The document provides an overview of mathematics in ancient Greece, covering important Greek mathematicians and their contributions. It discusses Thales of Miletus, who made early advances in geometry. It also covers Pythagoras and the Pythagorean school, whose discoveries included relationships between musical intervals and ratios. Euclid is discussed for writing the influential Elements, compiling earlier work. Archimedes made advances in areas and used a method of exhaustion to calculate pi. The document provides context on Plato, Aristotle, and others who helped develop mathematics.

1. MATHEMATICS
IN
ANCIENT GREECE

2. The Greek mathematician,
Pythagoras, used small rocks to
signify numbers while working on
mathematical equations.
Trivia

3.  Thales of Miletus
 Pythagoras of Samos
 Pythagorean School
 Plato and his influence on Mathematics and Science
 Aristotle’s Logic
 Eudoxus of Cnidus
 Euclid and his “Elements”
 Archimedes of Syracuse
 Method of Exhaustion
Discussions about…

4.  Often thought of as the first western scientist.
 Olive Trader based in Miletus, a city-state in
Anatolia
 Contact with Babylonian traders/scholars
probably led to his learning some geometry and
an attempt to organize his discoveries.
 Stated some of the first abstract
propositions.(Angles at the base of an isosceles
triangle are equal, any circle is bisected by it’s
diameter, and a triangle inscribed in a semi-circle
is right angled or a.k.a. “Thales’Theorem” )
THALES OF MILETUS
(624-546 BC)

5.  He is one of the earliest Greek
mathematicians and many historians regard
him as the first true mathematician.
 Very little is known about Thales; including
when he was born d when he died.
 He used geometry to solve problems including
determining the height of ancient Egyptian
pyramids based on the lengths of its shadows.
 The “Theorem of Thales” and the “Intercept
of Theorem”, an important theory in
geometry, are attributed to him.
THALES OF MILETUS
(624-546 BC)

6.  More of mystic/philosopher than a mathematician. Core belief that
number is fundamental to nature. Motto: “All is number”. Emphasized
form, pattern, proportion.
 Pythagoreans essentially practiced a mini-religion (they were
vegetarians, believed in the transmigration of souls, etc.).
 Using such intervals to tune ,musical instruments(in particular pianos) is
still known as the Pythagorean tuning.
-Identical strings whose lengths are in the ratio 2:1 vibrate an
octave part.
-A perfect fifth corresponds to the ratio 3:2
-A perfect fourth corresponds to the ratio 4:3
Pythagoras of samos
(572-497 BC)

7. THEOREMS 21-34 in Book IX of Euclid’s Elements are Pythagorean in
origin:
 Theorem(IX.21). A sum of even numbers is even. (2+2= 4 )
 Theorem(IX.27). Odd less odd is even (5 - 3 = 2)
 The Pythagoreans studied perfect numbers: equal to the sum of their
proper divisors (e.g. 6 = 1+2+3).
 Theorem(IX.36). If 𝟐 𝒏
− 𝟏 is prime then 𝟐 𝒏−𝟏
𝟐 𝒏
− 𝟏 𝒊s perfect
Pythagoras of samos
(572-497 BC)

8. The school of Pythagoras was every bit as much a religion as a
school of mathematics. For example, here are some rules:
 To abstain from beans.
 Not to pick up what has fallen.
 Not to touch a white cock.
 Not to stir the fire with iron.
 Do not look in a mirror beside the light.
Vegetarianism was strictly practiced probably because Pythagoras
preached the transmigration of souls.
Pythagorean
school

9. Life in the Pythagorean society was more –or-less egalitarian.
 The Pythagorean school regarded men and women equally.
 They enjoyed a common way of life.
 Property was communal.
 Even mathematical discoveries were communal and by
association attributed to Pythagoras himself – even from the
grave. Hence, exactly what Pythagoras discovered is difficult to
ascertain.
Pythagorean
school

10.  Plato was also one of the ancient Greece’s most important patrons of
mathematics.
 He was convinced that geometry was the key to unlocking the secrets of
the universe.
 There was sign above the entrance of his Academy “Let no-one ignorant
of geometry enter here”.
 The first 10 years of the 15 year course at the Academy involved the
study of mathematics and science, including plane, solid geometry,
astronomy, and harmonics.
 Plato became known as the “marker of mathematicians”.
 Platonic Solids ( Tetrahedron, octahedron, cube, icosahedron,
dodecahedron)
Plato and his influence on mathematics and science

11. ARISTOTLE’S LOGICAL WORKS
-Contain the earliest formal study of logic that we have. It
is therefore all the more remarkable that together they comprise a
highly developed logical theory, one that was able to command
immense respect for many centuries. In the last century, Aristotle’s
reputation as a logician has undergone two remarkable reversals.
Aristotle’s logic; today, very few would try to maintain that it is
adequate as a basis for understanding science, mathematics, or even
everyday reasoning.
Aristotle’s logic

