The Echoes of the Mind (48-5) Geometry


This knowledge at which geometry aims is of the eternal. Geometry will draw the soul toward truth. ~ Plato

Geometry arose in addressing the practical problems of measuring land areas and granary volumes. Whereas the Babylonians had an algebraic bent, the Egyptians were stronger with geometry. Both the Babylonians and ancient Indians anticipated the geometric work of ancient Greek mathematician and philosopher Pythagoras. The Pythagorean theorem was known in Mesopotamia long before Pythagoras.

Mathematicians in ancient China were also familiar with basic geometry but did not know all the correct modern formulas.

What Pythagoras had that those before him lacked was a cult of true believers: the Pythagoreans. Pythagoreanism was a brew of esoteric and mystical beliefs influenced by mathematics, astronomy, and music. Their lifestyle discipline included purification rites and vegetarianism: all designed to empower their souls.

The Pythagoreans considered the universe ordered by numbers. A person who fully understood the harmony in numerical ratios would become divine and immortal.

Numbers rule the universe. ~ Pythagoras

The Pythagoreans’ interest was not in manipulating numbers – arithmetic – but in understanding the properties of numbers. Geometry was a prominent means.

The famous Pythagorean theorem states that the square of the hypotenuse (c) of a right triangle is equal to the sum of the square of its 2 sides (a, b).

c2 = a2 + b2

In the instance of a right triangle with sides each equal to 1, the square of the hypotenuse is 2.

2 = 12 + 12

√2 is not a rational number, as it cannot be expressed as a ratio of 2 counting numbers.

At the time, the Greeks believed that all numbers were rational. The shocking discovery of irrational numbers by 5th-century-bce Pythagorean Hippasus was something that the secret society wanted to keep secret. Hippasus drowned at sea shortly after divulging this revelation; one of history’s lesser unsolved mysteries.

Not surprisingly given its history, the square root of a number (√x) is called a radical. The symbol √ is the radical sign, and x is the radicand.

Euclid is considered the father of geometry. His book Elements (~300 ce) developed all known mathematics into an integrated whole. Elements served as the primary textbook for teaching math, especially geometry, for nearly 2,000 years: from its original publication into the early 20th century.

In solving problems and providing proofs geometry is intertwined with formal logic, particularly deduction. Euclid developed the basic concepts which continue to be used: axioms, postulates, and theorems. An axiom is a self-evident truth requiring no proof. A postulate is an assumption. A theorem is a proposition proved via axioms and postulates.

Euclid’s 5th postulate – the parallel postulate – posits that parallel lines run to infinity. As an assumption its disparate complexity long troubled mathematicians.

I have had my results for a long time: but I do not yet know how I am to arrive at them. ~ Carl Friedrich Gauss

Relaxing the assumption underlying the parallel postulate resulted in non-Euclidean geometry, by way of hyperbolas and ellipses. This breakthrough was developed in the early 19th century.

Though others did earlier work that they kept to themselves, including Gauss (~1818), the first-published essays on hyperbolic geometry were ~1830 by Hungarian mathematician János Bolyai and Russian mathematician Nikolai Ivanovich.

In a famous 1854 lecture, German mathematician Bernhard Riemann presented higher-dimensional manifolds: a non-Euclidean geometry which in its simplest form is elliptic geometry. Riemannian geometry enabled Einstein’s general relativity theory. Bizarrely, Einstein refused to countenance extra-dimensionality.