‘Try to attend steadfastly to a dot on the paper or on the wall. You presently find that one or the other of two things has happened: either your field of vision has become blurred, so that you now see nothing distinct at all, or else you have involuntarily ceased to look at the dot in question, and are looking at something else. But, if you ask yourself successive questions about the dot,—how big it is, how far, of what shape, what shade of color, etc.; in other words, if you turn it over, if you think of it in various ways, and along with various kinds of associates,—you can keep your mind on it for a comparatively long time. This is what the genius does, in whose hands a given topic coruscates and grows.’

William James, ‘Attention’, Talks to Teachers.



‘Speaking for myself, I have already had more than my full measure of this exquisite enjoyment, so much that for many years my life was little short of continuous rapture. I am credited with being one of the hardest workers and perhaps I am, if thought is the equivalent of labour, for I have devoted it almost all of my waking hours. But if work is interpreted to be a definite performance in a specified time according to a rigid rule, then I may be the worst of idlers. Every effort under compulsion demands a sacrifice of life-energy. I have never paid such a price. On the contrary, I have thrived on my thoughts.’

Nikola Tesla, My Inventions.

‘Invention depends upon patience; contemplate your subject long; it will gradually unfold, till a sort of electric spark convulses for a moment the brain, and spreads down to the very heart a glow of irritation. Then come the luxuries of genius! the true hours for production and composition; hours so delightful that I have spent twelve and fourteen successively at my writing desk, and still been in  a state of pleasure.’

Comte de Buffon, as quoted by Isaac d’Israeli.

For the Tiny Remainder

‘We inevitably waste most of each day in eating and drinking, defecating, sleeping, talking and walking about. For the tiny remainder of our time, we do worthless things, speak worthless words, think worthless thoughts. And not only do we pass the moments in this way, but whole days, whole months, pass thus – a lifetime. This is supreme folly…Lose for a moment your grasp of the passing instant and you are as good as dead. You ask why time should be son precious? It is so that you may concentrate the mind on banishing all idle thoughts, refrain from engaging in worldly matters and meditate if this is what you choose, or perform austerities if that is your chosen path.’

Yoshida Kenko, Essays in Idleness, translated Meredith McKinney.

‘Were you to live three thousand years, or even thirty thousand, remember that the sole life which a man can lose is that which he is living at the moment; and furthermore, that he can have no life except the one he loses. This means that the longest life except and the shortest amount to the same thing. For the passing minute is every man’s equal possession, but what has gone by is not ours.

In the life of a man, his time is but a moment, his being an incessant flux, his senses a dim rushlight, his body a prey of worms, his soul an unquiet eddy, his fortune dark, and his fame doubtful. In short, all that is of the body is as coursing waters, all that is of the soul as dreams and vapours; life a warfare,a brief sojourning in an alien land; and after repute, oblivion.’

Marcus Aurelius, Meditations, translated by Maxwell Staniforth.

Thoughts about Thinking

‘The average student is unable to study from a book, unless the book is dealt out to him in small sections. In order to become proficient in mathematics, or in any other subject, he must realize that most topics involve only a small number of basic ideas, which, once grasped, give easy access to the mass of details with which they are inevitably surrounded. Reading a book, or a paper of any length, should not mean crawling along its outer circumference, but, by whatever method one finds best suited to his own temperament, aiming straight at the centre, from which the clearest view may be found of the whole panorama.’

André Weil, The Mathematics Curriculum, Collected Papers vol II.

‘It may sometimes happen that a truth, an insight, which you have slowly and laboriously puzzled out by thinking for yourself could easily have been found already written in a book; but it is a hundred times more valuable if you have arrived at it by thinking for yourself. For only then will it enter your thought-system as an integral part and living member, be perfectly and firmly consistent with it and in accord with all its other consequences and conclusions, bear the hue, colour and stamp of your whole manner of thinking, and have arrived at just the moment it was needed; it will thus stand firmly and forever lodged in your mind. ‘

‘For the man who thinks for himself becomes acquainted with the authorities on his opinions only after he has acquired them and merely as confirmation of them, while the book-philosopher starts with his authorities, in that he constructs his opinions by collecting together the opinions of others: his mind then compares with that of the former as an automaton compares with a living man.’

Arthur Schopenhauer, On Thinking for Yourself, translated by J. R. Hollingdale.

