Trigonometry A Complete Introduction Hugh Neill

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Digital Instant Download
Author(s): Hugh Neill
ISBN(s): 9781444191141, 1444191144
Edition: Paperback
File Details: PDF, 4.99 MB
Year: 2013
Language: english

TRIGONOMETRY
A complete introduction
Paul Abbott
Revised by Hugh Neill

Contents
Welcome to Trigonometry:
A complete introduction
Introduction
1 The tangent
1.1 Introduction
1.2 The idea of the tangent ratio
1.3 A definition of tangent
1.4 Values of the tangent
1.5 Notation for angles and sides
1.6 Using tangents
1.7 Opposite and adjacent sides
2 Sine and cosine
2.1 Introduction
2.2 Definition of sine and cosine
2.3 Using the sine and cosine
2.4 Trigonometric ratios of 45°, 30° and 60°
2.5 Using the calculator accurately
2.6 Slope and gradient
2.7 Projections
2.8 Multistage problems
3 In three dimensions
3.1 Introduction
3.2 Pyramid problems

3.3 Box problems
3.4 Wedge problems
4 Angles of any magnitude
4.1 Introduction
4.2 Sine and cosine for any angle
4.3 Graphs of sine and cosine functions
4.4 The tangent of any angle
4.5 Graph of the tangent function
4.6 Sine, cosine and tangent
5 Solving simple equations
5.1 Introduction
5.2 Solving equations involving sines
5.3 Solving equations involving cosines
5.4 Solving equations involving tangents
6 The sine and cosine formulae
6.1 Notation
6.2 Area of a triangle
6.3 The sine formula for a triangle
6.4 The ambiguous case
6.5 The cosine formula for a triangle
6.6 Introduction to surveying
6.7 Finding the height of a distant object
6.8 Distance of an inaccessible object
6.9 Distance between two inaccessible but visible objects
6.10 Triangulation
7 Radians
7.1 Introduction

7.2 Radians
7.3 Length of a circular arc
7.4 Converting from radians to degrees
7.5 Area of a circular sector
8 Relations between the ratios
8.1 Introduction
8.2 Secant, cosecant and cotangent
9 Ratios and compound angles
9.1 Compound angles
9.2 Formulae for sin(A + B) and sin(A − B)
9.3 Formulae for cos(A + B) and cos(A − B)
9.4 Formulae for tan(A + B) and tan(A − B)
9.5 Worked examples
9.6 Multiple angle formulae
9.7 Identities
9.8 More trigonometric equations
10 The forms a sin x and b cos x
10.1 Introduction
10.2 The form y = a sin x + b cos x
10.3 Using the alternative form
11 The factor formulae
11.1 The first set of factor formulae
11.2 The second set of factor formulae
12 Circles related to triangles
12.1 The circumcircle
12.2 The incircle

12.3 The ecircles
12.4 Heron’s formula: the area of a triangle
13 General solution of equations
13.1 The equation sin θ = sin α
13.2 The equation cos θ = cos α
13.3 The equation tan θ= tan α
Glossary
Summary of trigonometric formulae
Answers

Welcome to Trigonometry: A complete
introduction
Teach Yourself Trigonometry has been substantially revised and
rewritten to take account of modern needs and recent developments in
the subject.
It is anticipated that every reader will have access to a scientific
calculator which has sines, cosines and tangents, and their inverses. It
is also important that the calculator has a memory, so that intermediate
results can be stored accurately. No support has been given about how
to use the calculator, except in the most general terms. Calculators
vary considerably in the keystrokes which they use, and what is
appropriate for one calculator may be inappropriate for another.
There are many worked examples in the book, with complete, detailed
answers to all the questions. At the end of each worked example, you
will find the symbol to indicate that the example has been
completed, and what follows is text.
Some of the exercises from the original Teach Yourself Trigonometry
have been used in this revised text, but all the answers have been
reworked to take account of the greater accuracy available with
calculators.
I would like to thank Linda Moore for her help in reading and
correcting the text. But the responsibility for errors is mine.
Hugh Neill

