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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.
fluxions to the velocitys with which all other quantities
increase. Also from the moments of time I give the name
of moments to the parts of any other quantities generated
in moments of time. I expose time by any
quantity flowing uniformly & represent its fluxion by
an unit, & the fluxions of other quantities I represent
by any other fit symbols & the fluxions of their fluxions
by other fit symbols & the fluxions of those fluxions by
others, & their moments generated by those fluxions I
represent by the symbols of the fluxions drawn into the
letter o & its powers o2, o3,
&c: vizt their first moments
by their first fluxions drawn into the letter o, their
second moments by their second fluxions into o2, & so
on. And when I am investigating a truth or the solution
of a Probleme I use all sorts of approximations &
neglect to write down the letter o, but when I am
demonstrating a Proposition I always write down the
letter o & proceed exactly by the rules of Geometry
without admitting any approximations. And I found
the method not upon summs & differences, but upon
the solution of this probleme: By knowing the Quantities
generated in time to find their fluxions. And this is done
by finding not prima momenta but primas momentorum
nascentium rationes.
In an addendum (published only in 1969) to the
1671 Methodus fluxionum,49 Newton developed an
alternative geometrical theory of “first and last”
ratios of lines and curves. This was later partially
subsumed into the 1687 edition of the Principia,
section 1, book I, and in the introduction to the
Tractatus de quadratura curvarum (published by
Newton in 1704 as one of the two mathematical
appendixes to the Opticks). Newton had intended to
issue a version of his De quadratura with the Principia
on several occasions, both before and after the 1713
second edition, because, as he once wrote, “by the
help of this method of Quadratures I found the
Demonstration of Kepler's Propositions that the
Planets revolve in Ellipses describing . . . areas
proportional to the times,” and again, “By the inverse
Method of fluxions I found in the year 1677 the
demonstration of Kepler's Astronomical
Proposition....”50
Newton began De quadratura with the statement
that he did not use infinitesimals, “in this Place,”
considering “mathematical Quantities . . . not as
consisting of very small Parts; but as describ'd by a
continued Motion.”51 Thus lines are generated “not
by the Apposition of Parts, but by the continued
Motion of Points,” areas by the motion of lines,
solids by the motion of surfaces, angles by the rotation
of the sides, and “Portions of Time by a continual
Flux.” Recognizing that there are different rates of
increase and decrease, he called the “Velocities of the
Motions or Increments Fluxions, and the generated
Quantities Fluents,” adding that “Fluxions are very
nearly as the Augments of the Fluents generated in
equal but very small Particles of Time, and, to speak
accurately, they are in the first Ratio of the nascent
Augments; but they may be expounded in any Lines
which are proportional to them.”
As an example, consider that (as in Fig. 1) areas
ABC, ABDG are described by the uniform motion of
the ordinates BC, BD moving along the base in the
direction AB.
FIGURE 1
Suppose BC to advance to any new
position bc, complete the parallelogram BCEb, draw
the straight line VTH “touching the Curve in C, and
meeting the two lines bc and BA [produced] in T
and V.” The “augments” generated will be:
Bb, by AB;
Ec, by BC; and Cc, by “the Curve Line
ACc.” Hence,
“the Sides of the Triangle CET are in the first Ratio
of these Augments considered as nascent.” The
“Fluxions of AB, BC and AC” are therefore
“as the
Sides CE, ET and CT of that Triangle CET”
and “may
be expounded” by those sides, or by the sides of the
triangle VBC, which is similar to the triangle CET.
Contrariwise, one can “take the Fluxions in the
ultimate Ratio of the evanescent Parts.” Draw the
straight line Cc; produce it to K. Now let bc return
to its original position BC; when “C and c
coalesce,”
the line CK will coincide with the tangent CH; then,
“the evanescent Triangle CEc in its ultimate Form
will become similar to the Triangle CET, and its
evanescent Sides CE, Ec, and Cc will be
ultimately
among themselves as the sides CE, ET and CT of the
other Triangle CET, are, and therefore the Fluxions
of the Lines AB, BC and AC are in this same
Ratio.”
Newton concluded with an admonition that for the
line CK not to be “distant from the Tangent CH by
a small Distance,” it is necessary that the points C
