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Prof. Essam M. Wahba

The Contribution of Alexandria to

Mathematics and Engineering

Mechanical Engineering Department

Faculty of Engineering

Alexandria University, Egypt

The Ancient City of Alexandria

•The city of Alexandria was built in 331 B.C.

•It was named after Alexander the Great, King of Macedonia.

•Alexander the Great named more than twenty cities after his

name, the most famous of them all is Alexandria of Egypt.

•Alexander the Great envisioned two missions for this great city:

•The spread of Hellenic influence over the world.

•The return of the ancient land of Egypt to its former greatness and

glory.

•As such, Alexandria became the prime center of trade of the

world and the commercial junction point of Asia, Africa, and

Europe.

Alexander The Great

The Alexandrian Museum

•Upon the death of Alexander the Great, Egypt fell under the rule of

Ptolemy.

•The dynasty of Ptolemy was to rule the land for over 250 years.

•He envisaged the city as acenter of Greek culture, not merely as a

trading post.

•From early times in Greek history, philosophical schools were

developed, which were in reality communities of scholars. Their

housings were called museums.

•Ptolemy conceived the ambition to establish such a museum at

Alexandria.

Ptolemy

The Alexandrian Museum

•With large revenues at the hand of Ptolemy, he made the

museum attractive by the then novel means of granting

salary stipends, as well as board and residence, to

prominent scholars of his choice.

•This plan was an immediate success, and at about the

year 300 B.C. the Alexandrian museum became a reality.

•It included in its membership intellectuals of all sorts-

poets, philosophers, grammarians, mathematicians,

astronomers, geographers, physicians, historians,

artists, and many others.

•The best surviving description of the museum is by the

Greek geographer and historian Strabo, who mentions

that it was a large complex of buildings and gardens

with richly decorated lecture and banquet halls linked

by colonnaded walks.

Alexandria Museum

The museum would have been called a modern university today, and

with a great library. This picture shows one of a number of identical

side-by-side lecture rooms recently excavated in Alexandria.

The Library of Alexandria

•Ptolemy's constructiveness, which was manifested in his incorporation of the

Alexandrian Museum, was matched again by his founding of a public library.

•There had, of course, been libraries before that. The Phoenicians had had

them, as well as the Egyptians and the Greeks. But those had almost

invariably been archives of records or repositories for valued writings.

•The Library of Alexandria,by contrast, was conceived in the modern sense,

and was from the beginning intended to share the function of preservation of

writings with that of public service.

•It was intended to make the world's literature accessible to everyone.

•The Library was established almost simultaneously with the museum and

adjacent to it, and in time was to become monumentally important.

The Library of Alexandria

Alexandrian Scholars

•Euclid (330 BC –260 BC)

•Archimedes (287 BC –212 BC)

•Eratosthenes (276 BC –194 BC)

•Apollonius (262 BC –190 BC)

•Heron (10 AD –70 AD)

•Claudius Ptolemy (100 AD –170 AD)

•Diophantus (200 AD –284 AD)

•Pappus (290 AD –350 AD)

•Hypatia (355 AD –415 AD)

Euclid (330 BC –260 BC)

•The leading mathematician in this initial galaxy of the Alexandrian

Museum was the great Euclid.

•The Greek study of mathematics had begun three centuries before this

time. Initially it had been drawn largely from Egypt,in a state which was

rich only in potentialities.

•Its abstractions had appealed to the Greek genius, and under the

impingement of many able minds it had developed until the geometry

of plane figures, and of the simpler and regular solids, was well

advanced.

•Algebraic equations through the quadratics had been solved, and the

irrationalities arising in their solutions had been classified. Finally, the

concepts of rigorous proof, accurate enunciation of assumptions, and

the definition of terms had been fully evolved.

Euclid of Alexandria

Euclid (330 BC –260 BC)

•Euclid enters history as one of the greatest mathematicians of all time

and he is often referred to as the father of geometry.

•The standard geometry most of us learned in school is called Euclidian

Geometry.

•In 300 BC, Euclid gathered up all of the knowledge developed in Greek

mathematics at that time and created his great work “The Elements”.

•This treatise is unequaled in the history of science and mathematics,

and could safely lay claim to being the most influential scientific book of

all time.

Statue in honor of Euclid in the

Oxford University Museum of

Natural History

Euclid’s Elements

•The Elements consisted of thirteen books covering a vast body of

mathematical knowledge, spanning arithmetic, geometry and number

theory.

