relationship between more than just the squares on the sides of a right-angled triangle. The area of a semicircle on the hypotenuse, for example, is equal to the sum of the areas of the semicircles on the other two sides. A pentagon on the hypotenuse is equal to the sum of pentagons on the other two sides, and this holds for hexagons, octagons and, indeed, any regular or irregular shape. If, say, three Mona Lisas were drawn on a right-angled triangle, then the area of big Mona is equal to the combined area of the two smaller ones.
For me, the real delight in Pythagoras’s Theorem is in the realization of why it must be true. The simplest proof is as follows. It dates back to the Chinese, possibly to before even Pythagoras was born, and is one of the reasons why many doubt he came up with the theorem in the first place.
Stare at the two squares for a while before reading on. Square A is the same size as square B, and all the right-angled triangles inside the square are also the same size. Because the squares are equal, the white area inside them is also equal. Now, note that the big white square inside square A is the square of the hypotenuse of the riht-angled triangle. And the smaller white squares inside square B are the squares of the other two sides of the triangle. In other words, the square of the hypotenuse is equal to the square of the other two sides. Voilà.
Since we can construct a square like A and B for any shape or size of right-angled triangle, the theorem must be true in all cases.
The thrill of maths is the moment of instant revelation, from proofs such as this, when suddenly everything makes sense. It is immensely satisfying, an almost physical pleasure. The Indian mathematician Bhaskara was so taken by this proof that underneath a picture of it in his twelfth-century maths book Lilavati , he wrote no explanation, just the word ‘Behold!’
There are many other proofs of Pythagoras’s Theorem, and a particularly lovely one is found in the figure overleaf, credited to the Arabic mathematician Annairizi, and dated around 900 CE . The theorem is contained within the repeating pattern. Can you spot it? (If you can’t, some help is included as an appendix .)
In his 1940 book The Pythagorean Proposition , Elisha Scott Loomis published 371 proofs of the theorem, devised by a surprisingly diverse collection of people. One dating to 1888 was attributed to E.A. Coolidge, a blind girl, another to Ann Condit, a 16-year-old high-school student, dating to 1938, while others were attributed to Leonardo da Vinci and US President James A. Garfield. Garfield had stumbled on his proof during some mathematical amusements with colleagues when he was a Republican congressman. ‘We think it something on which the members of both houses can unite without distinction of party,’ he said when the proof was first published in 1876.
The diversity of proofs is a testament to the vitality of maths. There is never a ‘right’ way to attack a maths problem, and it’s intriguing to chart the different routes that different minds have taken in finding solutions. Above opposite are three different proofs from three different eras: one by Liu Hui, a Chinese mathematician from the third century ce, one by Leonardo da Vinci (1452–1519) and the third by Henry Dudeney, Britain’s most famous puzzlist, dated 1917. Both Liu Hui and Dudeney’s are ‘dissection proofs’ in which the two small squares are divided into shapes that can be reassembled perfectly in the big square. Leonardo’s needs a little more thought. (If you need help, see the appendix again.)
A particularly dynamic proof was devised by the mathematician Hermann Baravalle, shown below opposite. There is something more organic about this one – it shows how the big square, like an amoeba, divides itself into the two smaller ones. At each stage, the area shaded is the same. The only step that isn’t obvious is step 4. When a parallelogram
For me, the real delight in Pythagoras’s Theorem is in the realization of why it must be true. The simplest proof is as follows. It dates back to the Chinese, possibly to before even Pythagoras was born, and is one of the reasons why many doubt he came up with the theorem in the first place.
Stare at the two squares for a while before reading on. Square A is the same size as square B, and all the right-angled triangles inside the square are also the same size. Because the squares are equal, the white area inside them is also equal. Now, note that the big white square inside square A is the square of the hypotenuse of the riht-angled triangle. And the smaller white squares inside square B are the squares of the other two sides of the triangle. In other words, the square of the hypotenuse is equal to the square of the other two sides. Voilà.
Since we can construct a square like A and B for any shape or size of right-angled triangle, the theorem must be true in all cases.
The thrill of maths is the moment of instant revelation, from proofs such as this, when suddenly everything makes sense. It is immensely satisfying, an almost physical pleasure. The Indian mathematician Bhaskara was so taken by this proof that underneath a picture of it in his twelfth-century maths book Lilavati , he wrote no explanation, just the word ‘Behold!’
There are many other proofs of Pythagoras’s Theorem, and a particularly lovely one is found in the figure overleaf, credited to the Arabic mathematician Annairizi, and dated around 900 CE . The theorem is contained within the repeating pattern. Can you spot it? (If you can’t, some help is included as an appendix .)
In his 1940 book The Pythagorean Proposition , Elisha Scott Loomis published 371 proofs of the theorem, devised by a surprisingly diverse collection of people. One dating to 1888 was attributed to E.A. Coolidge, a blind girl, another to Ann Condit, a 16-year-old high-school student, dating to 1938, while others were attributed to Leonardo da Vinci and US President James A. Garfield. Garfield had stumbled on his proof during some mathematical amusements with colleagues when he was a Republican congressman. ‘We think it something on which the members of both houses can unite without distinction of party,’ he said when the proof was first published in 1876.
The diversity of proofs is a testament to the vitality of maths. There is never a ‘right’ way to attack a maths problem, and it’s intriguing to chart the different routes that different minds have taken in finding solutions. Above opposite are three different proofs from three different eras: one by Liu Hui, a Chinese mathematician from the third century ce, one by Leonardo da Vinci (1452–1519) and the third by Henry Dudeney, Britain’s most famous puzzlist, dated 1917. Both Liu Hui and Dudeney’s are ‘dissection proofs’ in which the two small squares are divided into shapes that can be reassembled perfectly in the big square. Leonardo’s needs a little more thought. (If you need help, see the appendix again.)
A particularly dynamic proof was devised by the mathematician Hermann Baravalle, shown below opposite. There is something more organic about this one – it shows how the big square, like an amoeba, divides itself into the two smaller ones. At each stage, the area shaded is the same. The only step that isn’t obvious is step 4. When a parallelogram
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