Euclid · Mathematics
Euclid’s proof of the most famous theorem in mathematics - that the square on the hypotenuse of a right triangle equals the sum of the squares on the other two sides - and how he proves it by pure geometry, without measuring anything.
Take any right-angled triangle - a triangle with one corner a perfect right angle (ninety degrees). The two sides that form the right angle are the legs; the long side opposite the right angle is the hypotenuse. The Pythagorean theorem states a remarkable relationship: the square built on the hypotenuse has exactly the same area as the two squares built on the legs added together. If the legs have lengths 3 and 4, then the squares on them have areas 9 and 16, summing to 25 - and the square on the hypotenuse has area 25, so the hypotenuse has length 5. This relationship holds for every right triangle, no matter its size or shape, so long as it has a right angle. In the algebraic shorthand we use today: a² + b² = c², where c is the hypotenuse. It is the single most famous theorem in all of mathematics - known to the Babylonians and the ancient Indians and Chinese as a practical rule, but it was the Greeks who first proved it must always be true, and Euclid who gave the proof its classic form as Proposition 47 of Book I.
Euclid’s proof of Proposition 47 is a masterpiece of geometric reasoning, and its central idea can be grasped without algebra. Picture the right triangle with the large square built on the hypotenuse below it. Euclid drops a line from the right-angle corner straight down, perpendicular to the hypotenuse, continuing it to divide the large square into two rectangles. His goal is to show that the left rectangle has exactly the area of the square on the left leg, and the right rectangle has exactly the area of the square on the right leg - so that the two rectangles together (the whole large square) equal the two leg-squares together. To prove each rectangle equals its corresponding leg-square, Euclid uses a clever chain: he shows that a certain triangle has half the area of the leg-square, and another triangle has half the area of the rectangle, and that these two triangles are congruent (identical in shape and size, proved using an earlier proposition about side-angle-side). Since the two triangles are equal, their doubles - the leg-square and the rectangle - are equal too. Apply the same argument on the other side, add the two results, and the square on the hypotenuse is shown to equal the sum of the squares on the legs. No ruler, no measurement - only the rigorous comparison of areas, justified at every step.
The Pythagorean theorem is far more than a fact about triangles; it is one of the load-bearing pillars of all mathematics. It is the fundamental link between length and area, and between the two perpendicular directions of space - it tells us how distance itself is measured. Every time we compute the straight-line distance between two points from their horizontal and vertical separations, we are using the Pythagorean theorem; it is the formula at the heart of coordinate geometry, of trigonometry, of navigation and surveying and engineering. It generalises to three dimensions (the diagonal of a box) and beyond, and in its most abstract form - the rule for measuring distance in a space - it becomes the definition of distance in the geometry of the universe itself, from Einstein’s spacetime to the abstract spaces of modern physics. When mathematicians wanted to describe curved space, the first thing they had to do was modify the Pythagorean rule. It is no exaggeration to say that the theorem Euclid proved in Book I, Proposition 47, is woven into the deep structure of how we understand space and measurement everywhere. And it carried a sting in its tail - for applied to the simplest case of all, the diagonal of a unit square, it produces a number that shattered the deepest belief of the Pythagoreans, as we will see.
This is the opening of the lesson. The rest — the dialogue, the primary source, and the recall — is in the app.
You learned the Pythagorean theorem and how Euclid proved it (Book I, Proposition 47). State the theorem and explain the idea behind Euclid’s proof.
Leads to Pythagoras.
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