The Pythagorean Theorem was the crown jewel of Euclid's Elements, appearing as Proposition 47 (out of 48) in Book I: "In right-angled triangles the square on the side opposite the right angle equals the sum of the squares on the sides containing the right angle."
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html
Note that the 48th and final proposition of the Elements, Book I was the converse of the Pythagorean Theorem.
The theorem itself was known to cultures across the world. The Babylonians, Chinese, and Egyptians knew it long before Pythagoras. However, Euclid's proof of the Pythagorean Theorem was the first formal proof of a theorem revered across the world. That is because Euclid invented the first ever axiomatic system for constructing formal proofs, building upon Plato's rules of geometric construction.
This page is dedicated to understanding the intuition that might have motivated Euclid's formal proof.
Consider the app shown below. Move Point A up and down, producing a vector from A to C. When the vector has length zero, there is only the green square, with side lengths of, say, a. However, when then vector has non-zero length (say, b), we get a right triangle, with sides a, b, and c (the hypotenuse).
https://mathcs.clarku.edu/~djoyce/java/elements/bookI/bookI.html
Note that the 48th and final proposition of the Elements, Book I was the converse of the Pythagorean Theorem.
The theorem itself was known to cultures across the world. The Babylonians, Chinese, and Egyptians knew it long before Pythagoras. However, Euclid's proof of the Pythagorean Theorem was the first formal proof of a theorem revered across the world. That is because Euclid invented the first ever axiomatic system for constructing formal proofs, building upon Plato's rules of geometric construction.
This page is dedicated to understanding the intuition that might have motivated Euclid's formal proof.
Consider the app shown below. Move Point A up and down, producing a vector from A to C. When the vector has length zero, there is only the green square, with side lengths of, say, a. However, when then vector has non-zero length (say, b), we get a right triangle, with sides a, b, and c (the hypotenuse).
To understand the Pythagorean Theorem is to understand how area is produced and transformed.
A length is produced when we sweep a point to another point, such as the segment of length a from B to C, or the segment of length b from C to A.
An area is produced when we sweep a segment in a new direction. For example, the square of area axa is constructed when we sweep the segment BC orthogonally by a distance of a.
So sweeps produce area, at least under certain conditions (i.e., sweeping a line segment in a new direction). Other actions preserve area. For example, rotations, reflections, and translations have no effect on the areas of the figures they transform. Shearing is another area-preserving transformation--one that is more challenging to visualize (see Proposition 36 in Book I of Euclid's Elements)..
Returning to the app, we can see the results of several different shears. For example, the green square is transformed into a rhombus as its right side is pushed up by the vector CA. Also, the rhombus is transformed into the blue rectangle by a second shear. All three figures--the green square, the rhombus, and the blue rectangle have the same area.
Likewise, the orange square, orange rhombus, and red rectangle all have the same area. In sum, we can see that the red rectangle and blue rectangle form a square with side length c. So, a^2 + b^2 = c^2.
However, we can make an even simpler argument for the Pythagorean Theorem if we understand the way area is produced, through orthogonal sweeps. The area of the green square was produced through a pair of orthogonal sweeps of length a. While BC is transformed into BA, by the sweep from C to A, of length b, the vertical side is transformed by an orthogonal sweep of length b so that we have added an area of b^2 to our original area of a^2 in order to produce a square of area c^2.
A length is produced when we sweep a point to another point, such as the segment of length a from B to C, or the segment of length b from C to A.
An area is produced when we sweep a segment in a new direction. For example, the square of area axa is constructed when we sweep the segment BC orthogonally by a distance of a.
So sweeps produce area, at least under certain conditions (i.e., sweeping a line segment in a new direction). Other actions preserve area. For example, rotations, reflections, and translations have no effect on the areas of the figures they transform. Shearing is another area-preserving transformation--one that is more challenging to visualize (see Proposition 36 in Book I of Euclid's Elements)..
Returning to the app, we can see the results of several different shears. For example, the green square is transformed into a rhombus as its right side is pushed up by the vector CA. Also, the rhombus is transformed into the blue rectangle by a second shear. All three figures--the green square, the rhombus, and the blue rectangle have the same area.
Likewise, the orange square, orange rhombus, and red rectangle all have the same area. In sum, we can see that the red rectangle and blue rectangle form a square with side length c. So, a^2 + b^2 = c^2.
However, we can make an even simpler argument for the Pythagorean Theorem if we understand the way area is produced, through orthogonal sweeps. The area of the green square was produced through a pair of orthogonal sweeps of length a. While BC is transformed into BA, by the sweep from C to A, of length b, the vertical side is transformed by an orthogonal sweep of length b so that we have added an area of b^2 to our original area of a^2 in order to produce a square of area c^2.