VI CONTENTS. 



Nomenclature, 22. Notation, ib. Advantages and disad 

 vantages of the two notations, 23. Finding fluxion (or diffe 

 rential) of a rectangle, 24. Square, ib. Solid, ib. Quantity 



of any power by analogy, ib Deduction of the rules from 



other principles, 25. Finding fluents (or integrals), ib. 

 Method of drawing tangents, 26. Normals, ib. Exemplified 

 in the conic sections, 27. Problems of maxima and minima, 

 ib. Example, ib. Quadrature of curves, 28. Example : 

 Parabola,, ib. Rectification of curves, ib. Example : Cir 

 cular arcs, ib. Measurement of solids, 29- Example : Cone, 

 sphere, and cylinder, ib. Finding radius of curvature, ib. 

 Example : Parabola, SO. Addition of constant quantity in 

 integration, ib. Method of investigation used by Sir Isaac 

 Newton, ib. 



Subjects of the Three Boohs, 31. 



(SECTION II. Principia.} Areas proportional to the times, round 

 a centre of forces, ib. Empirical discovery of Kepler, ib. 

 Proposition and its converse proved, 32. Corollaries to this 

 fundamental law of centripetal forces, 33. Law of circular 

 motion, the force as the square of the arc, and inversely as the 

 distance, 34. Demonstration, ib. Importance of this propo 

 sition, 35. Consequences in showing the laws of motion, ib. 

 Demonstrates the general law, of which Kepler s rule of the 

 sesquiplicate ratio is one case, 36. Demonstrates the law of 

 the inverse square of the distance, 37. Law extended to other 

 curves, ib. Consequence that bodies fall through portions of 

 the diameter, proportional to the squares of the times in which 

 they describe the corresponding arcs, 38. Moon being deflected 

 from the tangent of her orbit by gravitation proved from hence, 

 40. Reference to other proofs of it, 41. note. Investigation 

 of General Expressions for Centripetal Force, 42. Five for 

 mulas given, 43. Herrman s, 44. Laplace s, 46. Maclau- 

 rin s, ib. J. Bernouilli s, ib. Proof that this is taken from 

 Prop. VI. B. I., Principia, 47- Keill s imperfect acquaintance 

 with this subject, ib. Herrman s mistake, 48. Formulae 

 exemplified in the case of the parabola, 49. Ellipse and 

 hyperbola, ib. Centrifugal forces. Formulae of Huygens, 50. 



Subject of Centripetal forces divided into four heads, ib. i. The 

 force required to describe given conic sections. ii. The drawing 

 conic sections from points or tangents being given ; 1. When 

 one focus is given ; 2. When neither is given. iii. The find 

 ing the motion in trajectories that are given. iv. The finding 

 trajectories generally when the forces are given. 



i. The first head is treated of in the remainder of the Second, and 

 the whole of the Third Sections of the Principia. Central force 

 in a circle, when the centre of forces is the centre of the circle, 

 or any other point in the diameter, or in the circumference re- 



