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
CHAPTER IV.
THE DIFFERENTIAL AND INTEGRAL CALCULUS 120
Its two fundamental Divisions 120
Their Relations to each Other 121
Use of the Differential Calculus as preparatory to that of the Integral 123 2. Employment of the Differential Calculus alone 125 3. Employment of the Integral Calculus alone 125 Three Classes of Questions hence resulting 126
The Differential Calculus 127
Two Cases: Explicit and Implicit Functions 127 Two sub-Cases: a single Variable or several 129 Two other Cases: Functions separate or combined 130 Reduction of all to the Differentiation of the ten elementary Functions 131 Transformation of derived Functions for new Variables 132 Different Orders of Differentiation 133 Analytical Applications 133
The Integral Calculus 135
Its fundamental Division: Explicit and Implicit Functions 135 Subdivisions: a single Variable or several 136 Calculus of partial Differences 137 Another Subdivision: different Orders of Differentiation 138 Another equivalent Distinction 140 Quadratures 142 Integration of Transcendental Functions 143 Integration by Parts 143 Integration of Algebraic Functions 143 Singular Solutions 144 Definite Integrals 146 Prospects of the Integral Calculus 148