- NUMBER SYSTEMS 203
decimal system, with its easy approximations to irrational numbers, soothed the
consciences of mathematicians and gave them the confidence to proceed with their
development of the calculus. No one even seemed very concerned about the absence
of any good geometric construction of cube roots and higher roots of real numbers.
The real line answered the needs of algebra in that it gave a representation of any
real root there might be of any algebraic equation with real numbers as coefficients.
It was some time before anyone realized that geometry still had resources that even
algebra did not encompass and would lead to numbers for which pure algebra had
no use.
Those resources included the continuity of the geometric line, which turned out
to be exactly what was needed for the limiting processes of calculus. It was this
property that made it sensible for Euler to talk about the number that we now call
e, that is,
The intuitive notion of continuity assured mathematicians that there were points on
the line, and hence infinite decimal expansions, that must represent these numbers,
even though no one would ever know the full expansions. The geometry of the line
provided a geometric representation of real numbers and made it possible to reason
about them without having to worry about the decimal expansion.
The continuity of the line brought the realization that the real numbers had
more to offer than merely convenient representations of the solutions of equations.
They could even represent some numbers such as e and 7 that had not been found
to be solutions of any equations. The line was richer than it needed to be for algebra
alone. The concept of a real number had allowed arithmetic to penetrate into parts
of geometry where even algebra could not go. The sides and diagonals of regular
figures such as squares, cubes, pentagons, pyramids, and the like all had ratios
that could be represented as the solutions of equations, and hence are algebraic.
For example, the diagonal D and side S of a pentagon satisfy D^2 = S(D + S).
For a square the relationship is D^2 = 2S^2 , and for a cube it is D^2 — 3S^2. But
what about the number we now call ð, the ratio of the circumference C of a circle
to its diameter D? In the seventeenth century Leibniz noted that any line that
could be constructed using Euclidean methods (straightedge and compass) would
have a length that satisfied some equation with rational coefficients. In a number
of letters and papers written during the 1670s, Leibniz was the first to contrast
what is algebraic (involving polynomials with rational coefficients) with objects
that he called analytic or transcendental and the first to suggest that ð might be
transcendental. In the preface to his pamphlet De quadratura arithmetica circuit
(On the Arithmetical Quadrature of the Circle), he wrote:
A complete quadrature would be one that is both analytic and
linear; that is, it would be constructed by the use of curves whose
equations are of [finite] degrees. The brilliant Gregory [James Gre-
gory, 1638 1675], in his book On the Exact Quadrature of the Cir-
cle, has claimed that this is impossible, but, unless I am mistaken,n=0
and the other Euler constant