According to Al-Biruni, Hindu mathematical texts were written in verse so that it would be easier to memorise them.

-1200-1000 BC: In a Yajur-Veda hymn (7.2.20) the power of ten is expanded until 10^12. The hymn invokes the deities for the sake of a sacrificial rite. After having hailed various features of nature (like the moon and the sun), the hymn outlines the power of ten almost as a metaphor of the immensity of the Universe.

-800-400 BC (that’s when written down, but probably the oral tradition is more ancient): the Sulba-sutra contains a manual of geometry. The sutras do not show a formal axiomatic system of proofs, since they were intended for purely practical purposes (such as building altars, designing a wheel). The sutras contain the Indian version of Pythagoras’ theorem- there is also a section on how to to find the length of the diagonal of a square. The hindus also developed a method on how to approximate the value of the square root of 2.

-It is not clear when the Hindus employed the zero and decimal place value numerals. The earliest evidence that we have of the Hindu civilisation using decimal place value numerals are the following: a treatise on astrology, *Yavana-jātaka,* writing numbers in a positional form (dated in the third century AD). There also several inscriptions dated between the 6th and 11th century AD where years are written in a positional form.

Also a commentary of the philosopher Patanjali dated about the 4th century AD shows the awareness of decimal place value numerals: “Just as a line in the hundreds place [means] a hundred, in the tens place ten, and one in the ones place, so one and the same woman is called mother, daughter, and sister [by different people]”

-In the 6th Century AD the Pañca-siddhāntikā is written- a collection of astronomical treaties. The treatise gives rules on how to compute the sines. According to Kim Plofker “Indian astronomers appear to have been the first to think of replacing the rather clumsy Chord geometry of right triangles inscribed in a semicircle with the simpler Sine geometry of right triangles in a quadrant. ” [1] Mathematics would be used to predict the motion of space bodies- this field developed further after Alexander The Great’s invasion, which led to an encounter between Greek and Hindu astronomy (‘Jyotisha’ in Hindu). Greek sources on zodiacal signs and the horoscope influence Indian astronomy. In addition, according to, Plofker “The Sine tables present in those early texts also have apparent connections with Hellenistic trigonometry of chords.”

The Indian medieval siddhāntas containing epicycles and eccentrics also show familiarity with Greek sources. As of today it is not clear how far Ptolomey’s ‘Almagest’ influenced Indian astronomers such as Aryabatha.

Jyotisha was one of the six Vedangas- the ancillary subjects related to the Veda texts. Jyotisha was needed to establish when certain rituals should be held in accordance with specific astronomical events.

Astronomical calculations in the Rig-veda contain very large integers (billions).

Many astronomers of that time believed the Universe expanded from a single point and then gained the shape of a ‘cosmic egg’ (not to be confused with Georges Lemaître’s theory) in which the flat earth is enclosed. The earth was considered to be supported by a giant animal like a tortoise. The Universe is perpetually created and destroyed in cycles of 4 billion years.

The *Vaiśeṣika Sūtra *(6th-2nd Century BC) shows the theory of the Vaisheshika school on atomism and inductive reasoning.

-Some say that Panini’s *Ashtadhyayi *(written between 6th-4th Century BC) introduction of meta-rules to analyse Sansrkit may have led to the development of symbols in maths (thereby creating the first forms of algebra).

-The Buddhist philosopher Nagarjuna (2nd or 3rd Century BC) develops the ‘tetralemma’, a logical system according to which “Anything is either true, or not true, or both true and not true, or neither” [1].

-There was a lot of mathematical development also in Jainism. The Jainists believed that calculations could be ‘mundane’ (dealing with discrete, finite numbers) or super-mundane (transcendent) when dealing with non-finite entities (considered as ‘infinite’, ‘innumerable’). In the ‘Doors of Inquiry’, a Jainist text, there is a thought experiment on infinitesimals which is strikingly similar to Zeno’s paradoxes:

“[Suppose] there is a particular person, a son of a tailor, who is young, strong…He takes up a big piece of cotton cloth or a silken cloth and quickly tears a cubit of it. Here a questioner asked the teacher thus: Is the time taken by the son of a tailor quickly to tear a cubit … equal to one [time] instant? [The teacher replied:] Such an assertion is not possible. Why? Because one piece of cotton cloth is produced by the integration of the assemblage of groups of numerable numbers of threads…. The upper thread is cut at a time which is different from the time when the lower thread is cut. Therefore this is not one instant….”[1]

