Book:Elias Loomis/Elements of Analytical Geometry and of the Differential and Integral Calculus

Subject Matter

 * Analytic Geometry
 * Calculus

Contents

 * Preface

ANALYTICAL GEOMETRY

 * SECTION I. APPLICATION OF ALGEBRA TO GEOMETRY.
 * Geometrical Magnitudes represented by Algebraic Symbols
 * Solution of Problems


 * SECTION II. CONSTRUCTION OF EQUATIONS.
 * Construction of the Sum and Difference of two Quantities
 * Product of several Quantities
 * Fourth Proportional to three Quantities
 * Mean Proportional between two Quantities
 * Sum or Difference of two Squares
 * To inscribe a Square in a given Triangle
 * To draw a Tangent to two Circles


 * SECTION III. ON THE POINT AND STRAIGHT LINE.
 * Methods of denoting the position of a Point
 * Abscissa and Ordinate defined
 * Equations of a Point
 * Equations of a Point in each of the four Angles
 * Equation of a straight Line
 * Four Positions of the proposed Line
 * Equation of the first Degree containing two Variables
 * Equation of a straight Line passing through a given Point
 * Equation of a straight Line passing through two given Points
 * Distance between two given Points
 * Angle included between two Lines
 * Transformation of Co-ordinates
 * Formulas for passing from one System of Axes to a Parallel System
 * Formulas for passing from Rectangular Axes to Rectangular Axes
 * Formulas for passing from Rectangular to Oblique Axes
 * Formulas for passing from Rectangular to Polar Co-ordinates


 * SECTION IV. ON THE CIRCLE.
 * Equation of the Circle when the Origin is at the Center
 * Equation of the Circle when the Origin is on the Circumference
 * Most general form of the Equation
 * Equation of a Tangent to the Circle
 * Polar Equation of the Circle


 * SECTION V. ON THE PARABOLA.
 * Definitions
 * Equation of the Parabola
 * Equation of a Tangent Line
 * Equation of a Normal Line
 * The Normal bisects the Angle made by the Radius Vector and Diameter
 * Perpendicular from the Focus upon a Tangent
 * Equation referred to a Tangent and Diameter
 * Parameter of any Diameter
 * Polar Equation of the Parabola
 * Area of a Segment of a Parabola


 * SECTION VI. ON THE ELLIPSE.
 * Definitions
 * Equation of the Ellipse referred to its Center and Axes
 * Equation when the Origin is at the Vertex of the Major Axis
 * Squares of two Ordinates as Products of parts of Major Axis
 * Ordinates of the Circumscribed Circle
 * Every Diameter bisected at the Center
 * Supplementary Chords
 * Equation of a Tangent Line
 * Equation of a Normal Line
 * The Normal bisects the Angle formed by two Radius Vectors
 * Supplementary Chords parallel to a Tangent and Diameter
 * Equation of Ellipse referred to Conjugate Diameters
 * Squares of two Ordinates as Products of parts of a Diameter
 * Sum of Squares of two Conjugate Diameters
 * Parallelogram on two Conjugate Diameters
 * Polar Equation of the Ellipse
 * Area of the Ellipse


 * SECTION VII. ON THE HYPERBOLA
 * Definitions
 * Equation of the Hyperbola referred to its Center and Axes
 * Equation when the Origin is at the Vertex of the Transverse Axis
 * Squares of two Ordinates as Products of parts of Transverse Axis
 * Every Diameter bisected at the Center
 * Supplementary Chords
 * Equation of a Tangent Line
 * Equation of a Normal Line
 * The Tangent bisects the Angle contained by two Radius Vectors
 * Supplementary Chords parallel to a Tangent and Diameter
 * Equation referred to Conjugate Diameters
 * Squares of two Ordinates as the Rectangles of the Segments of a Diameter
 * Difference of Squares of Conjugate Diameters
 * Parallelogram on Conjugate Diameters
 * Polar Equation of the Hyperbola
 * Asymptotes of the Hyperbola
 * Equation of the Hyperbola referred to its Asymptotes
 * Parallelogram contained by Co-ordinates of the Curve
 * Equation of Tangent Line
 * Portion of a Tangent between the Asymptotes


 * SECTION VIII. CLASSIFICATION OF ALGEBRAIC CURVES.
 * Every Equation of the Second Degree is the Equation of a Conic Section
 * The Term containing the Product of the Variables removed
 * The Terms containing the first Power of the Variables removed
 * Lines divided into Classes
 * Number of Lines of the different orders
 * Family of Curves


 * SECTION IX. TRANSCENDENTAL CURVES.
 * Cycloid - Defined
 * Equation of the Cycloid
 * Logarithmic Curve - its Properties
 * Spiral of Archimedes - its Equation
 * Hyperbolic Spiral - its Equation
 * Logarithmic Spiral - its Equation

