2 Early Greek Mathematics Flashcards

Question 1

Q

ON EXAM

Answer

A

Euclid s elements 300BC
short question on mathematics before Euclids; Archimedes work on parabola!!!!

16 marks on Euclids elements:
the structure of the elements, begins abruptly with 23 defn then 5 postula/tes etc…

s/ay who said the quoatations

not just work on geometry but number theory too

Question 2

Q

Greeks are the supreme event’

Answer

A

proof:
Results proved deductively
geometry
Euclid, Archimedes, Apollonius

axiomatics
Euclid’s Elements

abstraction
Plato conceived a spiritual world of abstract ideas: timeless, changeless, indestructible beyond the imperfect, transitory world perceived by our senses.

EUCLID
ARCHIMEDES
PYTHAGORAS

Question 3

Q

GREEK SOURCES

Answer

A

*Virtually nothing physically written by ancient Greek mathematicians survives. Oldest extant Greek texts are copies of copies of copies etc., many centuries removed from originals.- NO ORIGINAL DOCUMENTS

*Main source for early Greek mathematics is the Eudemian Summary in Commentary on Euclid’s Elements Book I by Proclus (450), based on lost
History of Geometry by Eudemus (320 BC), pupil of Aristotle (350 BC).

Question 4

Q

THALES of MILETUS MILETUS (600 BC)

possibly on the exam?

Answer

A

earliest mathematician known by name
emphasized PROOF in mathematics
first person to whom specific mathematical discoveries are attributed
credited with predicting an eclipse of the sun

(no evidence but attributed to these results)
WHICH ARE:
1)circle is bisected by a diameter

2)base angles of an isosceles triangle are equal

3)vertically opposite angles are equal
AND
Thales’ Theorem: The angle in a semicircle is a right angle.

Question 5

Q

PYTHAGORAS of SAMOS (550 BC)

Answer

A

About 540 BC, Pythagoras founded a religious, scientific, philosophical brotherhood in Crotona, southern Italy. After his death around 500 BC, the Pythagoreans continued to flourish for over two centuries, spreading their master’s teaching. In contemplating the design of the universe, the
Pythagoreans recognized the special role played by whole numbers. Hence their motto:
ALL IS NUMBER which led to
NUMBER MYSTICISM
&
NUMBER THEORY

taught verbally
all accredited to pythagoreans not individuals

Question 6

Q

NUMBER MYSTICISM

Answer

A

Pythagoreans believed in a number mysticism that assigned to things material and spiritual a natural number. Odd numbers (exceeding one) were considered male, even ones female. In The Merry Wives of Windsor Shakespeare writes:
There is divinity in odd numbers

Question 7

Q

PYTHAGOREANS number mysticism 5

Answer

A

pythagoreans were interested in what gave order and harmony to the world, believing that natural numbers held the key to the universe.

1 REASON
2 OPINION first female number
3 HARMONY first male number
4 JUSTICE product of equals
5 MARRIAGE union of male &amp; female

Question 8

Q

FIGURE NUMBERS

Answer

A

NUMBERS REPRESENTED GEOMETRICALLY
numbers represented by dots in polygons
TRIANGLE NUMBERS
	•    	 •
                /  \
             • \_\_ •

     	     •
            / \
          •	•
         / \      \
	•\_\_•\_\_•
by drawing diagonals
1  3  6  10
SQUARE NUMBERS
	•
	 •-•
         |   |
         •-•
by drawing horizontal and vertical
1 4 9 16

PENTAGONAL NUMBERS

1 5 12 22

buy during pentagons at border

Question 9

Q

The nth m-gonal number can be seen as the sum of

Answer

A

the first n terms of an arithmetic progression with first term 1 and common difference m-2

eg n=4 m=3 gives the fourth triangular number 1+2+3+4=10

Question 10

Q

PYTHAGOREAN NUMBER THEORY

ABUNDANT DEFICIENT PERFECT

Answer

A

A number is abundant, deficient or perfect, according as the sum of its proper divisors exceeds, is less than or equals the number itself.

12 is abundant: 1 + 2 + 3 + 4 + 6 = 16
9 is deficient: 1 + 3 = 4
6 is perfect: 1 + 2 + 3 = 6

Greeks knew four perfect numbers: 6, 28, 496, 8128.

