Mathematicians from Greece

"Philosophy is the science of truth."

"My lectures are published and not published; they will be intelligible to those who heard them, and to none beside."

"Nature does not do anything in vain."

"Of things said without any combination, each signifies either substance or quantity or qualification or a relative or where or when or being-in-a-position or having or doing or being affected. To give a rough idea, examples of substance are man, horse; of quantity: four-foot, five-foot; of qualification: white, grammatical; of a relative: double, half, larger; of where: in the Lyceum, in the market-place; of when: yesterday, last-year; of being-in-a-position: is-lying, is sitting; of having: has-shoes-on, has-armour-on; of doing: cutting, burning; of being-affected: being-cut, being-burned."

"Knowledge of the fact differs from knowledge of the reason for the fact."

"The premises of demonstrative knowledge must be true, primary, immediate, more knowable than and prior to the conclusion, which is further related to them as effect to cause... The premises must be the cause of the conclusion, more knowable than it, and prior to it; its causes, since we possess scientific knowledge of a thing only when we know its cause; prior, in order to be causes; antecedently known, this antecedent knowledge being not our mere understanding of the meaning, but knowledge of the fact as well. Now 'prior' and 'more knowable' are ambiguous terms, for there is a difference between what is prior and more knowable in the order of being and what is prior and knowable to man. I mean that objects nearer to sense are prior and more knowable to man; objects without qualification prior and more knowable are those further from sense. Now the most universal causes are furthest from sense and particular causes are nearest to sense, and they are thus exactly opposed to each other."

"We may assume the superiority ceteris paribus [all things being equal] of the demonstration which derives from fewer postulates or hypotheses—in short from fewer premisses; for... given that all these are equally well known, where they are fewer knowledge will be more speedily acquired, and that is a desideratum. The argument implied in our contention that demonstration from fewer assumptions is superior may be set out in universal form..."

"The science which has to do with nature clearly concerns itself for the most part with bodies and magnitudes and their properties and movements, but also with the principles of this sort of substance, as many as they may be."

"The natural way of doing this [seeking scientific knowledge or explanation of fact] is to start from the things which are more knowable and obvious to us and proceed towards those which are clearer and more knowable by nature; for the same things are not 'knowable relatively to us' and 'knowable' without qualification. So in the present inquiry we must follow this method and advance from what is more obscure by nature, but clearer to us, towards what is more clear and more knowable by nature. Now what is to us plain and obvious at first is rather confused masses, the elements and principles of which became known to us by later analysis..."

"But it is better to assume principles less in number and finite, as Empedocles makes them to be. All philosophers... make principles to be contraries... (for Parmenides makes principles to be hot and cold, and these he demominates fire and earth) as those who introduce as principles the rare and the dense. But Democritus makes the principles to be the solid and the void; of which the former, he says, has the relation of being, and the latter of non-being. ...it is necessary that principles should be neither produced from each other, nor from other things; and that from these all things should be generated. But these requisites are inherent in the first contraries: for, because they are first, they are not from other things; and because they are contraries, they are not from each other."

"It is necessary that every thing which is harmonized, should be generated from that which is void of harmony, and that which is void of harmony from that which is harmonized. ...But there is no difference, whether this is asserted of harmony, or of order, or composition... the same reason will apply to all of these."

"[T]he ancient philosophers... all of them assert that the elements, and those things which are called by them principles, are contraries, though they establish them without reason, as if they were compelled to assert this by truth itself. They differ, however... that some of them assume prior, and others posterior principles; and some of them things more known according to reason, but others such as are more known according to sense: for some establish the hot and the cold, others the moist and the dry, others the odd and the even, and others strife and friendship, as the causes of generation. ...in a certain respect they assert the same things, and speak differently from each other. They assert different things... but the same things, so far as they speak analogously. For they assume principles from the same co-ordination; since, of contraries, some contain, and others are contained."

"[U]niversal is known according to reason, but that which is particular, according to sense..."

"This opinion... appears to be ancient... that the one, excess and defect, are the principles of things... It is not... probable that there are more than three principles... [E]ssence is one certain genus of being: so that principles will differ from each other in prior and posterior alone, but not in genus, for in one genus there is always one contrariety, and all contrarieties appear to be referred to one. That there is neither one element, therefore, nor more than two or three, is evident."

"[T]he first philosophers, in investigating the truth and the nature of things, wandered, as if led by ignorance, into a certain... path. Hence, they say that no being is either generated or corrupted, because it is necessary that what is generated should be generated either from being or non-being: but both these are impossible; for neither can being be generated, since it already is; and from nothing, nothing can be generated... And thus... they said that there were not many things, but that being alone had a subsistence. ...the ancient philosophers ...through this ignorance added so much to their want of knowledge, as to fancy that nothing else was generated or had a being; but they subverted all generation."

"[A]ll things as subsist from nature appear to contain in themselves a principle of motion and permanency; some according to place, others according to increase and diminuation; and others according to change in quality."

"According to one mode... nature is thus denominated, viz. the first subject matter to every thing which contains in itself the principle of motion and mutation. But after another mode it is denominated form, which subsists according to definition: for as art is called that which subsists according to art, and that which is artificial; so likewise nature is both called that which is according to nature, and that which is natural. ...that which is composed from these is not nature, but consists from nature; as, for instance, man. And this is nature in a greater degree than matter: for every thing is then said to be, when it is form in energy... entelecheia, rather than when it is incapacity."

