174
II - Convergence: Discrete variables
In studying algebraic plane curves, i.e. defined by a polynomial relation
between x and y, he is capable, for any point (a, b) of the curve, of calculating
with power series in x - a (or, if necessary, in a fractional power of x - a)
starting with b and which, substituted for y, satisfy the equation of the curve;
they allow one to study the curve in the neighbourhood of the point (a, b)
and to determine its singular points (multiple points where several branches
of the curve cross, cusps, etc.) or the asymptotes.
He explains this with the example y3 + y + xy - 2 - x 3 = 0, which
represents a plane curve of third degree. For x = 0, the equation possesses,
among others, the solution y = 1. Newton then sets y = 1 + p, where p is a
power series starting with a term in x; substituting in the initial equation,
the constant terms must cancel, and one finds an equation 4p + x + ... = 0,
where the unwritten terms contain, taking account of p, only powers ::::: 2
of x; in consequence, p = -x/4 + q where the series q starts with a term
in x 2 ; one then su bstitutes in the equation in p and one finds an equation
4q - x 2 /16 + ... = 0 where the unwritten terms all contain at least x 3 . One
has in consequence q = x 2 /64 + r where r starts with a term in x 3 , and so
on indefinitely. Newton thus finds, in the example considered,
y = 1 - x/4 + x 2 /64 + 131x 3 /512 + 509x 4 /16384 + ...
setting out the calculations fully in a very elegant way59.
The only problem is that these formal calculations prove nothing as to
the convergence of the series obtained, of which Newton never exhibits more
than the first terms. Here as always, it was the XIXth century, and quite
particularly Weierstrass, who invented a priori methods for majorising the
coefficients of a series so obtained without calculating them explicitly, and
showed that it converged otherwise than for x = 0; this in fact is what
Theorem 17 proves. One can see no other way, in the example just presented
a la Newton, of how, "without knowing anything", one might obtain an idea
of its convergence when its coefficients are of such complexity and do not
obey any obvious general law of formation.
More beautiful in this context is that the problem of convergence is of
hardly any interest. The theory of algebraic curves (or surfaces, or ... ), i.e.
defined by polynomial equations between the coordinates, lies in the domain
of algebra for the excellent reason that it still has a meaning if one replaces
~ or C by any commutative field, as discovered in the XXth century. Even
if the series obtained by Newton were completely divergent, in other words
reduced to formal series, his calculations retain an algebraic meaning, so that
his method is in fact applicable (and applied) to an arbitrary field: the four
59 See, in Vol. III, pp. 32-225, of the complete edition of the Mathematical Papers
by D. T. Whiteside, the "tract" De Methodis Serierum et Fluxionum of 16701671 where Newton systematically expounds his discoveries, especially pp. 55-57
for the example in question. Whiteside reproduces the Latin text and an English
translation from 1710.
II - Convergence: Discrete variables
In studying algebraic plane curves, i.e. defined by a polynomial relation
between x and y, he is capable, for any point (a, b) of the curve, of calculating
with power series in x - a (or, if necessary, in a fractional power of x - a)
starting with b and which, substituted for y, satisfy the equation of the curve;
they allow one to study the curve in the neighbourhood of the point (a, b)
and to determine its singular points (multiple points where several branches
of the curve cross, cusps, etc.) or the asymptotes.
He explains this with the example y3 + y + xy - 2 - x 3 = 0, which
represents a plane curve of third degree. For x = 0, the equation possesses,
among others, the solution y = 1. Newton then sets y = 1 + p, where p is a
power series starting with a term in x; substituting in the initial equation,
the constant terms must cancel, and one finds an equation 4p + x + ... = 0,
where the unwritten terms contain, taking account of p, only powers ::::: 2
of x; in consequence, p = -x/4 + q where the series q starts with a term
in x 2 ; one then su bstitutes in the equation in p and one finds an equation
4q - x 2 /16 + ... = 0 where the unwritten terms all contain at least x 3 . One
has in consequence q = x 2 /64 + r where r starts with a term in x 3 , and so
on indefinitely. Newton thus finds, in the example considered,
y = 1 - x/4 + x 2 /64 + 131x 3 /512 + 509x 4 /16384 + ...
setting out the calculations fully in a very elegant way59.
The only problem is that these formal calculations prove nothing as to
the convergence of the series obtained, of which Newton never exhibits more
than the first terms. Here as always, it was the XIXth century, and quite
particularly Weierstrass, who invented a priori methods for majorising the
coefficients of a series so obtained without calculating them explicitly, and
showed that it converged otherwise than for x = 0; this in fact is what
Theorem 17 proves. One can see no other way, in the example just presented
a la Newton, of how, "without knowing anything", one might obtain an idea
of its convergence when its coefficients are of such complexity and do not
obey any obvious general law of formation.
More beautiful in this context is that the problem of convergence is of
hardly any interest. The theory of algebraic curves (or surfaces, or ... ), i.e.
defined by polynomial equations between the coordinates, lies in the domain
of algebra for the excellent reason that it still has a meaning if one replaces
~ or C by any commutative field, as discovered in the XXth century. Even
if the series obtained by Newton were completely divergent, in other words
reduced to formal series, his calculations retain an algebraic meaning, so that
his method is in fact applicable (and applied) to an arbitrary field: the four
59 See, in Vol. III, pp. 32-225, of the complete edition of the Mathematical Papers
by D. T. Whiteside, the "tract" De Methodis Serierum et Fluxionum of 16701671 where Newton systematically expounds his discoveries, especially pp. 55-57
for the example in question. Whiteside reproduces the Latin text and an English
translation from 1710.