Yes, I did try Googling some math pages. I found a zillion different ways the theorem has been proven, but I couldn’t find exactly what I was looking for. What I want to know is three things:
First, how exactly is a right angle defined for purposes of geometric proofs. I guess I instinctively know what it is, but how is it actually defined? Is it half of the “angle” that’s a straight line? Is it the angle in a triangle that’s equal to the other two angles combined?
Second, what led up to the formulation of the theorem? I’m not talking specifically what proofs you already have to have in hand before you can prove the Pythagorean theorem; I mean more in the way of what led anyone to suppose there was something special or significant about triangles containing a right angle. Which leads to a related question:
Third, what’s the significance of the Pythagorean Theorem? Ok, so you have the interesting fact that you can relate the sides of a right triangle to each other. Why is that a big whoop? I know it’s extremely useful in many ways, and leads for instance to the fact that you can plot any point on a graph as a line segment from the origin forming the hypotenuse of a triangle with the X and Y coordinates as the other two sides. But in terms of the original Greek geometry, what did the theorem achieve that it’s formulators were looking for?
IIRC, a right angle is one where the line is perpendicular to the baseline.
As far as where it came from, people had noticed that triangles with a right angle often showed up in particular ratios. Thus, a right angle with one leg of 3 units and another leg of 4 units would have a hypoteneuse of 5 units. Same for multiples of these (e.g., 9, 12, and 15) and for a few other combinations (5, 12, 13 and 7, 24, 25, for example). The theorem went from the specific to the general.
Exactly what lead to the formulation isn’t known. One suggestion was that the idea was suggested by the tesselae of a mosaic – someone noticed that if you combined the little tiles from a square made using as a side one leg of a right triangle and those of a square based upon the other leg you had just enough to cover the area of a square made using as a side the hypoteneuse. That’s certainly one possibility.
the Greeks, of course, viewed the problem as one of equating the areas of the the squares based on the legs and the area of the square made from the hypoyeneuse, not as an algebraic problem as we do.
They weren’t “Looking” for anything. They didn’t have a Cartesian Coordinate system that this could give the lengths of diagonals in. They were interested in the relationships within geometry. This theorem has several implications, one of them being a proof that there are irrational numbers (ones which cannot be expressed as a simple ratio), since the length of the hypoteneuse of a right triangle whose legs are both of unit length cannot be expressed as a simple fraction. That was a pretty big deal in its day.
By the way, the Egyptians knew that a 3, 4, 5 triangle had a right angle in it a LONG time before Pythagoras. As reality Chuck mentions, the Pythagorean Theorem gave a general solution.
This only works if you have a notion of perpendicularity that isn’t defined in terms of right angles. Analytic geometry has a couple, but that’s not what the Greeks were working with. I don’t know the actual definition used in synthetic geometry, but defining a right angle as half of the angle formed by a straight line, or the angle that cuts off a quarter circle, is a very reasonable thing to do.
I don’t know if this counts as significant, but you can use the theorem to make sure the walls you are building are at a right angle, for a rectangular room.
The example that was put to me was this:
To make sure your walls are at a 90 degree angle, measure 6’ from the corner on one wall, and 8’ on the other. The distance between them should always be 10’ if your walls are straight, and a 90 degree angle.
And if it works for making sure a room is ‘square’ then a right angle can be verified by using known values for the theorem.
Couldn’t find the edit button… newbie here.
Anyway, of course the theorem only works with flat surfaces (euclidean geometry?). If you were to measure this on a balls exterior or interior, the angles would be different.
I don’t think that they were ‘looking’ for anything, in particular. They were more theoretical rather than practical scientists. But it does turn out to have a lot of practical applications, in things like surveying plots of land, construction of square, stable buildings, etc.
Yup. If it’s good enough for Euclid , it’s good enough for me!
Well in Euclidean geometry it turns out to be. But that’s because there are 180 degrees in a triangle. Which in turn is because alternate angles on a line crossing two parallels are equal. Which follows from Euclid’s fifth postulate (EFP).
In non-Euclidean geometries this is not the case, because EFP is replaced with other axioms. However the first definition of a right angle holds even in those geometries. (A bit confusingly, ‘non-Euclidean geometries’ traditionally only refer to geometries in which EFP is replaced, the rest of Euclid’s system is kept in!)
Then Fermat asserted in 1637 that there were no natural numbers other that 2 that satisfied the equation: x^n + y^b = z^n. He claimed that he had a proof thereof.
It was not until in recent years that it was proven.
I think Nancarrow’s link answers this one. In general, if you’re really interested in questions like this, you can look at Euclid’s Elements, which sets out all the definitions, theorems, and proofs in a logical and systematic order.
For what it’s worth, IIRC the Pythagorean theorem is the earliest theorem that has a person’s name associated with it.
You may be asking the wrong question, or mistakenly assuming that the Greek geometers were trying to achieve something useful. Arguably the Greeks’ greatest contribution to mathematics was appreciating mathematics for its own sake, for its beauty and intellectual satisfaction without any regard whatsoever for its usefulness. The attitude of the Greeks (or at least some of them) toward things like the Pythagorean theorem would have been, “Hey, neat! Is this always true? Why is it true? How can we prove that it’s true?” rather than developing and using it in trying to solve some practical problem. Or at least so I understand.
No, I wasn’t talking about practical real world applications; I meant what were the Greeks “looking for” in terms of unifying or underpinning their mathematical theory. In other words, was the Pythagorean Theorem the lynchpin of their system of geometry?
I guess I should find a good site for the original Elements.
I don’t think you can really say that. If anything was “underpinning” the Greeks’ geometry, it was the axioms that Euclid started with.
What the Pythagorean theorem does underpin is the standard notion of distance in the plane (or, by extension, in higher dimensions), called the Euclidean metric. (See here or here.) Other ways of defining the “distance” between two points—other metrics—are possible.
That is the legend; I don’t think we have any way of knowing whether it’s true or not, but you’re entitled to be suspicious. There’s a lot we don’t know for sure (if at all) about Pythagoras and the Pythagoreans.
