Not just that I don’t have the maths to understand, it’s that I don’t even think the right way to learn the maths to understand. But:
And since that particular field of study was discovered, mathematicians have maintained that any alleged proof of the Pythagorean theorem which uses trigonometry constitutes a logical fallacy known as circular reasoning, a term used when someone tries to validate an idea with the idea itself.
Johnson and Jackson’s abstract adds that the book with the largest known collection of proofs for the theorem – Elisha Loomis’s The Pythagorean Proposition – “flatly states that ‘there are no trigonometric proofs because all the fundamental formulae of trigonometry are themselves based upon the truth of the Pythagorean theorem’.”
OK, you can’t use the assertion of the thing to be the proof of the thing itself.
But, the abstract counters, “that isn’t quite true”. The pair asserts: “We present a new proof of Pythagoras’s Theorem which is based on a fundamental result in trigonometry – the Law of Sines – and we show that the proof is independent of the Pythagorean trig identity sin2x+cos2x=1.” In short, they could prove the theorem using trigonometry and without resorting to circular reasoning.
Excellent. Now, who is going to tell us whether these girls have something here or not?
Dunno about the maths, but I was shocked when I saw their picture.
I thought Maths was supposed to be racist. Someone should have a word with these two.
I think that the problem with “proving” Pythagoras is that it is essentially an arithmetic expression and they are very difficult to do algebraically.
@ Ottokring
The classic proof relies on algebra applied to areas
Without seeing the “proof” I cannot tell you whether it is circular – The Guardian doesn’t show it
Buggered if I know.
I’ll be interested to see if their proof does hold up.
I wish them luck.
Pythagoras’s Theorem is trivial to prove without using trigonometry. Bhaskara’s first proof from the 12th century does so elegantly using the simplest algebra.
No, no. The game is to prove it using trig but without using Pythag itself in the trig you are using to prove Pythag by trig.
It’ll be a screed of impenetrable text and formulae, terminated by
You Get Me?
No in all seriousness, very well done indeed. Even if it doesn’t stand up, to be able to even attempt this is light years beyond my tiny brain.
I was taught a proof based upon two squares. The larger square is a+b on a side, and sitting inside it is a rotated square of side c. Hence 4 right-angled triangles of sides a,b with hippopotomus c.
The area of the larger square is (a+b)*(a+b) = a2 + b2 + 2ab (superscript in your head!)
The area of the smaller square is c2
The area of the 4 trianges is 2ab
The area of the larger square is clearly equal to the smaller square plus the 4 triangles. QED.
No circular reasoning. Just basic definitions or area, and binomial expansion. Algebra.
My instinct is with people who say you can’t use trig because it’s all based on the truth of Pythagoras.
I might be able to form a better judgement if they put their proof online. I’d start, though, by asking whether it’s true that “the Law of Sines” (whatever exactly that may be) is independent of ol’ Pyth.
I’ve just googled: apparently The Law of Sines is an American expression for The Sine Rule. Only one extra word? C’mon, America, you can do better than that. Though it probably does let earnest souls discuss the difference between a Law and a Rule.
I was once shown a proof of Pyth which seemed to me more elegant than the construction-of-squares method used at school. You add a line of construction and the result leaps off the page at you. Wish I could remember it.
The pal who showed it to me said he’d seen it attributed to Einstein, but then Americans are keen on attributing all sorts of things to Einstein (or Mark Twain or Abraham Lincoln). We agreed that when we showed it to anyone in future we’d attribute it to Frankie Howerd.
I thought Maths was supposed to be racist.
Well, if it turns out they’re wrong . . .
You could draw a triangle and measure it.
Pythagoras theorem is involved in the geometry of the Christian fish symbol. If you draw the two intersecting arcs with a radius of 153mm the body of the fish will be 265mm long. 265÷153=1.732 which is the square root of 3.
@ Stonyground
You haven’t specified the distance between the centres of the two circles whose arcs you are using.
“ =1.732 which is the square root of 3.”
No, that’s an approximation, the SQRT(3) is SQRT(3)and for the purposes of what they were trying to prove accuracy is important.
I’d like to see a proof that any aliens sufficiently advanced to visit Earth would understand the concept of prime numbers. And would not understand the concept of mashed potato before getting here.
Back on topic – there’s a en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics and this claimed Pythagorean problem isn’t on it. Could be that the list is incomplete but you’d think it would be there considering the simplicity in describing it and length of time being unsolved.
Bongo, that’s because the problem is proven. Has been many times over.
The trick/mental exercise here is to prove it using trigonometry.
Anyhoo.. They Schmott Grrrls. Good on ’em.
“You haven’t specified the distance between the centres of the two circles whose arcs you are using.”
He’s using Noah’s arcs.