> The great mathematician Aryabhata (476-550), in his masterwork composed when he was only 23, covers square and cube roots, the properties of circles and triangles, algebra, quadratic equations and sines, and contains a decent approximation of the value of pi at 3.1416.
While we praise Aryabhatta man, i would like to shed some lights on Madhava of Sangamagrama c. 1340 - c. 1425 CE from India who less well known
Key Contributions
Infinite Series and Trigonometry
Discovered power series expansions for trigonometric functions:
Madhava's Sine Series: Infinite series representation for the sine function.
Madhava's Cosine Series: Infinite series representation for the cosine function.
Madhava–Gregory Series: Series for the arctangent function, predating James Gregory by over 200 years.
Calculus and Mathematical Analysis
Laid early foundations of calculus through: 200 years before Newton or leibniz
Methods of term-by-term integration and iterative techniques for solving transcendental equations.
Concepts related to the area under curves, similar to integral calculus.
Introduction of convergence tests for infinite series.
Creation of trigonometric tables with accurate sine and cosine values.
The Jesuit missionaries in India played a crucial role in the transmission of advanced Indian mathematical and astronomical knowledge to Europe by learning local languages, collaborating with local scholars, and documenting key works, thereby significantly influencing the development of mathematics in the West.
Which suggests a long oral and|or easily destroyed "document" tradition of teachings being passed down which came to Aryabhata who compiled such things in a manner that survived.
The sophistication of Babylonian mathematics boggles my mind. I am sure a credible science fiction story could be told where the Babylonians are a sophisticated alien race making Earth their home.
However, "the same kind of mathematics" rings dismissive. Trigonometry as we know it, came to its own and flourished in the middle ages in Indian, Arab and Persian civilizations. I am not aware of Babylonic trigonometry.
The story of the name of sin is itself quite interesting. It was half a 'jyay' (meaning chord subtended by an angle) in India. Through transliteration it became 'jayb' to the Arabs. Or the Europeans who were translating the Arabic mathematical literature derived from India, transliterate it as 'jayb', a phonetically similar bonafide Arabic word, that to this day is used to mean, a pocket/wallet/cavity. So pocket becomes sinus in Latin and then it evolves into just 'sin'. I think it was Napier who gave the name that we use.
Cultivation of geometry by Indian scholars go further in the past, to about 8th century BC as recorded in Sulbasutra.
You might be interested to know that combinatorics was also a hot topic among the Indian mathematicians. What we know as Fibonacci goes back far in the past, to Pingala (250 BC +/- 50). Pingala had worked out the binary numeral system and the 'Fibonacci' series.
What I am really keen to know is the mathematics of the Indus valley civilization, they were contemporaries of the Babylonians. Scarce little is known about their mathematics.
> However, "the same kind of mathematics" rings dismissive.
That's on you if you read it that way;
There are other sources, but sticking with the wikipedia article already linked (which references other sources):
* The Babylonian astronomers kept detailed records of the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.
"angular distances measured on spheres" is the domain of trigonometry, how deep is a matter of debate but trigonometry it is.
* They also used a form of Fourier analysis to compute an ephemeris (table of astronomical positions), which was discovered in the 1950s by Otto Neugebauer.
That seems reasonably advanced.
* Tablets kept in the British Museum provide evidence that the Babylonians even went so far as to have a concept of objects in an abstract mathematical space. The tablets date from between 350 and 50 B.C.E., revealing that the Babylonians understood and used geometry even earlier than previously thought. The Babylonians used a method for estimating the area under a curve by drawing a trapezoid underneath, a technique previously believed to have originated in 14th century Europe.
Neugebauer's books are a fantastic resource. I am mostly familiar with their contents.
I am aware of Babylonian's contribution to positional astronomy. If I remember right, Hipparchus, considered the father of trigonometry, was familiar with Babylonian astronomy. Contemporaneous (meaning Hipparchus's times) Greek astronomy was not as accurate in comparison.
However, I have missed the Fourier series that you mention or any record that shows trigonometric manipulation or working out the values of the trigonometric functions. If you have a reference I would love to read, perhaps a specific Neugebauer book. Almagest model too can be considered a crude Fourier decomposition. So I am quite keen to learn about this Fourieresque decomposition that you speak of.
If ancient trigonometry interests you, you should definitely checkout Indian scholars of the middle ages. Neugebauer covers some of that in his books. Glen Van Brummelen is another good resource.
Awareness of angle measurement is not yet trigonometry, that would amount to saying Euclid's Elements- I has trigonometry. Trigonometry, whether planar or spherical becomes trigonometry with the awareness of trigonometric functions, their evaluations and trigonometric identities.
> > However, "the same kind of mathematics" rings dismissive.
> That's on you if you read it that way;
Yes I do. The 'same kind of mathematics' is too broad a brush stroke that can easily sweep away any form of mathematical originality and novelty. As a characterization it is somewhere between 'vapid' and 'not very useful'.
We do agree about how mind-bogglingly sophisticated Babylonian math was. They had Algebra that the Greeks didn't, and as you noted had figured out the technique of area under the curve well before Archimedes ... another person who seems centuries ahead of his times.
I think it’s important to clarify that describing the Babylonian method of estimating the area under a curve using trapezoids as "proto-integration" or "pre-calculus" might be a bit misleading. While their approach demonstrates an impressive grasp of geometry and an early method for approximating areas, it doesn't quite align with the formal development of calculus that emerged centuries later.
Madhava of Sangamagrama, a 14th-century Indian mathematician, made groundbreaking contributions to calculus that were far more advanced. He is known for discovering infinite series expansions for trigonometric functions such as sine, cosine, and arctangent, as well as deriving power series for π. His work included innovative methods for numerically approximating π to remarkable precision. In comparison, Madhava's achievements represent a significant evolution in mathematical thought. While the Babylonians were certainly ahead of their time, their techniques were still relatively basic when juxtaposed with the sophisticated concepts introduced by Madhava. His work laid critical groundwork for the later development of calculus by figures like Newton and Leibniz.
The Babylonians, while advanced for their time, were still operating in a more primitive mathematical framework. So while the Babylonians showed an inkling of ideas that would later blossom into calculus, it's an overstatement to equate their methods directly with calculus. Madhava's work represents a much more mature and developed understanding of these concepts. The Babylonians were pioneers, but Madhava was a revolutionary in comparison. Let's give credit where it's due!
TIL...