The second book was completed by the summer of 1686. It treats of motion in a resisting medium and of various problems connected with waves. At the end of it, it is shown that the Cartesian theory of vortices is inconsistent with the laws of motion, and necessarily leads to incorrect results. This book opened another world to the application of mathematics and, in effect, created the science of hydrodynamics.
The third book was finished in March 1687. In this, the theorems previously established are applied to the chief phenomena of the universe, and briefly we may say that all the facts then known about the solar system and, in particular, the motion of the moon with its various inequalities, the figure of the earth, and the phenomena of the tides, were shown to be in accord with the theory. Much of the material for these calculations was collected by Flamsteed and Halley.
The Principia, as I have said, is a classic. Like other books to which that compliment is paid, it is rarely read: indeed, I doubt whether there are a [231] ]dozen men in Cambridge who have glanced all through it, even in a cursory manner. When I was an undergraduate the course for the Tripos involved five sections (1, 2, 3, 9, and 11) of the first book, but now, probably with good reason, even this slight acquaintance with the work is no longer required, and to-day the character of these investigations is unfamiliar to most mathematicians, while the fact that it is written in Latin tends to diminish the number of its readers. I will, then, with your permission, describe briefly its frame-work.
First, however, let me remark on how different was the knowledge of mathematics, even among experts, at the time it was written from that current to-day. In the geometry of the circle and conics mathematicians were familiar with the methods of Greek science, and in their application Newton was unrivalled among his contemporaries, but outside geometry methods of investigation were far to seek. Analysis had been but little developed; algebraic notation had only recently taken definite form; trigonometry was still used mainly as an adjunct to astronomy; analytical geometry had been invented by Descartes, but no text-books on it of modern type were available; while nothing about the calculus had been published. Mechanics, however, had recently been treated as a science—statics by Stevinus and dynamics by Galileo—and this paved the way for [232] ]Newton’s investigations. In particular, Galileo had established principles which foreshadowed the first two laws of motion, and had deduced formulae in linear motion like v² = 2fs, s = ½ft², and in circular motion like f = v² / r.
Newton prefaced the Principia by explaining that the earliest problems in natural philosophy which attract attention are connected with the phenomena of motion, and it was with motion). that the book dealt. To discuss motion effectively, it was necessary to give precision to the language used, and accordingly he propounded definitions of mass, momentum, inertia, and so on, which have settled the language of the subject. He next enunciated his three well-known laws of motion, and described the experiments on which he based them. He followed this up by deducing rules for the composition and resolution of forces, and discussed relative motion.
This preliminary matter is followed by the first book, concerned with the motion of bodies in an unresisting medium. It is divided into fourteen sections containing ninety-eight propositions with various interpolated lemmas, corollaries, and scholia.
The first section is on the method of prime and ultimate ratios, by the use of which Newton was able, in effect, to integrate. He applied this to the curvature and the areas of curves, and proved that, [233] ]at the very beginning of the motion of a body from rest under any force, the space described is proportional to the force and the square of the time.
The second section is concerned with the motion of a particle under a central force. It contains the well-known propositions that if the force is central the area swept out by the vector to the centre is proportional to the time, and conversely that if such area is proportional to the time the particle is acted on by a central force. Newton further discussed particular cases of circular, elliptic, and spiral motion. In the third section he dealt with motion in a conic under a central force to the focus, showed that in this case the force must vary inversely as the square of the distance, and conversely that if a particle be projected from any point in any direction with any velocity under such a force it must describe a conic about the centre of force as a focus, and that in such elliptic orbits the periodic times are in the sesquiplicate ratio of the major axes of the ellipses. He also explained how to treat the problem if disturbing forces are introduced. These two sections solved the problem of planetary motion if the planets could be treated as particles and did not disturb one another’s motions.
The fourth and fifth sections are given up to the proof of certain geometrical propositions in conics required for subsequent discussions: in particular [234] ]the construction of a conic when a focus and three other conditions or when five points on it or five tangents to it are given.
In the sixth section Newton returned to the problem of the motion of a particle in an ellipse under a central force to a focus, and discussed how to determine the position of the particle at any given time. (Kepler’s Problem.)