Euclid’s Plane Euclid’s Plane GeometryGeometry

The ElementsThe Elements

Euclid 300’s BCEEuclid 300’s BCE

► Teacher at Museum Teacher at Museum and Library in and Library in Alexandria, founded by Alexandria, founded by Ptolemy in 300 BCE.Ptolemy in 300 BCE.

► Best known for Best known for compiling and compiling and organizing the work of organizing the work of other Greek other Greek mathematicians mathematicians relating to Geometry.relating to Geometry.

Aristotle 384-322 BCEAristotle 384-322 BCE

Begin your scientific Begin your scientific work with definitions work with definitions and axioms.and axioms.

The ElementsThe Elements

► Consisted of 13 volumes of definitions, Consisted of 13 volumes of definitions, axioms, theorems and proofs.axioms, theorems and proofs.

► Compilation of knowledge.Compilation of knowledge.

► The Elements The Elements was first math book in which was first math book in which each theorem was proved using axioms and each theorem was proved using axioms and previously proven theorems – teaching how previously proven theorems – teaching how to think and develop logical arguments.to think and develop logical arguments.

► Second only to the Bible in publications.Second only to the Bible in publications.

►Books 1-6 Plane GeometryBooks 1-6 Plane Geometry►1-2 triangles, quadrilaterals, quadratics1-2 triangles, quadrilaterals, quadratics►3 - circles3 - circles►4 - inscribed and circumscribed polygons4 - inscribed and circumscribed polygons►5 – magnitudes and ratio, Euclidean Algorithm5 – magnitudes and ratio, Euclidean Algorithm►6 – applications of books 1-56 – applications of books 1-5

►Books 7-9 Number TheoryBooks 7-9 Number Theory►Book 10 Irrational NumbersBook 10 Irrational Numbers►Books 11-13 Three dimensional figures Books 11-13 Three dimensional figures

including 5 Platonic including 5 Platonic solidssolids

Book 1Book 1

►5 statements that Euclid believed were 5 statements that Euclid believed were obvious.obvious.

►5 postulates about Geometry that Euclid 5 postulates about Geometry that Euclid believed were intuitively true.believed were intuitively true.

►23 definitions to help clarify the 23 definitions to help clarify the postulates (point, line, plane, angle postulates (point, line, plane, angle etc…)etc…)

5 Common notions (obvious)5 Common notions (obvious)

1.1. Things equal to the same thing are equal.Things equal to the same thing are equal.

2.2. If equals are added to equals, the results If equals are added to equals, the results are equal.are equal.

3.3. If equals are subtracted to equals, the If equals are subtracted to equals, the results are equal.results are equal.

4.4. Things that coincide are equal.Things that coincide are equal.

5.5. The whole is greater than the part.The whole is greater than the part.

5 assumptions (intuitively true)5 assumptions (intuitively true)► Postulate 1 – a straight line can be drawn Postulate 1 – a straight line can be drawn

from from any point to any point. any point to any point. (assumes only one line)(assumes only one line)

► Postulate 2 – a line segment can be Postulate 2 – a line segment can be extended into extended into a line. a line.

► Postulate 3 – a circle can be formed with any Postulate 3 – a circle can be formed with any center and any radius center and any radius

(assumes only one circle)(assumes only one circle)

► Postulate 4 – all right angles are congruentPostulate 4 – all right angles are congruent

► Postulate 5 – if two lines are cut by a Postulate 5 – if two lines are cut by a transversal transversal and the consecutive and the consecutive interior angles interior angles are not are not supplementary then the lines supplementary then the lines intersect.intersect.

Book IBook IIncluded theorems such as:Included theorems such as:► Parallel Line PostulateParallel Line Postulate

► Pythagorean TheoremPythagorean Theorem

► construction of a square (using only a straight construction of a square (using only a straight edge and protractor)edge and protractor)

► SASSAS

► properties of parallelogramsproperties of parallelograms

► properties of parallel lines cut by a transversalproperties of parallel lines cut by a transversal

Inscribed PolygonsInscribed Polygons(Book IV)(Book IV)

► Euclid proved many theorems about circles Euclid proved many theorems about circles in Book III that allowed him to provide in Book III that allowed him to provide detailed constructions of inscribed and detailed constructions of inscribed and circumscribed polygons.circumscribed polygons.

► For example, to inscribe a pentagon, draw For example, to inscribe a pentagon, draw an isosceles triangle with the base angles an isosceles triangle with the base angles equal to twice the vertex angle. Bisect the equal to twice the vertex angle. Bisect the base angles and the 5 points together make base angles and the 5 points together make the pentagon.the pentagon.

Duplicate RatioDuplicate Ratio(Book V)(Book V)

►Book VII begins with a definition of Book VII begins with a definition of proportional which is based on the proportional which is based on the notion of duplicate ratio.notion of duplicate ratio.

►Duplicate ratio Duplicate ratio ‘‘When three magnitudes are When three magnitudes are

proportional, the first is said to have proportional, the first is said to have to the third the to the third the duplicate ratioduplicate ratio of of

that that which it has to the second.”which it has to the second.”►Ex. 2:6:18Ex. 2:6:18

Euclidean AlgorithmEuclidean Algorithm(Book VII)(Book VII)

► Process for finding the greatest common divisor.Process for finding the greatest common divisor.

► Given Given aa, , bb with with aa > > b, b, subtract subtract bb from from aa repeatedly repeatedly until get remainder until get remainder c.c.

