Euclid the Man

Euclid, the great Geometer and author of the Elements was born c. 325 B.C., and died c. 265 B.C. in Alexandria, Egypt. Proclus, the last major Greek philosopher wrote this¹ of Euclid:

Not much younger than these [pupils of Plato] is Euclid, who put together the "Elements", arranging in order many of Eudoxus's theorems, perfecting many of Theaetetus's, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato's circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole "Elements" the construction of the so-called Platonic figures.

Euclid wrote the Elements of Geometry 24 centuries ago in Ancient Greek. In thirteen books, he lays out the definitions, postulates, common notions, and propositions required to understand his system called Geometry.

Definitions, Postulates, and Common Notions

Definitions

A definition in the Elements is a word, followed by a meaning Euclid assigns for his system. For example from Definitions from book one: "A point is that which has no part." The word Euclid used which we translate point means sign or mark, he gave this Greek word a new definition for use in his system.

Postulates

Euclid's postulates are interesting. These, along with common notions only appear in book one. The greek word here translated postulate means to beg. Euclid begs us to accept these into his system even though he cannot prove them through propositions. The most interesting proposition is number five:

That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the agles less than the two right angles.

Common Notions

Common notions are ideas Euclid assumes his students "bring to the table," such as: "Things which are equal to the same things are also equal to each other."

Propositions

Propositions are the meat of Euclid's Elements. In the thirteen books he lays down propositions, and sets out to prove that they are true. There are construction propositions which end with "being what it was required to do", and proof propositions which end "being what it was required to prove." Also, the propositions are proved in different ways such as reductio ad absurdum or (i need help here, can't remember the term).

Constructions

An example of a construction is proposition one book one: "On a given finite line to construct an equilateral triangle." Euclid, using steps which refer only back to definitions, postulates, and common notions already set out, then constructs an equilateral triagle on a given finite line.

Proofs

The first three propositions are all constructions, but proposition four is a proof:

If triangles have the two sides equal to two sides respectively, and have the angles contained by the equal straight lines equal, they will also have the base equal to the base, the triangle will be equal to the triangle, and the remaining angles will be equal to the rmaining angles respectively, namely those which the equal sides subtend.

Types of Proofs

reductio ad absurdum

whatever the other kind is called