Introduction
Trigonometry is the study of the relationships between the sides and
angles in a triangle. It is one of the most practical branches of pure
mathematics and it has many applications in the real world.
Trigonometry is based on the principle that the ratio between two
sides of any similar right-angled triangle is a constant: this enables
you to calculate the size of any missing sides or angles in a right-
angled triangle. The trigonometric functions (sine, cosine and tangent)
are defined as the ratio between two sides of a right-angled triangle.
Using these functions, you can calculate the area of any triangle or
find missing angles or sides – these skills have obvious applications in
surveying and civil engineering. Trigonometry enables surveyors to
work out the height of buildings or use triangulation to work out the
exact location of a fixed point, which is vital for map making. The
triangle is the strongest shape to use in a structure and so trigonometry
underpins many of the calculations needed in civil engineering. Many
bridges, roofs and other structures are held up by a system of
triangular supports.
Trigonometry has many other real-life applications, for example it is
used extensively in mechanics to describe the motion of objects. To
work out the trajectory of a bullet fired from a gun or the motion of a
simple pendulum requires trigonometry – even the way that light or
sound waves move can be described using trigonometry.
Trigonometry was first used in astronomy and still has many
applications in this field today. For instance, it is used to accurately
work out the distance to nearer stars using the phenomenon of
parallax, which is the apparent motion of nearby stars relative to more
distant stars. You can see parallax for yourself by holding out a finger
and then looking at it with first just your right eye and then just your

left. The position of your finger will appear to shift, relative to the
more distant background. A nearby star appears to move slightly
against the background of more distant stars when viewed twice,
using observations made six months apart – once the earth has made
half of its orbit around the sun. Using the distance that the earth has
moved in this time as a base line, it is possible to construct a right-
angled triangle using the sun, the earth and the star in question as the
three vertices. The distance to the star can then be worked out using
the angle of parallax and the distance between the sun and the earth.
In fact, the earliest practical uses of trigonometry were in the fields of
astronomy and hence navigation.
However, it is when you extend the definitions of the trigonometric
functions so that they apply to angles of any size that even more
applications emerge. Trigonometry is the mathematics of oscillations
and waves – the graphs of sine and cosine are periodic (repeating)
waves and so these functions can be used to model waves that occur
in real-life situations. Modern technology is hugely reliant on waves;
electromagnetic waves such as radio waves, x-rays and microwaves
can all be modelled using sine and cosine functions. In optics, the sine
function is used in Snell’s law to work out the angle of refraction of
light entering a different medium. Modern power lines use alternating
current (where the flow of electric charge periodically reverses
direction) to deliver power over long distances, the voltage is
described mathematically using the sine function.
The periodic, wave-like nature of the trigonometric functions means
that they are incredibly useful in mathematical modelling: almost any
oscillating system can be described using a combination of sine or
cosine functions. For example, a musical note can be modelled by a
sine wave, and a chord (several notes played together) can be
modelled by several sine functions added together. Combining sine
and cosine functions allows us to produce louder or quieter tones, and

functions can even be added to together in order to cancel out
unwanted sound completely. Trigonometry is fundamental to the
principle of sound compression used in MP3 players. The sine
function also has applications in climatology as it can be used to
model the seasonal fluctuations of carbon dioxide in the atmosphere.
In fact, many real-life situations that display seasonal fluctuations,
including temperature, can be approximated by trigonometric
functions.
There is a seemingly endless list of uses for trigonometry. It is used in
medicine (it forms the basis of the mathematics behind CT (computed
tomography) scanning), cartography, astronomy, engineering,
surveying – even psychology and probability theory make use of
trigonometry. Trigonometry is not just to do with triangles, it is the
mathematics of waves and oscillations as well. Any problem to do
with angles, oscillations or waves can be modelled using
trigonometry, which must make trigonometry the most relevant
branch of mathematics there is.

1
The tangent
In this chapter you will learn:
• what a tangent is
• the meanings of ‘opposite’, ‘adjacent’and ‘hypotenuse’in right-
angled triangles
• how to solve problems using tangents.

1.1 Introduction
The method used by Thales to find the height of the pyramid in
ancient times is essentially the same as the method used today. It is
therefore worth examining more closely.
Figure 1.1
You can assume that the sun’s rays are parallel because the sun is a
long way from the earth. In Figure 1.1, it follows that the lines RC and
PB, which represent the rays falling on the tops of the objects, are
parallel.
Therefore, angle PBQ = angle ACB (they are corresponding angles).
These angles each represent the altitude of the sun.
As angles PQB and ABC are right angles, triangles PQB and ABC are
similar, so
The height PQ of the pyramid is independent of the length of the stick
AB. If you change the length AB of the stick, the length of its shadow
will be changed in proportion. You can therefore make the following
important general deduction.
For the given angle ACB, the ratio stays constant whatever the