Question 11

Q

AMICABLE PAIRS

**

Answer

A

Two numbers form an amicable pair, if each is the sum of the proper divisors of the other. Greeks knew only one amicable pair: 220 & 284.

This is amicable, since the sums of the proper divisors of 220 & 284 are, respectively,

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284,
1 + 2 + 4 + 71 + 142 = 220.

Question 12

Q

INCOMMENSURABILTY

Answer

A

The early Pythagoreans believed, given any two line segments, there is a third that can be laid off a whole number of times to fill out the other two, i.e. they have a common measure, whence the ratio of their lengths is a ratio of integers, rational number, 15 : 11 in case below.

-

Their discovery that the ratio of a diagonal of a square to a side was not a ratio of integers shattered their belief that all is number. From then on, geometrical quantities were treated differently from numbers. Legend has it Hippasus of Metapontum was drowned at sea for divulging the incommensurability of the diagonal of a square to a side, i.e. they have no common measure

Question 13

Q

Three Classical Problems of Antiquity

NOT ASSESSED?

Answer

A

problems are to do with a straight edge straight line Not a ruler as a ruler has markings on it

1)Duplication of the cube / Delian problem Given the edge of a cube, to construct the
edge of another cube, whose volume is twice that of the first.
Delians tried to double the volume of their cubical altar to
rid themselves of a plague.

2)Trisection of the angle
To trisect a general angle.

3)Squaring the circle: To construct a square equal in area to a given circle.

Question 14

Q

Delian Problem

NOT ASSESSED?

Answer

A

*called the Delian problem as island of DELOS

duplication of the cube volume given the edge using only a compass and a straight edge :
impossible using only these tools

HE REPHRASED IT AND HE WASN'T A DOCTOR
Hippocrates of Chios (430 BC) showed that the problem is equivalent to constructing two mean proportionals x, y
between a and 2a, i.e.
a : x = x : y = y : 2a, 
so x =  cube root(2) a

*EXAM:
“Arcyhtas of Tarentum (428 BC) solved the problem by finding an intersection point of a cone, cylinder, torus.”

*Menaechmus (350 BC) invented conic sections: parabola, ellipse, hyperbola to solve the problem, giving two solutions:

two parabolas
x^2=ay
y^2=2ax
meet in (cube root(2) a)

parabola y^2=2ax
hyperbola xy=2a^2
meet in (cube root(2)a)

(not archimedes’ spiral)

Question 15

Q

Trisection of the Angle of the Angle

NOT ASSESSED?

Answer

A

Hippias of Elis (460 BC) introduced into mathematics the first curve beyond the straight line and the circle, the
trisectrix or quadratrix, which can trisect, even multisect, an angle. He devised an instrument to draw it mechanically!

A TRISECTRIX
square with 2 parabolas and projections of angles
where the curves intersect

Archimedes (250 BC) trisected (multisected) angles using his Archimedean spiral.

*you can’t trisect a general angle using a compass and a straight edge

Question 16

Q

Squaring the circle:

NOT ASSESSED?

Answer

A

Hippocrates of CHios’ success in squaring certain lunes led hi, to think the circle could be squared

Shaded lune has same area as shaded square
(diagram)

Dinostratus (350 BC) squared circle using quadratrix
Archimedes (250 BC) squared circle using his spiral again

Question 17

Q

3 classical problems:

The Problems’ Legacy

essay?

Answer

A

Using only straightedge and compass, all three constructions were shown to be impossible by:
Wantzel (duplication, trisection) in 1837
Lindemann (squaring the circle) in 1882

Morris Kline wrote of the three classical problems:
The search for iron has often led to gold

trivial unimportant problems led to:
conics etc

*Enrichment of methods, proofs, tools of analysis
*Exhaustion method: finding areas. Pi approximations
*New curves: parabola, ellipse, hyperbola, quadratrix,
Archimedean spiral, cissoid, conchoid
*Theory of equations: Galois/algebraic number theory

Question 18

Q

PLATO

Answer

A

Born Athens 427 BC. Pupil of Socrates. Travelled far in search of wisdom. Founded Academy in 387 BC. Philospher not mathematician. Works in dialogue form. His enthusiasm for the subject and belief its
study was finest training for the mind encouraged others to pursue it. Died Athens 347 BC. Above the portals of his Academy was the inscription:

