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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 most important consequences of Newton's
analysis is that it must be one and the same law of
force that operates in the centrally directed acceleration
of the planetary bodies (toward the sun) and
of satellites (toward planets), and that controls the
linear downward acceleration of freely falling bodies.
This force of universal gravitation is also shown to be
the cause of the tides, through the action of the sun
and the moon on the seas.

Book I of the “Principia.”

Book I of the
Principia
contains the first of the two parts of De motu corporum.
It is a mathematical treatment of motion under the
action of impressed forces in free spaces—that is,
spaces devoid of resistance. (Although Newton
discussed elastic and inelastic impact in the scholium
to the laws, he did not reintroduce this topic in book I.)
For the most part, the subject of Newton's inquiries
is the motion of unit or point masses, usually having
some initial inertial motion and being acted upon by
a centripetal force. Newton thus tended to use the
change in velocity produced in a given time (the
“accelerative measure”) of such forces, rather than the
change in momentum produced in a given time (their
“motive” measure).151 He generally compared the
effects of different forces or conditions of force on
one and the same body, rather than on different
bodies, preferring to consider a mass point or unit
mass to computing actual magnitudes. Eventually,
however, when the properties and actions of force had
been displayed by an investigation of their
“accelerative” and “motive” measures, Newton was
able to approach the problem of their “absolute”
measure. Later in the book he considered the attraction
of spherical shells and spheres and of nonsymmetrical
bodies.

Sections 2 and 3 are devoted to aspects of motion
according to Kepler's laws. In proposition 1 Newton
proceeded by four stages. He first showed that in a
purely uniform linear (or purely inertial) motion, a
radius vector drawn from the moving body to any
point not in the line of motion sweeps out equal areas
in equal times.
FIGURE 8
The reason for this is clearly shown in
Figure 8, in which in equal times the body will move
through the equal distances AB, BC, CD, DE, ....
If a radius vector is drawn from a point PS, then
triangles ABS, BCS, CDS, DES, ... have equal
bases
and a common altitude h, and their areas are equal.
In the second stage, Newton assumed the moving
object to receive an impulsive force when it reaches
point B. A component of motion toward S is thereby
added to its motion toward C; its actual path is thus
along the diagonal Bc of a parallelogram (Figure 9).
FIGURE 9
Newton then showed by simple geometry that the
area of the triangle SBc is the same as the area of the
triangle SBC, so that area is still conserved. He
repeated the procedure in the third stage, with the
body receiving a new impetus toward S at point C,
and so on. In this way, the path is converted from a
straight line into a series of joined line segments,
traversed in equal intervals of time, which determine
triangles of equal areas, with S as a common vertex.

In Newton's final development of the problem, the
number of triangles is increased “and their breadth
diminished in infinitum”; in the limit the “ultimate
perimeter” will be a curve, the centripetal force “will
act continually,” and “any described areas” will be
proportional to the times. Newton thus showed that
inertial motion of and by itself implies an areaconservation
law, and that if a centripetal force is
directed to “an immovable centre” when a body has
such inertial motion initially, area is still conserved as
determined by a radius vector drawn from the moving
body to the immovable center of force. (A critical
examination of Newton's proof reveals the use of
second-order infinitesimals.)152 The most significant
aspect of this proposition (and its converse, proposition
2) may be its demonstration of the hitherto
wholly unsuspected logical connection, in the case
of planetary motion, between Descartes's law of