1.5: William Blake (1757-1827)
While comparatively unknown in his lifetime, William Blake shared Romantic ideals of independence, subjectivity, and imagination. He expressed these ideals in both his life and art. From his childhood on, William Blake was a visionary. He claimed to have seen God staring at him through a window. He saw a tree with branches spangled by angels. And he saw his brother’s soul leave his body and ascend to heaven. While homeschooled and self-taught—in Greek, Latin, Hebrew, and Italian—Blake was apprenticed as an engraver for seven years and studied art briefly at the Royal Academy.
He married Catherine Boucher in 1782 and taught her to read and write—a highly unconventional act in an age when women of the lower and lower middle classes were not educated at all. Their partnership extended to his work in that he also taught her draftsmanship. While he made a living as an engraver and book illustrator, Blake’s true vocation was his poetry which he illustrated through his unconventional artwork.
He synthesized text and image through his published poetry, in what he described as illuminated manuscripts. Much of his work functioned through synthesis, of good and evil, innocence and experience, thought and feeling, desire and restraint, imagination and reality. His radical thought was informed by such writers and philosophers as Thomas Paine; Mary Wollstonecraft, who advocated for educating women; and Emanuel Swedenborg (1688-1772), a visionary who claimed to see heaven and hell. Blake’s greatest source of knowledge and thought was the St. James Bible, which he “interpreted” with the freedom and power of a prophet.
While influenced by others, Blake determined to map his own course, to avoid being “enslaved to another man’s system” (Jerusalem). He developed his own artistic and poetic system, with his own verbal and visual vocabulary, through which he declared that so-called evil often challenges the status quo, that so-called deities reside in the human breast, that innocence (as opposed to ignorance) can be sustained along with experience, that the imagination is the God in man that is the immortal.
He expounded these views in such works as The Marriage of Heaven and Hell (1790), America, a Prophecy (1793), and Visions of the Daughters of Albion (1793). The selection of poems here includes those from his Songs of Innocence and of Experience (1794). Songs of Innocence was published first alone in 1789. Songs of Experience was published in 1794 and from there on after, the two works were always combined in a single work.
Innocence and Experience are states through which humans pass. Neither is an end-state on its own; both are necessary and exist in cyclic relation to each other. Innocence does not end with childhood, nor does experience dominate adulthood. They co-exist as dynamic contraries. This co-existence is highlighted by matching poems in both books, that is, poems with the same or similar titles. Blake engraved each poem of Songs of Innocence onto a copper plate. On the reverse side of the copper plate, he later engraved the matching poem for Songs of Experience . Each comments upon and completes the other.
1.6.1: Selections from Songs of Innocence and of Experience
<IP_ADDRESS>: “Introduction”
Piping down the valleys wild
Piping songs of pleasant glee
On a cloud I saw a child.
And he laughing said to me.
Pipe a song about a Lamb;
So I piped with merry chear,
Piper pipe that song again—
So I piped, he wept to hear.
Drop thy pipe thy happy pipe
Sing thy songs of happy chear,
So I sung the same again
While he wept with joy to hear
Piper sit thee down and write
In a book that all may read—
So he vanish’d from my sight.
And I pluck’d a hollow reed.
And I made a rural pen,
And I stain’d the water clear,
And I wrote my happy songs
Every child may joy to hear
<IP_ADDRESS>: “The Lamb”
Little Lamb who made thee
Dost thou know who made thee
Gave thee life & bid thee feed.
By the stream & o’er the mead;
Gave thee clothing of delight,
Softest clothing wooly bright;
Gave thee such a tender voice,
Making all the vales rejoice!
Little Lamb who made thee
Dost thou know who made thee
Little Lamb I’ll tell thee,
Little Lamb I’ll tell thee!
He is called by thy name,
For he calls himself a Lamb:
He is meek & he is mild,
He became a little child:
I a child & thou a lamb,
We are called by his name.
Little Lamb God bless thee.

So your chimneys I sweep & in soot I sleep.

And so he was quiet, & that very night,
As Tom was a-sleeping he had such a sight!
That thousands of sweepers, Dick, Joe, Ned, & Jack,
Were all of them locked up in coffins of black;
And by came an Angel who had a bright key,
And he opened the coffins & set them all free;
Then down a green plain, leaping, laughing they run,
And wash in a river and shine in the Sun.
Then naked & white, all their bags left behind,
They rise upon clouds, and sport in the wind.
And the Angel told Tom, if he’d be a good boy,
He’d have God for his father & never want joy.
And so Tom awoke; and we rose in the dark
And got with our bags & our brushes to work.
Though the morning was cold, Tom was happy & warm;
So if all do their duty, they need not fear harm.
<IP_ADDRESS>: “The Little Black Boy”
My mother bore me in the southern wild,
And I am black, but O! my soul is white;
White as an angel is the English child:
But I am black as if bereav’d of light.

Look on the rising sun: there God does live
And gives his light, and gives his heat away.
And flowers and trees and beasts and men receive
Comfort in morning joy in the noonday.
And we are put on earth a little space,
That we may learn to bear the beams of love,
And these black bodies and this sun-burnt face
Is but a cloud, and like a shady grove.
For when our souls have learn’d the heat to bear
The cloud will vanish we shall hear his voice.
Saying: come out from the grove my love & care,
And round my golden tent like lambs rejoice.

When I from black and he from white cloud free,
And round the tent of God like lambs we joy:
Ill shade him from the heat till he can bear,
To lean in joy upon our fathers knee.
And then I’ll stand and stroke his silver hair,
And be like him and he will then love me.
<IP_ADDRESS>: “Holy Thursday”
Twas on a Holy Thursday their innocent faces clean
The children walking two & two in red & blue & green
Grey-headed beadles walkd before with wands as white as snow,
Till into the high dome of Pauls they like Thames waters flow
O what a multitude they seemd these flowers of London town
Seated in companies they sit with radiance all their own
The hum of multitudes was there but multitudes of lambs
Thousands of little boys & girls raising their innocent hands
Now like a mighty wind they raise to heaven the voice of song
Or like harmonious thunderings the seats of Heaven among
Beneath them sit the aged men wise guardians of the poor
Then cherish pity, lest you drive an angel from your door
1.6.2: From Songs of Experience
<IP_ADDRESS>: “Introduction”
Hear the voice of the Bard!
Who Present, Past, & Future sees
Whose ears have heard,
The Holy Word,
That walk’d among the ancient trees.
Calling the lapsed Soul
And weeping in the evening dew:
That might controll,
The starry pole;
And fallen fallen light renew!
O Earth O Earth return!
Arise from out the dewy grass;
Night is worn,
And the morn
Rises from the slumberous mass.
Turn away no more:
Why wilt thou turn away
The starry floor
The watry shore
Is giv’n thee till the break of day.
<IP_ADDRESS>: “Earth’s Answer”
Earth rais’d up her head,
From the darkness dread & drear.
Her light fled:
Stony dread!
And her locks cover’d with grey despair.
Prison’d on watry shore
Starry Jealousy does keep my den
Cold and hoar
Weeping o’er
I hear the Father of the ancient men
Selfish father of men
Cruel, jealous, selfish fear
Can delight
Chain’d in night
The virgins of youth and morning bear.
Does spring hide its joy
When buds and blossoms grow?
Does the sower?
Sow by night?
Or the plowman in darkness plow?
Break this heavy chain,
That does freeze my bones around
Selfish! vain!
Eternal bane!
That free Love with bondage bound.
<IP_ADDRESS>: “Holy Thursday”
Is this a holy thing to see,
In a rich and fruitful land,
Babes reducd to misery,
Fed with cold and usurous hand?
Is that trembling cry a song?
Can it be a song of joy?
And so many children poor?
It is a land of poverty!
And their sun does never shine.
And their fields are bleak & bare.
And their ways are fill’d with thorns.
It is eternal winter there.
For where-e’er the sun does shine,
And where-e’er the rain does fall:
Babe can never hunger there,
Nor poverty the mind appall.
<IP_ADDRESS>: “The Chimney Sweeper”
A little black thing among the snow,
Crying “weep! ‘weep!” in notes of woe!
“Where are thy father and mother? say?”
“They are both gone up to the church to pray.
Because I was happy upon the heath,
And smil’d among the winter’s snow,
They clothed me in the clothes of death,
And taught me to sing the notes of woe.
And because I am happy and dance and sing,
They think they have done me no injury,
And are gone to praise God and his Priest and King,
Who make up a heaven of our misery.”
<IP_ADDRESS>: “The Tyger”
Tyger Tyger, burning bright,
In the forests of the night;
What immortal hand or eye,
Could frame thy fearful symmetry?
In what distant deeps or skies.
Burnt the fire of thine eyes?
On what wings dare he aspire?