LET NO ONE IGNORANT OF GEOMETRY ENTER MY DOORS

Question 19

Q

PLATO’S ASSOCIATES

Answer

A

Theaetetus (415 - 369 BC): regular polyhedra
Eudoxus (408 - 355 BC): proportions, exhaustion
Aristotle (384 - 322 BC): logic
Menaechmus (350 BC): conics, Delian problem
Dinostratus (350 BC): squaring the circle

Question 20

Q

how did the Greeks define conic sections

don’t think this will be assessed

Answer

A

a cone sectioned off at different angled sections to form a circle ellipse parabola hyperbola

Question 21

Q

ARCHIMEDES WORK ON PARABOLA

EXAM

Question 22

Q

EXAM QUESTION: Euclid

Answer

A

long question Euclids elements, Book 1 structure and contents

*comment on the structure and give an example of each

Question 23

Q

EUCLID**

Answer

A

300BC

we know of 2 quotations:

There is no royal road to geometry’

‘Give him a coin, for he must profit from what he learns’

*QUOTE: EUCLID TELLING THE KING THERE IS NO EASY ROAD,EVEN FOR YOU

*QUOTE: YOU COME HERE TO BE EDUCATED, NOT THE TYPE HE WANTS AT HIS SCHOOL
when euclid was I supposed to do that he might gain from studying the elements, he said to the servants, give this from what he learned, you could review resulted from his belief that knowledge was were studying for its own sake.

332 BC Alexander the Great founds Alexandria
323 BC Alexander dies - empire falls apart
306 BC Ptolemy I becomes ruler of Egypt
300 BC Alexandria: Museum & Library founded

EUCLID arrives

*we know nothing about euclid except for the elements of which we have a copy of a copy of a copy (NOT ORIGINAL) we don’t know what he looks like

Question 24

Q

THE ELEMENTS

notes

Answer

A

Unknown how much of work is Euclid’s as
Draws on earlier sources
13 books VOLUMES, 465 propositions & constructions
*
Most influential mathematics work ever written
Earliest application of axiomatic method
Superbly organized treatise shows Euclid’s genius
First landmark in mathematical organization
For 2000 years dominated geometry teaching
Bible apart, no work more widely studied
Thousands of editions printed since first one in Venice 1482
First English translation by Sir Henry Billingsley 1570
Definitive Greek edition by the Dane J L Heiberg 1888
Definitive English edition by Sir Thomas L Heath 1908

Euclid’s Elements
*First printed edition, Venice 1482

*Euclid’s definitions of points, lines etc.
LOGICAL MATHEMATICAL ORGANISATION

ABRAHAM LINCOLN- studied first 6 books of Euclid from entering congress? Aim to learn every proof in forst 6 to train his mind, improve powers of logic.

Question 25

Q

Euclid’s Elements

BOOK 1: STRUCTURE

Answer

A

Definitions
Opens abruptly with twenty-three definitions from plane geometry
then follow 5 postulates for his geometry (without proof) geometric in nature
5 axioms-

*48 PROPOSITIONS
34 proofs theorems and 14 constructions
*47 is pythagoras’ theorem so is 48

*doesn’t explain, was supposed to be a textbook for students in Alexandria

Question 26

Q

Euclid’s Elements
BOOK 1:
DEFINITIONS

Answer

A

three definitions from plane geometry

A point is that which has no part
A line is breadthless length
An acute angle is an angle less than a right angle
Parallel lines straight lines in same plane that being
produced indefinitely in both directions, do not meet.

Question 27

Q

Euclid’s Elements
BOOK 1:
POSTULATES

Answer

A

five postulates (self-evident truths) for his geometry- geometrical assumptions (unproven)

A straight line may be drawn from any point to any other
A straight line segment can be produced to any length
A circle may be described with any centre and radius
All right angles are equal
Parallel Postulate: If a straight line falls on two straight lines, making the interior angles on the same side less than two right angles, the two straight lines, if indefinitely produced, meet on the side on which the angles are less than two right angles.

