# Book:L. Harwood Clarke/A Note Book in Pure Mathematics

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## L. Harwood Clarke:

## L. Harwood Clarke: *A Note Book in Pure Mathematics*

Published $\text {1953}$, **William Heinemann Ltd.**

### Contents

- Foreword

- $\text {I}$. ALGEBRA

- Permutations and Combinations
- The value of ${}_n P_r$
- The number of ways of arranging $n$ things in a line
- The number of ways of seating $n$ people at a circular table
- The number of ways of threading $n$ beads on a wire
- The value of ${}_n C_r$
- The number of ways of arranging $n$ things in line if $p$ are alike of one kind and $q$ are alike of another kind
- The number of ways of dividing $\paren {p + q + r}$ things into three unequal groups, the first to contain $p$ things, the second $q$ things and the third $r$ things
- The number of ways of dividing $3 p$ things into three equal groups each containing $p$ things
- The number of selections from $n$ things if any number may be taken
- The number of selections from $n$ different things, $p$ similar things of one kind and $q$ similar things of another kind, if any number may be taken
- ${}^n C_r = {}_n C_{n - r}$
- ${}^n C_r + {}_n C_{r + 1} = {}_{n + 1} C_{r + 1}$

- Permutations and Combinations

- The Method of Induction

- The Binomial Theorem
- Relations between coefficients
- Negative and fractional indices
- Approximations
- The greatest term

- The Binomial Theorem

- Partial Fractions

- Summation of Series
- The sum of the squares of the first $n$ integers
- The sum of the cubes of the first $n$ integers
- Power series

- Summation of Series

- The Exponential and Logarithmic Series
- Applications to summing series

- The Exponential and Logarithmic Series

- The Remainder Theorem

- Roots of Equations
- The quadratic function
- The condition that $\paren {a x^2 + b x + c}$ should be positive for all values of $x$
- The condition for a common root
- The condition for a repeated root
- Relations between roots
- To form the equation whose roots are symmetrical functions of each of the roots of a given equation
- To form the equation whose roots are any symmetrical functions of the roots of a given equation
- The sum of the powers of the roots of a given equation
- Finding numerical roots
- The function $\dfrac {a x^2 + b x + c} {A x^2 + B x + C}$ when $x$ is real

- Roots of Equations

- $\text {II}$. CALCULUS

- Differentiation
- Differential coefficient
- Standard differential coefficients
- Double function
- Product
- Quotient
- Variable index
- Inverse ratios

- Differentiation

- Maximum, Minimum and Point of Inflection

- Velocity and Acceleration

- Tangent and Normal

- Parametric Equations

- Differentiation $n$ Times

- Leibnitz' Theorem

- Maclaurin's Theorem

- Taylor's Theorem

- Integration
- Algebraic integration
- Powers of cos and sine
- Useful substitutions
- Integration by parts

- Integration

- Definite Integrals

- Integral of $1 / x$

- Area, Volume and Centre of Gravity

- Arcs

- Polar Coordinates

- Moments of Inertia

- Reduction Formulae

- Hyperbolic Functions

- Differential Equations

- $\text {III}$. ANALYTICAL GEOMETRY

- The Straight Line
- Point dividing $AB$ in a given ratio
- Centre of gravity of a triangle
- The area of a triangle
- Forms for the equation of a straight line
- The angle between two lines
- The length of the perpendicular
- The equations of the angle bisectors
- Lines through the intersection of two given lines

- The Straight Line

- Pairs of Straight Lines
- Their equation
- The angle between the pair
- The equation of the angle bisectors
- The condition for a pair

- Pairs of Straight Lines

- The Circle
- Its equation
- The tangent of gradient $m$
- The tangent at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- Orthogonal circles
- The length of a tangent
- Coaxal circles and radical axes