And what shoulder, & what art,
Could twist the sinews of thy heart?
And when thy heart began to beat,
What dread hand? & what dread feet?
What the hammer? what the chain,
In what furnace was thy brain?
What the anvil? what dread grasp,
Dare its deadly terrors clasp!
When the stars threw down their spears
And water’d heaven with their tears:
Did he smile his work to see?
Did he who made the Lamb make thee?
Tyger Tyger burning bright,
In the forests of the night:
What immortal hand or eye,
Dare frame thy fearful symmetry?
<IP_ADDRESS> “London”
I wander thro’ each charter’d street,
Near where the charter’d Thames does flow.
And mark in every face I meet
Marks of weakness, marks of woe.
In every cry of every Man,
In every Infants cry of fear,
In every voice: in every ban,
The mind-forg’d manacles I hear
How the Chimney-sweepers cry
Every blackning Church appalls,
And the hapless Soldiers sigh
Runs in blood down Palace walls
But most thro’ midnight streets I hear
How the youthful Harlots curse
Blasts the new-born Infants tear
And blights with plagues the Marriage hearse
<IP_ADDRESS> “The Human Abstract”
Pity would be no more
If we did not make somebody Poor;
And Mercy no more could be
If all were as happy as we.
And mutual fear brings peace,
Till the selfish loves increase:
Then Cruelty knits a snare,
And spreads his baits with care.
He sits down with holy fears,
And waters the grounds with tears;
Then Humility takes its root
Underneath his foot.
Soon spreads the dismal shade
Of Mystery over his head;
And the Catterpiller and
Fly Feed on the Mystery.
And it bears the fruit of Deceit,
Ruddy and sweet to eat;
And the Raven his nest has made
In its thickest shade.
The Gods of the earth and sea
Sought thro’ Nature to find this Tree;
But their search was all in vain:
There grows one in the Human Brain.
1.6.3: The Marriage of Heaven and Hell
The Argument
Rintrah roars & shakes his fires in the burden’d air;
Hungry clouds swag on the deep.
Once meek, and in a perilous path,
The just man kept his course along
The vale of death.
Roses are planted where thorns grow,
And on the barren heath
Sing the honey bees.
Then the perilous path was planted:
And a river and a spring
On every cliff and tomb:
And on the bleached bones
Red clay brought forth.
Till the villain left the paths of ease,
To walk in perilous paths, and drive
The just man into barren climes.
Now the sneaking serpent walks
In mild humility,
And the just man rages in the wilds
Where lions roam.
Rintrah roars & shakes his fires in the burden’d air;
Hungry clouds swag on the deep.
As a new heaven is begun, and it is now thirty-three years since its advent: the Eternal Hell revives. And lo!  Now is the dominion of Edom, & the return of Adam into Paradise: see Isaiah XXXIV & XXXV Chap:
Without Contraries is no progression. Attraction and Repulsion, Reason and Energy, Love and Hate, are necessary to Human existence.
From these contraries spring what the religious call Good & Evil. Good is the passive that obeys Reason. Evil is the active springing from Energy.
Good is Heaven. Evil is Hell
The Voice of the Devil
All Bibles or sacred codes, have been the causes of the following Errors.
- That Man has two real existing principles Viz: a Body & a Soul.
- That Energy, call’d Evil, is alone from the Body, & that Reason, call’d Good, is alone from the Soul.
- That God will torment Man in Eternity for following his Energies.
But the following Contraries to these are True.
- Man has no Body distinct from his Soul; for that call’d Body is a portion of Soul discern’d by the five Senses, the chief inlets of Soul in this age.
- Energy is the only life and is from the Body and Reason is the bound or outward circumference of Energy.
- Energy is Eternal Delight.
Those who restrain desire, do so because theirs is weak enough to be restrained; and the restrainer of reason usurps its place & governs the unwilling.
And being restrain’d it by degrees becomes passive till it is only the shadow of desire.
The history of this written in Paradise Lost, & the Governor of Reason is call’d Messiah.
And the original Archangel or possessor of the command of the heavenly host, is call’d the Devil or Satan and his children are call’d Sin & Death.
But in the Book of Job Miltons Messiah is call’d Satan.
For this history has been adopted by both parties.
It indeed appear’d to Reason as if Desire was cast out, but the Devils account is that the Messiah fell, & formed a heaven of what he stole from the Abyss.
This is shewn in the Gospel, where he prays to the Father to send the comforter or Desire that Reason may have Ideas to build on, the Jehovah of the Bible being no other than he who dwells in flaming fire.
Know that after Christs death, he became Jehovah.
But in Milton’ the Father is Destiny, the Son, a Raio of the five senses, & the Holy-ghost, Vacuum!
Note. The reason Milton wrote in fetters when he wrote of Angels & God, and at liberty when of Devils & Hell, is because he was a true Poet and of the Devils party without knowing it
A Memorable Fancy
As I was walking among the fires of hell, delighted with the enjoyments of Genius; which to Angels look like torment and insanity, I collected some of their Proverbs; thinking that as the sayings used in a nation, mark its character, so the Proverbs of Hell, shew the nature in Infernal wisdom better than any description of buildings or garments,
When I came home: on the abyss of the five senses, where a flat sided steep frowns over the present world, I saw a mighty Devil folded in black clouds, hovering on the sides of the rock, with corroding fires he wrote the following sentence now percieved by the minds of men, & read by them on earth.
How do you know but ev’ry Bird that cuts the airy way,
Is an immense world of delight, clos’d by your senses five?
Proverbs of Hell
In seed time learn, in harvest teach, in winter enjoy.
Drive your cart and your plow over the bones of the dead.
The road of excess leads to the palace of wisdom.

He who desires but acts not, breeds pestilence.
The cut worm forgives the plow.
Dip him in the river who loves water.
A fool sees not the same tree that a wise man sees.

Eternity is in love with the productions of time.
The busy bee has no time for sorrow.
The hours of folly are measur’d by the clock, but of wisdom: no clock can measure.
All wholsom food is caught without a net or a trap.
Bring out number weight & measure in a year of dearth.
No bird soars too high, if he soars with his own wings.
A dead body, revenges not injuries.
The most sublime act is to set another before you.
If the fool would persist in his folly he would become wise.
Folly is the cloke of knavery.
Shame is Prides cloke.
Prisons are built with stones of Law, Brothels with bricks of Religion.
The pride of the peacock is the glory of God.
The lust of the goat is the bounty of God.
The wrath of the lion is the wisdom of God.
The nakedness of woman is the work of God.
Excess of sorrow laughs. Excess of joy weeps.
The roaring of lions, the howling of wolves, the raging of the stormy sea, and the destructive sword, are portions of eternity too great for the eye of man.
The fox condemns the trap, not himself.
Joys impregnate. Sorrows bring forth.
Let man wear the fell of the lion, woman the fleece of the sheep.
The bird a nest, the spider a web, man friendship.
The selfish smiling fool, & the sullen frowning fool, shall be both thought wise, that they may be a rod.
What is now proved was once, only imagin’d.
The rat, the mouse, the fox, the rabbit: watch the roots; the lion, the tyger, the horse, the elephant, watch the fruits.
The cistern contains; the fountain overflows.
One thought, fills immensity.
Always be ready to speak your mind, and a base man will avoid you.
Every thing possible to be believ’d is an image of truth.
The eagle never lost so much time, as when he submitted to learn of the crow.
The fox provides for himself, but God provides for the lion.
Think in the morning. Act in the noon. Eat in the evening. Sleep in the night.
He who has suffer’d you to impose on him knows you.
As the plow follows words, so God rewards prayers.
The tygers of wrath are wiser than the horses of instruction.
Expect poison from the standing water.
You never know what is enough unless you know what is more than enough.
Listen to the fools reproach! it is a kingly title!
The eyes of fire, the nostrils of air, the mouth of water, the beard of earth.
The weak in courage is strong in cunning.
The apple tree never asks the beech how he shall grow, nor the lion, the horse, how he shall take his prey.
The thankful reciever bears a plentiful harvest.
If others had not been foolish, we should be so.
The soul of sweet delight, can never be defil’d.
When thou seest an Eagle, thou seest a portion of Genius, lift up thy head!
As the catterpiller chooses the fairest leaves to lay her eggs on, so the priest lays his curse on the fairest joys.
To create a little flower is the labour of ages.
Damn, braces: Bless relaxes.
The best wine is the oldest, the best water the newest.
Prayers plow not! Praises reap not!
 Sorrows weep not!
The head Sublime, the heart Pathos, the genitals Beauty, the hands & feet Proportion.
As the air to a bird of the sea to a fish, so is contempt to the contemptible.
The crow wish’d every thing was black, the owl, that every thing was white.
Exuberance is Beauty.
If the lion was advised by the fox, he would be cunning.
Improvement makes strait roads, but the crooked roads without Improvement, are roads of Genius.
Sooner murder an infant in its cradle than nurse unacted desires.
Where man is not nature is barren.
Truth can never be told so as to be understood, and not be believ’d.
Enough! or Too much!
The ancient Poets animated all sensible objects with Gods or Geniuses, calling them by the names and adorning them with the properties of woods, rivers, mountains, lakes, cities, nations, and whatever their enlarged & numerous senses could percieve.
And particularly they studied the genius of each city & country, placing it under its mental deity.
Till a system was formed, which some took advantage of & enslav’d the vulgar by attempting to realize or abstract the mental deities from their objects; thus began Priesthood.
Choosing forms of worship from poetic tales.
And a length they pronounc’d that the Gods had order’d such things.
Thus men forgot that All deities reside in the human breast.
A Memorable Fancy
The Prophets Isaiah and Ezekiel dined with me, and I asked them how they dared so roundly to assert, that God spoke to them; and whether they did not think at the time, that they would be misunderstood, & so be the cause of imposition.
Isaiah answer’d, I saw no God, nor heard any, in a finite organical perception; but my senses discover’d the infinite in every thing, and as I was then perswaded, & remain confirm’d; that the voice of honest indignation is the voice of God, I cared not for consequences but wrote.
Then I asked: does a firm perswasion that a thing is so, make it so?
He replied, All poets that it does, & in ages of imagination this firm perswasion removed mountains; but many are not capable of a firm perswasion of any thing.
Then Ezekiel said, The philosophy of the east taught the first principles of human perception: some nations held one principle for the origin & some another; we of Israel taught that the Poetic Genius (as you now call it) was the first principle and all other others merely derivative, which was the cause of our despising the priests & Philosophers of other countries, and prophecying that all Gods would at last be proved to originate in ours & to be the tributaries of the Poetic Genius; it was this that our great poet King David desired so fervently & invokes so patheticly, saying by this he conquers enemies & governs kingdoms; and we so loved our God, that we cursed in his name all deities of surrounding nations, and asserted that they had rebelled; from these opinions the vulgar came to think that all nations would at last be subject to the jews.
This said he, like all firm perswasions, is come to pass, for all nations believe the jews code and worship the jews god, and what greater subjection can be?
I heard this with some wonder, & must confess my own conviction. After dinner I ask’d Isaiah to favour the world with his lost works, he said none of equal value was lost. Ezekiel said the same of his.
I also asked Isaiah what made him go naked and barefoot three years? he answer’d, the same that made our friend Diogenes the Grecian.
I then asked Ezekiel, why he eat dung, & lay so long on his right & left side? he answer’d, the desire of raising other men into a perception of the infinite; this the North American tribes practise, & is he honest who resists his genius or conscience only for the sake of present ease or gratification?
The ancient tradition that the world will be consumed in fire at the end of six thousand years is true, as I have heard from Hell.
For the cherub with his flaming sword is hereby commanded to leave his guard at tree of life, and when he does, the whole creation will be consumed, and appear infinite, and holy whereas it now appears finite & corrupt.
This will come to pass by an improvement of sensual enjoyment.
But first the notion that man has a body distinct from his soul, is to be expunged: this I shall do, by printing in the infernal method, by corrosives, which in Hell are salutary and medicinal, melting apparent surfaces away, and displaying the infinite which was hid.
If the doors of perception were cleansed every thing would appear to man as it is, infinite.

