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