•The books are organized by subjects, covering every area of mathematics

developed by the Greeks:

•Books I - IV, and Book VI: Plane Geometry

•Books XI - XIII: Solid Geometry

•Books V and X: Magnitudes and Ratios

•Books VII - IX: Whole Numbers

•The basic structure of the Elements begins with Euclid establishing axioms,

the starting point from which he developed 465 propositions, progressing

from his first established principles to the unknown in a series of steps, a

process that he called the 'Synthetic Approach‘.

•He looked at mathematics as a whole, but was concentrated on geometry

and that particular discipline formed the basis of his work.

A fragment of Euclid's "Elements" on

part of the Oxyrhynchus papyri

Euclid’s Axioms

•Euclid based his approach upon 10 axioms,

statements that could be accepted as truths.

•He called these axioms his 'postulates' and divided

them into two groups of five, the first set common to

all mathematics, the second specific to geometry.

•Euclid felt that anybody who could read and

understand words could understand his notions and

postulates but, to make sure, he included 23

definitions of common words, such as 'point' and

'line',to ensure that there could be no semantic

errors.

•From this basis, he built his entire theory of plane

geometry, which has shaped mathematics, science

and philosophy for centuries.

•Euclid's First Group of Postulates - the Common

Notions:

•Things which are equal to the same thing are also equal to

each other

•If equals are added to equals, the results are equal

•If equals are subtracted from equals, the remainders are

equal

•Things that coincide with each other are equal to each other

•The whole is greater than the part

•The remaining five postulates were related specifically

to geometry:

•A straight line can be drawn between any two points.

•Any finite straight line can be extended indefinitely in a

straight line.

•For any line segment, it is possible to draw a circle using the

segment as the radius and one end point as the center.

•All right angles are congruent (the same).

•If a straight line falling across two other straight lines results

in the sum of the angles on the same side less than two right

angles, then the two straight lines, if extended indefinitely,

meet on the same side as the side where the angle sums are

less than two right angles.

Euclid’s Influence

•The reason that Euclid was so influential is that his work is more than just an

explanation of geometry or even of mathematics.

•The way in which he used logic and demanded proof for every theorem shaped the

ideas of western philosophers right up until the present day.

•Great philosopher mathematicians such as Descartes and Newton presented their

philosophical works using Euclid's structure and format, moving from simple first

principles to complicated concepts.

•Abraham Lincoln was a fan of Euclid, and the US Declaration of Independence used

Euclid's axiomatic system. The writers of the Declaration of Independence allude to

Euclid by beginning their document with a short list of "self-evident truths" and

framing the rest of the Declaration as a logical argument based on those truths.

•Euclid's systematic development of his subject, from a small set of axioms to deep

results, and the consistency of his approach throughout the Elements, encouraged its

use as a textbook for about 2,000 years.

•The Elements still influences modern geometry books. Further, its logical axiomatic

approach and rigorous proofs remain the cornerstone of mathematics.

Euclid's Elements Title Page

Quotes and Proofs by Euclid

•The ruler of Egypt, Ptolemy, asked Euclid if there was

a shorter road to learning geometry than through

Euclid's Elements.

•Euclid’s answer to this question was

There is no royal road to geometry

•Let ACB be a right-angled triangle with right angle CAB.

•On each of the sides BC, AB, and CA, squares are drawn, CBDE, BAGF, and ACIH, in that

order.

•From A, draw a line parallel to BD and CE.It will perpendicularly intersect BC and DE

at K and L, respectively.

•Join CF and AD, to form the triangles BCF and BDA.

•Angles CAB and BAG are both right angles; therefore C, A, and G are collinear.

Similarly for B, A, and H.

•Angles CBD and FBA are both right angles; therefore angle ABD equals angle FBC,

since both are the sum of a right angle and angle ABC.

•Since AB is equal to FB and BD is equal to BC, triangle ABD must be congruent to

triangle FBC.

•Since A-K-L is a straight line, parallel to BD, then rectangle BDLK has twice the area of

triangle ABD because they share the base BD and have the same altitude BK, i.e., a line

normal to their common base, connecting the parallel lines BD and AL. (lemma 2)

•Since C is collinear with A and G, square BAGF must be twice in area to triangle FBC.

•Therefore, rectangle BDLK must have the same area as square BAGF = AB2.