"[L]et us consider, with respect to causes, what they are, and how many there are in number... this also must be done by us in discoursing concerning generation and corruption, and all physical mutation... knowing the principles of these... Cause... is after one manner said to be that, from which, being inherent, something is produced... But after another manner cause is form and paradigm (and this is the definition of the essence of a thing) and the genera of this. ...But it happens... that there are also many causes of the same thing, and this is not from accident. ...seed, a physician, he who consults, and, in short, he who makes, are all of them causes, as that whence the principle of mutation, or permanency, or motion is derived. ...It is, however, necessary always to investigate the supreme cause of every thing ...Further still, it is necessary to investigate the genera of genera; and the particulars of particulars... We should also explore the capacities of the capabilities, and the energizers of the things affected by energy."

"Fortune... and chance, are said to be in the number of causes... [W]ith some it is dubious whether these things have subsistence or not. For, say they, nothing is produced from fortune, but there is a definite cause of all such things... For if fortune were any thing, it would truly appear to be absurd; and some one might doubt why no one of the ancient wise men, when assigning the causes of generation and corruption, has ever defined any thing concerning fortune. ...[M]any things are produced, and have a subsistence, from fortune and chance... They did not, however, think that fortune was any thing belonging to friendship or strife, or fire, or intellect, or any thing else of things of this kind. They are chargeable, therefore, with absurdity, whether they did not conceive that it had a substance, or whether fancying that it had, they omitted it; especially since it was sometimes employed by them. Thus Empedocles says that the air...Thus it then chanc'd to run, tho' varying oft.He also says that the greater part of... animals were generated by fortune. But there are some who assign chance to the cause of this heaven, and of all mundane natures... [W]e must consider... whether chance or fortune are the same... or different from each other, and how they fall into definite causes."

"[S]ince causes are four in number, to know them all is the business of the natural philosopher, who also referring to the cause why a thing is to all of them, viz. to matter, form, that which moves, and for the sake of which a thing subsists, physically assigns a reason. Frequently, however, three of these causes pass into one: for the cause why a thing is, and that for the sake of which it is, are one. But that which motion first originates, is in species the same with these... [T]here are three treatises; once concerning that which is immoveable; another concerning that which is moved, indeed, but is incorruptible; and a third concerning corruptible natures. So the cause of why a thing is, is assigned by him who refers to matter, to essence, and to the first mover... But there are two principles which are naturally motive; of which, one is not physical, because it does not contain in itself the principle of motion. And if there is any thing which moves without being moved, it is of this kind; as is that which is perfectly immoveable, that which is the first of all things, together with essence and form: for it is the end, and that for the sake of which a thing subsists. So that since nature is for the sake of something, it is also necessary to know this cause."

"Since... nature is a principle of motion and mutation... it is necessary that we should not be ignorant of what motion is... But motion appears to belong to things continuous; and the infinite first presents itself to the view in that which is continuous. ...[F]requently ...those who define the continuous, employ the nature or the infinite, as if that which is divisible to infinity is continuous."

"[I]t is impossible for motion to subsist without place, and void, and time."

"There is... something which is in energy only; and there is something which is both in energy and capacity. ...of relatives, one is predicated as according to excess and defect: another according to the effective and passive, and, in short, the motive, and that which may be moved... Motion, however, has not a substance separate from things... But each of the categories subsists in a twofold manner in all things. Thus... one thing pertaining to it is form, and another privation. ...So the species of motion and mutation are as many as those of being. But since in every genus of things, there is that which is in entelecheia, and that which is in capacity; motion is the entelecheia of that which is in capacity... That there is energy, therefore, and that a thing then happens to be moved, when this energy exists, and neither prior nor posterior to it, is manifest. ... [N]either motion nor mutation can be placed in any other genus; nor have those who have advanced a different opinion concerning it spoken rightly. ...for by some motion is said to be difference, inequality, and non-being; though it is not necessary that any of these should be moved... Neither is mutation into these, nor from these, rather than from their opposites. ...The cause, however, why motion appears to be indefinite, is because it can neither be simply referred to the capacity, nor to the energy of beings. ...[I]t is difficult to apprehend what motion is: for it is necessary to refer it either to privation, or to capacity, or to simple energy; but it does not appear that it can be any of these. The above-mentioned mode, therfore remains, viz. that it is a certain energy; but... difficult to be perceived, but which may have a subsistence."

"Since the science of nature is conversant with magnitudes, motion, and time, each of which must necessarily be either infinite or finite...[we] should speculate the infinite, and consider whether it is or not; and if it is what it is. ...[A]ll those who appear to have touched on a philosophy of this kind... consider it as a certain principle of beings. Some, indeed, as the Pythagoreans and Plato, consider it, per se, not as being an accident to any thing else, but as having an essential subsistence... the Pythagoreans... consider the infinite as subsisting in sensibles; for they do not make number to be separate; and they assert that what is beyond the heavens is infinite; but Plato says that beyond the heavens there is not any body, nor ideas, because these are no where: he affirms, however, that the infinite is both in sensibles, and in ideas. ...Plato establishes two infinities, viz. the great and the small."

