The seventh and eighth sections are devoted to the motion of a particle under a central force which is any function of the distance. The geometrical treatment of these problems is ingenious, but necessarily more involved than when modern analysis is used.
In the ninth section Newton dealt with the motion of particles in orbits which are revolving about the centre of force, and on the motion of the apses of such orbits: this introduced the theory of disturbing forces. The tenth section is concerned with constrained motion, and particularly with the motion of pendulums. The eleventh section deals with the motion of particles under their mutual attractions and incidentally with the problem of three bodies. These three sections afford a notable illustration of Newton’s analytical powers.
The twelfth and thirteenth sections deal with the attraction under various laws of force of spherical bodies, circular laminae, and solids of revolution. [235] ]These sections brought the problem of the solar system, consisting of solid bodies of finite size and approximately spherical in form, into the domain of mathematics, and led up to the generalization that all particles of matter attract one another with a force proportional to the product of their masses and inversely proportional to the square of the distance between them, from which law it would seem that all the known phenomena of the motions of the solar system can be deduced.
The fourteenth section is concerned with the motion of minute corpuscles, with applications to the corpuscular theory of light.
The second book is devoted to the discussion of the motion of bodies in resisting mediums: there are fifty-three propositions besides lemmas, scholia, etc.
In the first section, Newton considered the motion of a particle or sphere moving in a medium whose resistance varies as the velocity of the particle: in the second section the resistance is assumed to vary as the square of the velocity: and in the third section the resistance is supposed to consist of two terms, one varying as the velocity and the other as the square of the velocity. The fourth section is on spiral motion caused by resistance of the medium.
The fifth section deals with the density and pressure of liquids and gases at rest (Hydrostatics).
[236]
]The sixth section treats of the motion of pendulums in a resisting medium; and the seventh section is concerned with the motion of fluids, and the resistance they offer to the motion of projectiles. The latter section contains the celebrated statement of the form of the solid of least resistance, whose demonstration proved a puzzle to mathematicians until the invention of the calculus of variations. Newton’s solution is in the Portsmouth papers, and elsewhere I have published it: it involves the use of fluxions, and it is probable that it was his failure to translate this demonstration into geometrical language that led him to give the result without a proof.
The eighth section deals with the motion of waves with applications to the theory of sound and the undulatory theory of light; and the ninth section deals with vortices; it is here shown that the theory of vortices suggested by Descartes to explain the motion of the solar system is untenable.
This book created the theory of hydrodynamics. Much of it is incomplete, but it is astonishing that Newton proved as much as he did; of course to-day no one would suggest that the best way of attacking these problems is by Newtonian geometrical methods.