•Similarly, it can be shown that rectangle CKLE must have the same area as square

ACIH = AC2.

•Adding these two results, AB2+AC2=BD ×BK +KL ×KC

•Since BD = KL, BD ×BK +KL ×KC = BD(BK + KC) =BD ×BC

•Therefore, AB2+AC2=BC2, since CBDE is a square.

Famous Quote by Euclid Proof of Pythagoras Theorem by Euclid

Archimedes (287 BC –212 BC)

•At the Museum the great tradition of Euclid passed on to his successors.

Among the students of that time there appeared at Alexandria as a

youth the all-overshadowing genius of Archimedes.

•Archimedes studied at the Museum for several years, and although the

locale of his later great work was not Alexandria but Syracuse in Sicily,

he carried through life the stamp of his early Alexandrian education and

training.

•His thought trend has been recognized as typical of the Alexandrian

school. Throughout his life he published his works in the form of

correspondence with the principal mathematicians of his time,

including the Alexandrian scholars Conon and Eratosthenes.

A Thoughtful Archimedes

Archimedes (287 BC –212 BC)

•Archimedes is known as the father of integral calculus.

•He derived the relation between the surface and volume of a sphere and its

circumscribing cylinder. Archimedes was proud enough of the latter discovery to

leave instructions for his tomb to be marked with a sphere inscribed in a cylinder.

•The principal results of his work “On the Sphere and Cylinder” are that the surface

area of any sphere of radius r is four times that of its greatest circle (in modern

notation, S = 4πr2)and that the volume of a sphere is two-thirds that of the cylinder

in which it is inscribed (leading to the formula, V = 4πr3/3).

•He is well known for developing a fundamental principle of hydrostatics, known as

Archimedes’ principle. This principle states that a body immersed in a fluid

experiences an upward buoyant force equal to the weight of the displaced fluid. As

such, he is considered by many to be the founder of hydrostatics.

•He is also known for inventing a device for raising water, still used in developing

countries, known as the Archimedes screw.

The Archimedes Screw

Archimedes’ Principle

The volume of the sphere to the

volume of the cylinder is 2 to 3

Famous Quotes by Archimedes

•Eureka! Eureka!

At one time, king Hieron ordered a golden crown and gave the goldsmith the exact amount of

metal to make it. When Hieron received it, the crown had the correct weight but the monarch

suspected that some silver had been used instead of the gold. Since he could not prove it, he

brought the problem to Archimedes. One day while considering the question, Archimedes

entered his bathtub and recognized that the amount of water that overflowed the tub was

proportional to the amount of his body that was submerged. This observation gave him the

means to solve the problem. He was so excited that he ran naked through the streets of

Syracuse shouting "Eureka! Eureka!" (I have found it!).

The expression “Eureka” is also the state motto of California, referring to the momentous

discovery of gold in 1848. The California State Seal has included the word “Eureka" since its

original design in 1850.

•Give me a place to stand and I will move the whole world

Said to be his assertion in demonstrating the principle of the lever as quoted by Pappus of

Alexandria.

•Do not disturb my circles!

Archimedes was active in defending the city of Syracuse from Roman invasion when he died.

He refused to obey an invading soldier's orders, saying that he would first finish solving the

problem on which he was working. "Do not disturb my circles" were said to be his last words

before the angered soldier killed him.

Eratosthenes (276 BC –194 BC)

•He was called to Alexandria in the first instance to direct its great Library.

•He wrote one of the earliest and most important works in the history of

geography, Geographika.

•In this work, which for the first time described the geography of the entire

inhabited world as it was then known, Eratosthenes invented the discipline

of geography as we understand it.As such, he is commonly referred to as

the "father of geography“.

•He is best known for being the first person to calculate the circumference of

the Earth.

•Eratosthenes proposed a simple algorithm for finding prime numbers. This

algorithm is known in mathematics as the Sieve of Eratosthenes.

Eratosthenes

First modern edition and English

translation of Geographika by

Princeton University Press (2010)

Eratosthenes’ Method for Measuring

the Circumference of the Earth

•At Syene (Aswan), some 5000 stadia (800 km) southeast of Alexandria,

the sun’s rays fall vertically at noon at the summer solstice.

Eratosthenes noted that at Alexandria,at the same date and time,

sunlight fell at an angle of about 7.2° from the vertical. (Writing before

the Greeks adopted the degree, a Babylonian unit of measure, he

actually said “a fiftieth of a circle.”)