I was in a Printing house in Hell & saw the method in which knowledge is transmitted from generation to generation.
In the first chamber was a Dragon-Man, clearing away the rubbish from a caves moth; within, a number of Dragons were hollowing the cave.
In the second chamber was a Viper folding round the rock & the cave, and others adorning it with gold, silver and precious stones.
In the third chamber was an Eagle with wings and feathers of air; he caused the inside of the cave to be infinite; around were numbers of Eagle like men, who built palaces in the immense cliffs.
In the fourth chamber were Lions of flaming fire raging around & melting the metals into living fluids.
In the fifth chamber were Unnam’d forms, which cast the metals into the expanse.
There they were reciev’d by Men who occupied the sixth chamber, and took the forms of books & were arranged in libraries.
The Giants who formed this world into its sensual existence and now seem to live in it in chains, are in truth, the causes of its life & the sources of all activity; but the chains are, the cunning of weak and tame minds, which have power to resist energy, according to the proverb, the weak in courage is strong in cunning.
Thus one portion of being, is the Prolific, the other, the Devouring: to the devourer it seems as if the producer was in his chains, but it is not so; he only takes portions of existence and fancies that the whole.
But the Prolific would cease to be Prolific unless the Devourer as a sea recieved the excess of his delights.
Some will say, Is not God alone the Prolific? I answer, God only Acts & Is, in existing beings or Men.
These two classes of men are always upon earth, & they should be enemies; whoever tries to reconcile them seeks to destroy existence.
Religion is an endeavour to reconcile the two.
Note. Jesus Christ did not wish to unit but to seperate them, as in the Parable of sheep and goats! & he says I came not to send Peace but a Sword.
Messiah or Satan or Tempter was formerly thought to be one of the Antediluvians who are our Energies.
An Angel came to me and said O pitiable foolish young man! O horrible! O dreadful state! consider the hot burning dungeon thou art preparing for thyself to all eternity, to which thou art going in such career.
I said, perhaps you will be willing to shew me my eternal lot & we will contemplate together upon it and see whether your lot or mine is most desirable.
So he took me thro’ a stable & thro’ a church & down into the church vault at the end of which was a mill: thro’ the mill we went, and came to a cave, down the winding cavern we groped our tedious way till a void boundless as a nether sky appear’d beneath us, & we held by the roots of trees and hung over this immensity, but I said, if you please we will commit ourselves to this void, and see whether providence is here also, if you will not, I will? but he answer’d, do not presume O young-man but as we here remain behold thy lot which will soon appear when the darkness passes away.
So I remain’d with him sitting in the twisted root of an oak; he was suspended in a fungus, which hung with the head downward into the deep.
By degrees we beheld the infinite Abyss, fiery as the smoke of a burning city; beneath us at an immense distance was the sun, black but shining; round it were fiery tracks on which revolv’d vast spiders, crawling after their prey; which flew or rather swum in the infinite deep, in the most terrific shapes of animals sprung from corruption, & the air was full of them, & seem’d composed of them; these are Devils, and arc called Powers of the air. I now asked my companion which was my eternal lot? he said, between the black & white spiders.
But now, from between the black & white spiders, a cloud and fire burst and rolled thro’ the deep, blackning all beneath, so that the nether deep grew black as a sea & rolled with a terrible noise; beneath us was nothing now to be seen but a black tempest, till looking east between the clouds & the waves, we saw a cataract of blood mixed with fire, and not many stones throw from us appear’d and sunk again the scaly fold of a monstrous serpent; at last to the east, distant about three degrees appear’d a fiery crest above the waves; slowly it reared like a ridge of golden rocks till we discover’d two globes of crimson fire, from which the sea fled away in clouds of smoke, and now we saw, it was the head of Leviathan; his forehead was divided into streaks of green & purple like those on a tygers forehead: soon we saw his mouth & red gills hang just above the raging foam tinging the black deep with beams of blood, advancing toward us with all the fury of a spiritual existence.
My friend the Angel climb’d up from his station into the mill; I remain’d alone, & then this appearance was no more, but I found myself sitting on a pleasant bank beside a river by moonlight hearing a harper who sung to the harp, & his theme was, The man who never alters his opinion is like standing water, & breeds reptiles of the mind.
But I arose, and sought for the mill & there I found my Angel, who surprised asked me how I escaped?
I answer’d, All that we saw was owing to your metaphysics; for when you ran away, I found myself on a bank by moonlight hearing a harper. But now we have seen my eternal lot, shall I shew you yours? he laugh’d at my proposal; but I by force suddenly caught him in my arms, & flew westerly thro’ the night, till we were elevated above the earths shadow; then I flung myself with him directly into the body of the sun; here I clothed myself in white, & taking in my hand Swedenborgs volumes, sunk from the glorious clime, and passed all the planets till we came to saturn; here I staid to rest, & then leap’d into the void, between saturn & the fixed stars.
Here, said I! is your lot, in this space, if space it may be call’d. Soon we saw the stable and the church, & I took him to the altar and open’d the Bible, and lo! it was a deep pit, into which I descended driving the Angel before me; soon we saw seven houses of brick; one we enter’d; in it were a number of monkeys, baboons, & all of that species, chain’d by the middle, grinning and snatching at one another, but witheld by the shortness of their chains; however I saw that they sometimes grew numerous, and then the weak were caught by the strong, and with a grinning aspect, first coupled with & then devour’d, by plucking off first one limb and then another till the body was left a helpless trunk; this after grinning & kissing it with seeming fondness they devour’d too; and here & there I saw one savourily picking the flesh off of his own tail; as the stench terribly annoy’d us both we went into the mill, & I in my hand brought the skeleton of a body, which in the mill was Aristotles Analytics.
So the Angel said: thy phantasy has imposed upon me & thou oughtest to be ashamed.
I answer’d: we impose on one another, & it is but lost time to converse with you whose works are only Analytics.
Opposition is true Friendship.
I have always found that Angels have the vanity to speak of themselves as the only wise; this they do with a confident insolence sprouting from systematic reasoning:
Thus Swedenborg boasts that what he writes is new; tho’ it is only the Contents or Index of already publish’d books.
A man carried a monkey about for a shew, & because he was a little wiser than the monkey, grew vain, and conciev’d himself as much wiser than seven men.
Now hear a plain fact: Swedenborg has not written one new truth:
Now hear another: he has written all the old falshoods.
And now hear the reason. He conversed with Angels who are all religious, & conversed not with Devils who all hate religion, for he was incapable thro’ his conceited notions.
Thus Swedenborgs writings are a recapitulation of all superficial, opinions, and an analysis of the more sublime, but no further.
Have now another plain fact: Any man of mechanical talents may from the writings of Paracelsus or Jacob Behmen, produce ten thousand volumes of equal value with Swedenborgs, and from those of Dante or Shakespear, an infinite number.
But when he has done this, let him not say that he knows better than his master, for he only holds a candle in sunshine.
Once I saw a Devil in a flame of fire, who arose before an Angel that sat on a cloud, and the Devil utter’d these words.
The worship of God is, Honouring his gifts in other men each according to his genius, and loving the greatest men best; those who envy or calumniate great men hate God, for there is no other God.
The Angel hearing this became almost blue, but mastering himself he grew yellow, & at last white pink & smiling, and then replied,
Thou Idolater, is not God One? & is not he visible in Jesus Christ? and has not Jesus Christ given his sanction to the law often commandments, and are not all other men fools, sinners, & nothings?
The Devil answer’d: bray a fool in a morter with wheat, yet shall not his folly be beaten out of him; if Jesus Christ is the greatest man, you ought to love him in the greatest degree; now hear how he has given his sanction to the law of ten commandments: did he not mock at the sabbath, and so mock the sabbaths God? murder those who were murder’d because of him? turn away the law from the woman taken in adultery? steal the labor of others to support him? bear false witness when he omitted making a defence before Pilate? covet when he pray’d for his disciples, and when he bid them shake off the dust of their feet against such as refused to lodge them? I tell you, no virtue can exist without breaking these ten commandments; Jesus was all virtue, and acted from impulse, not from rules.
When he had so spoken: I beheld the Angel who stretched out his arms embracing the flame of fire, & he was consumed and arose as Elijah.
Note. This Angel, who is now become a Devil, is my particular friend; we often read the Bible together in its infernal or diabolical sense which the world shall have if they behave well.
I have also: The Bible of Hell: which the world shall have whether they will or no.
One Law for the Lion & Ox is Oppression.
A Song of Liberty
- The Eternal Female groan’d! it was heard over all the Earth:
- Albions coast is sick silent; the American meadows faint!
- Shadows of Prophecy shiver along by the lakes and the rivers and mutter across the ocean. France rend down thy dungeon;
- Golden Spain burst the barriers of old Rome;
- Cast thy keys O Rome into the deep down falling, even to eternity down falling,
- And weep.
- In her trembling hands she took the new born terror howling;
- On those infinite mountains of light, now barr’d out by the atlantic sea, the new born fire stood before the starry king!
- Flag’d with grey brow’d snows and thunderous visages the jealous wings wav’d over the deep.
- The speary hand burned aloft, unbuckled was the shield, forth went the hand of jealousy among the flaming hair, and hurl’d the new born wonder thro’ the starry night.
- The fire, the fire, is falling!

