Mathematicians From Greece

"Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."

"Ὅπερ ἔδει δεῖξαι."

"ὅπερ ἔδει ποιῆσαι."

"We ought to regard the interests of the state as of far greater moment than all else, in order that they may be administered well; and we ought not to engage in eager rivalry in despite of equity, nor arrogate to ourselves any power contrary to the common welfare. For a state well administered is our greatest safeguard. In this all is summed up: When the state is in a healthy condition all things prosper; when it is corrupt, all things go to ruin."

"The Greeks elaborated several theories of vision. According to the Pythagoreans, Democritus, and others vision is caused by the projection of particles from the object seen, into the pupil of the eye. On the other hand Empedocles, the Platonists, and Euclid held the strange doctrine of ocular beams, according to which the eye itself sends out something which causes sight as soon as it meets something else emanated by the object."

"All who delight in the pleasures of the belly, exceeding all measure in eating and drinking and love, find that the pleasures are brief and last but a short while—only so long as they are eating and drinking—but the pains that come after are many and endure. The longing for the same things keeps ever returning, and whenever the objects of one's desire are realized forthwith the pleasure vanishes, and one has no further use for them. The pleasure is brief, and once more the need for the same things returns."

"Men in their prayers beg the gods for health, not knowing that this is a thing they have in their own power. Through their incontinence undermining it, they themselves become, because of their passions, the betrayers of their own health."

"The atomic theory was not generally accepted in the time of Democritus, largely because of its deterministic character, for it allows no chance, choice, or free will."

"Men achieve tranquillity through moderation in pleasure and through the symmetry of life. Want and superfluity are apt to upset them and to cause great perturbations in the soul. The souls that are rent by violent conflicts are neither stable nor tranquil. One should therefore set his mind upon the things that are within his power, and be content with his opportunities, nor let his memory dwell very long on the envied and admired of men, nor idly sit and dream of them. Rather, he should contemplate the lives of those who suffer hardship, and vividly bring to mind their sufferings, so that your own present situation may appear to you important and to be envied, and so that it may no longer be your portion to suffer torture in your soul by your longing for more. For he who admires those who have, and whom other men deem blest of fortune, and who spends all his time idly dreaming of them, will be forced to be always contriving some new device because of his [insatiable] desire, until he ends by doing some desperate deed forbidden by the laws. And therefore one ought not to desire other men's blessings, and one ought not to envy those who have more, but rather, comparing his life with that of those who fare worse, and laying to heart their sufferings, deem himself blest of fortune in that he lives and fares so much better than they. Holding fast to this saying you will pass your life in greater tranquillity and will avert not a few of the plagues of life—envy and jealousy and bitterness of mind."

"A life without a holiday is like a long journey without an inn to rest at."

"The pleasures that give most joy are the ones that most rarely come."

"A sensible man takes pleasure in what he has instead of pining for what he has not."

"Throw moderation to the winds, and the greatest pleasures bring the greatest pains."

"Of all things the worst to teach the young is dalliance, for it is this that is the parent of those pleasures from which wickedness springs."

"One thing of course was a fundamental necessity to the atomic world-view. There must be empty space for the atoms to move about in. The hallmark of Democritus’s thought, as Aristotle noted approvingly, was a determination to account for apparent fact and not be led astray by abstract argument. Hence he said that Parmenides’s denial of the existence of void could not be upheld. It was contrary to common sense. Aware however that he was flying in the face of that great authority, he made his denial with a kind of schoolboy daring, for according to Aristotle he put it in the form: ‘What is not does exist, no less than what is.’ If material atoms were the only real substance, then empty space was not real in the same sense. Dimly aware that there must be some way out, the atomists did not yet command a language capable of such a phrase as ‘not in the same sense’, and paradox was their only resource."

"An evil and foolish and intemperate and irreligious life should not be called a bad life, but rather, dying long drawn out."

"Fortune is lavish with her favors, but not to be depended on. Nature on the other hand is self-sufficing, and therefore with her feebler but trustworthy [resources] she wins the greater meed] of hope."

"In the weightiest matters we must go to school to the animals, and learn spinning and weaving from the spider, building from the swallow, singing from the birds,—from the swan and the nightingale, imitating their art."

"(Democritus said he would rather discover a single demonstration than win the throne of Persia.)"

"The right-minded man, ever inclined to righteous and lawful deeds, is joyous day and night, and strong, and free from care. But if a man take no heed of the right, and leave undone the things he ought to do, then will the recollection of no one of all his transgressions bring him any joy, but only anxiety and self-reproaching."

"Men have made an idol of luck as an excuse for their own thoughtlessness. Luck seldom measures swords with wisdom. Most things in life quick wit and sharp vision can set right."

"No one deserves to live who has not at least one good-man-and-true for a friend."

"Seek after the good, and with much toil shall ye find it; the evil turns up of itself without your seeking it."

"The friendship of one wise man is better than the friendship of a host of fools."

"Now as of old the gods give men all good things, excepting only those that are baneful and injurious and useless. These, now as of old, are not gifts of the gods: men stumble into them themselves because of their own blindness and folly."

"For a man petticoat government is the limit of insolence."

"Eudoxes... not only based the method [of exhaustion] on rigorous demonstration... but he actually applied the method to find the volumes (1) of any pyramid, (2) of the cone, proving (1) that any pyramid is one third part of the prism which has the same base and equal height, and (2) that any cone is one third part of the cylinder which has the same base and equal height. Archimedes, however, tells us the remarkable fact that these two theorems were first discovered by Democritus, though he was not able to prove them (which no doubt means, not that he gave no sort of proof, but that he was not able to establish the propositions by the rigorous methods of Eudoxes. Archimedes adds that we must give no small share of the credit for these theorems to Democritus... another testimony to the marvellous powers, in mathematics as well as in other subjects, of the great man who, in the words of Aristotle, "seems to have thought of everything". ...Democritus wrote on irrationals; he is also said to have discussed the question of two parallel sections of a cone (which were evidently supposed to be indefinitely close together), asking whether we are to regard them as equal or unequal... Democritus was already close on the track of infinitesimals."

"Making money is not without its value, but nothing is baser than to make it by wrong-doing."

"You can tell the man who rings true from the man who rings false, not by his deeds alone, but also by his desires."

"Fame and wealth without wisdom are unsafe possessions."

"Good means not [merely] not to do wrong, but rather not to desire to do wrong."

"False men and shams talk big and do nothing."

"There are many who know many things, yet are lacking in wisdom."

"Neither art nor wisdom may be attained without learning."

"It is better to correct your own faults than those of another."

"The hopes of the right-minded may be realized, those of fools are impossible."

"Those who have a well-ordered character lead also a well-ordered life."

"Strength of body is nobility in beasts of burden, strength of character is nobility in men."

"My enemy is not the man who wrongs me, but the man who means to wrong me."

"'Tis a grievous thing to be subject to an inferior."

"Many who have not learned wisdom live wisely, and many who do the basest deeds can make most learned speeches."

"He who does wrong is more unhappy than he who suffers wrong."

"Not from fear but from a sense of duty refrain from your sins."

"Fools learn wisdom through misfortune."

"Repentance for one's evil deeds is the safeguard of life."

"If one choose the goods of the soul, he chooses the diviner [portion]; if the goods of the body, the merely mortal."

"'Tis well to restrain the wicked, and in any case not to join him in his wrong-doing."

"One should emulate works and deeds of virtue, not arguments about it."

"'Tis not in strength of body nor in gold that men find happiness, but in uprightness and in fulness of understanding."

"The enmity of one's kindred is far more bitter than the enmity of strangers."