(5 is the most important: if less than 180 THEN they will meet)
postulate that is it cannot be proven

Question 28

Q

Euclid’s Elements
BOOK 1:
AXIOMS

Answer

A

A X I O M S
assumptions/logical
about logic rather than geometry
Now follow
five axioms (self-evident GENERAL truths)

Things equal to the same thing are equal to one another
If equals be added to equals, the wholes are equal
If equals be subtracted from equals, the remainders are equal
Things which coincide with one another are equal
The whole is greater than the part

Question 29

Q

a quote on the structure of book 1, Euclids elements

Russell Criticizes Euclid’s Basics

Answer

A

I had been told Euclid proved things, and was disappointed he started with axioms. At first, I refused to accept them unless my brother could offer me a reason for doing so, but he said “if you don’t accept them, we cannot go on” As I wished to go on, I reluctantly admitted them.

Bertrand Russell

His demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative form when no figure is drawn, but many of Euclid’s earlier proofs fail before this test.
HE USED DIAGRAMs

Question 30

Q

ELEMENTS I, PROPOSITION I

structure of a result

Answer

A

STATEMENT
always uses a diagram in his proof
diagram
QEF

Compass & straightedge construction only, equilateral triangle

Uses Definition 15 (circle),
Postulates 1, 3 and Axiom 1
Tacit assumption (reasoning from diagram) that two circles
intersect - not justifiable from basic assumptions- HE TOOK THIS FOR GRANTED; using a diagram as proof not from axioms.he wanted to prove everything from first principles however he assumed something not from his axioms
QEF ~ quod erat faciendum. That which was to be done.
Indicates Proposition I is a construction not a theorem.

Question 31

Q

ELEMENTS I, PROPOSITION 47

diagram squares drawn on triangle sides areas equal sum

Answer

A

In a right-angled triangle, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle

Proof
diagram
QED

Pythagoras’ Theorem
Euclid’s own proof (?) we think
Square on hypotenuse is divided into two rectangles, having areas, equal in turn, to the squares on the other two sides.
QED ~ quod erat demonstrandum. That which was to be demonstrated. PROVED Indicates Proposition 47 is a theorem not a construction.

uses definition 22 on a circle
postulates 1 and 4
axiom 2
and previous propositions 4 14 31 41 46

Question 32

Q

THOMAS HOBBES FALLS IN LOVE

Answer

A

(1628)
He was 40 years old before he looked on Geometry, which happened accidently. Being in a Gentlemen’s Library, Euclid’s Elements lay open, and ‘twas 47 El. libri I. He read the Proposition. By G—, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is imposssible. So he reads the Demonstration of it, which referred him back to such a Proposition; which he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that trueth. This made him in love with Geometry.

(Brief Lives by John Aubrey)

read book 1 by referring proposition to proposition referring back

Question 33

Q

ELEMENTS I, PROPOSITION 48

Answer

A

If the square on one side of a triangle equals the squares on the remaining two, then the angle contained by the remaining sides is right

Proof
diagram
QED

*CONVERSE of Pythagoras’ Theorem
* Final proposition in ELEMENTS I
* Proof uses Proposition 47 & Proposition 8.
* QED indicates Proposition 48 is theorem NOT construction
* Pythagoras’ Theorem & Converse thrilling climax to Book I
user’s previous propositions 3 8 and 11 47

Question 34

Q

BOOKS of the ELEMENTS

Answer

A

Elements textbook covering most elementary mathematics

Books I - IV: rectilinear figures, circles. Book V: Eudoxus’
theory of proportion.
Book VI: similarity. Books VII - IX: number theory.
Book X: incommensurables. Books X1-XIII: solid geometry.

Plane geometry: Propositions 47, 48 - Pythagoras, Converse
Geometrical Algebra: Propositions 12, 13 - Cosine Rule
Circles: Proposition 31 - Angle in semicircle is right
Construction of regular polygons: 3, 4, 5, 6, 15 sides
Theory of proportion: EUDOXUS
Similarity: Book V in action
Elementary properties of numbers: Euclid’s algorithm
Continued proportion: geometric progressions
Number theory: infinity of primes, formula for perfects
Incommensurables: basis for method of exhaustion
Lines, planes in space
Areas, volumes: circles, cones, pyramids
Regular polyhedra: only five regular polyhedra

