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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.

the sun, but significantly different in that the ellipse
implies periodic returns of the comet—and worked
out the details with Newton. In the second and third
editions of the Principia, Newton gave tables for both
the parabolic and elliptical orbits; he asserted unequivocally
that Halley had found “a remarkable comet”
appearing every seventy-five years or so, and added
that Halley had “computed the motions of the comet
in this elliptic orbit.” Nevertheless, Newton himself
remained primarily concerned with parabolic orbits.
In the conclusion to the example following proposition
41 (on the comet of 1680), Newton said that “comets
are a sort of planets revolved in very eccentric
orbits about the sun.” Even so, the proposition itself
states (in all editions): “From three given observations
to determine the orbit of a comet moving in a
parabola.”

Mathematics in the “Principia.”

The
Philosophiae naturalis principia mathematica is, as its title suggests,
an exposition of a natural philosophy conceived in
terms of new principles based on Newton's own
innovations in mathematics. It is too often described
as a treatise in the style of Greek geometry, since
on superficial examination it appears to have been
written in a synthetic geometrical style.132 But a close
examination shows that this external Euclidean form
masks the true and novel mathematical character of
Newton's treatise, which was recognized even in his
own day. (L'Hospital, for example—to Newton's
delight—observed in the preface to his 1696 Analyse
des infiniment petits, the first textbook on the
infinitesimal calculus, that Newton's “excellent
Livre intitulé Philosophiae Naturalis principia Mathematica
... est presque tout de ce calcul.”) Indeed, the
most superficial reading of the Principia must show
that, proposition by proposition and lemma by lemma,
Newton usually proceeded by establishing geometrical
conditions and their corresponding ratios and then at
once introducing some carefully defined limiting
process. This manner of proof or “invention,” in
marked distinction to the style of the classical Greek
geometers, is based on a set of general principles of
limits, or of prime and ultimate ratios, posited by
Newton so as to deal with nascent or evanescent
quantities or ratios of such quantities.

The doctrine of limits occurs in the Principia in a
set of eleven lemmas that constitute section 1 of book I.
These lemmas justify Newton in dealing with areas
as limits of sums of inscribed or circumscribed
rectangles (whose breadth ? 0, or whose number
? ?), and in assuming the equality, in the
limit, of arc, chord, and tangent (lemma 7), based on
the proportionality of “homologous sides of similar
figures, whether curvilinear or rectilinear” (lemma 5),
whose “areas are as the squares of the homologous
sides.” Newton's mathematical principles are founded
on a concept of limit disclosed at the very beginning
of lemma 1, “Quantities, and the ratios of quantities,
which in any finite time converge continually to
equality, and before the end of that time approach
nearer to each other than by any given difference,
become ultimately equal.”

Newton further devoted the concluding scholium of
section 1 to his concept of limit, and his method of
taking limits, stating the guiding principle thus:
“These lemmas are premised to avoid the tediousness
of deducing involved demonstrations ad absurdum,
according to the method of the ancient geometers.”
While he could have produced shorter (“more
contracted”) demonstrations by the “method of
indivisibles,” he judged the “hypothesis of indivisibles
”to be “somewhat harsh” and not geometrical:

I chose rather to reduce the demonstrations of the
following propositions to the first and last sums and
ratios of nascent and evanescent quantities, that is, to
the limits of those sums and ratios; and so to premise,
as short as I could, the demonstrations of those limits.
For hereby the same thing is performed as by the
method of indivisibles; and now those principles being
demonstrated, we may use them with greater safety.
Therefore if hereafter I should happen to consider
quantities as made up of particles, or should use little
curved lines for right ones, I would not be understood
to mean indivisibles, but evanescent divisible quantities;
not the sums and ratios of determinate parts, but
always the limits of sums and ratios; and that the force
of such demonstrations always depends on the method
laid down in the foregoing Lemmas.

Newton was aware that his principles were open to
criticism on the ground “that there is no ultimate
proportion of evanescent quantities; because the
proportion, before the quantities have vanished, is
not the ultimate, and when they are vanished, is none”;
and he anticipated any possible unfavorable reaction
by insisting that “the ultimate ratio of evanescent
quantities” is to be understood to mean “the ratio of
the quantities not before they vanish, nor afterwards,
but [that] with which they vanish.” In a “like manner,
the first ratio of nascent quantities is that with which
they begin to be,” and “the first or last sum is that
with which they begin and cease to be (or to be
augmented or diminished).” Comparing such ratios
and sums to velocities (for “it may be alleged, that a
body arriving at a certain place, and there stopping,
has no ultimate velocity; because the velocity, before
the body comes to the place, is not its ultimate
velocity; when it has arrived, there is none”), he
imagined the existence of “a limit which the velocity