246
III - Convergence: Continuous variables
(14.9)
df(a) = f'(a)dx,
or even, if f is differentiable at any x E I,
(14.10)
df(x) = J'(x)dx or simply df = f'(x)dx.
Hence the traditional formulation
(14.11)
J'(x) = dfldx
for derivatives; it has no meaning in this framework since df and dx are
functions and not numbers, but everyone uses it, not only to follow in the
tradition, but also, and above all, because of its convenience, particularly at
the elementary level.
The inventor of the notation dx, df and df / dx, namely Leibniz 26 , a metaphysician, interpreted them in quite another way; for him it was neither a
matter of a variable h nor of a linear function. For Leibniz and for those who
followed him, at least up to Cauchy, the symbol dx represented an "infinitely
small increase" in the variable x and df the "principal part", proportional to
dx, of the increase
f(x + dx) - f(x)
of f at the point x. These concepts, which rest on the "infinitely smalls" which
no one has been able to define, have put too many people to useless worry,
that one can only attribute them the role of an historical explanation the
differential notation. Newton, who had a positive outlook in what concerned
Mathematics, Astronomy, Physics, the minting of money and, in a lesser
measure, the Bible and Alchemy, did not care for them since "they are not
to be met in Nature".
But there is in fact an equivalent of the dx of Leibniz in his Treatise on
the methods of series and fluxions, composed in Latin in 1671 and never published, except in translation, in 1739 in England, and in 1740 in France by
Buffon, in order to legitimise the posthumous glory of the great man, somewhat eclipsed by that of Leibniz and of Bernoulli, who did not wait forty
26 See G. W. Leibnitz, Naissance du calcul difterentiel, papers translated and annotated by Marc Parmentier (Paris, Vrin, 1989 or 1995). For the history of derivatives from Galileo to Cauchy, see the excellent resume in Walter, Analysis 1,
pp. 221-240. The classic by H. G. Zeuthen, Geschichte der Mathematik im 16.
und 17. Jahrhundert (Teubner, 1903 or Johnson Reprint Co., 1966) is still very
usable and weighs much less than the blockbusters (ten ton bombs developed at
the end of the war, so, by extension: a book of 800 pages finding nevertheless
200000 readers in the USA, subject permitting) of Moritz Cantor, useful for
their abundance of detail.
III - Convergence: Continuous variables
(14.9)
df(a) = f'(a)dx,
or even, if f is differentiable at any x E I,
(14.10)
df(x) = J'(x)dx or simply df = f'(x)dx.
Hence the traditional formulation
(14.11)
J'(x) = dfldx
for derivatives; it has no meaning in this framework since df and dx are
functions and not numbers, but everyone uses it, not only to follow in the
tradition, but also, and above all, because of its convenience, particularly at
the elementary level.
The inventor of the notation dx, df and df / dx, namely Leibniz 26 , a metaphysician, interpreted them in quite another way; for him it was neither a
matter of a variable h nor of a linear function. For Leibniz and for those who
followed him, at least up to Cauchy, the symbol dx represented an "infinitely
small increase" in the variable x and df the "principal part", proportional to
dx, of the increase
f(x + dx) - f(x)
of f at the point x. These concepts, which rest on the "infinitely smalls" which
no one has been able to define, have put too many people to useless worry,
that one can only attribute them the role of an historical explanation the
differential notation. Newton, who had a positive outlook in what concerned
Mathematics, Astronomy, Physics, the minting of money and, in a lesser
measure, the Bible and Alchemy, did not care for them since "they are not
to be met in Nature".
But there is in fact an equivalent of the dx of Leibniz in his Treatise on
the methods of series and fluxions, composed in Latin in 1671 and never published, except in translation, in 1739 in England, and in 1740 in France by
Buffon, in order to legitimise the posthumous glory of the great man, somewhat eclipsed by that of Leibniz and of Bernoulli, who did not wait forty
26 See G. W. Leibnitz, Naissance du calcul difterentiel, papers translated and annotated by Marc Parmentier (Paris, Vrin, 1989 or 1995). For the history of derivatives from Galileo to Cauchy, see the excellent resume in Walter, Analysis 1,
pp. 221-240. The classic by H. G. Zeuthen, Geschichte der Mathematik im 16.
und 17. Jahrhundert (Teubner, 1903 or Johnson Reprint Co., 1966) is still very
usable and weighs much less than the blockbusters (ten ton bombs developed at
the end of the war, so, by extension: a book of 800 pages finding nevertheless
200000 readers in the USA, subject permitting) of Moritz Cantor, useful for
their abundance of detail.