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NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.
Mathematics. Any summary of Newton's contributions
to mathematics must take account not only
of his fundamental work in the calculus and other
aspects of analysis--including infinite series (and most
notably the general binomial expansion)--but also his
activity in algebra and number theory, classical and
analytic geometry, finite differences, the classification
of curves, methods of computation and approximation,
and even probability.
and c not be separated “by any small Distance.” If the
points C and c do not “coalesce and exactly
coincide,”
the lines CK and CH will not coincide, and “the
ultimate Ratios in the Lines CE, Ec, and Cc”
cannot
be found. In short, “The very smallest Errors in
mathematical Matters are not to be neglected.”52
This same topic appears in the mathematical
introduction (section 1, book I) to the Principia,
in which Newton stated a set of lemmas on limits of
geometrical ratios, making a distinction between the
limit of a ratio and the ratio of limits (for example,
as x ? 0, lim. xn/x ? 0; but lim.
xn/lim. x ? 0/0,
which is indeterminate).
The connection of fluxions with infinite series was
first publicly stated in a scholium to proposition 11 of
De quadratura, which Newton added for the 1704
printing, “We said formerly that there were first,
second, third, fourth, &c. Fluxions of flowing Quantities.
These Fluxions are as the Terms of an infinite
converging series.” As an example, he considered zn
to
“be the flowing Quantity” and “by flowing” to become
(z + o)n; he then demonstrated that the successive
terms of the expansion are the successive fluxions:
“The first Term of this Series zn will be that
flowing Quantity; the second will be the first
Increment or Difference, to which consider'd as
nascent, its first Fluxion is proportional ... and so on
in infinitum.” This clearly exemplifies the theorem
formally stated by Brook Taylor in 1715; Newton
himself explicitly derived it in an unpublished first
version of De quadratura in 1691.53 It should be noted
that Newton here showed himself to be aware of the
importance of convergence as a necessary condition
for expansion in an infinite series.
In describing his method of quadrature by “first and
last ratios,” Newton said:
Now to institute an Analysis after this manner in
finite Quantities and investigate the prime or ultimate
Ratios of these finite Quantities when in their nascent
or evanescent State, is consonant to the Geometry of
the Ancients: and I was willing [that is, desirous] to
show that, in the Method of Fluxions, there is no
necessity of introducing Figures infinitely small into
Geometry.54
Newton's statement on the geometry of the ancients
is typical of his lifelong philosophy. In mathematics
and in mathematical physics, he believed that the
results of analysis--the way in which things were
discovered--should ideally be presented synthetically,
in the form of a demonstration. Thus, in his review
of the Commercium epistolicum (published anonymously),
he wrote of the methods he had developed
in De quadratura and other works as follows:
By the help of the new Analysis Mr. Newton found
out most of the Propositions in his Principia Philosophiae:
but because the Ancients for making things
certain admitted nothing into Geometry before it was
demonstrated synthetically, he demonstrated the
Propositions synthetically, that the Systeme of the
Heavens might be founded upon good Geometry. And
this makes it now difficult for unskilful Men to see
the Analysis by which those Propositions were found
out.55
As to analysis itself, David Gregory recorded that
Newton once said “Algebra is the Analysis of the
Bunglers in Mathematicks.”56 No doubt! Newton did,
nevertheless, devote his main professorial lectures of
1673-1683 to algebra,57 and these lectures were
printed a number of times both during his lifetime and
after.58 This algebraical work includes, among other
things, what H. W. Turnbull has described as a
general method (given without proof) for discovering
“the rational factors, if any, of a polynomial in one
unknown and with integral coefficients”; he adds that
the “most remarkable passage in the book” is
Newton's rule for discovering the imaginary roots of
such a polynomial.59 (There is also developed a set
of formulas for “the sums of the powers of the roots
of a polynomial equation.”)60
Newton's preference for geometric methods over
purely analytical ones is further evident in his
statement that “Equations are Expressions of
Arithmetical Computation and properly have no place
in Geometry.” But such assertions must not be read
out of context, as if they were pronouncements about
algebra in general, since Newton was actually
discussing various points of view or standards
concerning what was proper to geometry. He
included the positions of Pappus and Archimedes
on whether to admit into geometry the conchoid for
the problem of trisection and those of the “new
generation of geometers” who “welcome” into
geometry many curves, conics among them.61
Newton's concern was with the limits to be set in
geometry, and in particular he took up the question
of the legitimacy of the conic sections in solid
geometry (that is, as solid constructions) as opposed
to their illegitimacy in plane geometry (since they
cannot be generated in a plane by a purely geometric
construction). He wished to divorce synthetic geometric
considerations from their “analytic” algebraic
counterparts. Synthesis would make the ellipse the
simplest of conic sections other than the circle;
analysis would award this place to the parabola.
“Simplicity in figures,” he wrote, “is dependent on the
simplicity of their genesis and conception, and it is
not its equation but its description (whether