§4. Differentiable functions
247
or fifty years to publish 27 . To understand him, one needs to know that for
Newton there was only one independent variable: time, an idea apparently
suggested by Isaac Barrow and conceived in an age when the concepts of
independent variable and of function 28 were still very badly understood, except in Mechanics and in Astronomy, where, precisely, everything depends
on time. This never appeared explicitly in Newton's formulae, though ever
present. All the other variables, the fluents , are functions of time, so that for
Newton the equation x 2 + y2 = a 2 represents a point moving as an a priori
arbitrary function of time on a circumference of radius a. The crucial passage
of his manuscript is the following 29 :
The moments of the fluent quantities (that is, their indefinitely small
parts, by addition of which they increase during each infinitely small period of time) are as their speeds of flow. Wherefore if the moment of any
particular one, say x, be expressed by the product of its speed 30 x and an
infinitely small quantity 0 (that is, by xo), then the moments of the others,
21 Newton, at least initially, was not entirely responsble for these delays. It was
clearly not his fault that his first papers were not published in the Transactions
of the Royal Society; his efforts, several years later, to find a publisher for his
Method fell through: the possible publishers had had their fingers burnt by their
losses on the books of Wallis and Mercator, books which, unsurprisingly, were
very far from being best-sellers. Moreover, Newton's publication of a report on
his experiments on the decomposition of white light brought violent attacks from
partisans of Descartes' theory (see Loup Verlet, La malle de Newton, Chaps. II
and III). Profoundly averse to polemic and to "disputation", Newton developed
a deep distrust of taking a public stance; his English students or disciples were
those who expounded his mathematical works, with a delay of many years. Newton made only two important exceptions, in favour of his Principia (1687) and
of his Opticks (1704); in the first he expounds the results of his research in Astronomy. This is the book which made him the hero of the French philosophers
of the XVlllth century (and of scientists, French or not) despite the fact that his
declared purpose was to demonstrate the Divine perfection of Creation. Having
presented his "Mecanique Celeste" to Napoleon, who was surprised at seeing
no mention of the name of God, Laplace replied that on the contrary his book
proved that it was not necessary. The standard argument - if one needs a Creator
to explain "Creation", how to explain the existence of a Creator who had not
been created? - clearly remains valid.
28 The word was introduced by Leibniz in 1694, but it was Jakob Bernoulli who
gave it its present meaning, in 1698, in very particular situations. The idea that
a function consists of arbitrarily associating a number y to each number x of an
interval, without any other condition, waited for the XIX th century and Dirichlet,
who invented the function equal to 1 for x is rational and to 0 if not.
29 D. T. Whiteside, The Mathematical Papers of Isaac Newton (CUP), III, p. 79-8l.
The passages between ( ) are in Newton.
30 In fact, Newton wrote p, q, r for what from 1691 he wrote x, y, z. The modern notation /' or x' was invented by Lagrange. Newton's usage of the term
"moment" to denote an infinitesimal increase of a fluent was inspired by the
analogous term relative to time: an infinitely small period (his 0). Newton naturally does not explain what is to be understood by x, as though everyone could
understand it intuitively.
247
or fifty years to publish 27 . To understand him, one needs to know that for
Newton there was only one independent variable: time, an idea apparently
suggested by Isaac Barrow and conceived in an age when the concepts of
independent variable and of function 28 were still very badly understood, except in Mechanics and in Astronomy, where, precisely, everything depends
on time. This never appeared explicitly in Newton's formulae, though ever
present. All the other variables, the fluents , are functions of time, so that for
Newton the equation x 2 + y2 = a 2 represents a point moving as an a priori
arbitrary function of time on a circumference of radius a. The crucial passage
of his manuscript is the following 29 :
The moments of the fluent quantities (that is, their indefinitely small
parts, by addition of which they increase during each infinitely small period of time) are as their speeds of flow. Wherefore if the moment of any
particular one, say x, be expressed by the product of its speed 30 x and an
infinitely small quantity 0 (that is, by xo), then the moments of the others,
21 Newton, at least initially, was not entirely responsble for these delays. It was
clearly not his fault that his first papers were not published in the Transactions
of the Royal Society; his efforts, several years later, to find a publisher for his
Method fell through: the possible publishers had had their fingers burnt by their
losses on the books of Wallis and Mercator, books which, unsurprisingly, were
very far from being best-sellers. Moreover, Newton's publication of a report on
his experiments on the decomposition of white light brought violent attacks from
partisans of Descartes' theory (see Loup Verlet, La malle de Newton, Chaps. II
and III). Profoundly averse to polemic and to "disputation", Newton developed
a deep distrust of taking a public stance; his English students or disciples were
those who expounded his mathematical works, with a delay of many years. Newton made only two important exceptions, in favour of his Principia (1687) and
of his Opticks (1704); in the first he expounds the results of his research in Astronomy. This is the book which made him the hero of the French philosophers
of the XVlllth century (and of scientists, French or not) despite the fact that his
declared purpose was to demonstrate the Divine perfection of Creation. Having
presented his "Mecanique Celeste" to Napoleon, who was surprised at seeing
no mention of the name of God, Laplace replied that on the contrary his book
proved that it was not necessary. The standard argument - if one needs a Creator
to explain "Creation", how to explain the existence of a Creator who had not
been created? - clearly remains valid.
28 The word was introduced by Leibniz in 1694, but it was Jakob Bernoulli who
gave it its present meaning, in 1698, in very particular situations. The idea that
a function consists of arbitrarily associating a number y to each number x of an
interval, without any other condition, waited for the XIX th century and Dirichlet,
who invented the function equal to 1 for x is rational and to 0 if not.
29 D. T. Whiteside, The Mathematical Papers of Isaac Newton (CUP), III, p. 79-8l.
The passages between ( ) are in Newton.
30 In fact, Newton wrote p, q, r for what from 1691 he wrote x, y, z. The modern notation /' or x' was invented by Lagrange. Newton's usage of the term
"moment" to denote an infinitesimal increase of a fluent was inspired by the
analogous term relative to time: an infinitely small period (his 0). Newton naturally does not explain what is to be understood by x, as though everyone could
understand it intuitively.