"All those... who discourse concerning nature, always subject a certain other nature of... elements, to the infinite... But no one of those who make the elements to be finite introduces infinity. Such, however, as make infinite elements, as Anaxagoras and Democritus, say that the infinite is continuous by contact. ...Rationally, too, do all philosophers consider the infinite as a principle; for it cannot be in vain, nor can any other power be present with it than that of a principle: for all things are either the principle, or from the principle; but of the infinite there is no principle, since otherwise it would have an end. ...it is also unbegotten and uncorruptible, as being a certain principle: for... end is the corruption of everything. ...It likewise appears to comprehend and govern all things, as those assert who do not introduce other causes beside the infinite... It would seem also that this is divine: for it is immortal and indestructible, as Anaximander says, and most of the physiologists."

"[B]ecause that which is finite is always bounded with reference to something... it is necessary that there should be no end... [N]umber also appears to be infinite, and mathematical magnitudes, and that which is beyond the heavens. And since that which is beyond is infinite, body also appears to be infinite, and it would seem that there are infinite worlds; for why is there rather void here than there? ...If also there is a vacuum, and an infinite place, it is necessary that there should be an infinite body: for in things which have a perpetual subsistence, capacity differs nothing from being. The speculation of the infinite is, however, attended with doubt: for many impossibilities happen both to those who do not admit that it has a subsistence, and to those who do. ...It is ...especially the province of a natural philosopher to consider if there be a sensible infinite magnitude."

"[T]hey pronounce absurdly who thus speak, as the Pythagoreans assert: for at the same time they make the infinite to be essence, and distribute it into parts."

"[I]t is impossible that each of the elements should be infinite. For that is body which has interval on all sides; and that is infinite which has extension without bound."

"[I]t's gravity is the cause; and that which is heavy abides in the middle, and the earth is in the middle: in like manner also, the infinite will abide in itself, through some other cause... and will itself support itself. ...[T]he places of the whole and the part are of the same species; as of the whole earth and a clod, the place is downward; and of the whole of fire, and a spark, the place is upward. So that if the place of the infinite is in itself, there will be the same place also of a part of the infinite."

"[H]ow will one part of the infinite be above, and another below? Or how will it have extremes or a middle? Further still, every sensible body is in place; but the species and differences of place are upward and downward, before and behind, to the right hand and to the left: and these things not only thus subsist with relation to us, and by position, but have a definite subsistence in the universe itself. But it is impossible that these things should be in the infinite: and... that there should be an infinite place. But every body is in place; and therefore it is also impossible that there should be an infinite body. ...[T]herefore ...there is not an infinite body in energy."

"[T]he infinite is in capacity. That, however, which is infinite in capacity is not to be assumed as that which is infinite in energy. ...[I]t has its being in capacity, and in division and diminution. ...[I]t is always possible to assume something beyond it. It does not, however, on this account surpass every definite magnitude; as in division it surpasses every definite magnitude, and will be less."

"Plato... introduces two infinities, because both in increase and diminution there appears to be transcendency, and a progression to infinity. Though... he did not use them: for neither is there infinity in numbers by diminution or division; since unity is a minimum: nor by increase; for he extends number as far as to the decad."

"The infinite... happens to subsist in a way contrary to what is asserted by others: for the infinite is not that beyond which there is nothing, but it is that of which there is always something beyond. ...But that pertaining to which there is nothing beyond is perfect and whole. ...that of which nothing is absent pertaining to the parts ...the whole is that pertaining to which there is nothing beyond. But that pertaining to which something external is absent, that is not all ...But nothing is perfect which has not an end; and the end is a bound. On this account... Parmenides spoke better than Melissus: for the latter says that the infinite is a whole; but the former, that the whole is finite, and equally balanced from the middle: for to conjoin the infinite with the universe and the whole, is not to connect line with line."

"The bodies of which the world is composed are solids, and therefore have three dimensions. Now, three is the most perfect number,—it is the first of numbers, for of one we do not speak as a number, of two we say both, but three is the first number of which we say all. Moreover, it has a beginning, a middle, and an end."

"The least initial deviation from the truth is multiplied later a thousandfold."

"...suppose α without weight, but β possessing weight; and let α pass over space γδ, but β in the same time pass over a space γε,—for that which has weight will be carried through the larger space. If now the heavy body be divided in the proportion that space γε bears to γδ, ... and if the whole is carried through the whole space γε, then it must be that a part in the same time would be carried through γδ..."

"That body is heavier than another which, in an equal bulk, moves downward quicker."

"Sound is the motion of that which is able to be moved, after the manner in which those things are moved, that rebound from smooth bodies, when any one strikes them. Not every thing... sounds... but it is necessary, that the body which is struck should be equable, that the air may collectively rebound, and be shaken. The differences, however, of bodies which sound, are manifested in the sound, which is in energy; for, as colours are not perceived without light, so neither are the sharp and the flat perceived without sound. But these things are asserted metaphorically, from those which pertain to the touch; for the sharp moves the sense much in a short time, but the flat a little in a long time. The sharp, therefore, is not rapid, and the flat slow; but such a motion is produced of the one, on account of celerity, and of the other on account of slowness, that, also, which is perceived in the touch, appears to be analogous to the acute and obtuse, for the acute, as it were, stings; but the obtuse, as it were, impels: because the one moves in a short, but the other in a long time. Hence it happens that the one is swift but the other slow. Let it therefore be thus determined concerning sound."

