Euclid’s Plane Geometry The Elements

Euclid 300’s BCE Teacher at Museum and Library in Alexandria, founded by Ptolemy in 300 BCE. Best known for compiling and organizing the work of other Greek mathematicians relating to Geometry.

Aristotle 384-322 BCE Begin your scientific work with definitions and axioms.

The Elements Consisted of 13 volumes of definitions, axioms, theorems and proofs. Compilation of knowledge. The Elements was first math book in which each theorem was proved using axioms and previously proven theorems – teaching how to think and develop logical arguments. Second only to the Bible in publications.

Book 10 Irrational Numbers Books 1-6 Plane Geometry 1-2 triangles, quadrilaterals, quadratics 3 - circles 4 - inscribed and circumscribed polygons 5 – magnitudes and ratio, Euclidean Algorithm 6 – applications of books 1-5 Books 7-9 Number Theory Book 10 Irrational Numbers Books 11-13 Three dimensional figures including 5 Platonic solids

Book 1 5 statements that Euclid believed were obvious. 5 postulates about Geometry that Euclid believed were intuitively true. 23 definitions to help clarify the postulates (point, line, plane, angle etc…)

5 Common notions (obvious) Things equal to the same thing are equal. If equals are added to equals, the results are equal. If equals are subtracted to equals, the results are equal. Things that coincide are equal. The whole is greater than the part.

(assumes only one line) 5 assumptions (intuitively true) Postulate 1 – a straight line can be drawn from any point to any point. (assumes only one line) Postulate 2 – a line segment can be extended into a line. Postulate 3 – a circle can be formed with any center and any radius (assumes only one circle) Postulate 4 – all right angles are congruent Postulate 5 – if two lines are cut by a transversal and the consecutive interior angles are not supplementary then the lines intersect.

Book I Included theorems such as: Parallel Line Postulate Pythagorean Theorem construction of a square (using only a straight edge and protractor) SAS properties of parallelograms properties of parallel lines cut by a transversal

Inscribed Polygons (Book IV) Euclid proved many theorems about circles in Book III that allowed him to provide detailed constructions of inscribed and circumscribed polygons. For example, to inscribe a pentagon, draw an isosceles triangle with the base angles equal to twice the vertex angle. Bisect the base angles and the 5 points together make the pentagon.

Duplicate Ratio (Book V) Book VII begins with a definition of proportional which is based on the notion of duplicate ratio. Duplicate ratio ‘When three magnitudes are proportional, the first is said to have to the third the duplicate ratio of that which it has to the second.” Ex. 2:6:18

Euclidean Algorithm (Book VII) Process for finding the greatest common divisor. Given a, b with a > b, subtract b from a repeatedly until get remainder c. Then subtract c from b repeatedly until get to m, then subtract m from c……when the result = 0, you have the greatest common divisor or the result = 1, which means a and b are relatively prime. ex. 80 and 18 ex. 7 and 32

Prime Numbers Consider these 3 statements about primes found in Book VII: “Any composite number can be divided by some prime number. “Any number is either prime or can be divided by a prime number.” “If a prime number can be divided into the product of two numbers, it can be divided into one of them. These statements form the Fundamental Theorem of Arithmetic – that any number can be expressed uniquely as a product of prime numbers. In Book IX, Euclid proves through induction that there are infinitely many prime numbers.

Geometric Series (Book IX) “If as many as we please are in continued proportion, and there is subtracted from the second and the last numbers equal to the first, then, as the excess of the second is to the first, so will the last be to all those before it.” a, ar, ar², ar³,...arn (ar – a) (arn-a) (ar – a):a = (arn-a):Sn

Solve this last equation for Sn (ar – a):a = (arn-a):Sn Ex. Find the sum of the first 5 terms when a =1 and r =2

Knowing how to think- who needs it? Lawyers, politicians, negotiators, programmers, and anyone dealing with social issues! Abraham Lincoln carried a copy of The Elements (and read it) to become a better lawyer. The Declaration of Independence is set up in the same format as The Elements (self-evident truths are axioms used to prove that the colonies are justified in breaking from England). 19th century Yale students studied The Elements for two years, at the end of which they participated in a celebration ritual called the Burial of Euclid. E.T. Bell wrote ‘Euclid taught me that without assumptions, there is no proof. Therefore, in any argument, examine the assumptions.”

High School Geometry Plane Geometry courses today are basically the content of Euclid’s Elements. Two-column proof appeared in the 1900’s to make proofs easier but led to rote memorization instead. 1970’s moved away from proofs because they were ‘ too painful’ and not fun. Now proofs are brief and irrelevant. They do not serve the purpose of developing logical thinking.

PSSA Standards: what they should know Anchors: what they are tested on

Timeline Prior to Euclid, Greek mathematicians such as Pythagorus, Theaetetus, Euxodus and Thales did work in Geometry. 384-322 BCE - Aristotle believed that scientific knowledge could only be gained through logical methods, beginning with axioms. 300 BCE- Euclid teaches at the Museum and Library at Alexandria 1880 J.L. Heiberg compiles Greek version of The Elements as close to original as possible. 1908 Thomas Heath translated Heiberg’s text. This version is the one most widely used and the basis for modern Geometry courses.

References 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