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CHAPTER VI.

ANALYSIS OF THE PRINCIPIA.

In the last chapter I mentioned in the briefest possible terms. the object and contents of the three books of the Principia. In this chapter I propose to indicate the subjects of the successive propositions. The work itself is so easily accessible that I shall not describe the proofs nor discuss the arrangement of the work, but shall confine myself to a mere list of the results established. Moreover, I wish explicitly to call the attention of the reader to the fact that in general the enunciations of the propositions and the prefatory and other notes are not translated literally.

A complete edition of the Principia should, I think, show the changes made in the second edition, and the further changes introduced in the third edition. I possess in manuscript a list of the additions and variations made in the second edition ; the changes are very numerous, in fact I find that of the 494 (i.e. 510-16) pages in the first edition 397 are more or less modified in the second edition. The most important alterations are the new preface by Cotes; the propositions on the resistance of fluids, book ii. section vii. props. 34-40 ; the lunar theory in book iii.; the proposition on the precession of the equinoxes, book iii. prop. 39; and the propositions on the theory of comets, book iii. props. 41, 42. I have not formed a list of the changes introduced into the third edition, but I believe that the bulk of them are given in the list by

Adams which is printed by Brewster*; the most important being the scholium on fluxions, book ii. lemma 2, and the addition of a new scholium on the motion of the moon's nodes, book iii. prop. 33, to which I may add an account of some additional experiments on the resistance of the air to bodies falling through it, and the use of some fresh astronomical observations in book iii.

Any of the three editions will serve almost equally well for drawing up a synopsis of the contents of the work where (as in this chapter) the proofs are not described, but in most cases if the proofs differ materially I have stated the fact.

PREFACE.

The first edition is preceded by a preface by Newton, to which, in the second edition, the date May 8, 1686, was added. Here Newton describes briefly the object of the work and his ideas of philosophy; he acknowledges his obligations to Halley and the Royal Society, and mentions a few facts connected with the preparation of the book for the press. To the second edition he prefixed a short note dated March 28, 1713, while Cotes added a long preface, dated May 12, 1713, in which he indicates the changes introduced. In the third edition Newton inserted a short prefatory note dated Jan. 12, 1725–6.

The preface to the first edition commences with a statement of the object of the work and a comparison of the methods used in mechanics and geometry.

Newton then explains that the book is entitled the mathematical principles of philosophy, because, says he, all the difficulty of philosophy seems to consist in this-to find from the phenomena of motion the laws of the forces in nature, and then from these laws to deduce other phenomena; this is the object of the general propositions in the first and second books, while in the third book the laws are applied to explain the phenomena of the solar system. I wish, he continues, that we could derive the rest of the phenomena in nature by the same kind of reasoning from mechanical principles; for I am

led by many reasons to suspect that all these phenomena may depend upon certain forces by which the particles of bodies (by some causes as yet unknown) are either mutually impelled towards each other and cohere in regular figures, or are repelled and recede from each other; which forces being unknown, philosophers have hitherto interrogated nature in vain. But I hope that the principles here laid down will afford some light either to this or some truer method of philosophy. In preparing this work, he proceeds, that most acute and learned scholar, Mr. Edmund Halley, has zealously assisted me by correcting the proof sheets and taking an interest in the general arrangement; nor is this all, for it was to his solicitations that its publication is due. For when he had obtained from me my proof of the figure of the celestial orbits, he continually urged me to communicate it to the Royal Society, who then by their requests and kind encouragement induced me to think of publishing it. But, after I had commenced to treat of the inequalities of the moon's motion, and had begun to consider other questions relating to the laws and measurement of gravity and other forces, the curves that would be described by bodies attracted according to given laws, the motion of several bodies moving among themselves, the motion of bodies in resisting mediums, the resistances, densities, and motions of the mediums themselves, the orbits of comets, and similar problems, I thought the publication should be deferred until I had investigated these questions, and could put the whole together in one book. What relates to the lunar theory (being imperfect) I have collected in one place in the corollaries of prop. 66, in order to avoid being obliged to demonstrate the several effects there enumerated by a method more prolix than the subject deserved, and thus interrupt the sequence of the propositions. Some things, discovered after the rest, I chose to insert in places less suitable, rather than change the numbers of the propositions and the references. I heartily beg, he concludes, that what I have here done may be read leniently; and that the defects in so difficult a subject be not so much blamed as kindly corrected and investigated afresh by my readers.

