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Transfer of mathematical knowledge from India to Europe (such as a positional number system with zero) allowed Europeans to develop arithmetic. But was there also a reverse direction (probably via Arab mathematicians) in which knowledge was transferred from Europe to India?

Especially I'm curious when Euclid's Elements (probably the most known ancient mathematical book) was introduced to India.

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    $\begingroup$ "Transfer of knowledge from India to Europe (positional number system with zero) allowed Europeans to develop arithmetic". This is highly doubtful and it has been discussed at length in this forum. Chinese had their own counting system, and so did the Arabs. This idea sounds like a part of recent wave of scam. BBC made a serious report on this topic bbc.com/news/world-asia-india-46778879, where statements such as "The head of a southern Indian university cited an old text as proof that stem cell research was discovered in India thousands of years ago." $\endgroup$ – M. Farooq Apr 28 at 15:24
  • $\begingroup$ @M.Farooq I have read, for example, " Hindu astronomers and mathematicians Aryabhata, born in 476, and Brahmagupta, born in 598, are both popularly believed to have been the first to formally describe the modern decimal place value system and present rules governing the use of the zero symbol." bbc.com/travel/story/20180807-how-india-gave-us-the-zero, but maybe there are some other concepts. However the question was about Euclid's Elements in India. $\endgroup$ – Widawensen Apr 28 at 15:47
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    $\begingroup$ This is a rather recent wave of nationalistic view of mathematical history. Such views when taken up by any culture (and then exaggerated) distort truth and history. Not all algebra was invented by Arab mathematicians, although the name al-jabr lingers today. Algorithm is an named after an Arab mathematician. Can Arabs claim they invented modern algorithms? Problem is with very old stuff, there is no primary information. I would ask where are books by Aryabhata? Where are the translations for Europeans? Even with the Euclid original work, was there a culture of learning Greek in ancient India? $\endgroup$ – M. Farooq Apr 28 at 16:16
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    $\begingroup$ @M.Farooq Yes, there are many connections between different ( also scientific) cultures and to identify all links it is not a simple task. $\endgroup$ – Widawensen Apr 28 at 16:24
  • $\begingroup$ @Widawensen What is this site hrdjfy.blogspot.com/2019/04/… ? $\endgroup$ – Paracosmiste May 13 at 19:52
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According to The Hindu Business Line, quoting the scholar TA Sarasvati Amma:

It was only in the 18th century, nearly 2,000 years after active contact of Indians with the Greeks, that Euclid’s Elements were translated into Sanskrit and even then perhaps the example of the Arabs provided the inspiration.


TL/DR;

Concerning Indian mathematics at about the time Euclid, according to Frits Staal, professor of Philosophy and South/Southeast Asian Studies at UC Berkley:

The ancient Greeks developed logic and a notion of rationality as deduction best exhibited by Euclid’s geometry. These discoveries contributed substantially to the development of Western science. Ancient Indian civilisation was an oral tradition and the oral transmission of the tradition became the first object of scientific inquiry.

Thus arose two human sciences, closely related to each other in their formal structure: the sciences of ritual and language.To begin with, while a number of key contributions were made by Indian mathematicians, they somehow remained in complete darkness about conic sections. These are simply the various dissections of an hourglass (or, a double cone) which are the ellipse, the parabola and the hyperbola. The importance of these curves in the history of science up to the time of Isaac Newton is unparalleled in geometry. Planets were found to move in elliptical orbits, cannonballs and projectiles fell in a parabolic arch under the influence of gravity, and shadows on sundials moved in a hyperbolic path.

The other omissions concern solid geometry, and the existence of only five Platonic solids, namely — tetrahedron, cube, octahedron, icosahedron and the dodecahedron. These five elemental solids were used since the time of Plato in pondering the structure of atoms, crystals and matter in general.

Thus, according to Staal, ancient Indian mathematicians were primarily concerned with a scientific exploration of ritual and mysticism, applying their geometry to things like the construction of elaborate altars.