length of AB. You can calculate this ratio beforehand for any angle
ACB. If you do this, you do not need to use the stick, because if you
know the angle and the value of the ratio, and you have measured the
length QB, you can calculate PQ.
Thus if the angle of elevation is 64° and the value of the ratio for this
angle had been previously found to be 2.05, then you have
1.2 The idea of the tangent ratio
The idea of a constant ratio for every angle is the key to the
development of trigonometry.
Let POQ (Figure 1.2) be any acute angle θ°. From points A, B, C on
one arm, say OQ, draw perpendiculars AD, BE, CF to the other arm,
OP. As these perpendiculars are parallel, the triangles AOD, BOE and
COF are similar.
Figure 1.2

Nugget
So if OE is double the length of OD then BE will be double the length of AD.
Now take any point Y, it does not matter which, on the arm OQ. For
that angle θ° the ratio of the perpendicular XY drawn from Y on the
arm OQ to the distance OX intercepted on the other arm OP is
constant (see Figure 1.3).
Figure 1.3
This is true for any angle; each angle θ° has its own particular ratio
corresponding to it. This ratio is called the tangent of the angle θ°. In
practice, the name tangent is abbreviated to tan.
Thus for θ° in Figures 1.2 and 1.3 you can write
1.3 A definition of tangent
There was a general discussion of the idea of the tangent ratio in
Section 1.2, but it is important to refine that discussion into a formal
definition of the tangent of an angle.
In Figure 1.4, the origin O is the centre of a circle of radius 1 unit.

Draw a radius OP at an angle θ° to the x-axis, where 0 ≤ θ° < 90. Let
the coordinates of P be (x, y).
Figure 1.4
Then the tangent of the angle θ°, written tan θ°, is defined by
You can see from the definition that if θ = 0. the y-coordinate of P is 0,
so tan θ = 0. If θ° = 45, then x = y, so
As θ increases, y increases and x decreases, so the tangents of angles
close to 90° are very large. You will see that when θ° = 90, the value
of x is 0, so is not defined; it follows that tan 90° does not exist, and
is undefined.
1.4 Values of the tangent
You can find the value of the tangent of an angle by using your
calculator. Try using it. You should find that the tangent of 45°,
written tan 45°, is 1, and tan 60° = 1.732…. If you have difficulty
with this, you should consult your calculator handbook, and make sure
that you can find the tangent of any angle quickly and easily.

Your calculator must be in the correct mode. There are other units,
notably radians or rads, for measuring angle, and you must ensure that
your calculator is in degree mode, rather than radian or rad mode.
Radians are widely used in calculus, and are the subject of Chapter 4.
Some calculators also give tangents for grades, another unit for angle.
There are 100 grades in a right angle; this book will not use grades.
Your calculator will also reverse this process of finding the tangent of
an angle. If you need to know which angle has a tangent of 0.9, you
look up the inverse tangent. This is often written as tan-1
0.9, or
sometimes as arctan 0.9. Check that tan−1
0.9 = 41.987…°. If it does
not, consult your calculator handbook.
In the work that follows, the degree sign will always be included, but
you might wish to leave it out in your work, provided there is no
ambiguity. Thus you would write tan 45° = 1 and tan 60° = 1.732….
Exercise 1.1
In questions 1 to 6, use your calculator to find the values of the tangents of the
angles. Give your answers correct to three decimal places.
1 tan 20°
2 tan 30°
3 tan 89.99°
4 tan 40.43°
5 tan 62°
6 tan 0.5°
In questions 7 to 12, use your calculator to find the angles with the following
tangents. Give your answer correct to the nearest one hundredth of a degree.
7 0.342
8 2

9 6.123
10 0.0001
11 1
1.5 Notation for angles and sides
Using notation such as ABC for an angle is cumbersome. It is often
more convenient to refer to an angle by using only the middle letter of
the three that define it. Thus, if there is no ambiguity, tan B will be
used in preference for tan ABC.
Single Greek letters such as α (alpha), β (beta), θ (theta) and Φ (phi)
are often used for angles.
Similarly, it is usually easier to use a single letter such as h to
represent a distance along a line, rather than to give the beginning and
end of the line as in the form AB.
1.6 Using tangents
Here are some examples which illustrate the use of tangents and the
technique of solving problems with them.
Example 1.1
A surveyor who is standing at a point 168 m horizontally distant from the foot of
a tall tower measures the angle of elevation of the top of the tower as 38.25°.
Find the height above the ground of the top of the tower.
You should always draw a figure. In Figure 1.5, P is the top of the tower and Q
is the bottom. The surveyor is standing at O which is at the same level as Q. Let
the height of the tower be h metres.