‘I never am really satisfied that I understand anything; because, understand it well as I may, my comprehension can only be an infinitesimal fraction of all I want to understand about the many connections and relations which occur to me, how the matter in question was first thought of or arrived at, etc.’

Ada Lovelace (apparently as quoted in Harry Henderson’s Modern Mathematicians).

Tout mouvement nous descouvre

‘All motion discovers us: the very same soul of Caesar, that made itself
so conspicuous in marshalling and commanding the battle of Pharsalia, was
also seen as solicitous and busy in the softer affairs of love and
leisure. A man makes a judgment of a horse, not only by seeing him when
he is showing off his paces, but by his very walk, nay, and by seeing him
stand in the stable.’

Montaigne, Of Democritus and Heraclitus, translated by Charles Cotton.

‘It is a wonderful thing how the individuality of every man (i.e. a certain particular character with a certain particular intellect) minutely determines his every thought and action and like penetrative dye permeates even the most insignificant part of them, so that the entire life-course, i.e. the inner and outer history, of each one differs fundamentally from that of all the others. As a botanist can recognize the whole plant from one leaf, as Cuvier can construct the whole animal from one bone, so an accurate knowledge of a man’s character can be arrived at from a single characteristic action; and that is true even when this action involves some trifle – indeed this is often better for the purpose, for with important things people are on their guard, while with trifles they follow their own nature without much reflection.’

Arthur Schopenhauer, On Ethics, translated by R. J. Hollingdale.

Some Barbarians

Now to return to the mere Irish. The lords, or rather chiefs of countries (for most of them are not lords from any grants of our kings, which English titles indeed they despise), prefix ‘O’ or ‘Mac’ before their names in token of greatness, being absolute tyrants over their people, themselves eating upon them and making them feed their kern, or footmen, and their horsemen. Also they, and gentlemen under them, before their names put nicknames, given them from the colour of their hair, from lameness, stuttering, diseases, or villainous inclinations, which they disdain not, being otherwise most impatient of reproach, though indeed they take it rather for a grace to be reputed active in any villainy, especially cruelty and theft. But it is strange how contrary they are to themselves, for in apparel, meat, fashions, and customs they are most base and abject, yet are they by nature proud and disdainful of reproach. In fighting they will run away and turn again to fight, because they think it no shame to run away and to make use of the advantage they have in swift running; yet have they great courage in fighting, and I have seen many of them suffer death with as constant resolution as ever Romans did. To conclude this point, they know not truly what honour is, but according to their knowledge no men more desire it, affecting extremely to be celebrated by their poets, or rather rhymers, and fearing more than death to have a rhyme made in their disgrace and infamy. So as these rhymers—pestilent members in that commonwealth—by animating all sorts by their rhymes to licentious living, to lawless and rebellious actions, are so much regarded by them as they grow very rich, the very women, when they are young and new married, or brought to bed, for fear of rhymes giving them the best apparel and ornaments they have.

The Irish are by nature very factious, all of a sept or name living together, and cleaving close one to another in all quarrels and actions whatsoever, in which kind they willingly suffer great men to eat upon them, and take whatsoever they have, proverbially saying ‘Defend me and spend me’; but this defence must be in all causes, just or unjust, for they are not content to be protected from wrong, except they may be borne out to do wrong.

They are by nature extremely given to idleness. The sea coasts and harbours abound with fish, but the fishermen must be beaten out before they will go to their boats. Theft is not infamous but rather commendable among them, so as the greatest men affect to have the best thieves to attend upon them; and if any man reprove them, they answer that they do as their fathers did, and it is infamy for gentlemen and swordsmen to live by labour and manual trades. Yea, they will not be persuaded that theft displeaseth God, because He gives the prey into their hands, and if He be displeased, they say, yet He is merciful and will pardon them for using means to live. This idleness makes them also slovenly and sluttish in their houses and apparel, so as upon every hill they lie lousing themselves, as formerly in the discourse of the Commonwealth. I have remembered four verses, of four beasts that plague Ireland, namely, lice upon their bodies, rats in their houses, wolves in their fields, and swarms of Romish priests tyrannising over their consciences. This idleness also makes them to love liberty above all things, and likewise naturally to delight in music, so as the Irish harpers are excellent, and their solemn music is much liked of strangers; and the women of some parts of Munster, as they wear Turkish heads and are thought to have come first out of these parts, so they have pleasant tunes of Moresco dances.

