Otdfbt

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.

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

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

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.