Figure 1.5
Then angle POQ is the angle of elevation and equals 38.25°.
The height of the tower is 132 m, correct to three significant figures.
In practice, if you are using a calculator, there is no need to write
down all the steps given above. You should write down enough so that
you can follow your own working, but you do not need to write down
the value of the tangent as an intermediate step. It is entirely enough,
and actually better practice, to write the calculation above as
However, in this chapter and the next, the extra line will be inserted as
a help to the reader.
Example 1.2
A person who is 168 cm tall had a shadow that was 154 cm long. Find the angle
of elevation of the sun.

In Figure 1.6 let PQ be the person and OQ be the shadow. Then PO is the sun’s
ray and θ is the angle of elevation of the sun.
Figure 1.6
Therefore the angle of elevation of the sun is approximately 47.49°.
Note once again that you can use the calculator and leave out a
number of steps, provided that you give enough explanation to show
how you obtain your result. Thus you could write
Example 1.3
Figure 1.7 represents a cross-section of a symmetrical roof in which AB is the
span, and OP the rise. P is the mid-point of AB.
The rise of the roof is 7 m and its angle of slope is 32°. Find the roof span.

Figure 1.7
As the roof is symmetrical, OAB is an isosceles triangle, so OP is perpendicular
to AB. Call the length AP w metres.
The roof span is 2w metres, that is approximately 22.4 m.
Exercise 1.2
1 The angle of elevation of the sun is 48.4°. Find the height of a flag staff whose
shadow is 7.42 m long.
2 A boat leaving a harbour travels 4 miles east and 5 miles north. Find the
bearing of the boat from the harbour.
3 A boat that is on a bearing of 038° from a harbour is 6 miles north of the
harbour. How far east is the boat from the harbour?
4 A ladder resting against a wall makes an angle of 69° with the ground. The
foot of the ladder is 7.5 m from the wall. Find the height of the top of the
ladder.
5 From the top window of a house that is 1.5 km away from a tower, it is
observed that the angle of elevation of the top of the tower is 3.6° and the
angle of depression of the bottom is 1.2°. Find the height of the tower in

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Just a whisper, half-heard,
But our heart knows the word;
Caresses that seem
Like love’s lips in a dream;
Yet we know she is here,
The desirèd, the dear,
The love of the year!
In the murmur of boughs,
In the softening of skies,
In the sun on the house,
In the daffodil’s green
(Half an inch, half-unseen
Mid the mournful brown mould
Where the rotten leaf lies)
Her story is told.
O Spring, darling Spring,
O sweet days of blue weather!
The thrushes shall sing,
Fields shall grow green again,
Daisies be seen again,
Hedges grow white;
Then down the lane,
Grown leafy again,
Shall go lovers together—
Lovers who see again
Sunshine and showers,
Perfume and flowers,
Dewy dear hours,
Dream and delight.
Warm shall nests be again,
Winter’s behind us;
Springtime shall find us,
Taking our hands,
Lead us away from the cold and the snow

Lead us away from the cold and the snow,
Into the green world where primroses grow.
Winter, hard winter, forgotten, forgiven;
All the old pain paid, to seventy times seven,
All the new glory a-glow.
Love, when Spring calls, will you still turn away?
Winter has wooed you in vain, and shall May?
Love, when Spring calls, will you go?
MEDWAY SONG
(Air: Carnaval de Venise)

Let Housman sing of Severn shore,
Of Thames let Arnold sing,
But we will sing no river more
Save this where crowbars ring.
Let others sing of Henley,
Of fashion and renown,
But we will sing the thirteen locks
That lead to Tonbridge town!
Then sing the Kentish river,
The Kentish fields and flowers,
We waste no dreams on other streams
Who call the Medway ours.
When on the level golden meads
The evening sunshine lies,
The little voles among the reeds
Look out with wondering eyes.
The patient anglers linger
The placid stream beside,
Where still with towering tarry prow
The stately barges glide.
On Medway banks the May droops white,
The wild rose blossoms fair,
O’er meadow-sweet and loosestrife bright,
For water nymphs to wear.
And mid the blowing rushes
Pan pipes a joyous song,
And woodland things peep from the shade
As soft we glide along.
The Kentish fields and flowers

