The Alexandrian School: EUCLID and Others

1
The Alexandrian School EUCLID and Others

• MATH 4-5620
• FALL 2004

2
Background

• Alexander the Great (334-323B.C.)
• City in Egypt still bears his name, Alexandria,
  had great harbor and docking facilities. Became
  the trading center of the world
• After the demise of trading center (by Arabs),
  intellectual life, initiated by the early
  Ptolemies, prospered.

3
The Museum

• Developed a center of learning called the
  Museum forerunner of modern university.
• Leading scholars - scientists, poets, artists,
  writers - came to Alexandria by invitation from
  the Ptolemies.
• Scholars had leisure to pursue their studies,
  access to finest libraries, and opportunity to
  talk to other resident experts.

4
Museum (cont.)

• Members were granted salary stipends RB (at
  Kings expense)
• Intended as an institution for research and
  learning rather than a place of education.
• Science and mathematics flourished critical
  time span from 300-100 B.C. in the history of
  mathematics.
• Alexandrian Library need for manuscripts said
  to have had 3-500,000 scrolls in time of Caesar

5
Distinguished Scholars

• Euclid
• Archimedes
• Eratosthenes
• Apollonius
• Pappus
• Claudius Ptolemy
• Diophantus

6
EUCLID (323-285 B.C.)

• Author of The Elements of Geometry
• Author of at least 10 other works covering a wide
  variety of topics
• Very little is known about his personal life he
  founded a school and taught in Alexandria is
  known but not much beyond that. Probable that he
  received his training from pupils of Plato (in
  Athens)

7
Take a look

• Also refer to page 144 in text, which illustrates
  a page taken from the Latin version (1482)

8
The Elements

• A compilation of the most important mathematical
  facts available at the time organized into 13
  books (6 were found.)
• Much material was drawn from earlier sources.
• The logical arrangement of theorems and
  development of proofs was evident.
• Ranks with the Bible for circulation

9
Classics of Mathematics (Calinger, 1995)

• Euclids fame
• Outline of the Elements
• Comment about the proofs of the propositions
  refer to p.122 for P.T.

10
Foundation of the Elements

• Tried to build all of geometry on 5 postulates
  (self evident truths) and 5 axioms (later called
  common notions)
• Refer to text p. 139 for postulates and p. 140
  for common notions.
• Postulate 5 better known as the parallel
  postulate has been one of the most famous and
  controversial statements in math. history (How
  does one derive it from the first 4 postulates?)

11
Comments

• Euclid tried to define all vocab. he used
• Failed to recognize the necessity of undefined
  terms
• Many proofs were based on reasoning from diagrams
  (which had visual evidence).
• David Hilbert (1862-1943) German mathematician
  credited with work in foundations of Geometry.
• http//babbage.clarku.edu/djoyce/hilbert/

12
Euclid and Number Theory

• Three of the books (Elements) VII, VIII, and IX
  contained 102 propositions devoted to
  arithmetic (natural s or int.s).
• Particular interest in divisibility and prime
  numbers.
• Euclid always represented numbers by line segments

13
Euclidean algorithm

• Based on the division theorem
• aqbr 0ltrltb
• Also known that for integers a and b, where a and
  b are non-zero, there exists integers x and y
  such that
• gcd (a,b) axby

14
Fundamental Theoremof Arithmetic

• Every positive integer ngt1 is either a prime or
  can be expressed as a product of primes this
  representation is unique, apart from the order in
  which the factors occur.
• (proof text, p. 172)
• i.e. 4725 33527

15
ERATOSTHENES(276-194 B.C.)

• Refer to Classics of Mathematics (Calinger, 1995)
  for background on Eratosthenes.

16
Eratosthenes

• His work included
• 3-volume Geographica (put geographical studies on
  a sound mathematical basis)
• As a mathematician, his chief work was a solution
  to the famous Delian problem of doubling the cube
  and the invention of a method for finding prime
  s.
• Now best known for work in calculating the
  earths circumference (most accurate to this
  point)

17
Claudius PTOLEMY (85-160 A.D.) native of Egypt

• Ptolemy did for astronomy what Euclid did for
  geometry reduced the work of his predecessors
  to a matter of historical interest
• His great work was the Almagest (authority on
  astronomy until Copernicuss De Revolutionibus
  1543).

18
Ptolemy

• He proposed eccentric solar motion
• His system (described by the Almagest) was noted
  to be as complicated (relative to the time frame)
  as Einsteins relativity theory is to our time.
• The chief flaw in Ptolemys system lay in its
  mistaken premise of an earth-centered universe
  (refer to text p. 182).

19
ARCHIMEDES (287-212 B.C.)

• Considered the greatest genius of ancient times
• Lived a generation or two after Euclid and was a
  contemporary of Eratosthenes
• Son of an astronomer
• Born in Syracuse - Greece

20
Archimedes

• Earned great deal of notoriety by his
  mathematical writings, his mechanical inventions,
  and the method in which he conducted the defense
  of his native city during the second Punic War
  (218-201 B.C.)
• One invention the Archimedean screw a pump
  still used in parts of the world raised canal
  water over levees into irrigated fields

21
Archimedes

• Was much more interested in theoretical studies
  rather than applications
• Of all of his achievements, he appeared to be
  most proud of his work contained in On the Sphere
  and Cylinder two books with about 53
  propositions. He indicated this work was for
  mathematicians to examine his proofs and judge
  their value (text, p. 190)

22
Archimedes

• Another work, The Measurement of a Circle, is a
  short but important treatise. Contains three
  propositions (thought to be part of a longer
  work).
• The most important proposition is the estimate of
  the numerical value of ? (ratio of circumference
  of a circle to its diameter). NOTE ? was not
  introduced until 1706 by an English writer
  (Wm.Jones)

23
Archimedes

• The Sand-Reckoner, a computational accomplishment
  contained a new system of notation for expressing
  numbers in excess of one hundred million (for
  which Greek mathematics had not yet developed).
  He used exponents for ordering classes of
  magnitude.

24
Archimedes

• On Spirals contains 28 propositions dealing with
  the properties of the curve now known as the
  spiral of Archimedes (text, p. 196).
• One of the most interesting results was his
  calculation of the area enclosed by the first
  loop of the spiral and the fixed line.
• Quadrature of a Parabola found the area of the
  segment formed by drawing any chord of a parabola.