sine traditions interacted within Islamic astronomy, and later Latin Europe inherited the resulting mathematical vocabulary and techniques.

Al-Khwarizmi's Zīj al-Sindhind is an especially clear case of Indian influence: the surviving evidence indicates that his astronomical tables were based substantially on an Indian astronomical work brought to Baghdad around 770, although Ptolemaic and other sources also influenced the form of the tables. The original Arabic work is lost, so historians reconstruct its contents from later Latin transmission and comparative evidence. That uncertainty must remain visible.

7. Indian astronomy and the Arabic scientific tradition

The Arabic scientific world was not merely a passive “receiver” of Indian science. It was a major creative center in its own right. But some early Arabic astronomical works explicitly belong to a transmission chain from Sanskrit siddhanta astronomy. The term sindhind itself is an Arabic rendering of Sanskrit siddhānta.

The evidence points to Indian influence on astronomical parameters, computational procedures, calendrical calculations, sine tables and planetary calculations. At the same time, Islamic astronomers compared these materials with Ptolemaic astronomy and Persian traditions, corrected inherited parameters, produced new tables and developed sophisticated observational astronomy. The proper historical model is therefore a multilingual scientific network, not a one-way ladder from “India to Arabs to Europe.”

8. Kerala mathematics: an extraordinary case, but not proof of European borrowing

The Kerala school, beginning with Mādhava of Sangamagrāma (c. 14th–15th century), developed infinite series for trigonometric functions. Modern histories rightly recognize that these series anticipate forms associated today with power-series expansions for sine, cosine and arctangent. Cambridge scholarship notes that Mādhava discovered infinite power series roughly two and a half centuries before Newton's rediscovery of comparable series in Europe.

This does not establish that Newton, Gregory, Leibniz or European calculus directly borrowed from Kerala. The documentary chain required to demonstrate such transmission has not been established. The responsible conclusion is stronger than either extreme: the Kerala school independently achieved important results that resemble later European analysis, but there is no demonstrated direct line from Mādhava to the European calculus tradition. Claims that “Newton stole calculus from India” therefore go beyond the evidence.

9. What about Pythagoras, Ptolemy, Euclid and the Greeks?

Some modern popular accounts reverse the older Eurocentric error and create a new error by declaring every Greek theorem Indian. That is not supported by evidence. The Pythagorean theorem has mathematical antecedents in Babylonian and Indian traditions, but the named theorem reflects the Greek mathematical tradition associated with Pythagoras and later Euclidean geometry; the Indian Śulbasūtras contain geometrical statements equivalent to the theorem, but one should not claim without evidence that Pythagoras copied India.

Likewise, Brahmagupta gave a theorem for cyclic quadrilaterals equivalent to what is commonly called Ptolemy's theorem. But Ptolemy's work predates Brahmagupta by centuries. The correct attribution is that Ptolemy has priority for his formulation, while Brahmagupta independently presented a closely related result in the Indian mathematical tradition. “Wrongly attributed to Greeks” would be an inaccurate description here.

Greek influence on Indian astronomy is also documented or seriously argued for particular periods and texts. This is precisely why the portal should not turn civilizational pride into a blanket rule. India both received influences and generated innovations. Historical maturity consists in identifying both.

10. What was transmitted through Arabic scholarship but later remembered mainly as “European”

European mathematics was transformed by Latin translations of Arabic scientific works, including works whose ultimate sources included Indian mathematics. The route was often India → Arabic/Persian scholarly world → al-Andalus/Sicily/other Mediterranean centers → Latin Europe. But each stage could add new mathematics. Fibonacci's Liber abaci is a particularly visible European endpoint for Indian-style numeration and computational methods.

This means that modern terminology can conceal provenance. “Algorithm” derives from the Latinized name of al-Khwarizmi, appropriately recognizing his role in systematic computation and transmission. But the arithmetic procedures described in his Sanatan Dharma-numeral work were based on an Indian place-value system. Similarly, “Arabic numerals” names the route by which Europeans encountered the numerals rather than the civilization in which the mature system originated.

11. A verified attribution ledger

12. The most important historical correction: “borrowed” requires evidence

The word “borrowed” should be reserved for cases where transmission can be supported. There are several grades of evidence. A surviving translation is very strong evidence. A later author explicitly citing an earlier source is also strong. Matching technical parameters combined with a known translation route is strong to moderate evidence. Mere resemblance is weak evidence. A modern Sanskrit interpretation of a modern scientific formula is not evidence of ancient transmission unless the ancient text actually contains the relevant mathematical structure and can be dated independently.

This hierarchy matters because some popular claims are based on retrospective translation. For example, calling a Vedic deity an “electromagnetic field,” or interpreting a cosmological hymn as a modern Big Bang equation, does not establish scientific priority. Conversely, Aryabhata's numerical astronomy, Brahmagupta's algebra, the Indian place-value system and Kerala series do not require such exaggeration: their mathematical content is independently documented and historically significant.

13. What modern science actually “borrowed”

For the physical sciences, the word “borrowed” is often too strong. Modern physics, chemistry, genetics and experimental biology are not demonstrably derived from Vedic scripture. Indian mathematical astronomy did, however, contribute to a wider premodern scientific tradition, and Indian numerical and computational methods entered Islamic scholarship and subsequently Europe. Mathematics is therefore where the clearest direct inheritance claims can be made.

There are also intellectual continuities in logic, grammar, medicine, metallurgy, architecture and astronomy, but these should be described discipline by discipline. A historical knowledge system can influence modern science without containing modern science in completed form. That distinction protects both the integrity of Indian intellectual history and the integrity of modern science.

14. Final conclusion: restore credit without rewriting history

The evidence supports a substantial correction to the simplistic story in which mathematics moves from Greece to the Arab world and then to Europe. India was not merely a recipient. Indian mathematicians developed the decimal place-value system, major treatments of zero and negative numbers, sophisticated algebraic and indeterminate-equation methods, sine tables and a major astronomical tradition. These ideas entered Arabic scholarship through identifiable channels, where they were translated, combined with Greek and Persian materials, criticized, extended and transmitted onward. Europe later received parts of this mathematical inheritance through Arabic and Mediterranean networks.

At the same time, a rigorous civilizational history must resist replacing one simplification with another. Greek mathematics was real and important; Babylonian mathematics was earlier in several areas; Chinese mathematics developed independently and powerfully; Islamic mathematicians were innovators rather than mere translators; and European mathematicians created genuinely new mathematics. The historical achievement of India becomes more—not less—impressive when it is stated precisely.