They are by nature very clamorous, upon every small occasion raising the hobou (that is a doleful outcry), which they take from one another’s mouth till they put the whole town in tumult. And their complaints to magistrates are commonly strained to the highest points of calamity, sometimes in hyperbolical terms, as many upon small violences offered them have petitioned to the Lord Deputy for justice against men for murdering them, while they stood before him sound and not so much as wounded.

The bodies of men and women are large for bigness and stature, because they are brought up in liberty and with loose apparel, but generally the very men are observed to have little and ladylike hands and feet, and the greatest part of the women are nasty with foul linen, and have very great dugs, some so big as they give their children suck over their shoulders. The women generally are not strait-laced, perhaps for fear to hurt the sweetness of breath, and the greatest part are not laced at all. Also the Irish are generally observed to be fruitful in generation, as at Dublin in the time of the last war, it was generally known for truth that one of the Segers, while she lodged in the house of Mistress Arglas, bore five children at one birth, and we all know an alderman’s wife that bore three at a birth, with many like examples.

The septs of one name carry deadly feud towards the man who kills any of their name, and towards all that are of the same name or sept of him who killed him.

Touching divers customs, they seldom eat wild fowl or fish, though they have great plenty of both, because they will not take pains in catching them, and so leave them all for the English. They gladly eat raw herbs, as watercresses and shamrocks, and most commonly eat flesh, many times raw; and if it be roasted or sodd, they seldom eat bread with it or any meat, holding him a churl who hath any bread left after Christmas, save that they keep most of their corn for their horses, whereof they take special care. They drink much of the usquebagh, the best aqua vitae in the world, and much sack, but seldom any claret wine. They swallow lumps of butter mixed with oatmeal, and often let their cows blood, eating the congealed blood with butter, and love no meat more than sour milk curdled. In their frequent drinkings and those feasts of flesh, not only the mere Irish, but also the old inhabitants of English-Irish have the German fashion of putting frolics about the table, as pinching and kissing over the shoulders, and many strange ways, and the manner is to sup where you dine.

Generally, or most commonly, the men go bare-headed, except they wear a steel helmet; but they wear long curled hair, which both men and women nourish long and take pride in it, especially if it be yellow. The men wear long and large shirts, coloured with saffron, a preservation against lice, they being seldom or never washed. The men wear short coats and straight trousers, or breeches, and both men and women wear long mantles for the uppermost garment, which the men at night cast into the water, and so upon the ground sleep in them cast over their heads. The women wear many yards of linen upon their heads, as the women do in Turkey; and wear so many bracelets and necklaces, as rather load than adorn. The men, as well mere Irish as the old inhabitants of the English-Irish, hold it a shame to go abroad or walk with their wives, and much more to ride before them on horseback. They hold it a disgrace to ride upon a mare.

As conquered nations seldom love their conquerors, so in those times Shane O’Neill, the great lord of the North, is said to have cursed his people, at his death, if any of them should build houses or shire towns, to invite the Englishmen to live among them. And in most customs they affected to be contrary to the English. Myself have heard a worthy old captain, who had served long in Ireland, relate some forty customs clean contrary to the English, which I have now forgotten and therefore will only instance one or two of them, namely that women took horse on the contrary side to the Englishmen, with their faces turned the contrary way, and that the Irish used no harness or traces for horses drawing in the plough or drawing sledges with carriage, but only fastened the plough and the carriage by withes to the tails of the horses (or garrans, for so they call them), whereby the tails of them are commonly pulled off, and the very rumps bared. To omit the rest which I cannot remember, we generally observed that not only the women of the mere Irish, but also the old English-Irish, who could speak English as well as ourselves, yet dared not speak it with us if their husbands or their fathers were present. They keep the old calendar, and only the cities have clocks, and keep them as we do in England.

– Fynes Moryson, An Itinerary, written by Fynes Moryson, gent., first in the Latine Tongue, and then translated by him into English: containing his Ten Yeeres Travell through the Twelve Dominions of Germany, Bohmerland, Sweitzerland, Netherland, Denmarke, Poland, Italy, Turky, France, England, Scotland, and Ireland, 1617.

Whenever they are not fighting, they pass much of their time in the chase, and still more in idleness, giving themselves up to sleep and to feasting, the bravest and the most warlike doing nothing, and surrendering the management of the household, of the home, and of the land, to the women, the old men, and all the weakest members of the family. They themselves lie buried in sloth, a strange combination in their nature that the same men should be so fond of idleness, so averse to peace. It is the custom of the states to bestow by voluntary and individual contribution on the chiefs a present of cattle or of grain, which, while accepted as a compliment, supplies their wants. They are particularly delighted by gifts from neighbouring tribes, which are sent not only by individuals but also by the state, such as choice steeds, heavy armour, trappings, and neckchains. We have now taught them to accept money also.

