Graphics Reference
In-Depth Information
Historical note
The 'Archimedean property' appears in Euclid in the form of a
definition of ratio (Book V, Def. 4), and thus pre-dates Archimedes (250
B.C.). It was important to Euclid because it precluded comparing a
measurement of length with a measurement of area or a measurement
of area with a measurement of volume. There is an equivalent
proposition in Euclid Book X.1 that if a half or moreof a quantity is
removed, then half or more of what is left, and so on, then after a finite
number of steps what remains is less than any predetermined quantity.
This proposition was used to great effect by Euclid in relation to the
area of a circle (Book XII.2) and by Archimedes in his calculations of
areas and volumes. It provided the classical way of establishing the
existence of a limit. The importance of the Archimedean property in
modern times dates from 1891, when G. Veronese recognised that it
eliminated constant infinitesimals. In Hilbert's
Foundations of Geometry
(1899) it is referred to as the 'axiom of continuity'.
The
in our definition of limit is foreshadowed in Euclid X.1, and
also by Newton who wrote 'Quantities, and the ratio of quantities,
which in any finite time converge continually to equality, and before the
end of that time approach
nearer to each other than by any given
difference
, become ultimately equal.' (
Principia
, Book 1, Lemma 1, 1687).
During the eighteenth century a great deal of work was done on
particular sequences and series, and sophisticated methods for
approximating and finding limits were developed. D'Alembert had
studied the binomial expansion of (1
x
)
when
m
is rational (1768)
and had established the boundedness of the series by comparison with
two geometric series when
x
1. Newton's statement on limits was
refined by d'Alembert (1765) who wrote 'One says that a quantity is the
limit of another quantity, if the second approaches the first closer than
any given quantity, however small.' There are indications of how to
provethat thelimit of a product is equal to theproduct of thelimits in
d'Alembert and de la Chapelle (1789) and of how to prove that the limit
of a quotient is the quotient of the limits (without attention to the
possibility of a zero denominator) in L'Huilier (1795). But formal
proofs, as in this chapter, had to await a working definition of
convergence and this came in Cauchy's
Cours d
'
analyse
(1821) in his
proof that (
a
/
n
)
k
. 'There exists an
N
such
that when
n
N
. . .' and the abbreviation 'for suMciently large
n
' are
both expressions which we owe to Cauchy. The suMx notation for
sequences stems from Lagrange (1759) who actually used superscripts
and not subscripts. ThesuMx notation propr is dueto Lacroix (1800),
but it was in Cauchy's hands that it became the powerful tool we know
a
)
k
implies (
a