- The fiery limbs, the flaming hair, shot like the sinking sun into the western sea.
- Wak’d from his eternal sleep, the hoary element roaring fled away;
- Down rush’d beating his wings in vain the jealous king; his grey brow’d councellors, thunderous warriors, curl’d veterans, among helms, and shields, and chariots, horses, elephants: banners, castles, slings, and rocks,
- Falling, rushing, ruining! buried in the ruins, on Urthona’s dens;
- All night beneath the ruins, then their sullen flames faded emerge round the gloomy King.
- With thunder and fire: leading his starry hosts thro’ the waste wilderness, he promulgates his ten commands, glancing: his beamy eyelids over the deep in dark dismay,
- Where the son of fire in his eastern cloud, while the morning plumes her Golden breast,
- Spurning the clouds written with curses, stamps the stony law to dust, loosing: the eternal horses from the dens of night, crying,
Empire is no more! and now the lion & wolf shall cease.
Chorus
Let the Priests of the Raven of dawn, no longer in deadly black, with hoarse note curse the sons of joy. Nor his accepted brethren, whom tyrant, he calls free: lay the bound or build the roof. Nor pale religious letchery call that virginity, that wishes but acts not!
For every thing that lives is Holy.


3.6: Research and Critical Reading (Part 2)
TWO SAMPLE STUDENT RESPONSES
To illustrate the principles laid out in this section, consider the following two reading responses. Both texts respond to a very well known piece, “A Letter from Birmingham Jail,” by Martin Luther King, Jr. In the letter, King responds to criticism from other clergymen who had called his methods of civil rights struggle “unwise and untimely.” Both student writers were given the same response prompt:
After reading King’s piece several times and with a pen or pencil in hand, consider what shapes King’s letter. Specifically, what rhetorical strategies is he using to achieve a persuasive effect on his readers? In making your decisions, consider such factors as background information that he gives, ways in which he addresses his immediate audience, and others. Remember that your goal is to explore King’s text, thus enabling you to understand his rhetorical strategies better.
Example Student A
Martin Luther King Jr’s “Letter from Birmingham Jail” is a very powerful text. At the time when minorities in America were silenced and persecuted, King had the courage to lead his people in the struggle for equality. After being jailed in Birmingham, Alabama, King wrote a letter to his “fellow clergymen” describing his struggle for civil rights. In the letter, King recounts a brief history of that struggle and rejects the accusation that it is “unwise and untimely.” Overall, I think that King’s letter is a very rhetorically effective text, one that greatly helped Americans to understand the civil rights movement.
Example Student B
King begins his “Letter from Birmingham Jail” by addressing it to his “fellow clergymen.” Thus, he immediately sets the tone of inclusion rather than exclusion. By using the word “fellow” in the address, I think he is trying to do two things. First of all, he presents himself as a colleague and a spiritual brother of his audience. That, in effect, says “you can trust me,” “I am one of your kind.” Secondly, by addressing his readers in that way, King suggests that everyone, even those Americans who are not directly involved in the struggle for civil rights, should be concerned with it. Hence the word “fellow.” King’s opening almost invokes the phrase “My fellow Americans” or “My fellow citizens” used so often by American Presidents when they address the nation.
King then proceeds to give a brief background of his actions as a civil rights leader. As I read this part of the letter, I was wondering whether his readers would really have not known what he had accomplished as a civil rights leader. Then I realized that perhaps he gives all that background information as a rhetorical move. His immediate goal is to keep reminding his readers about his activities. His ultimate goal is to show to his audience that his actions were non-violent but peaceful. In reading this passage by King, I remembered once again that it is important not to assume that your audience knows anything about the subject of the writing. I will try to use this strategy more in my own papers.
In the middle of the letter, King states: “The purpose of our direct-action program is to create a situation so crisis-packed that it will inevitably open the door to negotiation.” This sentence looks like a thesis statement and I wonder why he did not place it towards the beginning of the text, to get his point across right away. After thinking about this for a few minutes and re- reading several pages from our class textbook, I think he leaves his “thesis” till later in his piece because he is facing a not- so-friendly (if not hostile) audience. Delaying the thesis and laying out some background information and evidence first helps a writer to prepare his or her audience for the coming argument. That is another strategy I should probably use more often in my own writing, depending on the audience I am facing.
REFLECTING ON THE RESPONSES
To be sure, much more can be said about King’s letter than either of these writers have said. However, these two responses allow us to see two dramatically different approaches to reading. After studying both responses, consider the questions below.
- Which response fulfills the goals set in the prompt better and why?
- Which responses shows a deeper understanding of the texts by the reader and why?
- Which writer does a better job at avoiding binary thinking and creating a sophisticated reading of King’s text and why?
- Which writer is more likely to use the results of the reading in his or her own writing in the future and why?
- Which writer leaves room for response to his text by others and why?
CRITICAL READERS DO NOT READ ALONE AND IN SILENCE
One of the key principles of critical reading is that active readers do not read silently and by themselves. By this I mean that they take notes and write about what they read. They also discuss the texts they are working with, with others and compare their own interpretations of those texts with the interpretations constructed by their colleagues.
As a college student, you are probably used to taking notes of what you read. When I was in college, my favorite way of preparing for a test was reading a chapter or two from my textbook, then closing the book, then trying to summarize what I have read on a piece of paper. I tried to get the main points of the chapters down and the explanations and proofs that the textbooks’ authors used. Sometimes, I wrote a summary of every chapter in the textbook and then studied for the test from those summaries rather than from the textbook itself. I am sure you have favorite methods of note taking and studying from your notes, too.
But now it strikes me that what I did with those notes was not critical reading. I simply summarized my textbooks in a more concise, manageable form and then tried to memorize those summaries before the test. I did not take my reading of the textbooks any further than what was already on their pages. Reading for information and trying to extract the main points, I did not talk back to the texts, did not question them, and did not try to extend the knowledge which they offered in any way. I also did not try to connect my reading with my personal experiences or pre-existing knowledge in any way. I also read in silence, without exchanging ideas with other readers of the same texts. Of course, my reading strategies and techniques were dictated by my goal, which was to pass the test.
Critical reading has other goals, one of which is entering an on-going intellectual exchange. Therefore it demands different reading strategies, approaches, and techniques. One of these new approaches is not reading in silence and alone. Instead, critical readers read with a pen or pencil in hand. They also discuss what they read with others.
STRATEGIES FOR CONNECTING READING AND WRITING
If you want to become a critical reader, you need to get into a habit of writing as you read. You also need to understand that complex texts cannot be read just once. Instead, they require multiple readings, the first of which may be a more general one during which you get acquainted with the ideas presented in the text, its structure and style. During the second and any subsequent readings, however, you will need to write, and write a lot. The following are some critical reading and writing techniques which active readers employ as they work to create meanings from texts they read.
UNDERLINE INTERESTING AND IMPORTANT PLACES IN THE TEXT
Underline words, sentences, and passages that stand out, for whatever reason. Underline the key arguments that you believe the author of the text is making as well as any evidence, examples, and stories that seem interesting or important. Don’t be afraid to “get it wrong.” There is no right or wrong here. The places in the text that you underline may be the same or different from those noticed by your classmates, and this difference of interpretation is the essence of critical reading.
TAKE NOTES
Take notes on the margins. If you do not want to write on your book or journal, attach post-it notes with your comments to the text. Do not be afraid to write too much. This is the stage of the reading process during which you are actively making meaning. Writing about what you read is the best way to make sense of it, especially, if the text is difficult.
Do not be afraid to write too much. This is the stage of the reading process during which you are actively making meaning. Writing about what you read will help you not only to remember the argument which the author of the text is trying to advance (less important for critical reading), but to create your own interpretations of the text you are reading (more important).
Here are some things you can do in your comments
- Ask questions.
- Agree or disagree with the author.
- Question the evidence presented in the text
- Offer counter-evidence
- Offer additional evidence, examples, stories, and so on that support the author’s argument
- Mention other texts which advance the same or similar arguments
- Mention personal experiences that enhance your reading of the text
WRITE EXPLORATORY RESPONSES
Write extended responses to readings. Writing students are often asked to write one or two page exploratory responses to readings, but they are not always clear on the purpose of these responses and on how to approach writing them. By writing reading responses, you are continuing the important work of critical reading which you began when you underlined interesting passages and took notes on the margins. You are extending the meaning of the text by creating your own commentary to it and perhaps even branching off into creating your own argument inspired by your reading. Your teacher may give you a writing prompt, or ask you to come up with your own topic for a response. In either case, realize that reading responses are supposed to be exploratory, designed to help you delve deeper into the text you are reading than note-taking or underlining will allow.
When writing extended responses to the readings, it is important to keep one thing in mind, and that is their purpose. The purpose of these exploratory responses, which are often rather informal, is not to produce a complete argument, with an introduction, thesis, body, and conclusion. It is not to impress your classmates and your teacher with “big” words and complex sentences. On the contrary, it is to help you understand the text you are working with at a deeper level. The verb “explore” means to investigate something by looking at it more closely. Investigators get leads, some of which are fruitful and useful and some of which are dead-ends. As you investigate and create the meaning of the text you are working with, do not be afraid to take different directions with your reading response. In fact, it is important resist the urge to make conclusions or think that you have found out everything about your reading. When it comes to exploratory reading responses, lack of closure and presence of more leads at the end of the piece is usually a good thing. Of course, you should always check with your teacher for standards and format of reading responses.
Try the following guidelines to write a successful response to a reading:
Remember your goal—exploration. The purpose of writing a response is to construct the meaning of a difficult text. It is not to get the job done as quickly as possible and in as few words as possible.
As you write, “talk back to the text.” Make comments, ask questions, and elaborate on complex thoughts. This part of the writing becomes much easier if, prior to writing your response, you had read the assignment with a pen in hand and marked important places in the reading.
If your teacher provides a response prompt, make sure you understand it. Then try to answer the questions in the prompt to the best of your ability. While you are doing that, do not be afraid of bringing in related texts, examples, or experiences. Active reading is about making connections, and your readers will appreciate your work because it will help them understand the text better.
While your primary goal is exploration and questioning, make sure that others can understand your response. While it is OK to be informal in your response, make every effort to write in a clear, error-free language.
Involve your audience in the discussion of the reading by asking questions, expressing opinions, and connecting to responses made by others.
USE READING FOR INVENTION