They all wrap themselves in a cloak which is fastened with a clasp, or, if this is not forthcoming, with a thorn, leaving the rest of their persons bare. They pass whole days on the hearth by the fire. The wealthiest are distinguished by a dress which is not flowing, like that of the Sarmatae and Parthi, but is tight, and exhibits each limb. They also wear the skins of wild beasts; the tribes on the Rhine and Danube in a careless fashion, those of the interior with more elegance, as not obtaining other clothing by commerce. These select certain animals, the hides of which they strip off and vary them with the spotted skins of beasts, the produce of the outer ocean, and of seas unknown to us. The women have the same dress as the men, except that they generally wrap themselves in linen garments, which they embroider with purple, and do not lengthen out the upper part of their clothing into sleeves. The upper and lower arm is thus bare, and the nearest part of the bosom is also exposed.

In every household the children, naked and filthy, grow up with those stout frames and limbs which we so much admire. Every mother suckles her own offspring, and never entrusts it to servants and nurses. The master is not distinguished from the slave by being brought up with greater delicacy. Both live amid the same flocks and lie on the same ground till the freeborn are distinguished by age and recognised by merit. The young men marry late, and their vigour is thus unimpaired. Nor are the maidens hurried into marriage; the same age and a similar stature is required; well-matched and vigorous they wed, and the offspring reproduce the strength of the parents. Sister’s sons are held in as much esteem by their uncles as by their fathers; indeed, some regard the relation as even more sacred and binding, and prefer it in receiving hostages, thinking thus to secure a stronger hold on the affections and a wider bond for the family. But every man’s own children are his heirs and successors, and there are no wills. Should there be no issue, the next in succession to the property are his brothers and his uncles on either side. The more relatives he has, the more numerous his connections, the more honoured is his old age; nor are there any advantages in childlessness.

It is a duty among them to adopt the feuds as well as the friendships of a father or a kinsman. These feuds are not implacable; even homicide is expiated by the payment of a certain number of cattle and of sheep, and the satisfaction is accepted by the entire family, greatly to the advantage of the state, since feuds are dangerous in proportion to a people’s freedom. No nation indulges more profusely in entertainments and hospitality. To exclude any human being from their roof is thought impious; every German, according to his means, receives his guest with a well-furnished table. When his supplies are exhausted, he who was but now the host becomes the guide and companion to further hospitality, and without invitation they go to the next house. It matters not; they are entertained with like cordiality. No one distinguishes between an acquaintance and a stranger, as regards the rights of hospitality. It is usual to give the departing guest whatever he may ask for, and a present in return is asked with as little hesitation. They are greatly charmed with gifts, but they expect no return for what they give, nor feel any obligation for what they receive.

On waking from sleep, which they generally prolong to a late hour of the day, they take a bath, oftenest of warm water, which suits a country where winter is the longest of the seasons. After their bath they take their meal, each having a separate seat and table of his own. Then they go armed to business, or no less often to their festal meetings. To pass an entire day and night in drinking disgraces no one. Their quarrels, as might be expected with intoxicated people, are seldom fought out with mere abuse, but commonly with wounds and bloodshed. Yet it is at their feasts that they generally consult on the reconciliation of enemies, on the forming of matrimonial alliances, on the choice of chiefs, finally even on peace and war, for they think that at no time is the mind more open to simplicity of purpose or more warmed to noble aspirations. A race without either natural or acquired cunning, they disclose their hidden thoughts in the freedom of the festivity. Thus the sentiments of all having been discovered and laid bare, the discussion is renewed on the following day, and from each occasion its own peculiar advantage is derived. They deliberate when they have no power to dissemble; they resolve when error is impossible.

A liquor for drinking is made out of barley or other grain, and fermented into a certain resemblance to wine. The dwellers on the river-bank also buy wine. Their food is of a simple kind, consisting of wild-fruit, fresh game, and curdled milk. They satisfy their hunger without elaborate preparation and without delicacies. In quenching their thirst they are not equally moderate. If you indulge their love of drinking by supplying them with as much as they desire, they will be overcome by their own vices as easily as by the arms of an enemy.

