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

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

Published $1851$, Harper & Brothers.


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