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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.

dispute with Leibniz: he wished to demonstrate that
he had actually composed the Principia by analysis
and had rewritten the work synthetically. He affirmed
this claim, in and after 1713, in several manuscript
versions of prefaces to planned new editions of the
Principia (both with or without De quadratura as
a supplement). It is indeed plausible to argue that
much of the Principia was based upon an infinitesimal
analysis, veiled by the traditional form of Greek
synthetic geometry, but the question remains
whether Newton drew upon working papers in which
(in extreme form) he gave solutions in dotted fluxions
to problems that he later presented geometrically.
But, additionally, there is no evidence that Newton
used an analytic method of ordinary fluxional form
to discover the propositions he presented synthetically.

All evidence indicates that Newton had actually
found the propositions in the Principia in essentially
the way in which he there presented them to his
readers. He did, however, use algebraic methods to
determine the solid of least resistance. But in this case,
he did not make the discovery by analysis and then
recast it as an example of synthesis; he simply stated
his result without proof.139

It has already been mentioned that Newton did
make explicit use of the infinitesimal calculus in
section 2, book II, of the Principia, and that in that
work he often employed his favored method of
infinite series.140 But this claim is very different indeed
from such a statement of Newton's as: “... At length
in 1685 and part of 1686 by the aid of this method
and the help of the book on Quadratures I wrote the
first two books of the mathematical Principles of
Philosophy. And therefore I have subjoined a Book
on Quadratures to the Book of Principles.”141 This
“method” refers to fluxions, or the method of differential
calculus. But it is true, as mentioned earlier,
that Newton stated in the Principia that certain
theorems depended upon the “quadrature” (or
integration) of “certain curves”; he did need, for this
purpose, the inverse method of fluxions, or the
integral calculus. And proposition 41 of book I is,
moreover, an obvious exercise in the calculus.

Newton himself never did bring out an edition
of the Principia together with a version of De
quadratura.142 In the review that he published of the
Commercium epistolicum,143 Newton did announce in
print, although anonymously, that he had “found out
most of the Propositions in his Principia” by using
“the new Analysis,” and had then reworked the
material and had “demonstrated the Propositions
synthetically.” (This claim cannot, however, be substantiated
by documentary evidence.)

Apart from questions of the priority of Newton's
method, the Principia contains some problems of
notable mathematical interest. Sections 4 and 5 of
book I deal with conic sections, and section 6 with
Kepler's problem; Newton here introduced the method
of solution by successive iteration. Lemma 5 of
book III treats of a locus through a given number of
points, an example of Newton's widely used method
of interpolating a function. Proposition 71, book
I, contains Newton's important solution to
a major problem of integration, the attraction
of a sphere, called by Turnbull “the crown of
all.” Newton's proof that two spheres will mutually
attract each other as if the whole of their masses were
concentrated at their respective centers is posited on
the condition that, however the mass or density may
vary within each sphere as a function of that radius,
the density at any given radius is everywhere the same
(or is constant throughout any concentric shell).

The “Principia”: General Plan.

Newton's masterwork
was worked up and put into its final form in an
incredibly short time. His strategy was to develop the
subject of general dynamics from a mathematical
point of view in book I, then to apply his most
important results to solving astronomical and
physical problems in book III. Book II, introduced
at some point between Newton's first conception of
the treatise and the completion of the printer's manuscript,
is almost independent, and appears extraneous.

Book I opens with a series of definitions and axioms,
followed by a set of mathematical principles and
procedural rules for the use of limits; book III begins
with general precepts concerning empirical science
and a presentation of the phenomenological bases
of celestial mechanics, based on observation.

It is clear to any careful reader that Newton was,
in book I, developing mathematical principles of
motion chiefly so that he might apply them to the
physical conditions of experiment and observation
in book III, on the system of the world. Newton
maintained that even though he had, in book I, used
such apparently physical concepts as “force” and
“attraction,” he did so in a purely mathematical
sense. In fact, in book I (as in book II), he tended to
follow his inspiration to whatever aspect of any topic
might prove of mathematical interest, often going
far beyond any possible physical application. Only
in an occasional scholium in books I and II did he
raise the question of whether the mathematical
propositions might indeed be properly applied to the
physical circumstances that the use of such words as
“force” and “attraction” would seem to imply.

Newton's method of composition led to a certain
amount of repetition, since many topics are discussed
twice—in book I, with mathematical proofs, to