DIFFERENTIAL CALCULUS

 * SECTION I. DEFINITIONS AND FIRST PRINCIPLES - DIFFERENTIATION OF ALGEBRAIC FUNCTIONS.
 * Definitions - Variables and Constants
 * Functions - Explicit and implicit - increasing and decreasing
 * Limit of a Variable Quantity
 * Rate of Variation of the Area of a Square
 * Rate of Variation of the Solidity of a Cube
 * Differential defined - Differential Coefficient
 * Rule for finding the Differential Coefficient
 * Differential of any power of a Variable
 * Product of a Variable by a Constant
 * Differential of a Constant Term
 * General expression for the second State of a Function
 * Differential of the Sum or Difference of several Functions
 * Differential of the Product of several Functions
 * Differential of a Fraction
 * Differential of a Variable with any Exponent
 * Differential of the Square Root of a Variable
 * Differential of a Polynomial raised to any Power


 * SECTION II. SUCCESSIVE DIFFERENTIALS - MACLAURIN'S THEOREM - TAYLOR'S THEOREM - FUNCTIONS OF SEVERAL INDEPENDENT VARIABLES.
 * Successive Differentials - Second Differential Coefficient
 * Maclaurin's Theorem - Applications
 * Taylor's Theorem - Applications
 * Differential Coefficient of the Sum of two Variables
 * Differentiation of Functions of two or more independent Variables


 * SECTION III. SIGNIFICATION OF THE FIRST DIFFERENTIAL COEFFICIENT - MAXIMA AND MINIMA OF FUNCTIONS
 * Signification of the first Differential Coefficient
 * Maxima and Minima of Functions defined
 * Method of finding Maxima and Minima
 * Application of Taylor's Theorem
 * How the Process may be abridged
 * Examples


 * SECTION IV. TRANSCENDENTAL FUNCTIONS
 * Transcendental Functions
 * Differential of an Exponential Function
 * Differential of a Logarithm
 * Circular Functions
 * Differentials of Sine, Cosine, Tangent, and Cotangent
 * Differentials of Logarithmic Sine, Cosine, Tangent, and Cotangent
 * Differentials of Arc in terms of Sine, Cosine, etc.
 * Development of the Sine and Cosine of an Arc


 * SECTION V. APPLICATION OF THE DIFFERENTIAL CALCULUS TO THE THEORY OF CURVES.
 * Differential Equation of Lines of different Orders
 * Length of Tangent, Subtangent, Normal and Subnormal
 * Formulas applied to the Conic Sections
 * Subtangent of the Logarithmic Curve
 * Subtangent and Tangent of Polar Curves
 * Formulas applied to the Spirals
 * Differential of an Arc, Area, Surface, and Solid of Revolution
 * Differential of the Arc and Area of a Polar Circle
 * Asymptotes of Curves


 * SECTION VI. RADIUS OF CURVATURE - EVOLUTES OF CURVES
 * Curvature of Circles
 * Radius of Curvature at any Point of a Curve
 * Radius of Curvature of a Conic Section
 * Evolutes of Curves defined
 * Equation of the Evolute determined
 * Evolute of the common Parabola
 * Properties of the Cycloid
 * Expression for the Tangent, Normal etc., to the Cycloid
 * Radius of Curvature of the Cycloid
 * Evolute of the Cycloid


 * SECTION VII. ANALYSIS OF CURVE LINES
 * Singular Points of a Curve
 * Tangent parallel or perpendicular to Axis of Abscissas
 * Where Curve is Convex toward the Axis
 * Where Curve is Concave toward the Axis
 * To determine a Point of Inflection
 * To determine a Multiple Point
 * To determine a Cust
 * To determine an isolated Point

INTEGRAL CALCULUS

 * SECTION I. INTEGRATION OF MONOMIAL DIFFERENTIALS - OF BINOMIAL DIFFERENTIALS - OF THE DIFFERENTIALS OF CIRCULAR ARCS.
 * Integral Calculus defined
 * Integral of the Product of a Differential by a Constant
 * Integral of the Sum or Difference of any number of Differentials
 * Constant Term added to the Integral
 * Integration of Monomial Differentials
 * Integration of Logarithms
 * Integral of a Polynomial Differential
 * Integral of a Binomial Differential
 * Definite Integral
 * Integrating between Limits
 * Integration by Series
 * Integration of the Differentials of Circular Arcs
 * Integration of Binomial Differentials
 * When a Binomial Differential can be integrated
 * Integration by Parts
 * To diminish the Exponent of the Variable without the Parenthesis
 * When the Exponent of the Variable is Negative
 * To diminish the Exponent of the Parenthesis
 * When the Exponent of the Parenthesis is Negative


 * SECTION II. APPLICATIONS OF THE INTEGRAL CALCULUS
 * Rectification of Plane Curves
 * Quadrature of Curves
 * Area of Spirals
 * Area of Surfaces of Revolution
 * Cubature of Solids of Revolution


 * MISCELLANEOUS EXAMPLES