"It is not necessary to ask whether soul and body are one, just as it is not necessary to ask whether the wax and its shape are one, nor generally whether the matter of each thing and that of which it is the matter are one. For even if one and being are spoken of in several ways, what is properly so spoken of is the actuality."

"But voice is a certain sound of that which is animated; for nothing inanimate emits a voice; but they are said to emit a voice from similitude, as a pipe, and a lyre, and such other inanimate things, have extension, modulation, and dialect; for thus it appears, because voice, also, has these."

"The male has more teeth than the female in mankind, and sheep, and goats, and swine. This has not been observed in other animals."

"The brain is not responsible for any of the sensations.. the correct view [is] that the seat and source of sensation is the region of the heart….the motions of pleasure and pain, and generally all sensation plainly have their source in the heart."

"The essential nature (concerning the soul) cannot be corporeal, yet it is also clear that this soul is present in a particular bodily part, and this one of the parts having control over the rest (heart)."

"Nature flies from the infinite, for the infinite is unending or imperfect, and Nature ever seeks an end."

"Concerning the generation of animals akin to them, as hornets and wasps, the facts in all cases are similar to a certain extent, but are devoid of the extraordinary features which characterize bees; this we should expect, for they have nothing divine about them as the bees have."

"Just as it sometimes happens that deformed offspring are produced by deformed parents, and sometimes not, so the offspring produced by a female are sometimes female, sometimes not, but male, because the female is as it were a deformed male."

"Music directly represents the passion of the soul. If one listens to the wrong kind of music, he will become the wrong kind of person."

"All men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness they are loved for themselves; and above all others the sense of sight. For not only with a view to action, but even when we are not going to do anything, we prefer sight to almost everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things."

"But as more arts were invented, and some were directed to the necessities of life, others to recreation, the inventors of the latter were naturally always regarded as wiser than the inventors of the former, because their branches of knowledge did not aim at utility. ... This is why the mathematical arts were founded in Egypt; for there the priestly caste was allowed to be at leisure."

"For it is owing to their wonder that men both now begin and at first began to philosophize; they wondered originally at the obvious difficulties, then advanced little by little and stated difficulties about the greater matters, e.g. about the phenomena of the moon and those of the sun and of the stars, and about the genesis of the universe. And a man who is puzzled and wonders thinks himself ignorant (whence even the lover of myth is in a sense a lover of Wisdom, for the myth is composed of wonders); therefore since they philosophized order to escape from ignorance, evidently they were pursuing science in order to know, and not for any utilitarian end. — Metaphysics by Aristotle – Book 1, ClassicalWisdom.com"

"οὐ γὰρ δεῖν ἐπιτάττεσθαι τὸν σοφὸν ἀλλ᾽ ἐπιτάττειν, καὶ οὐ τοῦτον ἑτέρῳ πείθεσθαι, ἀλλὰ τούτῳ τὸν ἧττον σοφόν."

"That which is desirable on its own account and for the sake of knowing it is more of the nature of wisdom than that which is desirable on account of its results."

"τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν ἀδύνατον τῷ αὐτῷ καὶ κατὰ τὸ αὐτό （καὶ ὅσα ἄλλα προσδιορισαίμεθ᾽ ἄν, ἔστω προσδιωρισμένα πρὸς τὰς λογικὰς δυσχερείας）: αὕτη δὴ πασῶν ἐστὶ βεβαιοτάτη τῶν ἀρχῶν: ἔχει γὰρ τὸν εἰρημένον διορισμόν. ἀδύνατον γὰρ ὁντινοῦν ταὐτὸν ὑπολαμβάνειν εἶναι καὶ μὴ εἶναι, καθάπερ τινὲς οἴονται λέγειν Ἡράκλειτον."

"πάντων γὰρ ὅσα πλείω μέρη ἔχει καὶ μὴ ἔστιν οἷον σωρὸς τὸ πᾶν."

"εἰ οὖν οὕτως εὖ ἔχει, ὡς ἡμεῖς ποτέ, ὁ θεὸς ἀεί, θαυμαστόν: εἰ δὲ μᾶλλον, ἔτι θαυμασιώτερον. ἔχει δὲ ὧδε. καὶ ζωὴ δέ γε ὑπάρχει: ἡ γὰρ νοῦ ἐνέργεια ζωή, ἐκεῖνος δὲ ἡ ἐνέργεια: ἐνέργεια δὲ ἡ καθ᾽ αὑτὴν ἐκείνου ζωὴ ἀρίστη καὶ ἀΐδιος."

"Those who assert that the mathematical sciences say nothing of the beautiful or the good are in error. For these sciences say and prove a great deal about them; if they do not expressly mention them, but prove attributes which are their results or definitions, it is not true that they tell us nothing about them. The chief forms of beauty are order and symmetry and definiteness, which the mathematical sciences demonstrate in a special degree."