Further reading: History of Geometry - wikipedia

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    $\begingroup$ Alexander the Great, did visit (or rather conquer) areas which are now in Pakistan. One can find remnants of Greek culture in the Northern part and even very ancient graveyards. The views of Staal and Amma seem to be close to reality rather than web-based myths. $\endgroup$ – M. Farooq Apr 28 at 16:42
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    $\begingroup$ @M.Farooq Indeed. The Silk Road established shortly after the time of Alexander, would have, for over a millenium, introduced Greek culture far beyond what is today called Pakistan. $\endgroup$ – Nick R Apr 28 at 17:14
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Donald Knuth of Stanford University stated that the origins of binomial theorem were originally from India, for example. Like Theorems of euclidean geometry have logical equivalences with symbols in algebra or other topics of mathematics! The orbits of planets, the idea that earth is a sphere and not flat, and exact calculations of periodic events in the solar system, pyhthagorean theorem were known to the Indian mathematicians of lore. Pythagorus new the earth was round but believed it was the center of the universe. Nothing is known to exist of this belief of earth as central in India. To the contrary astronomical calculations were based on planetary motions including earth.

Refer Sulbasutras:

https://link.springer.com/referenceworkentry/10.1007%2F978-1-4020-4425-0_8618

An excerpt: An examination of the earliest known geometry in India, Vedic geometry, involves a study of the Śulbasūtras, conservatively dated as recorded between 800 and 500 BCE, though they contain knowledge from earlier times. Before what is conventionally known as the Vedic period (ca. 1500–500 BCE), there was the Harappan civilization dating back to the beginning of the third millennium BCE. Even a superficial study of the Harappan cities show its builders as extremely capable town planners and engineers requiring fairly sophisticated knowledge of practical geometry. An interesting conjecture has been suggested by a drawing on a seal found from Harappa (ca. 2500 BCE): was there an awareness then that the area of a polygon inscribed in a circle approaches the area of the circle as the number of sides of the polygon keeps increasing? This is the basic idea behind techniques that were developed for the mensuration of the circle in a number of mathematical traditions including Indian.

Deeper article on sulba sutras: https://www.academia.edu/4905149/_Sulba_Sutra_of_Vedic_India_and_Pythagorean_Principle_of_Mathematics_

Jagan Chidella

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    $\begingroup$ This answer appears to be unrelated to the original question which was specifically about Euclid's "Elements". $\endgroup$ – Moishe Kohan Apr 29 at 20:44
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You can read a book called Indian Mathematics by George Gheverghese Joseph. He has provided a comprehensive overview of Indian mathematical history with citations Additionally you can refer to works by Kim Plofker, Clemence Montelle and others

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    $\begingroup$ But does Joseph's book answer the specific question about Euclid's Elements? Same for the other references. $\endgroup$ – Moishe Kohan May 10 at 3:11
  • $\begingroup$ The author (George Gheverghese Joseph) is indian and the book, which seems one sided, only have one review on amazon (from an indian). Note: It'll have probably more reviews after this comment. $\endgroup$ – Paracosmiste May 10 at 21:12
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  1. Vedic geometry (Sulbasutras) existed around 800 and 500BC and Euclid is of the 300BC era.
  2. Equivalences in algebra of geometric principles also existed before Euclid in India.

From 1) and 2) it seems moot to discuss Euclid's elements to influence India or its arrival time in India since it suggests that Euclid influenced geometry in India.

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    $\begingroup$ Welcome on the site! Typically your posts are significantly improved, if you also give some references. $\endgroup$ – peterh May 3 at 18:56
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    $\begingroup$ As a person with an interest in science and history, we should be objective in analysis of historical data i.e. ask questions from ourselves. The key question is where are those books which written well before Euclid? Who is making those claims and if those claims have been authenticated by the rest or not? Otherwise, it becomes a joke like the one reported in BBC that aircrafts were invented centuries ago in India. bbc.com/news/world-asia-india-46778879 $\endgroup$ – M. Farooq May 3 at 22:08

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