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

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

Published $\text {1851}$, 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