"Every art, and every system, and in like manner every action and purpose aims, it is thought, at some good; for which reason a common and by no means a bad description of the good is, ‘that at which all things aim.’ (Bk I, Ch I)"

"But it is clear there is a difference in the ends proposed: for in some cases they are activities, and in others results beyond the mere activities, and where there are certain ends beyond and beside the actions, the results are naturally superior to the activities. Now, as there are numerous kinds of actions and numerous arts and sciences, it follows that the ends are also various. Thus the end of the healing art is health, of ship-building ships, of strategy victory, of economy wealth. (Bk I, Ch I)"

"If, then, in the sphere of action there is some one end which we desire for its own sake, and for the sake of which we desire every thing else; and if we do not choose every thing for the sake of something else, for this would go on without limit, and our desire would be idle and futile, it is clear that this must be the supreme good, and the best thing of all. (Bk I, Ch I)"

"And surely to know what this good is, is of great importance for the conduct of life, for in that case we shall be like archers shooting at a definite mark, and shall be more likely to do what is right. But, if this is the case, we must try to comprehend, in outline at least, what it is and to which of the sciences it belongs. (Bk I, Ch I)"

"Perhaps then we must begin with such facts as are known to us from individual experience. It is necessary therefore that the person who is to study, with any tolerable chance of profit, the principles of nobleness and justice and politics generally, should have received a good moral training. (Bk I, Ch II)"

"If a man knows what it is right to do, he does not require a formal reason. And a person that has been thus trained, either possesses these first principles already, or can easily acquire them. (Bk I, Ch II)"

"As for him who neither possesses nor can acquire them, let him take to heart the words of Hesiod: ‘ He is the best of all who thinks for himself in all things. He, too, is good who takes advice from a wiser (person). But he who neither thinks for himself, nor lays to heart another's wisdom, this is a useless man.’ (Bk. 1, Chapter II)"

"Now men seem, not unreasonably, to form their notions of the supreme good and of happiness from the lives of men."

"The majority of mankind and people who lack refinement conceive it to be pleasure, and hence they approve a life of sensual enjoyment. (Bk. 1, Chapter III)"

"There are three lines of life which stand out prominently to view: the life of pleasure, the political life, and the life of reflection."

"Now the mass of mankind are plainly... choosing a life like that of brute animals... (Bk. 1, Chapter III)"

"The refined and active, on the other hand, prefer honour, which I suppose may be said to be the end of the political life. Yet honour is plainly too superficial to be the object of our search, because it appears to depend rather on those who give than on those who receive it, whereas we feel instinctively that the good must be something proper to a man, which cannot easily be taken from him. (Bk. 1, Chapter III)"

"Men seem to pursue honour in order that they may believe themselves to be good. Accordingly, they seek to be honoured by the wise, and by those who know them well, and on the score of virtue; it is clear, therefore, that in their opinion at any rate, virtue is superior to honour. Perhaps, then, one ought to say that virtue rather than honour is the end of the political life; yet even virtue is plainly too imperfect: for it seems that a man might have all the virtues and yet be asleep, or fail to achieve anything all his life; moreover, such a person may suffer the greatest evils and misfortunes. And no one, in this case, would call a man, who passed his life in this manner, happy, except for argument's sake. (Bk. 1, Chapter III)"

"The third kind of life is the life of contemplation (Bk. 1, Chapter III)"

"As for the life of money-making, it is one of constraint, and wealth is manifestly not the good of which we are in search, for it is only useful as a means to something else, and for this reason there is less to be said for it than for the ends mentioned before, which are, at any rate, desired for their own sakes. (Bk. 1, Chapter III)"

"But it is better perhaps to examine next the universal good, and to enquire in what sense the expression is used. Though such an investigation is likely to be difficult, because the persons who have introduced these ideas are our friends. Yet it will perhaps appear the best, and indeed the right course, at least for the preservation of truth, to do away with private feelings, especially as we are philosophers; for since both are dear to us, we are bound to prefer the truth. (Bk. 1, Chapter III)"

"A person might fairly doubt also what in the world they mean by the ‘absolute’ this that or the other, since, as they would themselves allow, the account of the humanity is one and the same in the absolute man, and in any individual man: for so far as the individual and the absolute man are both man, they will not differ at all: and if so, then the essential good and any particular good will not differ, in so far as both are good. Nor will it do to say that the eternity of the absolute good makes it to be more good; for a white thing which has lasted white ever so long, is no whiter than that which only lasts for a day. (Bk. 1, Chapter III)"

"If there is some end of the things we do, which we desire for its own sake, clearly this must be the good. Will not knowledge of it, then, have a great influence on life? Shall we not, like archers who have a mark to aim at, be more likely to hit upon what we should? If so, we must try, in outline at least, to determine what it is."

"It is the mark of an educated man to look for precision in each class of things just so far as the nature of the subject admits; it is evidently equally foolish to accept probable reasoning from a mathematician and to demand from a rhetorician scientific proofs."

"The life of money-making is one undertaken under compulsion, and wealth is evidently not the good we are seeking; for it is merely useful and for the sake of something else."

"While both are dear, Piety requires us to honor truth above our friends."

