I intend to include a serialised guide to Euclid's first book of the elements
in this section of the website.
Your comments are welcome: garethlewis@freedom-in-education.co.uk
The first article in this series explained the importance of Euclid and Greek geometry to western culture.
In his system of Geometry, Euclid starts by asking the student to accept that certain things do exist, even though they cannot be defined. These things comprise the twenty-three ‘definitions’ described in last month’s article and include such things as lines, points, angles, circles, triangles, straight lines and parallel lines.
Postulates
Euclid’s next step is to put forward five postulates. Geometry seeks to understand the physical world through logic, but before this can be attempted certain things have to be accepted on an intuitive level, or simply on trust. This applies to the postulates – the student is asked to accept:
That it is possible to draw a straight line between any two points.
That straight lines can be extended from either end.
That a circle can be described (drawn) at any distance around any point.
That all right angles are equal to each other.
That if two straight lines are not parallel to each other, they will meet each other when extended indefinitely.
It is important to remember that pure geometry exists only in the imagination. It is possible to draw two points on a piece of paper and, with the aid of ruler, to connect them with a straight line. However, neither of the ‘points’ that have been drawn are really points – if you looked at them with the aid of a magnifying glass they would appear to be composed of many smaller points; the ‘line’ is not really a line because it has breadth as well as length; and it is not straight because it will have followed the microscopic bumps on the ruler and on the paper. Thus the straight line drawn on the page is simply a representation of something that we believe to be possible, but which we cannot reproduce.
We therefore accept the postulates because they seem to make sense. They do not themselves have any logical foundation and they cannot be proven to be true. This is an aspect of mathematics, and science, that is often hidden from children. Teachers present these subjects as though they are based upon incontrovertible facts but nothing could be further from the truth.
Every science and every branch of mathematics demands that the student accepts certain assumptions, in a very unscientific way, before they can begin. One of the beauties of Greek geometry is that these assumptions are set out clearly at the start: the student then has a chance to see how these assumptions come back to trip them up as the subject unfolds.
Next Instalment: 5 Common Notions