Use reading and your responses to start your own formal writing projects. Reading is a powerful invention tool. While preparing to start a new writing project, go back to the readings you have completed and your responses to those readings in search for possible topics and ideas. Also look through responses your classmates gave to your ideas about the text. Another excellent way to start your own writing projects and to begin research for them is to look through the list of references and sources at the end of the reading that you are working with. They can provide excellent topic-generating and research leads.
KEEP A DOUBLE-ENTRY JOURNAL
Many writers like double-entry journals because they allow us to make that leap from summary of a source to interpretation and persuasion. To start a double-entry journal, divide a page into two columns. As you read, in the left column write down interesting and important words, sentences, quotations, and passages from the text. In the right column, right your reaction and responses to them. Be as formal or informal as you want. Record words, passages, and ideas from the text that you find useful for your paper, interesting, or, in any, way striking or unusual. Quote or summarize in full, accurately, and fairly. In the right-hand side column, ask the kinds of questions and provide the kinds of responses that will later enable you to create an original reading of the text you are working with and use that reading to create your own paper.
DON’T GIVE UP
If the text you are reading seems too complicated or “boring,” that might mean that you have not attacked it aggressively and critically enough. Complex texts are the ones worth pursuing and investigating because they present the most interesting ideas. Critical reading is a liberating practice because you do not have to worry about “getting it right.” As long as you make an effort to engage with the text and as long as you are willing to work hard on creating a meaning out of what you read, the interpretation of the text you are working with will be valid.
IMPORTANT: So far, we have established that no pre-existing meaning is possible in written texts and that critical and active readers work hard to create such meaning. We have also established that interpretations differ from reader to reader and that there is no “right” or “wrong” during the critical reading process. So, you may ask, does this mean that any reading of a text that I create will be a valid and persuasive one? With the exception of the most outlandish and purposely-irrelevant readings that have nothing to do with the sources text, the answer is “yes.” However, remember that reading and interpreting texts, as well as sharing your interpretations with others are rhetorical acts. First of all, in order to learn something from your critical reading experience, you, the reader, need to be persuaded by your own reading of the text. Secondly, for your reading to be accepted by others, they need to be persuaded by it, too. It does not mean, however, that in order to make your reading of a text persuasive, you simply have to find “proof” in the text for your point of view. Doing that would mean reverting to reading “for the main point,” reading as consumption. Critical reading, on the other hand, requires a different approach. One of the components of this approach is the use of personal experiences, examples, stories, and knowledge for interpretive and persuasive purposes. This is the subject of the next section of this chapter.
ONE CRITICAL READER’S PATH TO CREATING A MEANING: A CASE STUDY
Earlier on in this chapter, we discussed the importance of using your existing knowledge and prior experience to create new meaning out of unfamiliar and difficult texts. In this section, I’d like to offer you one student writer’s account of his meaning- making process. Before I do that, however, it is important for me to tell you a little about the class and the kinds of reading and writing assignments that its members worked on.
All the writing projects offered to the members of the class were promoted by readings, and students were expected to actively develop their own ideas and provide their own readings of assigned texts in their essays. The main text for the class was the anthology Ways of Reading edited by David Bartholomae and Anthony Petrosky that contains challenging and complex texts. Like for most of his classmates, this approach to reading and writing was new to Alex who had told me earlier that he was used to reading “for information” or “for the main point”.
In preparation for the first writing project, the class read Adrienne Rich’s essay “When We Dead Awaken: Writing as Revision.” In her essay, Rich offers a moving account of her journey to becoming a writer. She makes the case for constantly “revising” one’s life in the light of all new events and experiences. Rich blends voices and genres throughout the essay, using personal narrative, academic argument, and even poetry. As a result, Rich creates the kind of personal-public argument which, on the one hand, highlights her own life, and on the other, illustrates that Rich’s life is typical for her time and her environment and that her readers can also learn from her experiences.
To many beginning readers and writers, who are used to a neat separation of “personal” and “academic” argument, such a blend of genres and styles may seem odd. In fact, on of the challenges that many of the students in the class faced was understanding why Rich chooses to blend personal writing with academic and what rhetorical effects she achieves by doing so. After writing informal responses to the essay and discussing it in class, the students were offered the following writing assignment:
Although Rich tells a story of her own, she does so to provide an illustration of an even larger story—one about what it means to be a woman and a writer. Tell a story of your own about the ways you might be said to have been named or shaped or positioned by an established or powerful culture. Like Rich (and perhaps with similar hesitation), use your own experience as an illustration of both your own situation and the situation of people like you. You should imagine that the assignment is a way for you to use (and put to the test) some of Rich’s terms, words like “re-vision,” “renaming,” and “structure.” (Bartholomae and Petrosky 648).
Notice that this assignment does not ask students to simply analyze Rich’s essay, to dissect its argument or “main points.” Instead, writers are asked to work with their own experiences and events of their own lives in order to provide a reading of Rich which is affected and informed by the writers’ own lives and own knowledge of life. This is critical reading in action when a reader creates his or her one’s own meaning of a complex text by reflecting on the relationship between the content of that text and one’s own life.
In response to the assignment, one of the class members, Alex Cimino-Hurt, wrote a paper that re-examined and re- evaluated his upbringing and how those factors have influenced his political and social views. In particular, Alex was trying to reconcile his own and his parents’ anti-war views with the fact than a close relative of his was fighting in the war in Iraq as he worked on the paper. Alex used such terms as “revision” and “hesitation” to develop his piece.
Like most other writers in the class, initially Alex seemed a little puzzled, even confused by the requirement to read someone else’s text through the prism of his own life and his own experiences. However, as he drafted, revised, and discussed his writing with his classmates and his instructor, the new approach to reading and writing became clearer to him. After finishing the paper, Alex commented on his reading strategies and techniques and on what he learned about critical reading during the project:
ON PREVIOUS READING HABITS AND TECHNIQUES
Previously when working on any project whether it be for a History, English, or any other class that involved reading and research, there was a certain amount of minimalism. As a student I tried to balance the least amount of effort with the best grade. I distinctly remember that before, being taught to skim over writing and reading so that I found “main” points and highlighted them. The value of thoroughly reading a piece was not taught because all that was needed was a shallow interpretation of whatever information that was provided followed by a regurgitation. [Critical reading] provided a dramatic difference in perspective and helped me learn to not only dissect the meaning of a piece, but also to see why the writer is using certain techniques or how the reading applies to my life.
ON DEVELOPING CRITICAL READING STRATEGIES
When reading critically I found that the most important thing for me was to set aside a block of time in which I would’t have to hurry my reading or skip parts to “Get the gist of it.” Developing an eye for…detail came in two ways. The first method is to read the text several times, and the second is to discuss it with my classmates and my teacher. It quickly became clear to me that the more I read a certain piece, the more I got from it as I became more comfortable with the prose and writing style. With respect to the second way, there is always something that you can miss and there is always a different perspective that can be brought to the table by either the teacher or a classmate.
ON READING RICH’S ESSAY
In reading Adrienne Rich’s essay, the problem for me was’t necessarily relating to her work but instead just finding the right perspective from which to read it. I was raised in a very open family so being able to relate to others was learned early in my life. Once I was able to parallel my perspective to hers, it was just a matter of composing my own story. Mine was my liberalism in conservative environments—the fact that frustrates me sometimes. I felt that her struggle frustrated her, too. By using quotations from her work, I was able to show my own situation to my readers.
ON WRITING THE PAPER
The process that I went through to write an essay consisted of three stages. During the first stage, I wrote down every coherent idea I had for the essay as well as a few incoherent ones. This helped me create a lot of material to work with. While this initial material doesn’t always have direction it provides a foundation for writing. The second stage involved rereading Rich’s essay and deciding which parts of it might be relevant to my own story. Looking at my own life and at Rich’s work together helped me consolidate my paper. The third and final stage involved taking what is left and refining the style of the paper and taking care of the mechanics.
ADVICE FOR CRITICAL READERS
The first key to being a critical and active reader is to find something in the piece that interests, bothers, encourages, or just confuses you. Use this to drive your analysis. Remember there is no such thing as a boring essay, only a boring reader.
- Reading something once is never enough so reading it quickly before class just won’t cut it. Read it once to get your brain comfortable with the work, then read it again and actually try to understand what’s going on in it. You can’t read it too many times.
- Ask questions. It seems like a simple suggestion but if you never ask questions you’ll never get any answers. So, while you’re reading, think of questions and just write them down on a piece of paper lest you forget them after about a line and a half of reading.
CONCLUSION
Reading and writing are rhetorical processes, and one does not exist without the other. The goal of a good writer is to engage his or her readers into a dialog presented in the piece of writing. Similarly, the goal of a critical and active reader is to participate in that dialog and to have something to say back to the writer and to others. Writing leads to reading and reading leads to writing. We write because we have something to say and we read because we are interested in ideas of others.
Reading what others have to say and responding to them help us make that all-important transition from simply having opinions about something to having ideas. Opinions are often over-simplified and fixed. They are not very useful because, if different people have different opinions that they are not willing to change or adjust, such people cannot work or think together. Ideas, on the other hand, are ever evolving, fluid, and flexible. Our ideas are informed and shaped by our interactions with others, both in person and through written texts. In a world where thought and action count, it is not enough to simply “agree to disagree.” Reading and writing, used together, allow us to discuss complex and difficult issues with others, to persuade and be persuaded, and, most importantly, to act.
Reading and writing are inextricably connected, and I hope that this chapter has shown you ways to use reading to inform and enrich you writing and your learning in general. The key to becoming an active, critical, and interested reader is the development of varied and effective reading techniques and strategies. I’d like to close this chapter with the words from the writer Alex Cimino-Hurt: “Being able to read critically is important no matter what you plan on doing with your career or life because it allows you to understand the world around you.”
References
- Pavel Zemilansky
- Bartholomae, David and Anthony Petrosky, Eds. Introduction. Ways of Reading . 8th Ed. New York: Bedford/St. Martin’s, 2008.
- Brent, Douglas. 1992. Reading as Rhetorical Invention. NCTE , Urbana, Illinois. Cimino-Hurt, Alex. Personal Interview. 2003.
- Martin, Janette. 2004. “Developing ‘Interesting Thoughts:’ Reading for Research.” In Research Writing Revisited: A Sourcebook for Teachers , eds. Pavel Zemliansky and Wendy Bishop, Heinemann, Portsmouth, NH. (3-13).
- Rich, Adrienne. 2002. “When We Dead Awaken: Writing as Re-vision.” In Ways of Reading , 6th ed. Eds. Bartholomae, David and Anthony Petrosky. Bedford/St. Martin’s Boston, (627-645).


