By the 3rd century BCE, mathematical breakthroughs were being made in Alexandria, Egypt which had become a major center of learning under the Ptolemies. Many influential mathematicians studied and taught there, including Euclid, Archimedes, Eratosthenes, and Diophantus. They made important advances in areas like geometry, astronomy, and early algebra. Meanwhile, other centers also contributed, such as Perga in modern-day Turkey where Apollonius did seminal work in the geometry of conic sections.

1. By the 3rd Century BCE, in the wake of the conquests of
Alexander the Great, mathematical breakthroughs
were also beginning to be made on the edges of the
Greek Hellenistic empire.

2. • In particular, Alexandria in Egypt became a great centre of
learning under the beneficent rule of the Ptolemies, and its
famous Library soon gained a reputation to rival that of the
Athenian Academy. The patrons of the Library were arguably
the first professional scientists, paid for their devotion to
research.

3. • Among the best known and most influential mathematicians who
studied and taught at Alexandria were Euclid, Archimedes,
Eratosthenes, Heron, Menelaus and Diophantus.

4. • During the late 4th and early 3rd Century BCE, Euclid was the
great chronicler of the mathematics of the time, and one of the
most influential teachers in history. He virtually invented classical
(Euclidean) geometry as we know it.

5. • Archimedes spent most of his life in Syracuse, Sicily, but also
studied for a while in Alexandria. He is perhaps best known as
an engineer and inventor but, in the light of recent discoveries,
he is now considered of one of the greatest pure
mathematicians of all time.

6. • Eratosthenes of Alexandria was a near contemporary
of Archimedes in the 3rd Century BCE. A mathematician,
astronomer and geographer, he devised the first system of
latitude and longitude, and calculated the circumference of the
earth to a remarkable degree of accuracy. As a
mathematician, his greatest legacy is the “Sieve of
Eratosthenes” algorithm for identifying prime numbers.

7. • In the 1st century BCE, Heron (or Hero) was another
great Alexandrian inventor, best known in
mathematical circles for Heronian triangles (triangles
with integer sides and integer area), Heron’s Formula
for finding the area of a triangle from its side
lengths, and Heron’s Method for iteratively computing
a square root. He was also the first mathematician to
confront at least the idea of √-1 (although he had no
idea how to treat it, something which had to wait
for Tartaglia and Cardano in the 16th Century).

8. • Menelaus of Alexandria, who lived in the 1st - 2nd Century CE,
was the first to recognize geodesics on a curved surface as the
natural analogues of straight lines on a flat plane. His book
“Sphaerica” dealt with the geometry of the sphere and its
application in astronomical measurements and calculations, and
introduced the concept of spherical triangle (a figure formed of
three great circle arcs, which he named "trilaterals“.

9. • In the 3rd Century CE, Diophantus of Alexandria was the first to
recognize fractions as numbers, and is considered an early
innovator in the field of what would later become known as
algebra. He applied himself to some quite complex algebraic
problems, including what is now known as Diophantine Analysis,
which deals with finding integer solutions to kinds of problems
that lead to equations in several unknowns (Diophantine
equations).

10. • But Alexandria was not the only centre of
learning in the Hellenistic Greek empire.
Mention should also be made of
Apollonius of Perga (a city in modern-day
southern Turkey) whose late 3rd Century
BCE work on geometry (and, in particular,
on conics and conic sections) was very
influential on later European
mathematicians.

11. • It was Apollonius who gave the ellipse, the parabola, and the
hyperbola the names by which we know them, and showed how
the Hipparchus, who was also from Hellenistic Anatolia and who
live in the 2nd Century BCE, was perhaps the greatest of all
ancient astronomersey could be derived from different sections
through a cone.

12. • He revived the use of arithmetic techniques first developed by
the Chaldeans and Babylonians, and is usually credited with the
beginnings of trigonometry. He calculated (with remarkable
accuracy for the time) the distance of the moon from the earth
by measuring the different parts of the moon visible at
different locations and calculating the distance using the
properties of triangles.

13. • He went on to create the first table of chords (side lengths
corresponding to different angles of a triangle). By the time of
the great Alexandrian astronomer Ptolemy in the 2nd Century
CE, however, Greek mastery of numerical procedures had
progressed to the point where Ptolemy was able to include in
his “Almagest” a table of trigonometric chords in a circle for
steps of ¼° which (although expressed sexagesimally in the
Babylonian style) is accurate to about five decimal places.

14. • By the middle of the 1st Century BCE and thereafter, however,
the Romans had tightened their grip on the old Greek empire.
The Romans had no use for pure mathematics, only for its
practical applications, and the Christian regime that followed it
even less so.

15. The final blow to the Hellenistic mathematical heritage at
Alexandria might be seen in the figure of Hypatia, the first
recorded female mathematician, and a renowned teacher who
had written some respected commentaries on Diophantus and
Apollonius. She was dragged to her death by a Christian mob in
415 CE.

16. • A Prime Number is:
• a whole number that cannot be made by multiplying other
whole numbers
• (if we can make it by multiplying other whole numbers it is
a Composite Number)
• And 1 is not prime and also not composite.
• Here we see it in action:
• 2 is Prime, 3 is Prime, 4 is Composite (=2×2), 5 is Prime, and so
on...

17. • Composite Number
• A whole number that can be made by multiplying other whole
numbers.
• Example: 6 can be made by 2 × 3 so is a composite number.
• But 7 can not be made by multiplying other whole numbers
(1×7 would work, but we said to use other whole numbers) so
is not a composite number, it is a prime number.

18. • Under Greco-Roman rule, Egypt hosted
several Greek settlements, mostly concentrated in Alexandria,
but also in a few other cities, where Greek settlers lived
alongside some seven to ten million native Egyptians.[2] Faiyum's
earliest Greek inhabitants were soldier-veterans
and cleruchs (elite military officials) who were settled by the
Ptolemaic kings on reclaimed lands.

19. • Native Egyptians also came to settle in Faiyum from all over the
country, notably the Nile Delta, Upper
Egypt, Oxyrhynchus and Memphis, to undertake the labor
involved in the land reclamation process, as attested by
personal names, local cults and recovered papyri.[5] It is
estimated that as much as 30 percent of the population of
Faiyum was Greek during the Ptolemaic period, with the rest
being native Egyptians.

20. • Native Egyptians also came to settle in Faiyum from all over the
country, notably the Nile Delta, Upper
Egypt, Oxyrhynchus and Memphis, to undertake the labor
involved in the land reclamation process, as attested by
personal names, local cults and recovered papyri.[5] It is
estimated that as much as 30 percent of the population of
Faiyum was Greek during the Ptolemaic period, with the rest
being native Egyptians.