history of mathematics - National STEM Centre

38
The Greeks
Activity 3.7
therefore the base AD is equal to the base FC, and the triangle ABD is
equal to the triangle FBC. [1.4]
Now the parallelogram BL is double of the triangle ABD, for they have the
same base BD and are in the same parallels BD, AL. [1.41]
And the square GB is double of the triangle FBC, for they again have the
same base FB and are in the same parallels FB, GC. [1.41]
[But the doubles of equals are equal to one another]
Therefore the parallelogram BL is also equal to the square GB.
Similarly, if AE, BK be joined, the parallelogram CL can also be proved
equal to the square HC;
therefore the whole square BDEC is equal to the two
squares GB, HC. [C.N.2]
And the square BDEC is described on BC, and the squares GB,
HC on BA, AC.
Therefore the square on the side BC is equal to the squares on the
sides BA, AC.
Axiomatic structure
The previous two activities involved two examples of the way in which Euclid built
a structure of propositions or theorems, each one depending on the previous ones.
This single deductive system is based upon a set of five initial assumptions, called
postulates, definitions and axioms. Euclid's geometry is like a firm building, based
on a foundation of five pillars (the postulates), in combination with a set of accepted
starting points, the common notions, and using definitions that explain the meaning
of some important ideas.
The first book of the Elements opens with a list of 23 definitions in which Euclid
attempts to define precisely the objects and ideas that he is going to use. Each
definition can make use of previously defined terms, but inevitably the first
definitions are difficult to understand.
Some definitions
It is difficult to write definitions for a point and a line. Try question 1 briefly before
looking at the answers.
1 Write definitions for 'a point' and 'a line' in your own words.
2 The answers to question 1 give Euclid's definitions of a point and a line. Do you
find them any more helpful than the ones you have written yourself?
3 The next definitions given by Euclid are the following.
3 The extremities of a line are points.
4 A straight line is a line which lies evenly with the points on itself.
5 A surface is that which has length and breadth only.