4.6E: Exercises
Practice Makes Perfect
Write the Augmented Matrix for a System of Equations
In the following exercises, write each system of linear equations as an augmented matrix.
ⓐ \(\left\{ \begin{array} {l} 3x−y=−1\\ 2y=2x+5\end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 4x+3y=−2\\ x−2y−3z=7 \\ 2x−y+2z=−6 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x+4y=−5\\ 3x−2y=2\end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 3x−2y−z=−2\\ −2x+y=5 \\ 5x+4y+z=−1 \end{array} \right.\)
- Answer
-
ⓐ \(\left[ \begin{matrix} 2 &4 &−5 \\ 3 &−2 &2 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 3 &−2 &−1 &−2 \\ −2 &1 &0 &5 \\ 5 &4 &1 &−1 \end{matrix} \right]\)
ⓐ \(\left\{ \begin{array} {l} 3x−y=−4 \\ 2x=y+2 \end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} x−3y−4z=−2 \\ 4x+2y+2z=5 \\ 2x−5y+7z=−8 \end{array} \right.\)
ⓐ \(\left\{ \begin{array} {l} 2x−5y=−3 \\ 4x=3y−1 \end{array} \right.\)
ⓑ \(\left\{ \begin{array} {l} 4x+3y−2z=−3 \\ −2x+y−3z=4 \\ −x−4y+5z=−2 \end{array} \right.\)
- Answer
-
ⓐ \(\left[ \begin{matrix} 2 &−5 &−3 \\ 4 &−3 &−1 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 4 &3 &−2 &−3 \\ −2 &1 &−3 &4 \\ −1 &−4 &5 &−2 \end{matrix} \right]\)
Write the system of equations that corresponds to the augmented matrix.
\(\left[ \begin{array} {cc|c} 2 &−1 &4 \\ 1 &−3 &2 \end{array} \right]\)
\(\left[ \begin{array} {cc|c} 2 &−4 &-2 \\ 3 &−3 &-1 \end{array} \right]\)
- Answer
-
\(\left\{ \begin{array} {l} 2x−4y=−2 \\ 3x−3y=−1 \end{array} \right.\)
\(\left[ \begin{array} {ccc|c} 1 &0 &−3 &-1 \\ 1 &−2 &0 &-2 \\ 0 &−1 &2 &3 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 2 &−2 &0 &-1 \\ 0 &2 &−1 &2 \\ 3 &0 &−1 &-2 \end{array} \right]\)
- Answer
-
\(\left\{ \begin{array} {l} 2x−2y=−1 \\ 2y−z=2 \\ 3x−z=−2 \end{array} \right.\)
Use Row Operations on a Matrix
In the following exercises, perform the indicated operations on the augmented matrices.
\(\left[ \begin{array} {cc|c} 6 &−4 &3 \\ 3 &−2 &1 \end{array} \right]\)
ⓐ Interchange rows 1 and 2
ⓑ Multiply row 2 by 3
ⓒ Multiply row 2 by \(−2\) and add row 1 to it.
\(\left[ \begin{array} {cc|c} 4 &−6 &-3 \\ 3 &2 &1 \end{array} \right]\)
ⓐ Interchange rows 1 and 2
ⓑ Multiply row 1 by 4
ⓒ Multiply row 2 by 3 and add row 1 to it.
- Answer
-
ⓐ \(\left[ \begin{matrix} 3 &2 &1 \\ 4 &−6 &−3 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 12 &8 &4 \\ 4 &−6 &−3 \end{matrix} \right]\)
ⓒ \(\left[ \begin{matrix} 12 &8 &4 \\ 24 &−10 &−5 \end{matrix} \right]\)
\(\left[ \begin{array} {ccc|c} 4 &−12 &−8 &16 \\ 4 &−2 &−3 &-1 \\ −6 &2 &−1 &-1 \end{array} \right]\)
\(\left[ \begin{array} {ccc|c} 6 &−5 &2 &3 \\ 2 &1 &−4 &5 \\ 3 &−3 &1 &-1 \end{array} \right]\)
- Answer
-
ⓐ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\)
ⓑ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ 3 &−3 &1 &−1 \end{matrix} \right]\)
ⓒ \(\left[ \begin{matrix} 2 &1 &−4 &5 \\ 6 &−5 &2 &3 \\ −4 &7 &−6 &7 \end{matrix} \right]\)
Perform the needed row operation that will get the first entry in row 2 to be zero in the augmented matrix: \(\left[ \begin{array} {cc|c} 1 &2 &5 \\ −3 &−4 &-1 \end{array} \right]\)
Perform the needed row operations that will get the first entry in both row 2 and row 3 to be zero in the augmented matrix: \(\left[ \begin{array} {ccc|c} 1 &−2 &3 &-4 \\ 3 &−1 &−2 &5 \\ 2 &−3 &−4 &1 \end{array} \right]\)
- Answer
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\(\left[ \begin{matrix} 1 &−2 &3 &−4 \\ 0 &5 &−11 &17 \\ 0 &1 &−10 &7 \end{matrix} \right]\)
Solve Systems of Equations Using Matrices
In the following exercises, solve each system of equations using a matrix.
\(\left\{ \begin{array} {l} 2x+y=2 \\ x−y=−2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x+y=2 \\ x−y=2 \end{array} \right.\)
- Answer
-
\((1,−1)\)
\(\left\{ \begin{array} {l} −x+2y=−2 \\ x+y=−4 \end{array} \right.\)
\(\left\{ \begin{array} {l} −2x+3y=3 \\ x+3y=12 \end{array} \right.\)
- Answer
-
\((3,3)\)
In the following exercises, solve each system of equations using a matrix.
\(\left\{ \begin{array} {l} 2x−3y+z=19 \\ −3x+y−2z=−1 \\ 5x+y+z=0 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x−y+3z=−3 \\ −x+2y−z=10 \\ x+y+z=5 \end{array} \right.\)
- Answer
-
\((−2,5,2)\)
\(\left\{ \begin{array} {l} 2x−6y+z=3 \\ 3x+2y−3z=2 \\ 2x+3y−2z=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+z=7 \\ 2x−5y−4z=3 \\ 3x−2y−2z=−7 \end{array} \right.\)
- Answer
-
\((−3,−5,4)\)
\(\left\{ \begin{array} {l} x+2z=0 \\ 4y+3z=−2 \\ 2x−5y=3 \end{array} \right.\)
\(\left\{ \begin{array} {l} 2x+5y=4 \\ 3y−z=3 \\ 4x+3z=−3 \end{array} \right.\)
- Answer
-
\((−3,2,3)\)
\(\left\{ \begin{array} {l} 2y+3z=−1 \\ 5x+3y=−6 \\ 7x+z=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} 3x−z=−3 \\ 5y+2z=−6 \\ 4x+3y=−8 \end{array} \right.\)
- Answer
-
\((−2,0,−3)\)
\(\left\{ \begin{array} {l} 2x+3y+z=1 \\ 2x+y+z=9 \\ 3x+4y+2z=20 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+6z=5 \\ −x+y−2z=3 \\ x−4y−2z=1 \end{array} \right.\)
- Answer
-
no solution
\(\left\{ \begin{array} {l} x+2y−3z=−1 \\ x−3y+z=1 \\ 2x−y−2z=2 \end{array} \right.\)
\(\left\{ \begin{array} {l} 4x−3y+2z=0 \\ −2x+3y−7z=1 \\ 2x−2y+3z=6 \end{array} \right.\)
- Answer
-
no solution
\(\left\{ \begin{array} {l} x−y+2z=−4 \\ 2x+y+3z=2 \\ −3x+3y−6z=12 \end{array} \right.\)
\(\left\{ \begin{array} {l} −x−3y+2z=14 \\ −x+2y−3z=−4 \\ 3x+y−2z=6 \end{array} \right.\)
- Answer
-
infinitely many solutions \((x,y,z)\) where \(x=12z+4;\space y=12z−6;\space z\) is any real number
\(\left\{ \begin{array} {l} x+y−3z=−1 \\ y−z=0 \\ −x+2y=1 \end{array} \right.\)
\(\left\{ \begin{array} {l} x+2y+z=4 \\ x+y−2z=3 \\ −2x−3y+z=−7 \end{array} \right.\)
- Answer
-
infinitely many solutions \((x,y,z)\) where \(x=5z+2;\space y=−3z+1;\space z\) is any real number
Writing Exercises
Solve the system of equations \(\left\{ \begin{array} {l} x+y=10 \\ x−y=6\end{array} \right.\) ⓐ by graphing and ⓑ by substitution. ⓒ Which method do you prefer? Why?
Solve the system of equations \(\left\{ \begin{array} {l} 3x+y=1 \\ 2x=y−8 \end{array} \right.\) by substitution and explain all your steps in words.
- Answer
-
Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After looking at the checklist, do you think you are well-prepared for the next section? Why or why not?


