When George Shoobridge Carr (1837–1914) wrote his Synopsis of Elementary Results he intended it as an aid to students preparing for degree-level examinations such as the Cambridge Mathematical Tripos, for which he provided private tuition. He would have been startled to see the two volumes, first published in 1880 and 1886 respectively, reissued more than a century later. Notably, in 1903 the work fell into the hands of the Indian prodigy Srinivasa Ramanujan (1887–1920) and greatly influenced his mathematical education. It is the interaction between a methodical teaching aid and the soaring spirit of a self-taught genius which gives this reissue its interest. Volume 1, presented here in its 1886 printing, contains sections on mathematical tables, algebra, the theory of equations, plane trigonometry, spherical trigonometry, elementary geometry and geometrical conics.
Author(s): George Shoobridge Carr
Series: Cambridge Library Collection - Mathematics
Publisher: Cambridge University Press
Year: 1886 (2013)
Language: English
Commentary: Combine Vol.1 (MD5: C190E5D2E12E4880CB815D5F07B0C2C3) + Vol.2 (MD5: 5F026E24F021DFF2D9490F3C54226F96), erase watermarks and add bookmarks. This book remains a true gem over 130 years.
Pages: 1015
Volume 1
PREFACE TO PART I
TABLE OF CONTENTS. PART I
INDEX TO PROPOSITIONS OF EUCLID
Book I
Book II, III
Book IV, VI
Book XI
TABLE OF CONTENTS. PART II
ERRATA. 1
ERRATA. 2
§1. MATHEMATICAL TABLES
Introduction
TABLE I-III
TABLE IV-V
TABLE VI-VIII
Burckhakdt's Factor Tables
1-9000
9000-18000
18000-27000
27000-36000
36000-45000
45000-54000
54000-63000
63000-72000
72000-81000
81000-90000
90000-99000
Log Γ(n)
§2. ALGEBRA
Factors
Newton's Rule for expanding a Binomial (12)
Multiplication and Division, by the Method of Detached Coefficients
Indices
Highest Common Factor
Lowest Common Multiple
Evolution
Quadratic Equations
Theory of Quadratic Expressions
Simultaneous Equations
Ratio and Proportion
Variation
Arithmetical Progression
Geometrical Progression
Harmonical Progression
Permutations and Combinations
Surds
Binomial Theorem
Multinomial Theorem
Logarithms
Exponential Theorem
Continued Fractions and Convergents
Indeterminate Equations
To reduce a Quadratic Surd to a Continued Fraction
General Theory(199)
Equations
Miscellaneous Equations and Solutions (214)
Imaginary Expressions
Method of Indeterminate Coefficients
Method of Proof by Induction
Partial Fractions
Convergency and Divergency of Series
Expansion of a Fraction
Recurring Series
Summation of Series by the Method of Differences
Direct Factorial Series
Inverse Factorial Series
Composite Factorial Series
Miscellaneous Series
Polygonal Numbers
Figurate Numbers
Hypergeometrical Series
Interest
Annuities
Probabilities
Inequalities
Scales of Notation
Theory of Numbers
§3. THEORY OF EQUATIONS
Factors of an Equation
Descartes' Rule of Signs
The Derived Functions of f(x)
Equal Roots of an Equation
Limits of the Roots
Newton's Method of Divisors
Reciprocal Equations
Binomial Equations
Cubic Equations
Biquadratic Equations
Commensurable Roots
Incommensurable Roots
Symmetrical Functions of the Roots of an Equation
Expansion of an Implicit Function of x
Determinants
General Theory (556)
Analysis of a Determinant (568)
Synthesis of a Determinant (569)
Product of Two Determinants of the n-th Order (570)
Partial and Complementary Determinants (576)
Elimination
I. Bezout's Method (586)
II. Sylvester's Dialytic Method (587)
III. Method of Elimination by Symmetrical Functions (588)
§4. PLANE TRIGONOMETRY
Angular Measurement
Trigonometrical Ratios
Formulae involving Two Angles, and Multiple Angles
Ratios of Certain Angles
Properties of the Triangle
Solution of Triangles
Regular Polygon and Circle
Use of Subsidiary Angles
de Moivre's Theorem
Additional Fobmulae
Examples of the Solution of Triangles (859)
§5. SPHERICAL TRIGONOMETRY
Introductory Theorems
Right-angled Triangles
Oblique-angled Triangles
Spherical Triangle and Circle
Spherical Areas
Polyhedrons
§6. ELEMENTARY GEOMETRY
