250
III - Convergence: Continuous variables
Following Leibniz, one replaces x and y by x + x'dt and y + y'dt (in fact
by x + dx and y + dy) and performs the same calculation, so obtaining, after
division by dt, the relation 0 = 2x.dx/dt+ 2y.dy/dt plus terms containing dt,
i.e. which are "infinitely small" and so can be eliminated.
It remains to understand what, from Newton's point of view, plays the
role of the derivative of a fluent y in terms of another fluent x on which it
depends, y = f(x). It is the ratio between an "infinitely small increment" of x
and the corresponding increment in y, i.e. what Newton denoted xo and yo
and Leibniz dx and dy; so
J'(x) = yo/xo = y/x,
a relation whose resemblance to the notation dy / dx of Leibniz shows well how
close their ideas were, if not identical. Whence fifty or so years of quarrels
about priority, Leibniz himself and his disciples being much more courteous than Newton's English partisans who accused the former of plagiarismwrongly, as we have known since the historians have examined the masses of
personal papers of Leibniz which slept in obscure corners of libraries for two
centuries. Newton's papers were absorbed by the descendants of the one who
had inherited them, namely an Earl of Portsmouth whose mother, having
married a Lord Lymington, was the daughter of a niece of Newton, practically adopted by the great man when he set up in London and to whom he
had entrusted the care of his establishment; she herself married John Conduitt MP, in 1717, and Newton lived with them until his death, so that they
inherited his papers which they transmitted to their daughter who handed
them on to Portsmouth. Conduitt was the deputy and became Newton's successor at the Mint, which Newton, leaving Cambridge, directed from 1695.
He organised it with redoubtable efficiency and integrity; there he recast all
the silver money, reduced to half of its value through the activities of sniders.
In 1872, after a fire, the Portsmouth family gave Newton's scientific papers,
in disorder and often damaged, to Cambridge. Most of the rest (theology,
history, alchemy, etc.) were sold at auction and dispersed in the 1930s. Some
purchasers, including libraries, refuse access. In contrast, the economist and
former mathematician John Maynard Keynes bought some and brought them
back to life by donating them to Cambridge.
Newton's calculations can be understood otherwise and in an ultramodern way by introducing what one calls dual numbers. To define the
complex numbers one introduced a mysterious symbol i such that i 2 = -1
and calculated with expressions of the form a + ib with a, b E R using this.
The dual numbers are similarly expressions a + bQ, where a and b are real
(or even complex) numbers with which one calculates in the usual way,
but requiring the symbol Q to satisfy the relation Q2 = 0, which is neither
more nor less strange than i 2 = -1. Then
(a + bQ) + (c + dQ) = (a + c) + (b + d)Q,
(a + bQ)(c + dQ) = ac + (ad + bC)Q,
III - Convergence: Continuous variables
Following Leibniz, one replaces x and y by x + x'dt and y + y'dt (in fact
by x + dx and y + dy) and performs the same calculation, so obtaining, after
division by dt, the relation 0 = 2x.dx/dt+ 2y.dy/dt plus terms containing dt,
i.e. which are "infinitely small" and so can be eliminated.
It remains to understand what, from Newton's point of view, plays the
role of the derivative of a fluent y in terms of another fluent x on which it
depends, y = f(x). It is the ratio between an "infinitely small increment" of x
and the corresponding increment in y, i.e. what Newton denoted xo and yo
and Leibniz dx and dy; so
J'(x) = yo/xo = y/x,
a relation whose resemblance to the notation dy / dx of Leibniz shows well how
close their ideas were, if not identical. Whence fifty or so years of quarrels
about priority, Leibniz himself and his disciples being much more courteous than Newton's English partisans who accused the former of plagiarismwrongly, as we have known since the historians have examined the masses of
personal papers of Leibniz which slept in obscure corners of libraries for two
centuries. Newton's papers were absorbed by the descendants of the one who
had inherited them, namely an Earl of Portsmouth whose mother, having
married a Lord Lymington, was the daughter of a niece of Newton, practically adopted by the great man when he set up in London and to whom he
had entrusted the care of his establishment; she herself married John Conduitt MP, in 1717, and Newton lived with them until his death, so that they
inherited his papers which they transmitted to their daughter who handed
them on to Portsmouth. Conduitt was the deputy and became Newton's successor at the Mint, which Newton, leaving Cambridge, directed from 1695.
He organised it with redoubtable efficiency and integrity; there he recast all
the silver money, reduced to half of its value through the activities of sniders.
In 1872, after a fire, the Portsmouth family gave Newton's scientific papers,
in disorder and often damaged, to Cambridge. Most of the rest (theology,
history, alchemy, etc.) were sold at auction and dispersed in the 1930s. Some
purchasers, including libraries, refuse access. In contrast, the economist and
former mathematician John Maynard Keynes bought some and brought them
back to life by donating them to Cambridge.
Newton's calculations can be understood otherwise and in an ultramodern way by introducing what one calls dual numbers. To define the
complex numbers one introduced a mysterious symbol i such that i 2 = -1
and calculated with expressions of the form a + ib with a, b E R using this.
The dual numbers are similarly expressions a + bQ, where a and b are real
(or even complex) numbers with which one calculates in the usual way,
but requiring the symbol Q to satisfy the relation Q2 = 0, which is neither
more nor less strange than i 2 = -1. Then
(a + bQ) + (c + dQ) = (a + c) + (b + d)Q,
(a + bQ)(c + dQ) = ac + (ad + bC)Q,