You see no freight on Medway boats
Of fashions fine and rare,
But happy men in shabby coats,
And girls with wind-kissed hair.
The world’s a pain forgotten,
And very far away,
The stream that flows, the boat that goes—
These are our world to-day.
CHAINS INVISIBLE

The lilies in my garden grow,
Wide meadows ring my garden round,
In that green copse wild violets blow,
And pale, frail cuckoo flowers are found.
For all you see and all you hear,
The city might be miles away,
And yet you feel the city near
Through all the quiet of the day.
Sweet smells the earth—wet with sweet rain—
Sweet lilac waves in moonlight pale,
And from the wood beyond the lane
I hear the hidden nightingale.
Though field and wood about me lie,
Hushed soft in dew and deep delight,
Yet can I hear the city’s sigh
Through all the silence of the night.
For me the skylark builds and sings,
For me the vine her garland weaves;
The swallow folds her glossy wings
To build beneath my cottage eaves.
But I can feel the giant near,
Can hear his slaves by daylight weep,
And, when at last the night is here,
I hear him moaning in his sleep.
Oh! for a little space of ground,
Though not a flower should make it gay,
Where miles of meadows wrapped me round,
And leagues and leagues of silence lay.
Oh! for a wind-lashed, treeless down,
A black night and a rising sea,
And never a thought of London town,
To steal the world’s delight from me.

AT EVENING TIME THERE SHALL BE LIGHT
The day was wild with wind and rain,
One grey wrapped sky and sea and shore,
It seemed our marsh would never again
Wear the rich robes that once it wore.
The scattered farms looked sad and chill,
Their sheltering trees writhed all awry,
And waves of mist broke on the hill
Where once the great sea thundered by.
Then God remembered this His land,
This little land that is our own,
He caught the rain up in His hand,
He hid the winds behind His throne,
He soothed the fretful waves to rest,
He called the clouds to come away,
And, by blue pathways, to the west,
They went, like children tired of play.
And then God bade our marsh put on
Its holy vestment of fine gold;
From marge to marge the glory shone
On lichened farm and fence and fold;
In the gold sky that walled the west,
In each transfigured stone and tree,
The glory of God was manifest,
Plain for a little child to see!
MAIDENHOOD

Through her fair world of blossoms fresh and bright,
Veiled with her maiden innocence, she goes;
Not all the splendour of the waxing light
She sees, nor all the colour of the rose;
And yet who knows what finer hues she sees,
Hid by our wisdom from our longing eyes?
Who knows what light she sees in skies and seas
Which is withholden from our seas and skies?
Shod with her youth the thorny paths she treads
And feels not yet the treachery of the thorn,
Her crown of lilies still its perfume sheds
Where Love, the thorny crown, not yet is borne.
Yet in the mystery of her peaceful way
Who knows what fears beset her innocence,
Who, trembling, learns that thorns will wound some day,
And wonders what thorns are, and why, and whence?

V
THE MONK
When in my narrow cell I lie,
The long day’s penance done at last,
I see the ghosts of days gone by,
And hear the voices of the past.
I see the blue-gray wood-smoke curled
From hearths where life has rhymed to love,
I see the kingdoms of the world—
The glory and the power thereof,
And cry, “Ah, vainly have I striven!”
And then a voice calls, soft and low:
“Thou gavest My Earth to win My Heaven;
But Heaven-on-Earth thou mayest not know!”
It is not for Thy Heaven, O Lord,
That I renounced Thy pleasant earth—
The ship, the furrow, and the sword—
The dreams of death, the dreams of birth!
Weary of vigil, fast, and prayer,
Weak in my hope and in my faith—
O Christ, for whom this cross I bear,
Meet me beside the gate of Death!
When the night comes, then let me rest
(O Christ, who sanctifiest pain!)
Falling asleep upon Thy breast,
And, if Thou wilt, wake never again!