The Jainists knew that the square root of 10 approximates the value of pi. By 4th and 5th Century BC they had developed their own way of calculating the area of a circle, and also classified mathematical operations in various types (square, cube, permutations etc)

-The Vedanga Jyotisha (6-7th Century BC) says that the knowledge of calculation is needed to understand astronomy, and that Jyotisa (astronomy) is the science of time (in the sense that by knowing how to make calculations, one will be able to predict when a certain astrological event will occur). This is related to the Vedas- astronomy’s purpose is about deciding the right time for a rite (such as the full moon or half-moon appearance).

Kalpa (the life time of the Universe) was considered to be 4.3 bn years (which is incorrect for the Universe, but almost accurate as our modern measure of the Age of the Earth).

-In the Middle-Ages various astronomical schools had different Siddhanta (doctrine) on the structure of the Universe.

-Brahmagupta (6th-5th Century BC): he wrote works on the motion of planets, eclipses, the use of negative numbers, equations of the second degree, equations with more than one unknown, algorithms (to compute square roots), cyclic quadrilaterals and the rules of the zero. He knew of the solutions to *a*^{2} + *b*^{2} = *c*^{2}

Negative numbers were called ‘debts’,positive numbers were called ‘fortune’/’property’ [a terminology that, in retrospect, would anticipate double-entry book-keeping]. Many formulas provided by Brahmagupta are not backed by proofs (unlike Euclid), hence little is known on how he derived them. In addition, generally, Sanskrit contexts rarely contain diagrams as in the case of Greek mathematicians.

Even though Brahmagupta believed in the static Earth theory, he almost gave an accurate calculation of one year (for him it was 365 days). His works on astronomy (with heavy use of trigonometry) would later influence the Arab astronomers.

Brahmagupta’s lemma and some of his work on interpolation formulas would be re-discovered centuries later.

-Some astronomers believed in the theory of the ‘cosmic wind’ (a force that makes the sky move with its stars and planets), others in the flat earth theory. However Aryabhata (6-5th century AD) claimed that the Earth rotates on its axis daily and that its shape is spherical. The idea that the Earth was spherical almost became universally accepted in India, though the idea that the Earth was in motion was met with scepticism [1]. However Aryabhata’s view of the Universe remained geocentric.

Aryabatha probably believed pi was an irrational numbers since he refers to his approximation as ‘approaching’ a value. His work on mathematics would later be influential in the history of Arab mathematics.

He is responsible for the name of trigonometric identities like jya, kojya, utkrama-jya and otkram jya. ‘Jya’ was translated into Arabic, later from Arabic into Latin (where it was given the ‘sine’ name). He tried to explain eclipses (understanding it was due to the alignment of the sun, moon and Earth in a given moment) rationally rather than religiously

Ahryabatha’s work also includes solutions to indeterminate equations.

He maintained the tradition of using Sanskrit letters to symbolise numbers.

Alike the pre-Socratic philosophers, Aryabatha believed that our planet was composed of four elements (earth, water, air, fire).He also believed the Earth was inside a ‘cage’ made of surrounding constellations. He criticises the idea that the Earth is sustained by giant animals (like serpents or turtles) on the basis that the idea is not rational- if the giant animals, Aryabatha suggests, can sustain themselves in space by the virtue of their essence why can’t that property belong to the Earth as well?

In the Aryabhatiya he shows: the power of ten in the decimal system; The cube and the square are defined; algorithms for extracting the square root and the cube roots; area of a triangle ( “half the base times the height”) and volume of a pyramid; area of a circle and volume of a sphere; pi (3.14…..) as the ratio of the circumference of a circle to its diameter.

Unfortunately much has been lost on the record of astronomical observations by the ancient astronomers in India. According to Plofker, “For the most part, all we have in the way of data in Indian texts are the orbital parameters themselves, celestial coordinates for some twenty or thirty fixed stars, mean longitude positions for epoch dates.”

