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Euclid – geometry

Euclid

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

Geometry

Euclid’s theorems are still true and his methods are still admired. For millenia his books have been studied and referenced, though they are no longer used as a school text-book.53 He entitled his principal work Elements, and it was intended to be a foundational work in the subject, a starting point. The same Greek word (stoikheia) also means the letters in the alphabet, and Euclid’s elements are to geometry what letters are to language: the building blocks or basic components.

One of the most outstanding features of Euclid’s work is its structure: the first book contains a number of definitions, postulates and common notions, and the following twelve books endeavour to introduce or assume no extraneous material as they progress, but only to construct from definitions and propositions already done. Thus, for any proposition one can trace back the reasoning for a particular result through earlier propositions until one comes back ultimately to the original postulates and common notions.

This trace can be illustrated by drawing a proof tree, of which an example is given below in Figure 3, to illustrate the reasoning for Pythagoras’ Theorem. Of course Euclid was not infallible, and there are occasionally holes in the arguments, but these should not be allowed to detract from the overall aim and success of his method. Another outstanding feature is the thoroughness with which propositions are proved, as will become apparent in the example given below. Let us first review the Elements.

Book 1 builds from twenty-three definitions, five postulates, and nine common notions.54 The definitions explain the basic terms of geometry, what is meant by words such as ‘point’ or ‘line’. The common notions are axioms or self-evident truths; statements that any sensible person would take as true, although it is not possible to prove them. For example, Common Notion 1 is ‘Things which are equal to the same thing are also equal to one another’. The postulates are unproved assertions about geometry. The first three postulates are assertions that amount to the possibility of doing geometry.

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

The fourth and fifth postulates are different: they are premises which the beginner must accept as given, and their validity, and their classification as postulates (rather than as e.g. axioms), have been subjects of contention among mathematicians since antiquity.

Postulate 4 ‘All right angles are equal to one another’.

Postulate 5 ‘If a straight line falling on two straight lines make the interior angles on the same sides less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles’.

Euclid thought that this last assertion was undemonstrable, and hence made it a postulate. The fifth (parallel) postulate is notorious. Attempts to prove it from the preceeding four started shortly after, and the problem attracted mathematicians for two millenia. This research led to many postulates which are equivalent with the fifth. In Gerolamo Saccheri’s Euclides ab omni naevo vindicatus of 1733 an equivalent postulate is rejected in the hope of deriving a contradiction, and thereby proving the fifth postulate by reductio ad absurdum. But the argument failed. Consequently Saccheri’s work produced the first theorems of what is called non-Euclidean geometry.55

After the definitions, common notions and postulates, Book 1 discusses triangles, parallels and parallelograms. Book 2 gives two more definitions, and then carries on to the transformation of a rectilinear area of any shape into a parallelogram of any shape. Book 3 gives eleven more definitions, to deal with the circle. Book 4 has another seven definitions and moves on to deal with triangles and regular polygons which are drawn inside and around circles. Book 5 gives eighteen more definitions and introduces the theory of ratios, most of which was worked out by Euclid’s famous predecessor Eudoxus.

Book 6 adds four more definitions and applies the theory of ratios to plane geometry. Books 7, 8 and 9 start with twenty-two definitions at the beginning of Book 7, and then deal with arithmetic and the theory of numbers.56 Book 10, after four more definitions, deals with the irrationals (alogoi or arrhtoi, lit. ‘inexpressible’). Books 11, 12 and 13 turn to solid geometry, introduced with twenty-eight definitions. So the entire work of 13 books, covering this huge range of mathematical topics, rests on a mere five postulates and nine common notions. On those few assertions which constitute a minimal undemonstrated bedrock, the framework of ancient geometry was constructed, in belt and braces fashion, proof by proof. It is a tremendous intellectual achievement.

Euclid's history

Born around 325 BC and died about 265 BC in Alexandria, Egypt. This is the most usually presented idea – that Euclid was an ordinary mathematician/scholar, who simply lived in Alexandria and wrote his Elements – a book which was as popular as Bible until the 19th century.

The real identity of Euclid is however not certain at all. The situation is best summed up by a historian of mathematics, Itard, who gave three possible hypotheses with regard to Euclid's identity:

• That Euclid was a historical character, known as Euclid of Alexandria, born about 325 BC and died about 265 BC in Alexandria, Egypt, who wrote the Elements and the other works attributed to him.
• That Euclid was the leader of a team of mathematicians working at Alexandria around 300 BC, who all contributed to writing The Elements, even continuing to write after Euclid's death.
• That Euclid was not a historical character and that The Elements were written by a team of mathematicians at Alexandria who took the name Euclid from the historical character of Euclid of Megara who had lived about 400 BC.

Euclid's achievement was so great that even in the Middle Ages, when mathematics was all but forgotten, and only handful of copies were preserved by the Arab mathematicians and later translated to Latin and even later to vernacular languages, myths were circulating among the masons and builders in England.

In some mediaeval manuscripts Euclid was described as someone who not only wrote the book in which all knowledge of mathematics was preserved, but also as someone who founded the craft of building.

"And the lords of the country (Egypt) grew together and took counsel how they might help their children who had no competent livelihood in order to provide for themselves and their children, for they had so many. And at the council amongst them was this worthy Clerk Euclid and when he saw that all of them could devise no remedy in the matter be said to them "Lay your orders upon your sons and I will teach them a science by which they may live as gentlemen, under the condition that they shall be sworn to me to uphold the regulations that I shall lay upon them." And both they and the king of the country and all the lords agreed thereto with one consent.

It is but reasonable that every many should agree to that which tended to profit himself; and so they took their sons to Euclid to be ruled by him and he taught them the Craft of Masonry and gave it the name of Geometry on account of the parcelling out of the ground which he had taught the people at the time of making the walls and ditches, as aforesaid, to keep out the water. And Isodoris says in Ethomologies that Euclid called the craft Geometry.

And there this worthy clerk Euclid gave it a name and taught it too the lord's sons of that land whom he had as pupils."