DIVISIONS OF THE CALCULUS 53 _The Calculus of Values, or Arithmetic_ 57 Its Extent 57 Its true Nature 59 _The Calculus of Functions_ 61 Two Modes of obtaining Equations 61 1. By the Relations between the given Quantities 61 2. By the Relations between auxiliary Quantities 64 Corresponding Divisions of the Calculus of Functions 67 CHAPTER II. ORDINARY ANALYSIS; OR, ALGEBRA. 69 Its Object 69 Classification of Equations 70 ALGEBRAIC EQUATIONS 71 Their Classification 71 ALGEBRAIC RESOLUTION OF EQUATIONS 72 Its Limits 72 General Solution 72 What we know in Algebra 74 NUMERICAL RESOLUTION OF EQUATIONS 75 Its limited Usefulness 76 Different Divisions of the two Systems 78 THE THEORY OF EQUATIONS 79 THE METHOD OF INDETERMINATE COEFFICIENTS 80 IMAGINARY QUANTITIES 81 NEGATIVE QUANTITIES 81 THE PRINCIPLE OF HOMOGENEITY 84 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