DEFINITIONS.

The Principia commences with eight definitions, each being followed by an explanation or illustration of its meaning. These definitions are to the following effect:

Def. 1. The mass of a body is measured by the product of its density and volume. Newton adds that practically the mass of a body is estimated by its weight, to which the mass is proportional.

Def. 2. The momentum of a body is measured by the product of its mass and velocity.

Def. 3. The inertia of a body is its tendency to continue in its state of rest or of uniform motion in a straight line.

Def. 4. A force impressed on a body is an action which tends to change its state of rest or of uniform motion in a straight line.

Def. 5. A force is said to be centripetal if it tend to move the body towards a point. In the second and third editions this is illustrated by a discussion of the action of gravity.

Def. 6. An absolute force or the absolute quantity of a centripetal force is proportional to and is measured by the efficacy of the cause that propagates it from the centre [and thus ex. gr. by its magnitude at a unit distance].

Def. 7. An accelerative force or the accelerative quantity of a centripetal force is proportional to and is measured by the velocity generated in a given time.

Def. 8. A moving force or the moving quantity of a centripetal force is proportional to and is measured by the momentum generated in a given time.

Note. To the above definitions Newton appends a memorandum that his object is to discuss the mathematical effects of forces, and that his language is not to be taken as implying any physical theory as to the cause or origin of force.

Scholium. On the conceptions of time, space, place, and motion; which ideas are classified into absolute and relative, true and apparent, mathematical and common.

AXIOMS OR LAWS OF MOTION.

The definitions are followed by three laws of motion, which are as follows:

Law 1. Every body continues in its state of rest or of uniform motion in a straight line unless compelled to change that state by forces impressed on it. [This seems to be a consequence of the second law, and if so it is not clear why it was enunciated as a separate law.]

Law 2. The change of momentum [per unit of time] is always proportional to the moving force impressed, and takes place in the direction in which the force is impressed.

Law 3. To every action of one body on another there is always opposed an equal and opposite reaction of the second body on the first.

To these laws six corollaries are added, namely, (i) the parallelogram of forces; (ii) on the composition and resolution of forces; (iii)

the total momentum of a system of bodies in any direction is unaffected by their mutual actions; (iv) the state of motion or rest of the centre of gravity of a system of bodies is unaffected by their mutual actions; (v) the relative motions of bodies in a given space are the same whether that space is at rest or in uniform motion in a straight line; and (vi) the relative motions of bodies are unaffected by imposing equal and parallel accelerations on each of them.

Scholium. On the history of the laws of motion; the relations between the space described, the time occupied, and the velocity acquired in uniformly accelerated motion; the evidence for the truth of the laws of motion; and the meaning to be assigned to them. This scholium contains also an account of various experiments made by Newton, and an indication of the principle of conservation of energy as far as machines are concerned.

Book I. ON THE MOTION OF BODIES IN UNRESISTING MEDIUMS.

Newton divides the Principia into three books: the first on the motion of bodies in unresisting mediums, the second on the motion of bodies in resisting mediums, and the third on the application of the results of the first two books to the explanation of the solar system.

The first book is divided into fourteen sections as follows:

SECTION I.-On the method of prime and ultimate ratios.

Lemma 1. On the definition of a limit.

Lemmas 2, 3, 4. On the quadrature of curves.

Lemma 5. On similar figures.

Lemmas 6, 7, 8. On the ultimate equality of the evanescent arc, chord, and tangent of a continuous curve.

Lemma 9. On the ratio of the areas of certain evanescent triangles. Lemma 10. The spaces which a body describes [from rest] under any finite force, whether constant or continually increasing or continually decreasing, are in the very beginning of the motion in the duplicate ratio of the times of description.

Lemma 11. On the measure of curvature.

Scholium. In this is included a comparison of the method of limits, the method of exhaustions, and the method of indivisibles.

SECTION II. On the determination of centripetal forces.

Prop. 1. If a body describe an orbit under forces to a fixed point, the areas which it describes by radii drawn to the fixed centre of force