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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.

geometrical or mechanical) by which a figure is
generated and rendered easy to conceive.”62

The “written record of [Newton's] first researches
in the interlocking structures of Cartesian coordinate
geometry and infinitesimal analysis”63
shows him to have been establishing “the foundations
of his mature work in mathematics”
and reveals “for the first time the true magnitude
of his genius.”64 And in fact Newton did contribute
significantly to analytic geometry. In his
1671 Methodis fluxionum, he devoted “Prob. 4: To
draw tangents to curves” to a study of the different
ways in which tangents may be drawn “according to
the various relationships of curves to straight lines,”
that is, according to the “modes” or coordinate
systems in which the curve is specified.65

Newton proceeded “by considering the ratios of
limit-increments of the co-ordinate variables (which
are those of their fluxions).”66 His “Mode 3”
consists
of using what are now known as standard bipolar
coordinates, which Newton applied to Cartesian
ovals as follows: Let x, y be the distances from a pair
of fixed points (two “poles”); the equation
a ? (e/d)x - y = 0 for Descartes's “second-order
ovals” will then yield the fluxional relation
?(e/d)? - ? = 0 (in dot notation) or
?em/d - n = 0
(in the notation of the original manuscript, in which
m, n are used for the fluxions ?, ?
of x, y). When
d = e, “the curve turns out to be a conic.” In
“Mode 7,” Newton introduced polar coordinates for
the construction of spirals; “the equation of an
Archimedean spiral” in these coordinates becomes
(a/b)x = y, where y is the radius vector (now usually
designated r or p) and x the angle (θ or
φ).

Newton constructed equations for the transformation
of coordinates (as, for example, from polar
to Cartesian), and found formulas in both polar and
rectangular coordinates for the curvature of a variety
of curves, including conics and spirals. On the basis
of these results Boyer has quite properly referred to
Newton as “an originator of polar coordinates.”67

Further geometrical results may be found in
Enumeratio linearum tertii ordinis, first written in 1667
or 1668, and then redone and published, together with
De quadratura, as an appendix to the Opticks
(1704).68
Newton devoted the bulk of the tract to classifying
cubic curves into seventy-two “Classes, Genders,
or Orders, according to the Number of the Dimensions
of an Equation, expressing the relation between
the Ordinates and the Abscissae; or which is much at
one [that is, the same thing], according to the Number
of Points in which they may be cut by a Right Line.”

In a brief fifth section, Newton dealt with “The
Generation of Curves by Shadows,” or the theory of
projections, by which he considered the shadows
produced “by a luminous point” as projections “on
an infinite plane.” He showed that the “shadows”
(or projections) of conic sections are themselves conic
sections, while “those of curves of the second genus
will always be curves of the second genus; those of the
third genus will always be curves of the third genus;
and so on ad infinitum.” Furthermore, “in the same
manner as the circle, projecting its shadow, generates
all the conic sections, so the five divergent parabolae,
by their shadows, generate all the other curves of the
second genus.” As C. R. M. Talbot observed, this
presentation is “substantially the same as that which
is discussed at greater length in the twenty-second
lemma [book III, section 5] of the Principia, in which
it is proposed to ‘transmute’ any rectilinear or curvilinear
figure into another of the same analytical order
by means of the method of projections.”69

The work ends with a brief supplement on “The
Organical Description of Curves,” leading to the
“Description of the Conick-Section by Five Given
Points” and including the clear statement, “The Use
of Curves in Geometry is, that by their Intersections
Problems may be solved” (with an example of an
equation of the ninth degree). Newton in this tract
laid “the foundation for the study of Higher Plane
Curves, bringing out the importance of asymptotes,
nodes, cusps,” according to Turnbull, while Boyer
has asserted that it “is the earliest instance of a work
devoted solely to graphs of higher plane curves in
algebra,” and has called attention to the systematic
use of two axes and the lack of “hesitation about
negative coordinates.”70

Newton's major mathematical activity had come
to a halt by 1696, when he left Cambridge for London.
The Principia, composed in the 1680's, marked the
last great exertion of his mathematical genius,
although in the early 1690's he worked on porisms and
began a “Liber geometriae,” never completed, of
which David Gregory gave a good description of the
planned whole.71 For the most part, Newton spent the
rest of his mathematical life revising earlier works.

Newton's other chief mathematical activity during
the London years lay in furthering his own position
against Leibniz in the dispute over priority and
originality in the invention of the calculus. But he did
respond elegantly to a pair of challenge problems set
by Johann [I] Bernoulli in June 1696. The first of
these problems was “mechanico-geometrical,” to
find the curve of swiftest descent. Newton's answer was
brief: the “brachistochrone” is a cycloid. The second
problem was to find a curve with the following
property, “that the two segments [of a right line
drawn from a given point through the curve], being