Miscellaneous Propositions
The Nine-Point Circle
Collinear and Concurrent Systems of Points and Lines
Triangles of Constant Species Circumscribed to a Triangle
Radical Axis
The Method of Inversion
Pole and Polar
Coaxal Circles
Centres and axes of similitude
Anharmonic Ratio
Homographic Systems of Points
Involution
The Method of Projection
On Perspective Drawing
Orthogonal Projection
Projections of the Sphere
Additional Theorems
§7. GEOMETRICAL CONICS
The Sections of the Cone
The Ellipse and Hyperbola
The Parabola
Volume 2
TABLE OF CONTENTS. PART II.
PREFACE TO PART II.
§8. DIFFERENTIAL CALCULUS
Introduction
Differentiation
Successive Differentiation
Partial Differentiation
Theory of Operations
Expansion of Explicit Functions
Expansion of Implicit Functions
Indeterminate Forms
Jacobians
Quantics
Implicit Functions
Change of the Independent Variable
Maxima and Minima
Maxima and Minima values of a function of threeor more variables (1852)
Continuous Maxima and Minima (1866)
§9. INTEGRAL CALCULUS
Introduction
Methods of Integration
Standard Integrals
Various Indefinite Integrals
Integration by Rationalization
Integrals Reducible to Elliptic Integrals
Successive Integration
Hyperbolic Functions
Definite Integrals
Theorems Respecting the Limits of Integration
Methods of Evaluating Definite Integeals
Differentiation under the Sign of Integration
Approximate Integration
The Integrals B(l,m) and Γ(n)
Integration of Algebraic Forms
Integration of Logarithmic and Exponential Forms
Integration of Circular Forms
Integration of Circular Logarithmic and Exponential Forms
Miscellaneous Theorems
Frullani's Formula (2700)
Poisson's Formulae (2702)
Abel's Formula (2705)
Kummer's Formula (2706)
Cauchy's Formula (2712)
Finite Variation of a Parameter
Fourier's Formula
The Function ψ(x)
Numerical Calculation of log Γ(x)
Change of the Variables in a Definite Multiple Integral
Multiple Integrals
Expansions of Functions in Converging Series
Lagrauge's Method (2857)
Miscellaneous Expansions (2911)
Legendre's Function X_n (2936)
Formulae for the Expansion of Functions in Trigonometrical Series
Approximate Integeation
Cotes's Method (2995)
Gauss's Method (2997)
§10. CALCULUS OF VARIATIONS
Functions of One Independent Variable
Particular Cases (3033)
Other Exceptional Cases (3045)
Functions of Two Dependent Vaeiables
Relative Maxima and Minima (3069)
Geometrical Applications (3070)
Functions of Two Independent Variables
Geometrical Applications (3078)
Appendix
On the General Object of the Calculus of Variations (3084)
Successive Variation (3087)
Immediate Intesuability of the Function V (3090)
§11. DIFFERENTIAL EQUATIONS
Generation of Differential Equations
Definitions and Rules
Singular Solutions
Geometrical Meaning of a Singular Solution (3072)
Determination of a Singular Solution from the Differential Equation (3075)
Rules for Discriminating a Singular Solution of the Envelope Species (3078)
First Order Linear Equations
First Order Non-linear Equations
Higher Order Linear Equations
Higher Order Non-linear Equations
Exact Differential Equations
Miscellaneous Methods
Singular Solutions of Higher Order Equations
Equations with more than Two Variables
Simultaneous Equations with One Independent Variable
Partial Differential Equations
Second Order P. D. Equations
Poisson's Equation (3441)
Laplace's Reduction of the Equation (3442)
Law of Reciprocity [Boole, ch. xv
Symbolic Methods
Solution of Linear Differential Equations by Series
Solution by Definite Integrals*
P. D. Equations with more than Two Independent Variables
Differential Resolvents of Algebraic Equations
§12. CALCULUS OF FINITE DIFFERENCES
Introduction
Formulae for First and n-th Differences
Interpolation
Mechanical Quadrature
Summation of Series
Appeoximate Summation
§13. PLANE COORDINATE GEOMETRY
Systems of Coordinates
ANALYTICAL CONICS IN CARTESIAN COORDINATES
Lengths and Areas
Transformation of Coordinates
The Right Line
General Methods
The Circle
The Parabola
The Ellipse and Hyperbola (See also p.233, et seq.)