"Life seems to be common even to plants, but we are seeking what is peculiar to man. Let us exclude, therefore, the life of nutrition and growth. Next there would be a life of perception, but it also seems to be common even to the horse, the ox, and every animal. There remains, then, an active life of the element that has a rational principle; of this, one part has such a principle in the sense of being obedient to one, the other in the sense of possessing one and exercising thought. And, as "life of the rational element" also has two meanings, we must state that life in the sense of activity is what we mean; for this seems to be the more proper sense of the term. Now if the function of man is an activity of soul which follows or implies a rational principle, and if we say "so-and-so" and "a good so-and-so" have a function which is the same in kind, e.g. a lyre, and a good lyre-player, and so without qualification in all cases, eminence in respect of goodness being added to the name of the function (for the function of a lyre-player is to play the lyre, and that of a good lyre-player is to do so well): if this is the case, and we state the function of man to be a certain kind of life, and this to be an activity or actions of the soul implying a rational principle, and the function of a good man to be the good and noble performance of these, and if any action is well performed when it is performed in accordance with the appropriate excellence: if this is the case, human good turns out to be activity of soul in accordance with virtue, and if there are more than one virtue, in accordance with the best and most complete. But we must add "in a complete life." For one swallow does not make a summer, nor does one day; and so too one day, or a short time, does not make a man blessed and happy."

"Let this serve as an outline of the good; for we must presumably first sketch it roughly, and then later fill in the details. But it would seem that any one is capable of carrying on and articulating what has once been well outlined, and that time is a good discoverer or partner in such a work; to which facts the advances of the arts are due; for any one can add what is lacking. And we must also remember what has been said before, and not look for precision in all things alike, but in each class of things such precision as accords with the subject-matter, and so much as is appropriate to the inquiry. For a carpenter and a geometer investigate the right angle in different ways; the former does so in so far as the right angle is useful for his work, while the latter inquires what it is or what sort of thing it is; for he is a spectator of the truth. We must act in the same way, then, in all other matters as well, that our main task may not be subordinated to minor questions. Nor must we demand the cause in all matters alike; it is enough in some cases that the fact be well established, as in the case of the first principles; the fact is the primary thing or first principle. Now of first principles we see some by induction, some by perception, some by a certain habituation, and others too in other ways. But each set of principles we must try to investigate in the natural way, and we must take pains to state them definitely, since they have a great influence on what follows. For the beginning is thought to be more than half of the whole, and many of the questions we ask are cleared up by it."

"For some identify happiness with virtue, some with practical wisdom, others with a kind of philosophic wisdom, others with these, or one of these, accompanied by pleasure or not without pleasure; while others include also external prosperity. Now ... it is not probable that these should be entirely mistaken, but rather that they should be right in at least some one respect or even in most respects."

"τὸ μὲν γὰρ ἥδεσθαι τῶν ψυχικῶν, ἑκάστῳ δ᾽ ἐστὶν ἡδὺ πρὸς ὃ λέγεται φιλοτοιοῦτος... Τοῖς μὲν οὖν πολλοῖς τὰ ἡδέα μάχεται διὰ τὸ μὴ φύσει τοιαῦτ᾽ εἶναι, τοῖς δὲ φιλοκάλοις ἐστὶν ἡδέα τὰ φύσει ἡδέα: τοιαῦται δ᾽ αἱ κατ᾽ ἀρετὴν πράξεις... Ἄριστον ἄρα καὶ κάλλιστον καὶ ἥδιστον ἡ εὐδαιμονία, καὶ οὐ διώρισται ταῦτα κατὰ τὸ Δηλιακὸν ἐπίγραμμα: “κάλλιστον τὸ δικαιότατον, λῷστον δ᾽ ὑγιαίνειν: ἥδιστον δὲ πέφυχ᾽ οὗ τις ἐρᾷ τὸ τυχεῖν.""

"Everything that depends on the action of nature is by nature as good as it can be, and similarly everything that depends on art or any rational cause, and especially if it depends on the best of all causes. To entrust to chance what is greatest and most noble would be a very defective arrangement."

"The truly good and wise man will bear all kinds of fortune in a seemly way, and will always act in the noblest manner that the circumstances allow."

"May not we then confidently pronounce that man happy who realizes complete goodness in action, and is adequately furnished with external goods? Or should we add, that he must also be destined to go on living not for any casual period but throughout a complete lifetime in the same manner, and to die accordingly, because the future is hidden from us, and we conceive happiness as an end, something utterly and absolutely final and complete? If this is so, we shall pronounce those of the living who possess and are destined to go on possessing the good things we have specified to be supremely blessed, though on the human scale of bliss."

"For the things we have to learn before we can do, we learn by doing."

"For legislators make the citizens good by forming habits in them, and this is the wish of every legislator, and those who do not effect it miss their mark, and it is in this that a good constitution differs from a bad one."

".... In a word, acts of any kind produce habits or characters of the same kind. Hence we ought to make sure that our acts are of a certain kind; for the resulting character varies as they vary. It makes no small difference, therefore, whether a man be trained in his youth up in this way or that, but a great difference, or rather all the difference."

"It is well said, then, that it is by doing just acts that the just man is produced, and by doing temperate acts the temperate man; without doing these no one would have even a prospect of becoming good. But most people do not do these, but take refuge in theory and think they are being philosophers and will become good in this way, behaving somewhat like patients who listen attentively to their doctors, but do none of the things they are ordered to do."

"Again, it is possible to fail in many ways (for evil belongs to the class of the unlimited ... and good to that of the limited), while to succeed is possible only in one way (for which reason also one is easy and the other difficult—to miss the mark easy, to hit it difficult); for these reasons also, then, excess and defect are characteristic of vice, and the mean of virtue; For men are good in but one way, but bad in many."

"The vices respectively fall short of or exceed what is right in both passions and actions, while virtue both finds and chooses that which is intermediate."