-The Bakhshali manuscript (dated between 200-400 AD) found in Peshawar, modern Pakistan, shows one of the first proofs of the use of the decimal system. It also shows the use of fractions in a way remarkably similar to ours. A large dot would be used to express the unknown (x) in an equation or zero. The equations found involve practical problems like calculating wages. There is also a formula to calculate square roots. It also contains negative numbers.

-A medieval text called “Epitome of the essence of calculation” is attributed to a Jainist follower. The book begins with a phrase attributed to Mahavira: “the lamp of the knowledge of numbers by whom the whole universe is enlightened.” (a philosophy similar to the Pythagoreans). The author adds that ‘ganita’ (mathematics, ‘gana’ means ‘to count’) can be used in all the fields of human knowledge. Since the Jainists had an atomic view of matter, they also believed there were infinitely small numbers like atoms. The text contains explanations on how operations work and on how negative numbers work.

Bhaskara (7th Century AD), a follower of Aryabhata: one of the firsts employ symbols for numbers (like the circle for zero) rather than letters or words. Before, for example, ‘eyes’ was used to refer to number two, while ‘moon’ for number one because at that time it was believed there was just one moon.

Bhaskara II (12th Century AD) used letter symbols to express the unknown in algebric equations. One of the first to make systematic use of the decimal system in his books. Some of his solutions to indeterminate and Diophantine equations would be discovered in Europe centuries later (in the Renaissance and in the 1600s). For instance the solution to 61x^2 + 1 = y^2 was unknown in Europe until Euler solved it.

He stresses the importance of demonstration when using mathematical models.

Plofker believes studies in Sanskrit grammar encouraged mathematical studying:” mathematical texts were grounded in the basics of the fundamental Sanskrit disciplines such as grammar and logic…. We can infer, at least generally and tentatively, that mathematics remained mostly a technical speciality for members of those jātis professionally concerned in it, on the periphery of the core disciplines of Sanskrit learning.”.

In the late 15th Century, in Kerala, the Madhava school of mathematics and astronomy produced a remarkable list of achievements. Madhava introduced concepts similar to classical analysis like the infinitesimal, the infinite series and the limit. According to Plofker, *“The crest-jewel of the Kerala school is generally considered to be the infinite series for trigonometric quantities discovered by its founder, Mādhava”*. Keral was a hub for trade with arab merchants, therefore it is highly likely there were exchange of ideas between the Madhava school and Islamic knowledge of astronomy. There seems to be

In the 7th Century, the Syrian Monophysite bishop Severus Sebokht wrote how more advanced Indian mathematics was compared to Greek mathematics due to the advantage of using the decimal system (in reality Sebokht thought the Indians only used ‘nine signs’, probably omitting the zero).

The Islamic mathematicians adopted the Hindu decimal system. The zero was called ‘sifr’ (‘void’, ’emptiness’ in Arabic).

It is worth citing the passage from Plofker explaining how, due to a misunderstanding, Westerners now use the ‘sine’ expression in trigonometry:

*“The standard Arabic word for Sine, jayb (literally “cavity,” “pocket”), is apparently a misinterpretation of an earlier word jība using the same consonants j-y-b. This term jība, being meaningless in Arabic, was read as the more familiar word jayb. (It is the literal sense of jayb as “pocket” or “fold” that was later translated into Latin as “sinus,” whence our “sine.”)But where did the mysterious word jība itself come from? Evidently from transliterating the Sanskrit word jīvā, or “bowstring,” a synonym of the standard Sanskrit term jyā for the Sine. As the eleventh-century Muslim scientist al-Bīrūnī …. explained:People [who know] this [trigonometry] call its scientific books zījes, from al-zīq which in Persian is zih, i.e., “chord.” And they call the half-Chords juyūb [plural of jayb], for the name of the Chord in the Indian [language] was jībā, and [the name] of its half jībārd. But since the Indians use only the half-Chords, they applied the name of the full [Chord] to the half, for ease of expression.”*

The Islamic mathematicians also adopted the Hindu trigonometric functions, but maintained Greek astronomy (mainly Ptolemaic).

Al-Biruni’s book on India “Verifying All That the Indians Recount, the Reasonable and the Unreasonable”

Bhaskara’s II texts on astronomy became the standard reference for the next centuries in India. Some claim that Bahaskara’s canonical status led to stagnation in Indian astronomy.

SOURCES:

[1] ‘Mathematics in India’ by Kim Plofker- Princeton University Press 2008