The Hyperbola referred to the Asymptotes
The General Equation
Similar Conics
Circle of Curvature
Confocal Conics
ANALYTICAL CONICS IN TRILINEAR COORDINATES
The Right Line
Anharmonic Ratio
The General Equation of a Conic
Particular Conics
The Circumscribing Conic of the Trigon (4724)
The Circumscribing Circle of the Trigon (4738)
The Inscribed Conic of the Trigon (4739)
The Inscribed Circle of the Trigon (4747)
General Equation of the Circle (4751)
The Nine-point Circle (4754)
The Triplicate-ratio Circle (4754b)
The Seven-point Circle* (4754c)
Conic and Self-conjugate Triangle
Important Theorems
Carnot's Theorem (4778)
Pascal's Theorem (4781)
Brianchon's Theorem (4783)
The Conic Referred to Two Tangents and the Chord of Contact
Anharmonic Pencils of Conics
Construction of Conics
The Method of Reciprocal Polars
General Rules for Reciprocating (4846)
Tangential Coordinates
On the Intersection of Two Conics
The Method of Projection
Invariants and Covariants
To Find the Foci of the General Conic (4656)
Note on Tangential Coordinates (5030)
THEORY OF PLANE CURVES
Tangent and Normal
Radius of Curvature and Evolute
Inveese Problem and Intrinsic Equation
Asymptotes
Singularities of Curves
Contact of Curves
Envelopes
Integrals of Curves and Areas
Inverse Curves
Pedal Curves
Roulettes
Transcendental and other Curves
Linkages and Linkwork
Mechanical Calculators
Appendix on Biangular Coordinates*
§14. SOLID COORDINATE GEOMETRY
Systems of Coordinates
Transformation of Coordinates
The Sphere
Cylindrical and Conical Surfaces
Conicoids
Central Quadric Surface
The General Equation of a Quadric
Reciprocal Polars
Rules for Reciprocating (5705)
Theory of Tortuous Curves
General Theory of Surfaces
Invariants
Integrals for Volumes and Surfaces
Centre of Mass
Moments and Products of Inertia
Perimeters, Areas, Volumes, Centres of Mass, and Moments of Inertia of Various Figures
JOINT INDEX
Key to the Index
Explanation of Abbreviations, &c.
Index
A
B
C
D
E
F
G
H
I
JK
L
M
N
O
P
Q
R
S
T
UV
WZ
Fig. 1-15
Fig. 16-30
Fig. 31-45
Fig. 46-51
Fig. 52-62
Fig. 63-75
Fig. 76-86
Fig. 87-88
Fig. 89-93
Fig. 94-110
Fig. 111-115
Fig. 116-123
Fig. 124-130
Fig. 133-141
Fig. 142-152
Fig. 153-164
Fig. 165-167
Fig. 168-177
Fig. 178-182
Fig. 183-193
