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today. Most of the limits which were studied in the eighteenth century
were one-sided, as was mentioned in the section preceding qn 25, and
there was also a presumption (explicit in d'Alembert) that limits were
not to be reached.
Thenotation for absolutevaluewas not availableto Cauchy,
though he sometimes wrote
(
x
) where we would write
x
. Where
absolute value was pertinent, early nineteenth century mathematicians
would declare that an inequality holds 'without regard to sign'. The
symbolism of absolute value and the notion of neighbourhood were
introduced by K. Weierstrass in his lectures in 1859.
Thesymbol
was introduced by J. Wallis in 1655, but he
manipulated it like a number. The 'lim' notation was due to L'Huilier
in 1786, and mid-nineteenth-century mathematicians, including
Weierstrass, used expressions such as
lim
n
.
This only became unacceptable with the greater precision about what is
and what is not a real number in the latter part of the nineteenth
century, and it was in 1905 that the Cambridge mathematician J. G.
Leathem proposed the use of '
n
'. G. H. Hardy (1908) vehemently
exhorts his readers to forgo writing
n
, though even he allowed the
statement lim
a
.
The language of bounds and boundedness, introduced by Pasch
(1882), was adopted at the beginning of the twentieth century. For most
of the nineteenth century a bound was called a limit (as in common
non-mathematical usage) and the boundedness of a set described by
saying that its elements were finite.
Thesymbol [
x
] for theintegral part of
x
was used for certain
special cases by Dirichlet in 1849, and in the sense that we now use it
by F. Mertens in 1874. G. Peano wrote
Ex
for [
x
] in 1899 (
E
being the
first letter of
entier
, theword for
whole number
in French).
Both recurring (1657) and non-recurring (1693) infinite decimals
appear first in the work of J. Wallis.