"In cases of this sort, let us say adultery, rightness and wrongness do not depend on committing it with the right woman at the right time and in the right manner, but the mere fact of committing such action at all is to do wrong."

"οὕτω δὲ καὶ τὸ μὲν ὀργισθῆναι παντὸς καὶ ῥᾴδιον, καὶ τὸ δοῦναι ἀργύριον καὶ δαπανῆσαι· τὸ δ᾽ ᾧ καὶ ὅσον καὶ ὅτε καὶ οὗ ἕνεκα καὶ ὥς, οὐκέτι παντὸς οὐδὲ ῥᾴδιον"

"κατὰ τὸν δεύτερον, φασί, πλοῦν τὰ ἐλάχιστα ληπτέον τῶν κακῶν"

"Therefore only an utterly senseless person can fail to know that our characters are the result of our conduct."

"What it lies in our power to do, it lies in our power not to do."

"μεταβολὴ δὲ πάντων γλυκύ"

"ἄνευ γὰρ φίλων οὐδεὶς ἕλοιτ᾽ ἂν ζῆν, ἔχων τὰ λοιπὰ ἀγαθὰ πάντα"

"When people are friends, they have no need of justice, but when they are just, they need friendship in addition."

"The best friend is he that, when he wishes a person's good, wishes it for that person's own sake."

"After these matters we ought perhaps next to discuss pleasure. For it is thought to be most intimately connected with our human nature, which is the reason why in educating the young we steer them by the rudders of pleasure and pain; it is thought, too, that to enjoy the things we ought and to hate the things we ought has the greatest bearing on virtue of character. For these things extend right through life, with a weight and power of their own in respect both to virtue and to the happy life, since men choose what is pleasant and avoid what is painful; and such things, it will be thought, we should least of all omit to discuss, especially since they admit of much dispute."

"And happiness is thought to depend on leisure; for we are busy that we may have leisure, and make war that we may live in peace."

"Now the activity of the practical virtues is exhibited in political or military affairs, but the actions concerned with these seem to be unleisurely. Warlike actions are completely so (for no one chooses to be at war, or provokes war, for the sake of being at war; any one would seem absolutely murderous if he were to make enemies of his friends in order to bring about battle and slaughter); but the action of the statesman is also unleisurely, and-apart from the political action itself—aims at despotic power and honours, or at all events happiness, for him and his fellow citizens—a happiness different from political action, and evidently sought as being different. So if among virtuous actions political and military actions are distinguished by nobility and greatness, and these are unleisurely and aim at an end and are not desirable for their own sake, but the activity of reason, which is contemplative, seems both to be superior in serious worth and to aim at no end beyond itself, and to have its pleasure proper to itself (and this augments the activity), and the self-sufficiency, leisureliness, unweariedness (so far as this is possible for man), and all the other attributes ascribed to the supremely happy man are evidently those connected with this activity, it follows that this will be the complete happiness of man, if it be allowed a complete term of life."

"Misfortune shows those who are not really friends."

"For well-being and health, again, the homestead should be airy in summer, and sunny in winter. A homestead possessing these qualities would be longer than it is deep; and its main front would face the south."

"Rhetoric is the counterpart of Dialectic. Both alike are concerned with such things as come, more or less, within the general ken of all men and belong to no definite science. Accordingly, all men make use, more or less, of both; for to a certain extent all men attempt to discuss statements and to maintain them, to defend themselves and to attack others."

"It is absurd to hold that a man ought to be ashamed of being unable to defend himself with his limbs but not of being unable to defend himself with reason when the use of reason is more distinctive of a human being than the use of his limbs."

"Evils draw men together."

"Thus every action must be due to one or other of seven causes: chance, nature, compulsion, habit, reasoning, anger, or appetite."

"The young have exalted notions, because they have not been humbled by life or learned its necessary limitations; moreover, their hopeful disposition makes them think themselves equal to great things—and that means having exalted notions. They would always rather do noble deeds than useful ones: Their lives are regulated more by moral feeling than by reasoning.... All their mistakes are due to excess and vehemence and their neglect of the maxim of Chilon [The maxim was Μηδὲν ἄγαν, Ne quid nimis, Never go to extremes.]. They overdo everything; they love too much, hate too much, and the same with everything else. And they think they know everything, and confidently affirm it, and this is the cause of their excess in everything."

"Wit is cultured insolence."

"It is simplicity that makes the uneducated more effective than the educated when addressing popular audiences."

"A tragedy, then, is the imitation of an action that is serious and also, as having magnitude, complete in itself; in language ... not in a narrative form; with incidents arousing pity and fear, wherewith to accomplish its catharsis of such emotions."

"A whole is that which has beginning, middle, and end."

"διὸ καὶ φιλοσοφώτερον καὶ σπουδαιότερον ποίησις ἱστορίας ἐστίν: ἡ μὲν γὰρ ποίησις μᾶλλον τὰ καθόλου, ἡ δ᾽ ἱστορία τὰ καθ᾽ ἕκαστον λέγει."

"διὸ εὐφυοῦς ἡ ποιητική ἐστιν ἢ μανικοῦ"

"But the greatest thing by far is to have a command of metaphor. This alone cannot be imparted by another; it is the mark of genius, for to make good metaphors implies an eye for resemblances."

