202 8. NUMBERS AND NUMBER THEORY IN MODERN MATHEMATICSThe existence of incommensurables throws doubt on certain oversimplified
proofs of geometric proportion, and this question is discussed in detail in Chap-
ter 10. At present we are concerned with its effect on the concept of a number. At
the beginning, one would have to say that the effect was almost nil. Geometry and
arithmetic were separate subjects in the Greek tradition. But when algebra arose
and the Persian mathematician Omar Khayyam (1050-1130) discovered that some
cubic equations that could not be solved arithmetically had geometric solutions,
the idea of a real number as a ratio of lines began to take shape.
The idea of using a line to stand for a number, the numbers being regarded
as the length of the line, is very familiar to us and has its origin in the work of
the ancient Greeks and medieval Muslim mathematicians. In Europe this idea re-
ceived some development in the work of the fourteenth-century Bishop of Lisieux,
Nicole d'Oresme, whose graphical representation of relationships was a forerun-
ner of our modern analytic geometry. Oresme was familiar with the concept of
incommensurable lines, a subject that was missing from earlier medieval work in
geometry, and he was careful to keep the distinction between commensurable and
incommensurable clear. Indeed, Oresme was even more advanced than the average
twentieth-century person, in that he recognized a logical difficulty in talking about
a power of, say, \ that equals |, whereas modern students are taught how to use
the rules of exponents, but not encouraged to ask what is meant by expressions
such as \/2.
A great advance came in the seventeenth century, when analytic geometry as
we know it today was invented by Descartes and Fermat. Fermat's work seems
somewhat closer to what we know, in the sense that he used a pair of mutually
perpendicular axes; on the other hand, he believed that only dimensionally equiv-
alent expressions could be added. This is the restriction that led Omar Khayyam
to write a cubic equation in the form equivalent to x^2 + ax^2 + b^2 x = b^2 c, in which
each term is of degree 3. In his Geometrie, Descartes showed how to avoid this
complication. The difficulty lay in the geometric representation of the operation
of multiplication. Because ratios of lines were not always numbers, Euclid did not
make the association of a line with a number called its length. The product of
two numbers is a number, but Euclid did not speak of the product of two lines.
He spoke instead of the rectangle on the two lines. That was the tradition Omar
Khayyam was following. Stimulated by algebra, however, and the application of
geometry to it, Descartes looked at the product of two lengths in a different way.
As pure numbers, the product ab is simply the fourth proportional to 1 : a : b.
That is, ab : b :: a : 1. He therefore fixed an arbitrary line that he called / to
represent the number 1 and represented ab as the line that satisfied the proportion
ab : b :: a: I, when á and b were lines representing two given numbers.
The notion of a real number had at last arisen, not as most people think of it
today—an infinite decimal expansion—but as a ratio of line segments. Only a few
decades later Newton defined a (real) number to be "the ratio of one magnitude
to another magnitude of the same kind, arbitrarily taken as a unit." Newton
classified numbers as integers, fractions, and surds (Whiteside, 1967, Vol. 2, p.
7). Even with this amount of clarity introduced, however, mathematicians were
inclined to gloss over certain difficulties. For example, there is an arithmetic rule
according to which %/ab = y/a\/b. Even with Descartes' geometric interpretation
of these results, it is not obvious how this rule is to be proved. The use of the