In-Depth Notes on the Alexandria School and Euclidean Mathematics

Founded by Alexander the Great circa 331 B.C. in Egypt.
Known for his conquests across Egypt, the Eastern Mediterranean, the Middle East, and parts of Asia, spreading Greek culture (Hellenistic Age).
Alexandria became a center of Hellenistic culture for nearly a thousand years, surpassing Athens.
The Greek Ptolemaic dynasty ruled Alexandria, which was later conquered by Arabs in A.D. 641.
The city's population decreased to about 4,000 post-Arab seizure, largely due to Cairo's establishment and new shipping routes.

The Ptolemaic dynasty prioritised intellectual life, establishing a Museum (a seat of the Muses) for renowned scholars.
Notable scholars had to receive invitations from the Ptolemies and were provided luxurious living conditions and free accommodations.
Facilities included extensive libraries, with reports of nearly 500,000 scrolls during Caesar’s time and collections reputed to be Aristotle's.

Alexandria’s School was considered superior to its main competitor, Pergamon (in Western Asia Minor).
Notable scholars: Euclid, Archimedes, Eratosthenes, Apollonius, Pappus, Claudius Ptolemy, and Diophantus.

Born circa 300 B.C. during the Hellenistic Age; attended the Alexandria School.
Authored "Euclid’s Elements of Geometry"—a 13-volume work containing fundamental theorems.
Elements was first printed in 1482; was among the most published works, second only to the Bible.

Euclid established the use of mathematical proofs, questioning established techniques to confirm accuracy.
Introduced proofs, axioms, postulates, and theorems, legitimizing mathematical knowledge.

Division Theorem: For integers $a$ and $b$ , there exist unique integers $q$ and $r$ such that $a = qb + r$ , with 0 \leq r < b.
Fundamental Theorem of Arithmetic: "Every positive integer n > 1 is either prime or can be expressed as a product of primes; this representation is unique apart from the order of factors."

Born circa 276–194 AD, in Cyrene.
Chief librarian at the Alexandria Museum for 40 years; known for significant contributions in geography.
Authored "Geographica", detailing geographical studies and proposing the spherical nature of Earth.

Alexandrian astronomer who authored "Syntaxis Mathematica", a significant astronomy blueprint used for 14 centuries.
Promoted the geocentric model of the solar system, which posited Earth at its center.
Created a geographical dictionary, "Geographica Syntaxis".

Mathematician and inventor, known for work in mathematics and mechanics.
Famous invention: Archimedes’ screw for raising water; utilized for irrigation and mines.
Authored work on sphere and cylinder, providing propositions related to geometric measurements.

The surface area of a sphere is four times that of a great circle.
A cylinder circumscribing a sphere with height equal to the sphere's diameter has a volume three halves that of the sphere.

Entering the 3rd century B.C., the era of great Greek mathematics came to a close due to political conflicts.
The rise of Christianity shifted focus from mathematics to theology, contributing to the decline of Alexandrian scholarship.
Prominent figures like Hypatia emerged during this decline, notable for her contributions in mathematics and philosophy.

Hypatia (c. 370-415 AD): 1st known female mathematician, authored commentaries on Diophantus's works.

Influential in algebra, having authored "Arithmetica" containing Diophantine equations.