12.  Eudoxus was arguably the most prolific of the pre- Euclidean
mathematicians.
 He is famous for explaining how to calculate with ratios of lengths .
e.g.
A : B is greater than C : D if there are positive integers m , n, such that
mA > nB and mC ≤ nD.
Eudoxus of cnidus
(390-337BC)

13.  Euclid worked in the library of Alexandria, now on the north coast of
Egypt.
 The library is thought to have been constructed around 320BC as a
means of organizing the knowledge of the world and for the
demonstration of Greek power.
 Although the library was destroyed, the knowledge was preserved.
 Euclid’s Elements was the most influential mathematics text in history.
 it is better thought of as a compilation of the work of earlier
mathematicians rather than an original work.
 Partly due to its fame it was edited and added to over the centuries,
eclipsing and subsuming other important works.
 Theon of Alexandria and his daughter Hypatia famously edited the
elements in an attempt to improve the work and make it easier to follow.
Euclid and his “elements”

14. Euclid and his “elements”
The earliest (almost) complete copies of Euclid date from the
ninth century. The pictured version, written in greek, is at the
Vatican, and does not contain some of Theon’s edits, thus
showing that multiple versions of the text were circulating during
the first millennium AD.

15. EDITIONS AND VARIATIONS THAT HAVE BEEN
PRODUCED

16. BOOK I:
 Consists of 48 theorems, the final two being Pythagoras’ and its
converse.
 Euclid organized this book with the goal of proving this important
result in a thorough manner.
 It begins with a list of definitions, axioms, and postulate.
1. Given any two points, a straight line can be drawn between
them
2. Any line may be indefinitely extended
3. Given a center and a radius, a circle may be drawn
4. All right angles are equal to each other
5.If a straight line crosses two others so that the angles on the
same
side make less than two right angles, then the two lines meet on
that side of the original.
Euclid and his “elements”

17. BOOK II:
 Geometric solutions to problems: constructing a length, even if
incommensurate with the unit
BOOK III:
 Mostly theorems regarding circles(e.g. Thales’ theorem) and tangency
 Probably most of the material came from Hippocrates
BOOK IV:
 Construction of regular 3, 4, 5, 6, 15-sided polygons in and around a
circle
 Also from Hippocrates
BOOK V:
 Ratios/ magnitudes á la Eudoxus
 If a : b = c : d and c : d = e : f then a : b = e : f
Euclid and his “elements”

18. BOOK VI:
 Ratios of magnitudes applied to geometry
 Triangles with equal angles have corresponding sides proportional
BOOK VII:
 Divisibility and the Euclidean algorithm
BOOK VIII:
 Number and progressions, geometric sequences
BOOK IX:
 Numbers: even/odd + perfect numbers
BOOK X:
 Discussion of commensurable and incommensurable lines
 Long and difficult, possibly derived from Theaetetus
Euclid and his “elements”

19. BOOK XI:
 Solid geometry (lines/planes in 3D)
BOOK XII:
 Ratios of areas volumes (Eudoxus)
BOOK XIII:
 Construction of regular polyhedra inside a sphere and their
classification.
Euclid and his “elements”

20.  Archimedes is arguably the greatest of the ancient Greek
mathematicians.
 His place of birth and residence (Syracuse) is in modern-day Sicily,the
large island at the foot of the Italian peninsula.
 He famously helped defend Syracuse against the Romans using
catapults and he died at the hands of the Romans after they captured
the city.
 He travelled to Alexandria in his youth and perhaps met and studied
with eminences such as Eratosthenes.
 He is rare among ancient Greek mathematicians for being highly
practical.
 He is credited with a large number of inventions and technical
innovations, including Archimedes’ screw.
Archimedes of syracuse
(287-212 BC)

21.  Is a technique that the classical Greek mathematicians used to prove
results that would now be dealt with by means of limits.
1. Archimedes ‘ formula for the area of a circle
-the way Archimedes formulated his Proposition about the area of a circle is
that it is equal to the area of a triangle whose height is equal to its radius and
whose base is equal to its circumference: (1/2)(r x 2πr)= 𝝅𝒓 𝟐
2. Convexity
-
The method of exhaustion

22. 3. Archimedes’Axioms
-Axioms for Archimedes are assumptions, not to be proven.
4. The proof Archimedes formula
-The basic idea is to approximate the area of a circle from the
above and below by circumscribing and inscribing regular polygons of
a larger and number of sides.
The method of exhaustion

23. Luke Mastin,(2010);The story of
mathematics;https://www.storyofmathematics.com/
https://www.math.uci.edu/~ndonalds/math184/greece.pdf
References: