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Euclid's Elements
One of the greatest powers of scientific thinking is the ability to uncover truths that are visible only "to the eyes of the mind," as Plato says, and to develop ways and means to handle them. This is what Euclid does [..]. [I]n the Elements we find so many pieces of beautiful mathematics that are easily accessible and can be studied in detail by anybody with a minimal training in mathematics.
[Artmann], p. vi
Elements is considered by many to be the most important historical mathematical source of all time (some even say it is the most influential book in the history of mankind). It is believed to have been written around 300 BCE by the Greek mathematician Euclid, and it consists of thirteen chapters (also called books). These chapters contain the mathematics that was developed in Greece from c. 600 BCE to c. 300 BCE, this being mainly geometry, but with some number theory too.
The image shows some of the first pages of the oldest surviving copy of Euclid's Elements (click to enlarge). It was written in the year 888 CE by Stephen the Clerk in Constantinople. You can look through the book yourself here.
Euclid's Elements is famous for introducing the axiomatic and deductive approach to mathematics - an approach that has now been used with great success for over 2000 years. This fact is what makes Elements one of the most influential books of all time. Although some of the results mentioned in Elements can seem a bit "elementary" now, they have not always been viewed as such, and they only appear "elementary" because we are very much trained to think Euclidian in our modern school education.
An axiom is a mathematical proposition that is considered self-evident, hence true without proof. Axioms are the very building blocks of mathematical theory.
Deductive reasoning uses logical arguments to prove specific conclusions based on generally known premises (e.g. axioms).
During the 1800s, mathematicians realized that Euclidean geometry was not the only possible geometry. On the curved surface of a sphere for example, a consistent non-Euclidean geometry can be developed (great circles, the "parallel" lines on a sphere, intersect, hence violating Eucld's fifth postulate). In the beginning of the 1900s physicists discovered that the geometry of our universe is actually a 4-dimensional curved non-Euclidean geometry, and that it only looks and feels Euclidean over short distances. Hence it can be argued that non-Euclidean geometry is actually more "real" than Euclidean geometry. Euclid would be very interested in hearing that!
References:
- [Artmann], The creation of mathematics, Springer.