- The Circle

- The Parabola
- Its equation
- The tangent of gradient $m$
- The normal
- The equation of a chord
- The tangent at $\tuple {x', y'}$
- The locus of the foot of the perpendicular from the focus to a tangent
- The locus of the intersection of perpendicular tangents
- The polar of $\tuple {x', y'}$
- The feet of the normals from a point
- The locus of the mid points of parallel chords
- The chord of mid point $\tuple {x', y'}$

- The Parabola

- The Ellipse and Hyperbola
- Their equations
- The tangent of gradient $m$
- The auxiliary circle
- The director circle
- The tangent at $\tuple {x', y'}$
- The normal at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- The pole of $l x + m y + n = 0$
- The locus of the mid points of parallel chords
- Conjugate diameters
- The chord of mid point $\tuple {x', y'}$

- The Ellipse and Hyperbola

- The Ellipse
- The eccentric angle
- The line joining $\alpha$ and $\beta$
- Eccentric angles at the ends of conjugate diameters

- The Ellipse

- The Hyperbola
- The asymptotes
- Parametric representation

- The Hyperbola

- The Rectangular Hyperbola
- Its equation
- The tangent at $\tuple {x', y'}$
- The polar of $\tuple {x', y'}$
- The chord joining $m$ and $l$
- The locus of the mid points of parallel chords
- The chord of mid point $\tuple {x', y'}$
- The equation of a hyperbola with given asymptotes

- The Rectangular Hyperbola

- The General Conic
- The conditions for: a pair of straight lines, a circle, a parabola, an ellipse, a hyperbola, a rectangular hyperbola
- A conic through the intersections of two given conics
- The polar of $\tuple {x', y'}$

- The General Conic

- $\text {IV}$. PURE GEOMETRY

- Plane Geometry
- The incentre
- The circumcentre
- The orthcentre
- The centre of gravity
- The centre of similitude
- Apollonius' Circle
- Ptolemy's Theorem
- The Euler Line
- The Nine-Point Circle
- Ceva's Theorem
- Menelaus' Theorem
- The Simson Line
- The radical axis and coaxal circles

- Plane Geometry

- Solid Geometry
- The plane
- Skew lines
- Generators
- The angle between a line and a plane
- The angle between two planes
- If a line is perpendicular to each of two intersecting lines, it is perpendicular to their plane
- The tetrahedron
- Its circumscribing parallelepiped
- Its circumscribing sphere
- The common perpendicular to two skew lines
- The intersection of a sphere and a plane
- The intersection of two spheres

- Solid Geometry

- Orthogonal Projection
- Lines
- Centre of Gravity
- Areas
- The circle
- Geometrical properties of the ellipse

- Orthogonal Projection

- $\text {V}$. TRIGONOMETRY

- Definitions of the Ratios

- Special Angles

- Complementary Angles

- Angles Larger than $90 \degrees$

- The Cosine Formula and the Sine Formula

- The Addition Formulae

- Tangents of Sum and Difference

- The Product Formulae

- Factorization of Sums of Sines and Cosines

- The Auxiliary Angle

- Solution of Equations

- Solution of Triangles

- Inverse Ratios

- Half Angle Formulae

- Area of the Triangle

- The Median and the Centre of Gravity

- The Orthocentre

- The Angle Bisector

- The Pedal Triangle

- The Circumcircle

- The Incircle

- The Ex-circles

- ANSWERS

### Click here for errata

## Source work progress

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous): $\text V$. Trigonometry: The ex-circles

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous): $\text V$. Trigonometry: Area of the triangle

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text V$. Trigonometry: Solution of triangles

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text {IV}$. Pure Geometry: Plane Geometry: The Centre of Similitude

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines: Example $\text{(ii)}$

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text {II}$. Calculus: Exercises $\text {XV}$: $3$

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- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Maximum, Minimum and Point of Inflection

- From the start:

- 1953: L. Harwood Clarke:
*A Note Book in Pure Mathematics*... (previous) ... (next): $\text I$. Algebra: The Binomial Theorem: Exercises $\text {III}$: $1 \ \text {(d)}$