Chapter III: Transcendental Analysis

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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