is ‘sheared’, or moved in such a way that preserves the base and altitude, its area stays the same.
Baravalle’s proof is similar to the most established one in mathematical literature, that set out by Euclid around 300 BC .
Euclid, the next most famous Greek mathematician after Pythagoras, lived in Alexandria, the city founded by the man who never abbreviated his name to Alex the Great. His chef-d’œuvre , The Elements , contained 465 theorems that summarized the extent of Greek knowledge at that time. Greek mathematics was almost entirely geometry – derived from their words for ‘earth’ and ‘measurement’ – although The Elements was not concerned with the real world. Euclid was operating in an abstract domain of points and lines. All he allowed in his toolkit was a pencil, a ruler and a compass, which is why they have been the fundamental components of children’s pencil cases for centuries.
Euclid’s first task (Book 1, Proposition 1) was to show that, given any line, he could make an equilateral triangle, i.e. a triangle with three equal sides, with that line as one side:
Step 1: Put the compass point on one end of the given line and draw a circle that passes through the other end of the line.
Step 2: Repeat the first step with the compass on the other end of the line. You now have two intersecting circles.
Step 3: Draw a line from one of the intersections of the circle to the end points of the original line.
The Elements , Proposition 1.
He then meticulously progressed from proposition to proposition, revealing a host of properties of lines, triangles and circles. For example, Proposition 9 shows how to ‘bisect’ an angle, that is construct an angle that is exactly half of a given angle. Proposition 32 states that the internal angles of a triangle always add up to two right angles, or 180 degrees. The Elements is a magnum opus of pedantry and rigour. Nothing is ever assumed. Every line follows logically from the line before. Yet from only a few basic axioms, Euclid amassed an impressive body of compelling results.
The grand finale of the first book is Proposition 47. The commentary from a 1570 edition of the first English translation reads: ‘This most excellent and notable Theoreme was first invented of the greate philosopher Pithagoras, who for the exceeding ioy conceived of the invention thereof, offered in sacrifice an Oxe, as recorde Hierone, Proclus, Lycius, & Vitruvius. And it hath bene commonly called of barbarous writers of the latter time Dulcarnon.’ Dulcarnon means two-horned, or ‘at wit’s end’ – possibly because the diagram of the proof has two horn-like squares, or possibly because understanding it is indeed horribly difficult.
There is nothing pretty about Euclid’s proof of Pythagoras’s Theorem. It is long, meticulous and convoluted, and requires a diagram full of lines and superimposed triangles. Arthur Schopenhauer, the nineteenth-century German philosopher, said it was so unnecessarily complicated that it was a ‘brilliant piece of perversity’. To be fair to Euclid, he was not trying to be playful (as was Dudeney), or aesthetic (as was Annairizi) or intuitive (as was Baravalle). Euclid’s driving concern was the of his deductive system.
While Pythagoras saw wonder in numbers, Euclid in The Elements reveals a deeper beauty, a watertight system of mathematical truths. On page after page he demonstrates that mathematical knowledge is of a different order than any other. The propositions of The Elements are true in perpetuity. They do not become less certain, or indeed less relevant with time (which is why Euclid is still taught at school and why Greek playwrights, poets and historians are not). The Euclidean method is awe-inspiring. The seventeenth-century English polymath Thomas Hobbes is said to have glanced at a copy of The Elements that lay open in a library when he was a 40-year-old man. He read one of the propositions
Baravalle’s proof is similar to the most established one in mathematical literature, that set out by Euclid around 300 BC .
Euclid, the next most famous Greek mathematician after Pythagoras, lived in Alexandria, the city founded by the man who never abbreviated his name to Alex the Great. His chef-d’œuvre , The Elements , contained 465 theorems that summarized the extent of Greek knowledge at that time. Greek mathematics was almost entirely geometry – derived from their words for ‘earth’ and ‘measurement’ – although The Elements was not concerned with the real world. Euclid was operating in an abstract domain of points and lines. All he allowed in his toolkit was a pencil, a ruler and a compass, which is why they have been the fundamental components of children’s pencil cases for centuries.
Euclid’s first task (Book 1, Proposition 1) was to show that, given any line, he could make an equilateral triangle, i.e. a triangle with three equal sides, with that line as one side:
Step 1: Put the compass point on one end of the given line and draw a circle that passes through the other end of the line.
Step 2: Repeat the first step with the compass on the other end of the line. You now have two intersecting circles.
Step 3: Draw a line from one of the intersections of the circle to the end points of the original line.
The Elements , Proposition 1.
He then meticulously progressed from proposition to proposition, revealing a host of properties of lines, triangles and circles. For example, Proposition 9 shows how to ‘bisect’ an angle, that is construct an angle that is exactly half of a given angle. Proposition 32 states that the internal angles of a triangle always add up to two right angles, or 180 degrees. The Elements is a magnum opus of pedantry and rigour. Nothing is ever assumed. Every line follows logically from the line before. Yet from only a few basic axioms, Euclid amassed an impressive body of compelling results.
The grand finale of the first book is Proposition 47. The commentary from a 1570 edition of the first English translation reads: ‘This most excellent and notable Theoreme was first invented of the greate philosopher Pithagoras, who for the exceeding ioy conceived of the invention thereof, offered in sacrifice an Oxe, as recorde Hierone, Proclus, Lycius, & Vitruvius. And it hath bene commonly called of barbarous writers of the latter time Dulcarnon.’ Dulcarnon means two-horned, or ‘at wit’s end’ – possibly because the diagram of the proof has two horn-like squares, or possibly because understanding it is indeed horribly difficult.
There is nothing pretty about Euclid’s proof of Pythagoras’s Theorem. It is long, meticulous and convoluted, and requires a diagram full of lines and superimposed triangles. Arthur Schopenhauer, the nineteenth-century German philosopher, said it was so unnecessarily complicated that it was a ‘brilliant piece of perversity’. To be fair to Euclid, he was not trying to be playful (as was Dudeney), or aesthetic (as was Annairizi) or intuitive (as was Baravalle). Euclid’s driving concern was the of his deductive system.
While Pythagoras saw wonder in numbers, Euclid in The Elements reveals a deeper beauty, a watertight system of mathematical truths. On page after page he demonstrates that mathematical knowledge is of a different order than any other. The propositions of The Elements are true in perpetuity. They do not become less certain, or indeed less relevant with time (which is why Euclid is still taught at school and why Greek playwrights, poets and historians are not). The Euclidean method is awe-inspiring. The seventeenth-century English polymath Thomas Hobbes is said to have glanced at a copy of The Elements that lay open in a library when he was a 40-year-old man. He read one of the propositions
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