Question 35

Q

A Gem of Number Theory

the most beautiful theorem

**

Answer

A

Elements IX, Proposition 20: Prime numbers are more
than any assigned multitude of prime numbers.
(a) Assertion: Given any (finite) set of primes, there is always
another prime not in the set.
Modern terminology: there are infinitely many primes
(b) Euclid only shows: given any three primes, a fourth can be found. Assumes his method generalizes to any finite set of primes.
(c) Numbers represented by line segments: UNIT NOT thought a number. (generates the others)

(d) References to previously established results: Book VII, 31.
(another number theory book)

(e) Euclid’s proof is by contradiction (reductio ad absurdum).
(f) QED ~ quod erat demonstrandum, it is a theorem being proved

PAGE 9 problem studies
*sheet handed out

look at numbers and generate a prime number by multiplying together and adding 1

(last years exam question)

Question 36

Q

Book IX, Proposition 35

geometric progression

Answer

A

If numbers are in continued proportion, and there be subtracted from the second and last numbers equal to the first, then as the excess of the second is to the first, so will be the excess of the last to those
before it.

Numbers in continued proportion: a, ar, . . . , ar^{n−1}, ar^n
Interpretation:
(ar − a)/a
=
(ar^n − a)/(a + ar + · · · + arn−1)

whence
a + ar + · · · + ar^{n−1} = a(r^n − 1)/(r − 1)

Tacit assumptions: a not equal to 0, r not equal to 1.
Example: 1 + 2 + 2^2 + · · · + 2^{n−1} = 2^{n} − 1.

Question 37

Q

Book IX, Proposition 36

FINAL PROP OF boOK 9

Answer

A

If numbers beginning with a unit be set out in double proportion, until their sum becomes prime, and if the sum is multiplied into the last, the product will be perfect.

Interpretation:
(1 + 2 + · · · + 2^{n−1})2^{n−1}
perfect when 1 + 2 + · · · + 2n−1 prime,

i.e.
(2^n − 1)2^{n−1} perfect when 2^{n} − 1 prime.

(by previous result we can write and simplify)

If we have a number n st 2^{n-1} is a prime we get a perfect number if we multiply by the last term in the progression 2^{n-1}

*comment on the understanding and importance IN EXAM

Question 38

Q

*how suitable the elements are for different readers exam question:

Answer

A

*RESEARCH MATHEMATICIAN:
not rigourous in SOME places, sometimes assumed things from diagrams which isnt a proper proof. Although the results are very interesting

FOR STUDENTS: no motivating examples, no introduction, not a student friendly book
HISTORIANS OF MATHEMATICS: no dates, no names of people, no attributed people to any of the results.

Question 39

Q

NOTE ON EXAMS

Answer

A

*unchanged is Mesopotamia, Early Greek, Euclid, Greek calc

cubics

Question 40

Q

BErtrand russell

Answer

A

Russell Criticizes Euclid’s Basics
His demonstrations require many axioms of
which he is quite unconscious. A valid proof
retains its demonstrative form when no
figure is drawn, but many of Euclid’s earlier
proofs fail before this test

On the structure:
I had been told Euclid proved things, and was disappointed he started with axioms. At first, I refused to accept them unless my brother could offer me a reason for doing so, but he said “if you don’t accept them, we cannot go on” As I wished to go on, I reluctantly admitted them.

‘At eleven, I began Euclid, one of the great events
of my life, dazzling as first love. I had not imagined
there was anything so delicious in the world.’
Bertrand Russell

Question 41

Q

Weakness of EUclids

Answer

A

doesn’t explain, was supposed to be a textbook for students in Alexandria

Begins abruptly
not very reader friendly
begins with 2 useless definitions, what is a part
what is a line?
NO STUDENT FRIENDLY
BORING FOR STUDENTS no motivation for students

Question 42

Q

mark scheme euclid’s elementsPobook 1

Answer

A

48 propositions each of which is meticulously proved by back reference to the initial definitions, postulates, axioms and propositions already established. Propositions themselves comprise 14 straight edge and Compass constructions and 34 theorems the proofs of the former and with qts and Latter qet. Most proofs are illustrated by diagrams. Book one concern plane rectilinear geometry, points, lines, lengths, perpendiculars, parallel, angles, congruences, triangles, quadrilaterals, parallelograms, areas etc. Examples of a construction and 0 are respectively proposition 1 and proposition 47

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2 Early Greek Mathematics Flashcards

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