Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later theorems. His first postulates were

1. To draw a straight line from any point to any other.

2. To produce a finite straight line continuously in a straight line.

3. To describe a circle with any center and any distance.

4. That all right angles are equal to each other.

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

It has been stated than Euclid was not satisfied with the last postulate, and even tried to derive it from the other four. His first 28 theorems in The Elements are not proven with the last postulate—it is not used until needed. Many other authors have written of Euclid’s and others’ attempts at deducing the fifth postulate, as Proclus stated in his commentary. He mentions when another author, Ptolemy, who produces a false "proof" of the fifth postulate. He gives an alternative postulate that is the same in principle to Euclid’s fifth postulate.

Playfair’s Axiom: Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.

This postulate became Playfair’s after an Englishman named John Playfair wrote commentaries on Euclid in 1795. Another "proof" was written in 1663 by Wallis who thought he had proved the fifth postulate. He actually restated equivalently the postulate as

To each triangle, there exists a similar triangle of arbitrary magnitude.

Several other attempts to prove or disprove the fifth postulate have followed, notably that of Girolamo Saccheri. He assumed the fifth postulate to be false and attempted to derive a contradiction. Another mathematician, Gauss, started working on the postulate as early as 1792 while 15 years old. In 1813, after making little progress, he wrote

In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics…

In 1817, Gauss stated that the fifth postulate was independent from the other postulates, and therefore needed no proof from the others, and begun to work on a different geometry in which multiple lines can be parallel to another line through a given point. In fact, the fifth postulate has been called "the one sentence in the history of science that has given rise to more publication than any other."

Euclid deduced many theorems and other conjectures from his five original postulates. Many porisms, now called corollaries, and many lemmas, or something assumed in the proof of a theorem, were used.  Furthermore, many propositions in the later books were based on previous theorems proven true.

Book - Previous books or propositions upon which it depends

I - (independent)

II - I

III - I; II.5,6

IV - I; II.11; III

V - (independent)

VI - I; III.27,31; V

VII - (independent)

VIII - V.Def.; VII

IX - II.3,4; VII; VIII

X - I.44,47; II; III.31; V; VI; VII.4,11,26; IX.1,24,26

XI - I; III.31; IV.1; V; VI

XII - I; III; IV.6,7; V; VI; X.1; XI

XIII - I; II.4; III; IV; V; VI; X; XI

     This leads to the conclusion that if one of the early theorems were subsequently proven false, many of the latter theorems may not be true either.

Many things now considered essential to geometry are omitted from The Elements, such as the formulas for the areas of figures. Euclid left calculations totally out of The Elements, as well as principles that can not be expressed with straight-edge and compass alone. In Book I of The Elements Euclid defines many of the terms used commonly in modern geometry, such as point, line, and figure.

Many things used in Euclid’s proofs are not proven. However, they are not postulates, either. They were called common notions by Euclid, and today we call them axioms. Some of Euclid’s axioms are

1. Things that are equal to the same thing are also equal to one another.

2. If equals be added to equals, the wholes are equal.

3. If equals be subtracted from equals, the remainders are equal.

4. Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

This would suggest than many mathematicians believed that the postulates were not enough to solve all problems, and have asserted these what he considered "common" sense. Some other axioms added by later authors are

6. Two lines do not enclose a space.

7. If equals be added to unequals, the wholes are unequal.

8. If equals be subtracted from unequals, the remainders are unequal.

9. Doubles of the same thing are equal to one another.

10. Halves of the same thing are equal to one another.

However, all of these axioms can be derived from Euclid’s axioms and theorems and therefore are obsolete. Also note that axiom six should in fact be a postulate because it deals with geometric figures. Other propositions proved in The Elements include S-A-S, S-S-S, A-S-A, A-A-S, that the sum of a triangle’s angles equals that of two right angles, and the Pythagorean Theorem. The proof of the latter is believed to be originally from Euclid and is called the Bride’s Chair.