Chapter 2: Euclid’s Proof of the Pythagorean Theorem

1. Chapter 2: Euclid’s Proof of the Pythagorean Theorem Math 402Elaine RobanchoGrant Weller

2. Outline • Euclid and his Elements • Preliminaries: Definitions, Postulates, and Common Notions • Early Propositions • Parallelism and Related Topics • Euclid’s Proof of the Pythagorean Theorem • Other Proofs

3. Euclid • Greek mathematician – “Father of Geometry” • Developed mathematical proof techniques that we know today • Influenced by Plato’s enthusiasm for mathematics • On Plato’s Academy entryway: “Let no man ignorant of geometry enter here.” • Almost all Greek mathematicians following Euclid had some connection with his school in Alexandria

4. Euclid’s Elements • Written in Alexandria around 300 BCE • 13 books on mathematics and geometry • Axiomatic: began with 23 definitions, 5 postulates, and 5 common notions • Built these into 465 propositions • Only the Bible has been more scrutinized over time • Nearly all propositions have stood the test of time

5. Preliminaries: Definitions • Basic foundations of Euclidean geometry • Euclid defines points, lines, straight lines, circles, perpendicularity, and parallelism • Language is often not acceptable for modern definitions • Avoided using algebra; used only geometry • Euclid never uses degree measure for angles

6. Preliminaries: Postulates • Self-evident truths of Euclid’s system • Euclid only needed five • Things that can be done with a straightedge and compass • Postulate 5 caused some controversy

7. Preliminaries: Common Notions • Not specific to geometry • Self-evident truths • Common Notion 4: “Things which coincide with one another are equal to one another” • To accept Euclid’s Propositions, you must be satisfied with the preliminaries

8. Early Propositions • Angles produced by triangles • Proposition I.20: any two sides of a triangle are together greater than the remaining one • This shows there were some omissions in his work • However, none of his propositions are false • Construction of triangles (e.g. I.1)

9. Early Propositions: Congruence • SAS • ASA • AAS • SSS • These hold without reference to the angles of a triangle summing to two right angles (180˚) • Do not use the parallel postulate

10. Parallelism and related topics • Parallel lines produce equal alternate angles (I.29) • Angles of a triangle sum to two right angles (I.32) • Area of a triangle is half the area of a parallelogram with same base and height (I.41) • How to construct a square on a line segment (I.46)

11. Pythagorean Theorem: Euclid’s proof • Consider a right triangle • Want to show a2 + b2 = c2

12. Pythagorean Theorem: Euclid’s proof • Euclid’s idea was to use areas of squares in the proof. First he constructed squares with the sides of the triangle as bases.

13. Pythagorean Theorem: Euclid’s proof • Euclid wanted to show that the areas of the smaller squares equaled the area of the larger square.

14. Pythagorean Theorem: Euclid’s proof • By I.41, a triangle with the same base and height as one of the smaller squares will have half the area of the square. We want to show that the two triangles together are half the area of the large square.

15. Pythagorean Theorem: Euclid’s proof • When we shear the triangle like this, the area does not change because it has the same base and height. • Euclid also made certain to prove that the line along which the triangle is sheared was straight; this was the only time Euclid actually made use of the fact that the triangle is right.

16. Pythagorean Theorem: Euclid’s proof • Now we can rotate the triangle without changing it. These two triangles are congruent by I.4 (SAS).

17. Pythagorean Theorem: Euclid’s proof • We can draw a perpendicular (from A to L on handout) by I.31 • Now the side of the large square is the base of the triangle, and the distance between the base and the red line is the height (because the two are parallel).

18. Pythagorean Theorem: Euclid’s proof • Just like before, we can do another shear without changing the area of the triangle. • This area is half the area of the rectangle formed by the side of the square and the red line (AL on handout)

19. Pythagorean Theorem: Euclid’s proof • Repeat these steps for the triangle that is half the area of the other small square. • Then the areas of the two triangles together are half the area of the large square, so the areas of the two smaller squares add up to the area of the large square. • Therefore a2 + b2 = c2 !!!!

20. Pythagorean Theorem: Euclid’s proof • Euclid also proved the converse of the Pythagorean Theorem; that is if two of the sides squared equaled the remaining side squared, the triangle was right. • Interestingly, he used the theorem itself to prove its converse!

21. Other proofs of the Theorem Mathematician Proof • Chou-pei Suan-ching (China), 3rd c. BCE • Bhaskara (India), 12th c. BCE • James Garfield (U.S. president), 1881

22. Further issues • Controversy over parallel postulate • Nobody could successfully prove it • Non-Euclidean geometry: Bolyai, Gauss, and Lobachevski • Geometry where the sum of angles of a triangle is less than 180 degrees • Gives you the AAA congruence