The days, the doubts, the dreams of pain
Are over, not to come again,
And from the menace of the night
Has dawned the day-star of delight:
My baby lies against me pressed—
Thus, Mother of God, are mothers blessed!
His little head upon my arm,
His little body soft and warm,
His little feet that cannot stand
Held in the heart of this, my hand.
His little mouth close on my breast—
Thus, Mary’s Son, are mothers blessed.
All dreams of deeds, all deeds of day
Are very faint and far away,
Yet you some day will stand upright
And fight God’s foes, in manhood’s might,
You—tiny, worshipped, clasped, caressed—
Thus, Mother of God, are mothers blessed.
Whatever grief may come to be
This hour divine goes on for me.
All glorious is my little span,
Since I, like God, have made a man,
A little image of God’s best—
Thus, Mary’s Son, are mothers blessed.
Come change, come loss, come worlds of tears,
Come endless chain of empty years;
They cannot take away the hour
That gives me You—my bird, my flower!
Thank God for this! Leave God the rest!—
Thus, Mother of God, are mothers blessed.

This is Christ’s birthday: long ago
He lay upon His Mother’s knee,
Who kissed and blessed Him soft and low—
God’s gift to her, as you to me.
My baby dear, my little one,
The love that rocks this cradling breast
Is such as Mary gave her Son:
She was more honoured, not more blest.
He smiled as you smile: not more sweet
Than your eyes were those eyes of His,
And just such little hands and feet
As yours Our Lady used to kiss.
The world’s desire that Mother bore:
She held a King upon her knee:
O King of all my world, and more
Than all the world’s desire to me!
I thank God on the Christmas morn,
For He has given me all things good:
This body which a child has borne,
This breast, made holy for his food.
High in high heaven Our Lady’s throne
Beside her Son’s stands up apart:
I sit on heaven’s steps alone
And hold my king against my heart.
Across dark depths she hears your cry;
She sees your smile, through worlds of blue
Who was a mother, even as I,
And loved her Child, as I love you.
And to her heart my babe is dear,
B h b th B b Di i

Because she bore the Babe Divine,
And all my soul to hers draws near,
And loves Him for the sake of mine!
EVENING PRAYER
Not to the terrible God, avenging, bright,
Whose altars struck their roots in flame and blood,
Not to the jealous God, whose merciless might
The infamy of unclean years withstood;
But to the God who lit the evening star,
Who taught the flower to blossom in delight,
Who taught His world what love and worship are
We pray, we two, to-night.
To no vast Presence too immense to love,
To no enthronèd King too great to care,
To no strange Spirit human needs above
We bring our little, intimate, heart-warm prayer;
But to the God who is a Father too,
The Father who loved and gave His only Son
We pray across the cradle, I and you,
For ours, our little one!
CHRISTMAS HYMN

O Christ, born on the holy day,
I have no gift to give my King;
No flowers grow by my weary way;
I have no birthday song to sing.
How can I sing Thy name and praise,
Who never saw Thy face divine;
Who walk in darkness all my days,
And see no Eastern stars a-shine?
Yet, when their Christmas gifts they bring,
How can I leave Thy praise unsung?
How stay from homage to the King,
And hold a silent, grudging tongue?
Lord, I found many a song to sing,
And many a humble hymn of praise
For Thy great Miracle of Spring,
The wonder of the waxing days.
When I beheld Thy days and years,
Did I not sing Thy pleasant earth?
The moons of love, the years of tears,
The mysteries of death and birth?
Have I not sung with all my soul
While soul and song were mine to yield,
Thy lightning crown, Thy cloud-control,
The dewy clover of Thy field?
Have I not loved Thy birds and beasts,
Thy streams and woods, Thy sun and shade;
Have I not made me holy feasts
Of all the beauty Thou hast made?
What though my tear-tired eyes, alas!
W Th f t ?

Won never grace Thy face to see?
I heard Thy footstep on the grass,
Thy voice in every wind-blown tree.
No music now I make or win,
Yet, Lord, remember I have been
The lover of Thy world, wherein
I found nought common or unclean.
Grown old and blind, I sing no more,
Thy saints in heaven sing sweet and strong,
Yet take the songs I made of yore
For echoes to Thy birthday song.
ABSOLUTION

