38 A History ofMathematics
have worried commentators, but they are not—perhaps you will agree—very important unless you
suppose theElementsto be perfect.
As you see—and this is Euclid’s most famous and influential characteristic—every statement
is backed up by a reference back to a previous result, or in this case to a ‘common notion’.
(If you like, like Thomas Hobbes in our opening quote, you can read ‘backwards’ from an apparently
unbelievable statement until you arrive at something you can believe.) For example, Common
Notion 1, clearly important in the proof, states that ‘if equals are added to equals, the results will be
equal’. This deductive structure (statement of proposition→proof, justified by reference back→
Q.E.D.) is still in use today. It has been usual to treat Greek mathematics as part of our Western tra-
dition, and so as similar to our own, and indeed the style of exposition and the material are familiar;
but the actual aim of the proposition is unusual from our viewpoint, as it would have been for the
Egyptians; it tells us only that two parallelograms have equal areas, instead of saying what those
areas are. Does this mean that Euclid did not think of areas as numbers? There is evidence that
he did not. To do so, you have to have a fixed unit of measure (one square foot or one square metre,
or whatever), and this theElementsis unwilling to do.^2
In fact, theElementsis not a user-friendly text, to put it mildly; there are no examples, no explan-
ations of aims and objectives, or of the uses to which any result can be put. All this the reader has
to invent. And, because Euclid did not explain his aim, we have to use other sources to try and find
out what he was trying to do and why. Here are two other examples of the thinking in theElements
to illustrate what I mean:- Today, we usually think of ‘Pythagoras’s theorem’ as a statement about numbers; again, this
theorem was known in much the same sense by the Egyptians and Babylonians. In these terms,
the theorem says that if a right-angled triangle has short sides of lengthsaandb, and long side
(hypotenuse)c, thena^2 +b^2 =c^2. But the theorem as stated in Euclid I.47 (see Fauvel and Gray
3.B.5, p. 115–6) is about actualsquares: if you draw the squares on the three sides, then you can
cut up the squares on the two short sides and piece them together to make up the square on the
hypotenuse. - Again, when we want to describe the area of a circle, we—like the Egyptians—describe it by
the formulaπr^2 , whereris the radius andπis a number which we can approximate in various
ways, depending on how accurate we want to be (^227 , for example, or 3.14159.; the Egyptian
approximation was worse, but not too bad.) There is no such formula in Euclid; the only statement
about areas of circles is (XII.2—see Fauvel and Gray 3.E.3, p. 136–7): ‘Circles are to one another
as the squares on their diameters’.
This rather enigmatic sentence means in plain—slightly simplified—terms that the area of a
circle C is in a fixed ratio to the area of the square S on its diameter. (Which is four times the square
on the radius.) We could recover the usual statement if we called the ratio ‘π/4’, but that would
not be in agreement with the whole use of the term ‘ratio’ in Euclid—a ‘ratio’, whatever it is, is not
the same as a number, as we shall explain later.
Confused? It is not surprising; and theElements, despite their prestige, have probably generated
more confusion than any other Greek mathematical work, among readers in the Islamic, medieval,
and modern European worlds. How the ancient Greeks understood them is rather unclear to us.
These examples should give some sense of the difference between Greek mathematics and our
own; and this is a useful starting point, since our problems in understanding it help us to begin- However, for comparison, we have Socrates’ use of numbers for areas in theMeno—‘the four-foot square’ and so on.