Euclid

Euclid was an ancient Greek mathematician born in 365 BC in Alexandria, Egypt. In ancient Greece, mathematics and philosophy were not as distinct as they are today. Mathematical relationships were considered fundamental to understanding the nature of reality and held in high regard by Pythagoras and Plato. In Athens, Euclid studied at Plato’s Academy, where it was written on top of the entrance “Let no one ignorant of geometry enter here.” Euclid famously demonstrated the basic principles of geometry from just a few initial axioms in his work the Elements, which served as the essential textbook on mathematics until the 19th century.

Mathematics and geometry had been practiced in ancient Greece, Egypt and Babylonia. In the Elements, Euclid was the first to show that all geometric theorems could be logically deduced from a simple set of initial axioms (what is known as an axiomatic system). He showed how the principles of geometry were true both for plane geometry (two dimensions), and solid geometry (three dimensions). His geometrical system, which came to be known as Euclidean geometry, was seen as explaining the nature of the universe. It served as the basis for all mathematics and related philosophy until in the 19th century it was recognized that in the real world one of his axioms did not hold, which led to the creation of non-Euclidean geometry.

Euclid’s system starts from five simple axioms. The first axiom is that between any two points a straight line can be drawn. The second axiom is that a line can be extended infinitely. The third axiom says that any circle can be created if it has a known center and distance (i.e. radius). The fourth axiom is that all right angles are equal. The fifth axiom explains that only one line can be drawn through a point parallel to a given line. This fifth axiom was always considered doubtful, because it is less obvious than the other four, and numerous attempts were made to show that it followed from the earlier four axioms. Euclid also assumed some basic notions, for example, that two things which are equal to the same thing will be equal to each other and that the whole is always greater than one of its parts.

Euclid’s system is based on classical logic, as presented by Aristotle, which states that something cannot be and not be at the same time (also known as the law of noncontradiction). Euclid used indirect proofs (also known as proof by contradiction) for his theorems by showing how if one were false it would imply a contradiction, and therefore it must be true. He also employed the method of exhaustion to determine the area of a shape by inscribing within it a sequence of smaller shapes that together add up to the larger shape. This is seen as a precursor to calculus. Much of what Euclid wrote in the Elements is now considered algebra and number theory.

In the Elements, Euclid provides formal proofs for many well known theorems including the Pythagorean theorem, Thales theorem, and what he refers to as the Bridge of Asses. The Pythagorean theorem (a2 + b2 = c2) states that the area of a square whose side is the hypotenuse (i.e. the hypotenuse squared) is equal to the sum of the areas of two squares made from it’s sides. Thales theorem explains that any triangle inscribed in a circle using the diameter as its hypotenuse will be a right triangle. The Bridge of Asses, Euclid’s first real test of the skill of his readers, explains how the angles at the base of an isosceles triangle will be equal, as will the angles created if the triangles two equal sides are extended further. He also showed that two triangles are equal if both their sides and angles are equal (known as the congruency of triangles) and that the sum of a triangle’s angles is always 180 degrees. Euclid also used his geometry to explain how things can appear at a greater angle to appear larger, while at a lesser angle to appear smaller (i.e. geometric optics).

Euclidean geometry was seen as the only possible type of geometry until the 19th century, when it was discovered that Euclid’s fifth axiom only holds in Euclidean space (i.e. space that isn’t curved). Euclid’s fifth axiom stated that only one line could be drawn through a point that is parallel to a given line, since any other line (i.e. another straight line with an angle) would intersect with the given line and therefore not be parallel. Non-Euclidean geometries (such as elliptical and hyperbolic geometry, where lines can curve downwards or upwards) realized that on a curved surface, such as a globe, more than one line could meet at such a point, as is the case at the top and bottom of the earth. Einstein‘s theory of relativity recognized that light is bent by gravity, and therefore that the universe is actually non-Euclidean in nature.

Euclidean geometry has been tremendously influential on developments in mathematics and philosophy ever since. The certainty provided by Euclid’s system was the basis upon which modern philosophers such as Descartes, Newton, and Kant would base their beliefs. In the 20th century, Bertrand Russell recognized the profound effects that the new scientific discoveries were having on philosophers’ and mathematician’s beliefs in certainty. Nevertheless, Euclid’s system of geometry continues to be used in practical ways ranging from architecture to origami.