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Pythagoras' Theorem
Background
Pythagoras, the Man
Born on the Aegean island of Samos in around 580 BC, Pythagoras travelled as a young man to Egypt and Babylon. He eventually settled in Crotona, a Greek colony in southern Italy. There he founded a brotherhood that studied religion, politics and philosophy as well as mathematics and science. Actually Pythagoras’ most famous ‘discovery’ was already known to the Babylonians, the Chinese and the Egyptians. But the Pythagorean brotherhood may have provided the first general proof of the theorem.
Chronology
(by Roger Webster, University of Sheffield)
- 2000 BC: According to the archaeologist Alexander Thom, megalithic man built stone ellipses based on the Pythagorean triples (3,4,5), (5,13,13) and (12,35,37).
- 1800: The Babylonians knew many Pythagorean triples, and the theorem itself.
- 540: Pythagoras is often credited with the first proof of the theorem.
- 500: Hindu sulvasutras show acquaintance with the theorem and the Pythagorean triples (3,4,5), (5,12,13), (8,15,17) and (12,35,37).
- 440: Hippocrates of Chios used the theorem to prove that certain lunes had areas that could be found exactly by the methods of Greek geometry.
- 300: Euclid gives the classical Greek proofs of the theorem and its converse (Elements I: Propositions 47, 48).
- 200: Chou-pei boasts the first diagrammatical proof of the theorem.
- 100: Chiu-chang suan-shu contains a chapter explaining some impressive applications of the theorem, including the Pythagorean triples (48,55,73) and (60,91,109).
- AD 150: Ptolemy’s Almagest presents a spherical version of the theorem.
- 300: Pappus’ ‘Mathematical Collection’ extends the theorem to parallelograms described on any two sides of any triangle.
- 870: Thabit Ibn Qurra gives a further generalisation of the theorem.
- 1783: De Gua announces a 3-D version of the theorem: the square of the hypotenuse of a right-tetrahedron equals the sum of the squares of its other three faces.
- 1876: James Garfield, twentieth president of the United States, publishes a nice proof of the theorem in the New England Journal of Education.
- 1917: W Lietzman devotes his Der Pythagorische Lehrsat to the theorem.
- 1940: E T Loomis’s The Pythagorean Proposition provides 370 proofs of its title.
- 1955: Greece commemorates the theorem with a set of four stamps.
- 1971: Nicaragua issues a series of stamps honouring the ‘ten most important mathematical formulae’: the formula A2 + B2+ C2 is represented.
- 1995: The Guinness Book of Records declares Pythagoras’ Theorem the most proved theorem of all time.
Models to Make
1.
Take two strips of card, 15cm by 3cm, draw lines on them as shown in the above illustration, and cut along the lines.
In the centre of a blank piece of paper, accurately draw the right-angled triangle shown below.
Take one of the sets of five pieces, place the square against the 3cm side of the triangle, and arrange the other four pieces to make a square against the 6cm side.
Take the other set of five pieces and assemble them to make a square on the longest side of the triangle.
This demonstrates Pythagoras’ Theorem.
2.
Take a sheet of card and draw a square on it. Mark off an equal distance along each side and draw lines as in the diagram below to make two congruent triangles.
Cut out two more copies of the same triangle from another piece of card, leaving a bit of extra space around the corners as shown below.
Place these on the square and use paper fasteners to pivot them. Rotate each of the bottom triangles by a quarter turn, one clockwise, the other anticlockwise (see below).
In the first diagram there is a square in the middle with four triangles on the outside (the square on the hypotenuse). In the second diagram there are two smaller squares on the smaller sides of the triangles (the squares on the other two sides). The two diagrams are really the same big square, but with the triangles in different positions. This demonstrates Pythagoras’ Theorem.
3. Perigal’s Dissection
In 1873 the mathematican Perigal showed that two squares lying next to each other could be cut into five pieces: four congruent quadrilaterals and a small square.
These five pieces can then be rearranged into one large square. This provides an illustrative proof of Pythagoras’ Theorem:
4. Pythagorean Spiral
In the centre of a blank page draw a right-angled triangle with two sides exactly one unit long:
Draw a new right-angled triangle on the hypotenuse of the first triangle, keeping the outer side one unit long:
Continue this process to form the Pythagorean Spiral:
Pythagorean Triples
Groups of three numbers (a,b,c) which satisfy a2 + b2 = c2 are called Pythagorean triples. Here are the first 12 primitive triples (triples that are not multiples of smaller triples):
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3
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4
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5
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5
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12
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13
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8
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15
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17
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7
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24
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25
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20
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21
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29
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12
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35
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37
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9
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40
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41
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28
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45
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53
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11
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60
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61
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16
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63
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65
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33
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56
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65
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48
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55
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73
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The Greeks used formulas to generate Pythagorean triples. Choose two numbers m and n, and then let:
a = m2 — n2
b = 2mn
c = m2 + n2
By using these formulas it is possible to find some interesting patterns in Pythagorean triples. Investigate what happens when m and n are consecutive numbers. What if m and n are triangular numbers?
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