Introduction to Euclidean

Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to thesolid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.

For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only where the gravitational field is weak.

India (1500 BC - 200 BC)

Everything that we know about ancient Indian (Vedic) mathematics is contained in:

The Sulbasutras
These are appendices to the Vedas, and give rules for constructing sacrificial altars. To please the gods, an altar's measurements had to conform to very precise formula, and mathematical accuracy was very important. It is not historically clear whether this mathematics was developed by the Indian Vedic culture, or whether it was borrowed from the Babylonians. Like the Babylonians, results in the Sulbasutras are stated in terms of ropes; and "sutra" eventually came to mean a rope for measuring an altar. Ultimately, the Sulbasutras are simply construction manuals for some basic geometric shapes. It is noteworthy, though, that all the Sulbasutras contain a method to square the circle (one of the infamous Greek problems) as well as the converse problem of finding a circle equal in area to a given square.

Baudhayana (800 BC)

Baudhayana was the author of the earliest known Sulbasutra. Although he was a priest interested in constructing altars, and not a mathematician, his Sulbasutra contains geometric constructions for solving linear and quadratic equations, plus approximations of   construct circles.  It also gives, often approximate, geometric area-preserving  transformations from one geometric shape to another. These include  transforming a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle, and finally transforming a circle into a square. Further, he gives the special case of the “Pythagorean theorem” for the diagonal of a square, and also a method to derive “Pythagorian triples”. But he also has a construction (for a square with the same area as a rectangle) that implies knowing the more general “Pythagorian theorem”. Some historians consider the Baudhayana as the discovery of the “Pythagorian theorem”.  However, the Baudhayana descriptions are all empirical methods, with no proofs, and were likely predated by the Babylonians.

Manava (750-690 BC)
contains approximate constructions of circles from rectangles, and squares from circles, which give an approximation of  pi = 25/8 = 3.125.

Apastamba (600-540 BC)
considers the problems of squaring the circle, and of dividing a segment into 7 equal parts. It also gives an accurate approximation of root 2= 577 / 408 = 1.414215686, correct to 5 decimal places.

Katyayana (200-140 BC)
states the general case of the Pythagorean theorem for the diagonal of any rectangle.