INTRODUCTION TO THE AMERICAN EDITION.

THAT the PRINCIPIA of Newton should have remained so generally unknown in this country to the present day is a somewhat remarkable fact; because the name of the author, learned with the very elements of science, is revered at every hearth-stone where knowledge and virtue are of chief esteem, while, abroad, in all the high places of the land, the character which that name recalls is held up as the noblest illustration of what MAN may be, and may do, in the possession and manifestation of pre-eminent intellectual and moral worth; because the work is celebrated, not only in the history of one career and one mind, but in the history of all achievement and human reason itself; because of the spirit of inquiry, which has been aroused, and which, in pursuing its searchings, is not always satisfied with stopping short of the fountain-head of any given truth; and, finally, because of the earnest endeavour that has been and is constantly going on, in many sections of the Republic, to elevate the popular standard of education and give to scientific and other efforts a higher and a better aim.

True, the PRINCIPIA has been hitherto inaccessible to popular use. A few copies in Latin, and occasionally one in English may be found in some of our larger libraries, or in the possession of some ardent disciple of the great Master. But a dead language in the one case, and an enormous price in both, particularly in that of the English edition, have thus far opposed very sufficient obstacles to the wide circulation of the work. It is now, however, placed within the reach of all. And in performing this labour, the utmost care has been taken, by collation, revision, and otherwise, to render the First American Edition the most accurate and beautiful in our language. "Le plus beau monument que l'on puisse élever à la gloire de Newton, c'est une bonne édition de ses ouvrages:" and a monument like unto that we would here set up. The PRINCIPIA, above all, glows with the immortality of a transcendant mind. Marble and brass dissolve and pass away; but the true creations of genius endure, in time and beyond time, forever: high upon the adamant of the indestructible, they send forth afar and near, over the troublous waters of life, a pure, unwavering, quenchless light whereby the myriad myriads of barques, richly laden with reason, intelligence and various faculty, are guided through the night and the storm, by the beetling shore and the hidden rock, the breaker and the shoal, safely into havens calm and secure.

To the teacher and the taught, the scholar and the student, the devotee of Science and the worshipper of Truth, the PRINCIPIA must ever continue to be of inestimable value. If to educate means, not so much to store the memory with symbols and facts, as to bring forth the faculties of the soul and develope them to the full by healthy nurture and a hardy discipline, then, what so effective to the accomplishment of that end as the study of Geometrical Synthesis? The Calculus, in some shape or other, is, indeed, necessary to the successful prosecution of researches in the higher branches of philosophy. But has not the Analytical encroached upon the Synthetical, and Algorithmic Formulæ been employed when not requisite, either for the evolution of truth, or even its apter illustration? To each method belongs, undoubtedly, an appropriate use. Newton, himself the inventor of Fluxions, censured the handling of Geometrical subjects by Algebraical calculations; and the maturest opinions which he expressed were additionally in favour of the Geometrical Method. His preference, so strongly marked, is not to be reckoned a mere matter of taste; and his authority should bear with preponderating weight upon the decision of every instructor in adopting what may be deemed the best plan to insure the completest mental development. Geometry, the vigorous product of remote time; blended with the earliest aspirations of Science and the earliest applications of Art; as well in the measures of music as in the movement of spheres; as wholly in the structure of the atom as in that of the world; directing MOTION and shaping APPEARANCE; in a word, at the moulding of the created all, is, in comprehensive view, the outward form of that Inner Harmony of which and in which all things are. Plainly, therefore, this noble study has other and infinitely higher uses than to increase the power of abstraction. A more general and thorough cultivation of it should be strenuously insisted on. Passing from the pages of Euclid or Legendre, might not the student be led, at the suitable time, to those of the PRINCIPIA wherein Geometry may be found in varied use from the familiar to the sublime? The profoundest and the happiest results, it is believed, would attend upon this enlargement of our Educational System.

Let the PRINCIPIA, then, be gladly welcomed into every Hall where a TRUE TEACHER presides. And they who are guided to the diligent study of this incomparable work, who become strengthened by its reason, assured by its evidence, and enlightened by its truths, and who rise into loving communion with the great and pure spirit of its author, will go forth from the scenes of their pupilage, and take their places in the world as strong-minded, right-hearted men—such men as the Theory of our Government contemplates and its practical operation absolutely demands.

Life of Sir Isaac Newton 9

THE PRINCIPIA.

The Author's Preface lxvii

BOOK I.

Definitions 73

Axioms, or Laws of Motion 83

OF THE MOTION OF BODIES

Section I: Of the method of first and last ratios of quantities, by the help whereof we demonstrate the propositions that follow 95

Section II: Of the Invention of Centripetal Forces 103

Section III: Of the motion of bodies in eccentric conic sections 116

Section IV: Of the finding of elliptic, parabolic, and hyperbolic orbits, from the focus given 125

Section V: How the orbits are to be found when neither focus is given 131

Section VI: How the motions are to be found in given orbits 153

Section VII: Concerning the rectilinear ascent and descent of bodies 159

Section VIII: Of the invention of orbits wherein bodies will revolve, being acted upon by any sort of centripetal force 168

Section IX: Of the motion of bodies in movable orbits; and of the motion of the apsides 172

Section X: Of the motion of bodies in given superficies; and of the reciprocal motion of funependulous bodies 182

Section XI: Of the motions of bodies tending to each other with centripetal forces 194

Section XII: Of the attractive forces of sphærical bodies 218

Section XIII: Of the attractive forces of bodies which are not of a sphærical figure 233

Section XIV: Of the motion of very small bodies when agitated by centripetal forces tending to the several parts of any very great body 243

BOOK II.

OF THE MOTION OF BODIES

Section I: Of the motion of bodies that are resisted in the ratio of the velocity 251

Section II: Of the motion of bodies that are resisted in the duplicate ratio of their velocities 258

Section III: Of the motions of bodies which are resisted partly in the ratio of the velocities, and partly in the duplicate of the same ratio 279

Section IV: Of the circular motion of bodies in resisting mediums 287

Section V: Of the density and compression of fluids; and of hydrostatics 293

Section VI: Of the motion and resistance of funependulous bodies 303

Section VII: Of the motion of fluids and the resistance made to projected bodies 323

Section VIII: Of motion propagated through fluids 356

Section IX: Of the circular motion of fluids 370

BOOK III.

Book III 383

Rules of Reasoning in Philosophy 384

Phænomena, or Appearances 386

Propositions I-IX (Force of gravity) 390

Propositions X-XXIV (Motions of celestial bodies and the sea) 400

Propositions XXV-XXXIII (Quantity of lunar motions) 419

Propositions XXXVI-XXXVIII (Forces to move the sea) 449

Lemmas I-III, Proposition XXXIX (Precession of equinoxes) 455

Lemmas IV-XI, Propositions XL-XLII (Comets) 460

General Scholium 503

The System of the World. 511

Index to the Principia. 575