> 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!
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.
This is proto-integration, pre-caclulus, etc.