Tacitus, Germania, circa 98 ad.

Day and Night

‘True knowledge comes down to vigils in the darkness: the sum of our insomnias alone distinguishes us from the animals and from our kind. What rich or strange idea was ever the work of a sleeper? Is your sleep sound? Are your dreams sweet? You swell the anonymous crowd. Daylight is hostile to thoughts, the sun blocks them out; they flourish only in the middle of the night…Conclusion of nocturnal knowledge: every man who arrives at a reassuring conclusion about anything at all gives evidence of imbecility or false charity. Who ever found a single joyous truth which was valid?’

Emil Cioran, A Short History of Decay.

‘A hostility towards the sun was my only rebellion against the spirit of the age. I hankered after Novalis’s night and Yeatsian Irish twilights. However, from the time the war ended, I gradually sensed that an era was approaching in which to treat the sun as an enemy would be tantamount to following the herd.

The literary works written or put before the public around that time were dominated by night thoughts—though their night was far less aesthetic than mine. To be really respected at that time, moreover, one’s darkness had to be rich and cloying, not thin. Even the rich honeyed night in which I myself had wallowed in my boyhood seemed to them, apparently, very thin stuff indeed.

Little by little, I began to feel uncertain about the night in which I had placed such trust during the war, and to suspect that I might have belonged with the sun worshippers all along. It may well have been so. And if it was indeed so—I began to wonder—might not my persistent hostility towards the sun, and the continued importance I attached to my own small private night, be no more than than a desire to follow the herd?

The men who indulged in nocturnal thought, it seemed to me, had without exception dry, lustreless skins and sagging stomachs. They sought to wrap up a whole epoch in a capacious night of ideas, and rejected in all its forms the sun that I had seen. They rejected both life and death as I had seen them, for in both of these the sun had had a hand.’

Yukio Mishima, Sun and Steel.

Experiencing with a Healthy Soul

‘New forms of exercise, including geometrical exercise, could provide the means to come to a baseline of spiritual health…Descartes’s new geometry hardly offered a totalizing and algorythmic means for mechanically gaining knowledge of a mathematical world but rather gave exemplary practice in seeing and thinking clearly, in experiencing with a healthy soul.’

– Matthew L. Jones, ‘Descartes’s Geometry as a Spiritual Exercise’, Critical Enquiry, 28, (2001), 40-71 at p 44.

‘Charles Howard Hinton was a professional mathematician but for him formal mathematics was never an end in itself. Hinton’s touchstone was rather direct and intuitive knowledge of four-dimensional space. The bulk of his writings are aimed at developing in the reader the power to think about 4d space; and the rest of his work focuses on using a knowledge of higher space to solve various problems in physics and metaphysics.

Hinton was born in London in 1853, the first son in his family. He was schooled at Rugby, and matriculated at Oxford in 1871. From a letter written to him by his father in 1869, we learn that already while at Rugby, Hinton evidenced an interest in “studying geometry as an exercise of direct perception.”


Of this period he writes in A New Era of Thought that, “I found myself in respect to knowledge like a man who is in the midst of plenty and yet who cannot find anything to eat. All around me were the evidences of knowledge-the arts, the sciences, interesting talk, useful inventions-and yet I myself was profited nothing at all; for somehow, amidst all this activity, I was left alone, I could get nothing which I could know.” Desperate for some absolute knowledge, Hinton hit upon the plan of memorizing a cubic yard of one-inch cubes. That is, he took a 36 X 36 X 36 block of cubes, assigned a two-word Latin name (e.g. Collis Nebula) to each of the 46,656 units, and learned to use this network as a sort of “solid paper.” Thus when he: wished to visualize some solid structure, he would do so by adjusting its size so that it fit into his cubic yard. Then he could describe the structure by listing the names of the occupied cells. Hinton maintains that he thereby obtained a sort of direct and sensuous appreciation of space.’

Rudy Rucker in the introduction to Speculations on the Fourth Dimension, Selected Writings of Charles H. Hinton.

‘And I feel that I have a great advantage in this project, inasmuch as many of the thoughts which spring up in the mind of one who studies higher space, and many of the conceptions to which he is driven, turn out to be nothing more nor less than old truths ” the property of every mind that thinks and feels ” truths which are not generally associated with the scientific apprehension of the world, but which are not for that reason any the less valuable.’

Charles Howard Hinton, A New Era of Thought.