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

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## Elias Loomis:

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

Published $\text {1851}$, **Harper & Brothers**

### Subject Matter

### 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