Unbind thine eyes, with thine own soul confer,
Look on the sins that made thy life unclean,
Behold how poor thy vaunted virtues were,
How weak thy faith, thy deeds how small and mean,
How far from thy high dreams thy life hath been,
How poor thy use of all thou hast received,
How little of all God’s glory thou hast seen,
How misconstrued that which thou hast perceived.
Turn not thine eyes away from thine unworth,
The cup of shame drink to the bitter lees;
And when thou art lowerèd to the least on earth,
And in the dust makest common cause with these,
Then shall kind arms enfold thee, bringing peace,
The Earth, thy Mother, shall assuage thy pain,
Her woods and fields, Her quiet streams and seas
Shall touch thy soul, and make thee whole again.
But if thy heart holds fast one secret sin,
If one vile script thy soul shrinks to erase,
The mighty Mother cannot bring thee in
Unto the happy, holy, healing place;
But thou shalt weep in darkness, out of grace,
And miss the light of beauty undefiled;
For he who would behold Her, face to face,
Must be in spirit as a little child.
NOW BEING PUBLISHED
The New Popular Edition

OF THE
Works of
George Meredith
Crown 8vo, 6s. each.
With Frontispieces by Bernard Partridge, Harrison
Miller, and others.
THE ORDEAL OF RICHARD FEVEREL
EVAN HARRINGTON
SANDRA BELLONI
VITTORIA
RHODA FLEMING
THE ADVENTURES OF HARRY RICHMOND
BEAUCHAMP’S CAREER
THE EGOIST
DIANA OF THE CROSSWAYS
ONE OF OUR CONQUERORS
LORD ORMONT AND HIS AMINTA
THE AMAZING MARRIAGE
THE SHAVING OF SHAGPAT
THE TRAGIC COMEDIANS
SHORT STORIES
SELECTED POEMS

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By FLORAANNIE STEEL
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6s.
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the salient features of modern fiction; but ‘In the
Tideway,’ although dealing entirely with England and
Scotland, presents the same keen and unerring grasp of
character, the same faculty of conveying local atmosphere
and colour, the same talent for creating strong and
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Dracula
By BRAM STOKER
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The Folly of Pen Harrington
By JULIAN STURGIS. 6s.
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The Dark Way of Love
From the French of M. Charles le Goffic.
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Some Observations of a Foster Parent
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may be assured the book is never dull.”—Glasgow
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“A series of readable and discursive essays on
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The Amazing Marriage
By GEORGE MEREDITH
Crown 8vo, 6s.
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London City Churches
BY
A. E. DANIELL
WITH NUMEROUS ILLUSTRATIONS BY

LEONARD MARTIN
With a Map showing the position of each Church
Imperial 16mo, 6s.
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This volume is profusely illustrated from drawings
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“The author of this book knows the City churches one
and all, and has studied their monuments and archives
with the patient reverence of the true antiquary, and, armed
with the pen instead of the chisel, he has done his best to
give permanent record to their claims on the nation, as
well as on the man in the street.”—Leeds Mercury.
“His interesting text is accompanied by numerous
illustrations, many of them full-page, and altogether his
book is one which has every claim to a warm welcome

from those who have a taste for ecclesiastical
archæology.”—Glasgow Herald.
“This is an interesting and descriptive account of the
various churches still extant in London, and is illustrated
by several excellent photographs.... His work will be of
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observer.”—Western Morning News.
“Mr. Daniell’s work will prove very interesting reading,
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facts concerning the City churches, their history and
associations.”—London.
“The illustrations to this book are good, and it deserves
to be widely read.”—Morning Post.
Crown 8vo, 3s. 6d.
The Shoulder of Shasta
By BRAM STOKER
Author of “Dracula.”

“Will be one of the most popular romances, in one
volume, of the season now opening. It is chiefly
remarkable for the very marked and superior descriptive
power displayed by the author in his rich and inspiring
picture of the scenery of the Shasta Mountain.... So
entirely unconventional, humorous, and bizarre, as to be
quite unique.... The composition is bold and lucid.... He is
an accomplished artist, and shows here at his best.... Mr.
Bram Stoker will add widely to his reputation by this.”—
Irish Times.
“A pure and well-told story.”—Glasgow Herald.
“The story is charmingly written, and deserves to be
read for its brilliant open-air passages, and the portrait it
contains of Grizzly Dick.”—Daily News.
“Mr. Bram Stoker has given the reading world one of
the breeziest and most picturesque tales of life on the
Pacific slope that has been penned for many a long
day.”—Daily Telegraph.
“Mr. Stoker seems quite at home in picturing the wild
beauty of Californian scenery.... ‘The Shoulder of Shasta’
is eminently fresh and readable.”—Globe.
“It is a capital story.”—Bristol Times and Mirror.
“The story is gracefully conceived, and wrought out
with considerable skill.... A readable and entertaining
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