See https://infoproc.blogspot.com/2020/12/the-celestial-empire.html#disqus_thread and https://infoproc.blogspot.com/2020/12/hot-and-cold-wars-in-21st-century.html#disqus_thread for originals. See https://gmachine1729.wpcomstaging.com/disqus%e8%af%84%e8%ae%ba%e5%91%98/gmachine1729/ for all my Disqus comments up to Dec 19, 2020, and use www.disqussearch.com if you want to search them.
Something I didn’t like about reading popular books on history of math or science in high school was the often gross exaggeration and cult of genius in them. For instance, I remember some book mentioned that Euler effortlessly calculated 6th powers of all of the numbers up to 100 in his head. Isaac Newton was portrayed as a genius saint too. This doesn’t help at all for someone trying to actually learn math or science. In high school, the callow and ignorant me could only view those great men of science with awe.
Now, I’m well aware for instance that Newton’s figuring out series for sine in 1670s, 300+ years later than Madhava in India was much influenced by the popularization of the decimal system by Simon Stevin. Simon Stevin’s system for decimal notation was still rather redundant by today’s standards.
Simon Stevin is also credited for being the first in the West to discover that equal temperament in music is connected to 12th root of 2. There is the qualification “in the West” because a bit before Stevin, Chinese by the name of Zhu Zaiyu had figured out the same thing, and not only that more accurate and thorough in his calculation and exposition. Again, this has to do with decimal system’s having originated in China over a millennia ago.
Worth noting is that Europe around 1600 had 1/3 to 1/2 the Chinese population and printing press on an alphabet with mass manufactured types. The latter was an enormous advantage for West vis-a-vis the outside world, especially in development of theoretical knowledge, wherein West visibly advanced much faster than it did in practical matters, with the scientific revolution’s preceding the industrial revolution, when it could have been the other way round (take as an example that the industrial revolution does not require knowledge of analytic solution of cubic and quartic, which West figured out in 16th century). Taking this into consideration, the rapid advances in the West especially in theoretical science was more or less expected, and the catalyst was the invention and proliferation of the printing press, possible because of alphabetic writing and advances in metallurgy, the latter of which was at least indirectly influenced by China. So these historical phenomena are much more explainable than people think.
Those white Americans over paranoid about Chinese espionage and prone to baseless accusation are idiots. They do not realize that transmission of knowledge and expertise takes time, no matter how smart you are. It was like a high school math teacher who sort of regarded certain results in math as “magic” when by then, I had already figured out how one would reasonably derive them under minimal prior knowledge and hypotheses, though I had never heard of Occam’s razor.
The Chinese and Japanese in 19th century after opening up under duress could only view West with awe. But gradually, a better understanding of all that science and technology was attained, after which the Japanese even began to treat Westerners with some contempt. Hantaro Nagaoka published his Saturnian model of the atom with positive charge concentrated at the center, refuting Thomson’s plum putting model, but Rutherford did not want to take this Japanese guy seriously, despite seeing his paper. Japanese militarism was much triggered by West’s interfering in its treaties with China in 1895 and Russia in 1905 and also the Anglo world’s rejection of the racial equality clause that Japan proposed to the League of Nations after WWI. Japan in the Southeast Asia part of WWII and China in Korean War both exposed to some degree the incompetence of the US and British militaries.
Back in those days, it was so much easier to make a big scientific discovery. Even two Chinese as physics PhD students in America in 1920s did work in scattering experimental physics that was at or close to Nobel caliber that entered the history books. Both returned to China despite that their advisors wanted them to stay, and were instrumental towards developing the later school of physics in China.
Given that math and physics at the fundamental level is pretty much a solved problem now, certainly studying the history deeply would be more worthwhile. I am certainly eager for connections and potential collaborators in this regard.
Another rebuttal of Douglas Knight.
Maybe classical Greece wasn’t superhuman, but it took 1500 years or more for the rest of the world to match 200 years of Hellenistic math and science, even with Hellenistic texts to read.
Note that you used the word “match” as opposed to “exceed”. It’s obvious to anyone who knows math that the ability to calculate trigonometric functions by infinite series vastly exceeded what the Greeks could do.
I only wonder how Madhava came up with it at that time. The only way I can think of devoid of calculus that would have been natural and feasible at that time was to estimate the coefficients of an polynomial approximation up to a certain order by plugging in enough known values (deducible via , , double angle and half angle formulas) and then solving a linear systems of equations which would for sin would yield values close zero for even terms and for odd terms, etc. Or seeing that https://bhavyabharatam.blogspot.com/2020/06/madhava-sine-and-cosine-series.html is more of a recipe than a derivation, he might well have been less systematic, guessing the terms one by one with known values, such as with , then on , and so on, to the point in which he recognized the factorial pattern. Again, much of the reason why the Indians could this when the Greeks and medieval West could not was because Indians already knew decimal arithmetic by 800 AD.
A Mongol in early 18th century China did a somewhat complicated geometric derivation of it based on chord length in terms of arc length after seeing those series from a French Jesuit. The way he did so was suggestive of his already knowing the series.