Why did Euclid prove his first four postulates? Why didn't he prove his fifth postulate (parallelism)?

Euclid's first four postulates are often referred to as the "common notions," as they are basic assumptions that are generally accepted as self-evident. These postulates deal with concepts such as points, lines, angles, and basic geometrical relationships, and they serve as the foundation for Euclidean geometry.

In contrast, the fifth postulate, also known as the "parallel postulate," deals specifically with the concept of parallel lines. It states that if a line intersects two other lines and the sum of the interior angles on one side is less than two right angles, then the two other lines, if extended indefinitely, will eventually intersect on that side. This postulate is more complex and less immediately self-evident than the first four, and it was also not universally accepted by all mathematicians and philosophers of Euclid's time.

Euclid did not provide a proof for the fifth postulate, but rather used it as an assumption to build upon in the development of Euclidean geometry. Later mathematicians, including John Playfair and Nikolai Lobachevsky, would explore alternative postulates and non-Euclidean geometries that did not rely on the fifth postulate.

Overall, Euclid's decision to accept the first four postulates as self-evident and use the fifth postulate as an assumption was likely influenced by the mathematical and philosophical thinking of his time, as well as his own approach to mathematical reasoning and logic.