► Then subtract Then subtract cc from from bb repeatedly until get to repeatedly until get to mm, , then subtract then subtract mm from from cc……when the result = 0, you ……when the result = 0, you have the greatest common divisor or the result = 1, have the greatest common divisor or the result = 1, which means which means aa and and bb are relatively prime. are relatively prime.

► ex. 80 and 18ex. 80 and 18► ex. 7 and 32ex. 7 and 32

Prime NumbersPrime Numbers► Consider these 3 statements about primes found in Consider these 3 statements about primes found in

Book VII:Book VII: ““Any composite number can be divided by some Any composite number can be divided by some

prime number.prime number. ““Any number is either prime or can be divided by Any number is either prime or can be divided by

a prime number.”a prime number.” ““If a prime number can be divided into the If a prime number can be divided into the

product of two numbers, it can be divided into product of two numbers, it can be divided into one of them. one of them.

► These statements form the Fundamental Theorem These statements form the Fundamental Theorem of Arithmetic – that any number can be expressed of Arithmetic – that any number can be expressed uniquely as a product of prime numbers.uniquely as a product of prime numbers.

► In Book IX, Euclid proves through induction that In Book IX, Euclid proves through induction that there are infinitely many prime numbers.there are infinitely many prime numbers.

Geometric SeriesGeometric Series(Book IX)(Book IX)

a, ar, ar², ar³,...ara, ar, ar², ar³,...arnn

(ar – a)(ar – a) (ar(arnn--a)a)

(ar – a):a = (ar(ar – a):a = (arnn-a):S-a):Snn

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

Solve this last equation for SSolve this last equation for Snn

(ar – a):a = (ar(ar – a):a = (arnn-a):S-a):Snn

Ex. Find the sum of the first 5 terms when Ex. Find the sum of the first 5 terms when a a =1 and =1 and r r =2=2

n

n

S

aar

a

aar

1

)1(

r

raS

n

n

Knowing how to think- who needs it?Knowing how to think- who needs it?

► Lawyers, politicians, negotiators, programmers, and anyone Lawyers, politicians, negotiators, programmers, and anyone dealing with social issues!dealing with social issues!

► Abraham Lincoln carried a copy of Abraham Lincoln carried a copy of The ElementsThe Elements (and read (and read it) to become a better lawyer.it) to become a better lawyer.

► The Declaration of Independence is set up in the same The Declaration of Independence is set up in the same format as format as The ElementsThe Elements (self-evident truths are axioms (self-evident truths are axioms used to prove that the colonies are justified in breaking used to prove that the colonies are justified in breaking from England).from England).

► 1919thth century Yale students studied century Yale students studied The ElementsThe Elements for two for two years, at the end of which they participated in a celebration years, at the end of which they participated in a celebration ritual called the Burial of Euclid.ritual called the Burial of Euclid.

► E.T. Bell wrote ‘Euclid taught me that without assumptions, E.T. Bell wrote ‘Euclid taught me that without assumptions, there is no proof. Therefore, in any argument, examine the there is no proof. Therefore, in any argument, examine the assumptions.”assumptions.”

High School GeometryHigh School Geometry

► Plane Geometry courses today are basically the Plane Geometry courses today are basically the content of Euclid’scontent of Euclid’s Elements Elements..

► Two-column proof appeared in the 1900’s to make Two-column proof appeared in the 1900’s to make proofs easier but led to rote memorization instead.proofs easier but led to rote memorization instead.

► 1970’s moved away from proofs because they were 1970’s moved away from proofs because they were ‘ too painful’ and not fun.‘ too painful’ and not fun.

► Now proofs are brief and irrelevant. They do not Now proofs are brief and irrelevant. They do not serve the purpose of developing logical thinking.serve the purpose of developing logical thinking.

PSSAPSSA

►Standards: what they should knowStandards: what they should know►Anchors: what they are tested onAnchors: what they are tested on

TimelineTimeline

► Prior to Euclid, Greek mathematicians such as Prior to Euclid, Greek mathematicians such as Pythagorus, Theaetetus, Euxodus and Thales did Pythagorus, Theaetetus, Euxodus and Thales did work in Geometry.work in Geometry.

► 384-322 BCE - Aristotle believed that scientific 384-322 BCE - Aristotle believed that scientific knowledge could only be gained through logical knowledge could only be gained through logical methods, beginning with axioms.methods, beginning with axioms.

► 300 BCE- Euclid teaches at the Museum and Library 300 BCE- Euclid teaches at the Museum and Library at Alexandriaat Alexandria

► 1880 J.L. Heiberg compiles Greek version of 1880 J.L. Heiberg compiles Greek version of The The ElementsElements as close to original as possible. as close to original as possible.

► 1908 Thomas Heath translated Heiberg’s text. This 1908 Thomas Heath translated Heiberg’s text. This version is the one most widely used and the basis version is the one most widely used and the basis for modern Geometry courses.for modern Geometry courses.

ReferencesReferences

Berlinghoff, F. & Gouvêa. Math Through the Ages: A Gentle History for Teachers and Others. . Farmington, Maine: Oxton House, 2002.

Heath, T. History of Greek Mathematics, Volume 2. . New York, 1981.

Katz, V. The History of Mathematics. Boston, MA: Pearson, 2004.

http://scienceworld.wolfram.com

http://www.groups.dcs.stand.ac.uk/~history/mathematicians/Euclid

www.pde.state.pa.us