Electronic edition published by Cultural Heritage Langauge Technologies (with permission from Charles Scribners and Sons) and funded by the National Science Foundation International Digital Libraries Program. This text has been proofread to a low degree of accuracy. It was converted to electronic form using data entry.
NEWTON, ISAAC (b. Woolsthorpe, England,
25 December 1642; d. London, England, 20 March
1727), mathematics, dynamics, celestial mechanics,
astronomy, optics, natural philosophy.
The “Principia”: Definitions and Axioms.
Motion.” The first two entities defined are
“quantity
of matter,” or “mass,” and “quantity of
motion.”
The former is said to be the measure of matter
proportional to bulk and density conjunctively.
“Mass” is, in addition, given as being generally
known by its weight, to which it is proportional
at any given place, as shown by Newton's experiments
with pendulums, of which the results are more exact
than Galileo's for freely falling bodies. Newton's
“quantity of motion” is the entity now known as
momentum; it is said to be measured by the velocity
and mass of a body, conjunctively.
Definition 3 introduces vis insita (probably best
translated as “inherent force”), a concept of which the
actual definition and explanation are both so difficult
to understand that much scholarly debate has been
expended on them.146 Newton wrote that the vis insita
may be known by “a most significant name, vis
inertiae.” But this “force” is not like the
“impressed
forces” of definition 4, which change the state of rest
or uniform rectilinear motion of a body; the vis inertiae
merely maintains any new state acquired by a body,
and it may cause a body to “resist” any change in
state.147
Newton then defined “centripetal force” (vis
centripeta), a concept he had invented and named to
complement the vis centrifuga of Christiaan
Huygens.148 In definitions 6 through 8, Newton gave
three “measures” of centripetal force, of which the
most important for the purposes of the Principia is that
one “proportional to the velocity which it generates in
a given time” (for point masses, unit masses, or for
comparing equal masses). There follows the famous
scholium on space and time, in which Newton opted
for concepts of absolute space and absolute time,
although recognizing that both are usually reckoned
by “sensible measures”; time, especially, is usually
“relative, apparent, and common.” Newton's belief
in absolute space led him to hold that absolute motion
is sensible or detectable, notably in rotation, although
contemporaries as different in their outlooks as
Huygens and Berkeley demurred from this view.
The “Axioms” or “Laws of Motion” are three in
number: the law of inertia, a form of what is today
known as the second law, and finally the law that
“To every action there is always opposed an equal
and opposite reaction.” There is much puzzlement
over the second law, which Newton stated as
a proportionality between “change in motion”
(in momentum) and “the motive force impressed”
(a change “made in the direction ..., in which
that force is impressed”); he did not specify
“per unit time” or “in some given time.” The
second law thus seems clearly to be stated for
an impulse, but throughout the Principia (and, in
a special case, in the antecedent definition 8), Newton
used the law for continuous forces, including gravitation,
taking account of time. For Newton, in fact,
the concepts of impulse and continuous force were
infinitesimally equivalent, and represented conditions
of action “altogether and at once” or “by degrees and
successively.”149 There are thus two conditions of
“force” in the second law; accordingly, this Newtonian
law may be written in the two forms f?d(mv) and
f?d(mv)/dt, in which both concepts of force
are taken
account of by means of two different constants of
proportionality. The two forms of the law can be
considered equivalent through Newton's concept of a
uniformly flowing time, which makes dt a kind of
secondary constant, which can arbitrarily be absorbed
in the constant of proportionality.
There may be some doubt as to whether or not
Newton himself was unclear in his own mind about
these matters. His use of such expressions as “vis
impressa” shows an abiding influence of older
physics, while his continued reference to a “vis” or a
“force” needed to maintain bodies in a state of motion
raises the question of whether such usage is one of a
number of possibly misleading “artifacts left behind
in the historical development of his [Newton's]
dynamics.”150 It must be remembered, of course, that
throughout the seventeenth and much of the eighteenth
century the word “force” could be used in a number
of ways. Most notably, it served to indicate the
concept now called “momentum,” although it could
also even mean energy. In Newton's time there were
no categories of strict formalistic logic that required
a unitary one-to-one correspondence between names
and concepts, and neither Newton nor his contemporaries
(or, for that matter, his successors) were
always precise in making such distinctions.
The careful reader of books I-III should not be
confused by such language, however, nor by the
preliminary intrusion of such concepts. Even the idea
of force as a measure of motion or of change of
motion (or of change per se, or rate of change)
is not troublesome in practice, once Newton's own
formulation is accepted and the infinitesimal level of
his discourse (which is not always explicitly stated)
understood. In short, Newton's dynamical and
mathematical elaboration of the three books of the
Principia is free of the errors and ambiguities implicit
in his less successful attempt to give a logically simple
and coherent set of definitions and axioms for
dynamics. (It is even possible that the definitions and
axioms may represent an independent later exercise,
since there are, for example, varying sets of definitions
and axioms for the same system of dynamics.) One of