CHAPTER III.

TRANSCENDENTAL ANALYSIS:

Page

ITS DIFFERENT CONCEPTIONS 88

Preliminary Remarks 88 Its early History 89

METHOD OF LEIBNITZ 91 Infinitely small Elements 91 _Examples_: 1. Tangents 93 2. Rectification of an Arc 94 3. Quadrature of a Curve 95 4. Velocity in variable Motion 95 5. Distribution of Heat 96 Generality of the Formulas 97 Demonstration of the Method 98 Illustration by Tangents 102

METHOD OF NEWTON 103 Method of Limits 103 _Examples_: 1. Tangents 104 2. Rectifications 105 Fluxions and Fluents 106

METHOD OF LAGRANGE 108 Derived Functions 108 An extension of ordinary Analysis 108 _Example_: Tangents 109 _Fundamental Identity of the three Methods_ 110 _Their comparative Value_ 113 That of Leibnitz 113 That of Newton 115 That of Lagrange 117