•He correctly assumed the Sun’s distance to be very great; its rays

therefore are practically parallel when they reach Earth. Given the

distance between the two cities, he was able to calculate the

circumference of Earth, obtaining 250,000 stadia.

•The exact length of the units (stadia) he used is doubtful, and the

accuracy of his result is therefore uncertain. His measurement of Earth’s

circumference may have varied by 0.5 to 17 percent from the value

accepted by modern astronomers.

By knowing the length of an arc (l) and

the size of the central angle (α) that it

subtends, one can obtain the relation

that the proportion of the length of arc l

to Earth’s circumference, 2πR (where R

is Earth’s radius) equals the proportion

of the central angle αto the angle

subtended by the whole circumference

(360°)—i.e., l : 2πR =α:360.

Apollonius (262 BC –190 BC)

•Apollonius is a mathematician who was known by his contemporaries as “the Great

Geometer”.

•Apollonius gave the ellipse, the parabola, and the hyperbola their modern names.

•His treatise “Conics” is one of the greatest scientific works from the ancient world.

•As a youth, Apollonius studied in Alexandria under the pupils of Euclid and subsequently

taught at the university there.

•The Apollonius crater on the Moon is named in his honor.

The conic sections result from intersecting a plane

with a double cone, as shown in the figure. There

are three distinct families of conic sections: the

ellipse (including the circle); the parabola (with one

branch); and the hyperbola (with two branches). Modern edition of the treatise on

Conic Sections of Apollonius by

Cambridge University Press (1896)

The Roman Era of Alexandria

•Upon the death of Cleopatra in the year 31 BC, the Ptolemaic dynasty came to

its end, and Egypt became a province of the Empire of Rome.

•The development of the Museum had proceeded, until it resembled in almost

all of its aspects the modern university.

•There were regularly scheduled courses of graded lectures, through which

systematic higher instruction was imparted in various academic subjects.

Moreover, there was a large body of students, more or less regularly enrolled.

•Meanwhile, the library also had been evolving and enlarging its range of

functions. The art of copying manuscripts had been developed. Calligraphers

were now especially trained for this purpose.

•The repute of Alexandria as the world's intellectual focal point was enhanced

by this activity. Teachers had need go there for training, and from there also

came the books.

CLEOPATRA (1963)

Heron of Alexandria (10 AD –70 AD)

•Heron was a brilliant geometer and mathematician, but he is most

commonly remembered as a truly great inventor.

•He was known as Mechanikos or the Machine Man.

•Heron’s contributions to mathematics include:

•Hero described a method of iteratively computing the square root.

•Today, however, his name is most closely associated with Heron's Formula for

finding the area of a triangle from its side lengths.

Heron’s formula

Heron of Alexandria

Heron’s Inventions

•Heron’s inventions included the following:

•The first-recorded steam engine,it was created almost two

millennia before the industrial revolution.

•The first vending machine was also one of his

constructions. When a coin was introduced via a slot on

the top of the machine, a set amount of holy water was

dispensed.

•A windwheel operating an organ, marking the first

instance of wind powering a machine in history.

•Heron’s fountain, which is a standalone fountain that

operates under self-contained hydrostatic energy.

Heron’s wind-powered organ

Heron’s Engine Heron’s Vending Machine

Heron’s Fountain

Heron’s Fountain

•Heron’s fountain operates as follows:

•Initially, the air supply container (C) should contain only air; the water

supply container (B) should contain only water.

•To start the fountain, pour water into the basin (A).

•The water from the basin (A) flows by gravity into the air supply

container (C).

•This water forces the air in (C) to move into the water supply

container (B), where the increased air pressure in (B) forces the water

in (B) to issue out the top as a fountain into the basin (A).

•The fountain water caught in the basin (A) will drain back to the air

supply container (C).

•The flow will stop when the water supply container (B) is empty.

Heron’s Fountain

A

B

C

Claudius Ptolemy (100 AD –170 AD)

•Like Euclid, Ptolemy lives in history as a mind and hardly at all as a man. Euclid excelled

as a selector, organizer and expositor of knowledge, and precisely so did Ptolemy.

•The most important of Ptolemy's writings is a work in thirteen books of mathematical

and astronomical content. Ptolemy called it the mathematical syntax. Since Arabian

times, however, it has been generally known as The Almagest.

