_Sect._ 1. Establishment of the First Law of Motion. 322 _Sect._ 2. Formation and Application of the Motion of Accelerating Force. Laws of Falling Bodies. 324 _Sect._ 3. Establishment of the Second Law of Motion.--Curvilinear Motions. 330 _Sect._ 4. Generalization of the Laws of Equilibrium.--Principle of Virtual Velocities. 331 _Sect._ 5. Attempts at the Third Law of Motion.--Notion of Momentum. 334 CHAPTER III.--SEQUEL TO THE EPOCH OF GALILEO.--PERIOD OF VERIFICATION AND DEDUCTION. 340 CHAPTER IV.--DISCOVERY OF THE MECHANICAL PRINCIPLES OF FLUIDS. _Sect._ 1. Rediscovery of the Laws of Equilibrium of Fluids. 345 _Sect._ 2. Discovery of the Laws of Motion of Fluids. 348 CHAPTER V.--GENERALIZATION OF THE PRINCIPLES OF MECHANICS. _Sect._ 1. Generalization of the Second Law of Motion.--Central Forces. 352 _Sect._ 2. Generalization of the Third Law of Motion.--Centre of Oscillation.--Huyghens. 356 CHAPTER VI.--SEQUEL TO THE GENERALIZATION OF THE PRINCIPLES OF MECHANICS.--PERIOD OF MATHEMATICAL DEDUCTION.--ANALYTICAL MECHANICS. 362 1. Geometrical Mechanics.--Newton, &c. 363 2. Analytical Mechanics.--Euler. 363 3. Mechanical Problems. 364 4. D'Alembert's Principle. 365 5. Motion in Resisting Media.--Ballistics. 365 6. Constellation of Mathematicians. 366 7. The Problem of Three Bodies. 367 8. Mécanique Céleste, &c. 371 9. Precession.--Motion of Rigid Bodies. 374 10. Vibrating Strings. 375 11. Equilibrium of Fluids.--Figure of the Earth.--Tides. 376 12. Capillary Action. 377 13. Motion of Fluids. 378 14. Various General Mechanical Principles. 380 15. Analytical Generality.--Connection of Statics and Dynamics. 381 {19} BOOK VII. HISTORY OF PHYSICAL ASTRONOMY. CHAPTER I.--PRELUDE TO THE INDUCTIVE EPOCH OF NEWTON. 385 CHAPTER II.--THE INDUCTIVE EPOCH OF NEWTON.--DISCOVERY OF THE UNIVERSAL GRAVITATION OF MATTER, ACCORDING TO THE LAW OF THE INVERSE SQUARE OF THE DISTANCE. 399