Euclid’s Elements consists of a chain of theorems each proceeding from the ones before. The starting point of this chain is a set of axioms and postulates, one of which is far from obvious.
Any logical chain of reasoning must start from somewhere, and in the case of Euclid it is a set of common notions and postulates. The common notions apply to all reasoning and are uncontroversial. They contain obvious statements such as the first one which states: Things which are equal to the same thing are equal to each other.
The postulates are specifically about geometry. The first four are deemed unexceptionable, for example the fourth states: …all right angles are equal to each other.
The fifth postulate is much longer and more complicated than the others:
That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
In the diagram on the opposite page, XY and PQ are the two straight lines. Another straight line crosses them, and the interior angles on the same side are a and b.
Suppose that the sum of these angles is less than two right angles, that is, a + b < 90° + 90° = 180°. Then, if XY and PQ are extended in both directions, they will meet on the left-side of the diagram.
This is also called the parallel postulate. If XY and PQ are parallel, then they will never meet on either side. Hence a + b must be exactly 180°.
Euclid manages to avoid this postulate until Proposition 29, which proves well-known results about alternate and corresponding angles. He has to use it to prove, for example, that the sum of the angles of a triangle is 180°. The fifth postulate is also necessary to prove Pythagoras’s Theorem and many other standard results of geometry.
There were many attempts to prove the fifth postulate from the other four but these attempts were always shown to contain some other assumption.
It was a long time before people began to suspect that it was impossible to prove the fifth postulate and that there are many possible geometries, some assuming the fifth postulate and some denying it.
See: Non-Euclidean Geometry, page 113–114
3rd century BC Greece
Sum of the Angles in a Triangle
The sum of the angles in a triangle is two right angles, or 180°.
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For any triangle, the sum of its angles is always 180°. This result relies on a disputed axiom of geometry and is familiar from school geometry.
Take a triangle ABC and draw a line DAE parallel to BC.
Then ∠ABC = ∠DAB and ∠ACB = ∠EAC.
So ∠BAC + ∠ABC + ∠ACB = ∠BAC + ∠DAB + ∠EAC.
The left-hand side of this equation, ∠BAC + ∠ABC + ∠ACB, is the sum of the angles of the triangle.
The right-hand side, ∠BAC + ∠DAB + ∠EAC, is the sum of the angles along a straight line, which is 180°. So the sum of the angles of the triangle is 180°.
In the proof above we stated that:
∠ABC = ∠DAB and ∠ACB = ∠EAC
These pairs of angles are known as “alternate angles”. The equality of alternate angles is a consequence of the fifth postulate of Euclid concerning parallel lines. A different postulate could give a different result: it might be that the sum of the angles is less than 180°, or greater than 180°.
Summing the angles.
3rd century BC Greece
The Fundamental Theorem of Arithmetic
Euclid (c. 325–265 BC)
Every whole number can be written as a product of prime numbers.
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It is possible to factorize any whole number until one is left with prime numbers which, by definition, cannot be factorized any further.
A prime number is a whole number whose only factors are 1 and itself. For example 11 is a prime number, as it cannot be written as the product of smaller numbers. The number 15 is not a prime number, as it can be written as 3 x 5. The first few prime numbers are 2, 3, 5, 7, and 11. Prime numbers are of supreme importance in the theory of numbers.
Many branches of mathematics have a “fundamental theorem.” Often this is an arbitrary choice but when there is a result from which all the other results flow, it is natural to choose it as the fundamental one. In the case of arithmetic, the fundamental theorem says that you can factorize whole numbers into prime numbers. Furthermore, for each number there is only one possible factorization.
For example:
12 = 2 x 2 x 3
35 = 5 x 7
1001 = 7 x 11 x 13
To draw an analogy with chemistry, prime numbers are like atomic particles, that is, they cannot be split up and every other number can be expressed in terms of them.
3rd century BC Greece
The Infinity of Prime Numbers
Euclid (c. 325–265 BC)
The number of primes is infinite.
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However many prime numbers are written down, there will always be another one.
A prime is a number which has exactly two divisors – 1 and itself. The first few prime numbers are 2, 3, 5, 7, 11, and 13.
This sequence goes on for ever. The proof is in Euclid’s Elements. His proof shows that there are more than three primes, but it can be extended to any number.
The proof goes by contradiction. Suppose that the only primes are a, b, and c. Then consider abc + 1. This is one greater than a multiple of a, and so it cannot be divisible by a. Similarly it cannot be divisible by b or c. By the fundamental theorem of arithmetic, any number can be written as a product of primes.
