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