•The Almagest presents Greek trigonometry in the definitive form it was to retain, and

over which no effective improvement was to be achieved for a thousand years.

•On the side of astronomy, it contains the exposition of a great theory of geocentric

motion of the heavenly bodies which came to be universally known as the Ptolemaic

System.In this system, the Earth is at the center of all the celestial bodies.

•Ptolemy's theory remained unchallenged until the introduction of the Copernican

system by Copernicus in 1543. Copernicus proposed that the Earth and the other

planets instead revolved around the Sun.

The Ptolemaic System

Diophantus (200 AD –284 AD)

•He is sometimes called the "father of algebra" and is the author of a

series of books called Arithmetica. These texts deal with solving

algebraic equations.

•He made important advances in mathematical notation, becoming the

first person known to use algebraic notation and symbolism. Before him

everyone wrote out equations completely.

•Mathematical historian Kurt Vogel states: “Diophantus took a

fundamental step from verbal algebra towards symbolic algebra.”

•In modern use, Diophantine equations usually denote algebraic

equations with integer coefficients, for which integer solutions are

sought.

Title page of the 1621 edition

of Diophantus' Arithmetica

Diophantus and Fermat’s Last Theorem

•The 1621 edition of Arithmetica gained fame after Pierre de Fermat wrote his

famous “Fermat’s Last Theorem” in 1637 in the margins of his copy:

“No three positive integers a, b, and c satisfy the equation an+ bn= cnfor any integer

value of nstrictly greater than two.”

•Fermat's proof was never found,and the problem of finding a proof for the

theorem went unsolved for centuries.

•After 358 years of effort by mathematicians, a proof was finally found in 1994,

by Andrew Wiles after working on it for seven years.

•Sir Andrew Miles is a British mathematician and a Royal Society Research

Professor at the University of Oxford. He received the 2016 Abel Prize for

proving Fermat’s last theorem.

•It is believed that Fermat did not actually have the proof he claimed to have.

Title page of the 1621 edition

of Diophantus' Arithmetica

Pappus of Alexandria (290 AD –350 AD)

•Pappus was one of the last great Alexandrian mathematicians of Antiquity.

•He is known for his Synagoge or Collection, which was written in 340 AD.

•He is also known for Pappus' hexagon theorem in projective geometry.

Pappus's Mathematicae Collectiones,

translated into Latin (1589)

Pappus’hexagon theorem states that given one set of

collinear points A, B, C, and another set of collinear

points a, b, c, then the intersection points X, Y, Zof

line pairs Ab and aB,Ac and aC,Bc and bC are

collinear, lying on the Pappus line.

Hypatia of Alexandria (355 AD –415 AD)

•Theon and his daughter Hypatia taught mathematics at the Museum.

•Theon wrote an excellent commentary upon the Almagest of Ptolemy,

and is still prominently known for his revised edition of Euclid's

Elements.

•Hypatia,on her part, wrote upon the first six books of the Arithmetica

of Diophantus and the Conics of Apollonius.

•She is the earliest female mathematician of whose life and work

reasonably detailed knowledge exists.

•Hypatia was the last of the Alexandrian mathematicians and probably

the last of the Museum's scholars.

Hypatia of Alexandria

Ethnicity of Alexandrian Scholars

A quote from Victor J. Katz (1998). A History of Mathematics: An Introduction.

"But what we really want to know is to what extent the Alexandrian mathematicians of the

period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek

and were part of the Greek intellectual community of Alexandria. And most modern studies

conclude that the Greek community coexisted. So should we assume that Ptolemy and

Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from

Greece at some point in the past but had remained effectively isolated from the Egyptians?

Ethnicity of Alexandrian Scholars

A quote from Victor J. Katz (1998). A History of Mathematics: An Introduction.

"It is, of course, impossible to answer this question definitively. But research in papyri dating

from the early centuries of the common era demonstrates that a significant amount of

intermarriage took place between the Greek and Egyptian communities. And it is known that

Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from

the founding of Alexandria, small numbers of Egyptians were admitted to the privileged classes

in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the

Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the

Alexandrian mathematicians mentioned here were active several hundred years after the

founding of the city, it would seem at least equally possible that they were ethnically Egyptian

as that they remained ethnically Greek.In any case, it is unreasonable to portray them with

purely European features when no physical descriptions exist."

Thank you for your time