11.1: Neurological Assessment Introduction
Learning Objectives
- Perform a neurological assessment, including mental status, cranial nerves, sensory function, motor strength, cerebellar function, and reflexes
- Modify assessment techniques to reflect variations across the life span
- Document actions and observations
- Recognize and report significant deviations from norms
The neurological system is a complex and intricate system that affects all body functions. A neurological assessment includes collecting subjective and objective data through an interview and detailed physical examination of the central nervous system and the peripheral nervous system. Let’s begin by reviewing the anatomy of the neurological system.


11.2: Pedigree Analysis
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- Ying Liu
- City College of San Francisco
Pedigree charts are diagrams that show the phenotypes and/or genotypes for a particular organism and its ancestors. While commonly used in human families to track genetic diseases, they can be used for any species and any inherited trait. Geneticists use a standardized set of symbols to represent an individual’s sex, family relationships and phenotype. These diagrams are used to determine the mode of inheritance of a particular disease or trait, and to predict the probability of its appearance among offspring. Pedigree analysis is therefore an important tool in both basic research and genetic counseling .
Each pedigree chart represents all of the available information about the inheritance of a single trait (most often a disease) within a family. The pedigree chart is therefore drawn using factual information, but there is always some possibility of errors in this information, especially when relying on family members’ recollections or even clinical diagnoses. In real pedigrees, further complications can arise due to incomplete penetrance (including age of onset) and variable expressivity of disease alleles, but for the examples presented in this book, we will presume complete accuracy of the pedigrees. A pedigree may be drawn when trying to determine the nature of a newly discovered disease, or when an individual with a family history of a disease wants to know the probability of passing the disease on to their children. In either case, a tree is drawn, as shown in Figure \(\PageIndex{2}\), with circles to represent females, and squares to represent males. Matings are drawn as a line joining a male and female, while a consanguineous mating (closely related is two lines.
The affected individual that brings the family to the attention of a geneticist is called the proband (or propositus). If an individual is known to have symptoms of the disease ( affected ), the symbol is filled in. Sometimes a half-filled in symbol is used to indicate a known carrier of a disease; this is someone who does not have any symptoms of the disease, but who passed the disease on to subsequent generations because they are a heterozygote. Note that when a pedigree is constructed, it is often unknown whether a particular individual is a carrier or not, so not all carriers are always explicitly indicated in a pedigree. For simplicity, in this chapter we will assume that the pedigrees presented are accurate, and represent fully penetrant traits.


5.12: Variance Sum Law I - Uncorrelated Variables
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Learning Objectives
- Compute the variance of the sum of two uncorrelated variables
- Compute the variance of the difference between two uncorrelated variables
As you will see in later sections, there are many occasions in which it is important to know the variance of the sum of two variables. Consider the following situation:
- you have two populations,
- you sample one number from each population, and
- you add the two numbers together.
The question is, "What is the variance of this sum?" For example, suppose the two populations are the populations of \(8\)-year-old males and \(8\)-year-old females in Houston, Texas, and that the variable of interest is memory span. You repeat the following steps thousands of times:
- sample one male and one female
- measure the memory span of each
- sum the two memory spans.
After you have done this thousands of times, you compute the variance of the sum. It turns out that the variance of this sum can be computed according to the following formula:
\[ \sigma_{sum}^2 = \sigma_{M}^2 + \sigma_{F}^2\]
where the first term is the variance of the sum, the second term is the variance of the males and the third term is the variance of the females. Therefore, if the variances on the memory span test for the males and females were \(0.9\) and \(0.8\) respectively, then the variance of the sum would be \(1.7\).
The formula for the variance of the difference between the two variables (memory span in this example) is shown below. Notice that the expression for the difference is the same as the formula for the sum.
\[ \sigma_{difference}^2 = \sigma_{M}^2 + \sigma_{F}^2\]
More generally, the variance sum law can be written as follows:
\[ \sigma_{Z \pm Y}^2 = \sigma_{X}^2 + \sigma_{Y}^2\]
which is read: The variance of \(X\) plus or minus \(Y\) is equal to the variance of \(X\) plus the variance of \(Y\).
These formulas for the sum and difference of variables given above only apply when the variables are independent.
In this example, we have thousands of randomly-paired scores. Since the scores are paired randomly, there is no relationship between the memory span of one member of the pair and the memory span of the other. Therefore the two scores are independent. Contrast this situation with one in which thousands of people are sampled and two measures (such as verbal and quantitative SAT) are taken from each. In this case, there would be a relationship between the two variables since higher scores on the verbal SAT are associated with higher scores on the quantitative SAT (although there are many examples of people who score high on one test and low on the other). Thus the two variables are not independent and the variance of the total SAT score would not be the sum of the variances of the verbal SAT and the quantitative SAT. The general form of the variance sum law is presented in Section 4.7 in the chapter on correlation.


29.2.3: Mosses
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- Boundless
- Boundless
Learning Objectives
- Describe the distinguishing traits of mosses
Mosses
More than 10,000 species of mosses have been cataloged. Their habitats vary from the tundra, where they are the main vegetation, to the understory of tropical forests. In the tundra, the mosses’ shallow rhizoids allow them to fasten to a substrate without penetrating the frozen soil. Mosses slow down erosion, store moisture and soil nutrients, and provide shelter for small animals as well as food for larger herbivores, such as the musk ox. Mosses are very sensitive to air pollution and are used to monitor air quality. They are also sensitive to copper salts. Such salts are a common ingredient of compounds marketed to eliminate mosses from lawns.
Mosses form diminutive gametophytes, which are the dominant phase of the life cycle. Green, flat structures resembling true leaves, but lacking vascular tissue are attached in a spiral to a central stalk or seta. The plants absorb water and nutrients directly through these leaf-like structures. The seta (plural, setae) contains tubular cells that transfer nutrients from the base of the sporophyte (the foot) to the sporangium. Some mosses have small branches. Some primitive traits of green algae, such as flagellated sperm, are still present in mosses that are dependent on water for reproduction. Other features of mosses are adaptations to dry land. For example, stomata are present on the stems of the sporophyte and a primitive vascular system runs up the sporophyte’s stalk. Additionally, mosses are anchored to the substrate, whether it is soil, rock, or roof tiles, by multicellular rhizoids. These structures are precursors of roots. They originate from the base of the gametophyte, but are not the major route for the absorption of water and minerals. The lack of a true root system explains why it is so easy to rip moss mats from a tree trunk.
The moss life cycle follows the pattern of alternation of generations. The most familiar structure is the haploid gametophyte, which germinates from a haploid spore and forms first a protonema: usually, a tangle of single-celled filaments that hug the ground. Cells akin to an apical meristem actively divide and give rise to a gametophore, consisting of a photosynthetic stem and foliage-like structures. Rhizoids form at the base of the gametophore. Gametangia of both sexes develop on separate gametophores. The male organ (the antheridium) produces many sperm, whereas the archegonium (the female organ) forms a single egg. At fertilization, the sperm swims down the neck to the venter and unites with the egg inside the archegonium. The zygote, protected by the archegonium, divides and grows into a sporophyte, still attached by its foot to the gametophyte.
A structure called a peristome increases the spread of spores after the tip of the capsule falls off at dispersal. The concentric tissue around the mouth of the capsule is made of triangular, close-fitting units, a little like “teeth”; these open and close depending on moisture levels, periodically releasing spores.
Contributions and Attributions
- OpenStax College, Biology. October 22, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/?collection=col11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. November 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/?collection=col11448/latest . License : CC BY: Attribution
- bryophyte. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/bryophyte . License : CC BY-SA: Attribution-ShareAlike
- sporangium. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/sporangium . License : CC BY-SA: Attribution-ShareAlike
- tracheid. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/tracheid . License : CC BY-SA: Attribution-ShareAlike
- Moss on a dry stone wall. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Moss_on_a_dry_stone_wall.jpg . License : CC BY: Attribution
- Adventist Youth Honors Answer Book/Nature/Lichens, Liverworts & Mosses. Provided by : Wikibooks. Located at : en.wikibooks.org/wiki/Adventi...worts_&_Mosses . License : CC BY-SA: Attribution-ShareAlike
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/?collection=col11448/latest . License : CC BY: Attribution
- OpenStax College, Biology. October 23, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/?collection=col11448/latest . License : CC BY: Attribution
- Marchantiophyta. Provided by : Wikipedia. Located at : en.Wikipedia.org/wiki/Marchantiophyta%23Life_cycle . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/definition/pseudoelater . License : CC BY-SA: Attribution-ShareAlike
- Boundless. Provided by : Boundless Learning. Located at : www.boundless.com//biology/definition/gemmae . License : CC BY-SA: Attribution-ShareAlike
- alternation of generation. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/alternation_of_generation . License : CC BY-SA: Attribution-ShareAlike
- Moss on a dry stone wall. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Moss_on_a_dry_stone_wall.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest..._25_03_03f.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest...e_25_03_04.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest...e_25_03_05.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest...e_25_03_02.jpg . License : CC BY: Attribution
- OpenStax College, Biology. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/?collection=col11448/latest . License : CC BY: Attribution
- rhizoid. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/rhizoid . License : CC BY-SA: Attribution-ShareAlike
- peristome. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/peristome . License : CC BY-SA: Attribution-ShareAlike
- seta. Provided by : Wiktionary. Located at : en.wiktionary.org/wiki/seta . License : CC BY-SA: Attribution-ShareAlike
- Moss on a dry stone wall. Provided by : Wikimedia. Located at : commons.wikimedia.org/wiki/File:Moss_on_a_dry_stone_wall.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/Figure_25_03_03f.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/Figure_25_03_04.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/Figure_25_03_05.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest/Figure_25_03_02.jpg . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest...e_25_03_06.png . License : CC BY: Attribution
- OpenStax College, Bryophytes. October 17, 2013. Provided by : OpenStax CNX. Located at : http://cnx.org/content/m44638/latest...e_25_03_07.jpg . License : CC BY: Attribution
Key Points
- Mosses slow down erosion, store moisture and soil nutrients, and provide shelter for small animals and food for larger herbivores.
- Mosses have green, flat structures that resemble true leaves, which absorb water and nutrients; some mosses have small branches.
- Mosses have traits that are adaptations to dry land, such as stomata present on the stems of the sporophyte.
- Mosses are anchored to the substrate by rhizoids, which originate from the base of the gametophyte.
- The moss life cycle follows the pattern of alternation of generations where gametophytes form male and female gametophores, which fertilize to form the sporophyte; spores are released from the sporophyte to produce new gametophytes.
- The concentric tissue around the mouth of the capsule is made of triangular, close-fitting units that open and close to release spores, and the peristome increases the spread of spores after the tip of the capsule falls off at dispersal.
Key Terms
- peristome : one or two rings of tooth-like appendages surrounding the opening of the capsule of many mosses that aid in spreading spores
- rhizoid : a rootlike structure that acts as support and anchors the plant to its substrate
- seta : the stalk of a moss sporangium, or occasionally in a liverwort


0.12: Taylor Series
A Taylor series of a function \(f(x)\) about a point \(x = a\) is a power series representation of \(f(x)\) developed so that all the derivatives of \(f(x)\) at \(a\) match all the derivatives of the power series. Without worrying about convergence here, we have
\[f(x)=f(a)+f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2+\frac{f'''(a)}{3!}(x-a)^3+\ldots .\nonumber \]
. Notice that the first term in the power series matches \(f(a)\), all other terms vanishing, the second term matches \(f'(a)\), all other terms vanishing, etc. Commonly, the Taylor series is developed with \(a = 0\). We will also make use of the Taylor series in a slightly different form, with \(x = x_* + e\) and \(a = x_*\):
\[f(x_*+\epsilon)=f(x_*)+f'(x_*)\epsilon +\frac{f''(x_*)}{2!}\epsilon^2+\frac{f'''(x_*)}{3!}\epsilon^3+\ldots .\nonumber \]
Another way to view this series is that of \(g(\epsilon) = f(x_* + \epsilon )\), expanded about \(\epsilon = 0\).
Taylor series that are commonly used include
\[\begin{aligned}e^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots , \\ \sin x&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots ,\\ \cos x&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots , \\ \frac{1}{1+x}&=1-x+x^2-\cdots ,\quad\text{for }|x|<1, \\ \ln (1+x)&=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots ,\quad\text{for }|x|<1.\end{aligned} \nonumber \]