"Homer has taught all other poets the art of telling lies skillfully."

"For the purposes of poetry a convincing impossibility is preferable to an unconvincing possibility."

"The roots of education ... are bitter, but the fruit is sweet."

"I have gained this by philosophy ... I do without being ordered what some are constrained to do by their fear of the law."

"Liars ... when they speak the truth they are not believed."

"Hope is the dream of a waking man."

"A friend is one soul abiding in two bodies."

"Amicus Plato, sed magis amica veritas."

"The single harmony produced by all the heavenly bodies singing and dancing together springs from one source and ends by achieving one purpose, and has rightly bestowed the name not of "disordered" but of "ordered universe" upon the whole."

"Remember that time slurs over everything, let all deeds fade, blurs all writings and kills all memories. Except are only those which dig into the hearts of men by love."

"It is the mark of an educated mind to be able to entertain a thought without accepting it."

"Man is a goal-seeking animal. His life only has meaning if he is reaching out and striving for goals."

"Happiness depends upon ourselves"

"What is the essence of life? To serve others and to do good."

"We are what we repeatedly do. Excellence, then, is not an act, but a habit."

"The aim of art is to represent not the outward appearance of things, but their inward significance."

"We live in deeds, not years; in thoughts not breaths; // In feelings, not in figures on a dial. // We should count time by heart throbs. He most lives // Who thinks most, feels the noblest, acts the best."

"The worst form of inequality is to try to make unequal things equal. (Whilst a paraphrase this is based on Aristotle's writings as Aristotle stated "For instance, it is thought that justice is equality, and so it is, though not for everybody but only for those who are equals; and it is thought that inequality is just, for so indeed it is, though not for everybody, but for those who are unequal" in https://www.loebclassics.com/view/aristotle-politics/1932/pb_LCL264.211.xml Politics, III. V. 8."

"There is only one way to avoid criticism: do nothing, say nothing and be nothing."

"Suffering becomes beautiful when anyone bears great calamities with cheerfulness, not through insensibility but through greatness of mind."

"Those who can, do, those who cannot, teach."

"Humour is the only test of gravity, and gravity of humour. For a subject which would not bear raillery is suspicious; and a jest which would not bear a serious examination is certainly false wit."

"Tolerance and apathy are the last virtues of a dying society."

"I was very fortunate. I was curious and handicapped as a young person. And so I read everything I could get my hands on and I have a good memory. And I have a lot of energy. It's a blessing. So I continued to learn. I'm hungry for knowledge still. Not every young person is blessed or visited with that combination. So he or she desperately needs to go to a university and be introduced to some of the great ideas of humankind. One needs to worry over the question of "Why am I here, what am I doing here of all things in this place, this life?" One needs to know Aristotle and Plato. One needs it desperately. One must have Leopold and Pascal. Must! I mean desperately, if one is to be at ease anywhere. One should have read the African folk tale to see what the West African calls deep thinking. One must worry over ideas that if I come forward how far do we have to go before we meet? And when we meet will I go through you and you go through me and continue until we meet someone else? This is an African concept. Do we stay once we meet or do I actually go right through you and pass through you and continue on that road. Is that what life is? All this knowledge is available at universities and one is more likely to run into a great teacher at a university than one is at a pool hall. It just follows."

"Once upon a time, Aristotle taught Alexander that he should restrain himself from frequently approaching his wife, who was very beautiful, lest he should impede his spirit from seeking the general good. Alexander acquiesed to him. The queen, when she perceived this and was upset, began to draw Aristotle to love her. Many times she crossed paths with him alone, with bare feet and disheveled hair, so that she might entice him. At last, being enticed, he began to solicit her carnally. She says, "This I will certainly not do, unless I see a sign of love, lest you be testing me. Therefore, come to my chamber crawling on hand and foot, in order to carry me like a horse. Then I'll know that you aren't deluding me." When he had consented to that condition, she secretly told the matter to Alexander, who lying in wait apprehended him carrying the queen. When Alexander wished to kill Aristotle, in order to excuse himself, Aristotle says, If thus it happened to me, an old man most wise, that I was deceived by a woman, you can see that I taught you well, that it could happen to you, a young man." Hearing that, the king spared him, and made progress in Aristotle's teachings."

"Aristotle, notwithstanding that for political reasons of his own he maintained a prudent silence as to certain esoteric matters, expressed very clearly his opinion on the subject. It was his belief that human souls are emanations of God, that are finally re-absorbed into Divinity. (p. 13)"

"Robert [Grosseteste] became much interested in science and scientific method... He was conscious of the dual approach by means of induction and deduction (resolution and composition); i.e., from the empirical knowledge one proceeds to probable general principles, and from these as premises one them derives conclusions which constitute verifications or falsifications of the principles. This approach to science was not that far removed from Aristotle..."

"Some 2,000 years ago Aristotle declared that eels were generated spontaneously from mud."

"[Aristotle] totally misrepresents Plato's doctrine of "Ideas." ... It is also pertinent to inquire, what is the difference between the "formal cause" of Aristotle and the archetypal ideas of Plato? ... Yet Aristotle is forever congratulating himself that he alone has properly treated the "formal" and the "final cause"!"

"Aristotle was the first genuine scientist in history. . . . Every scientist is in his debt."

"According to Aristotle, scientific investigation and explanation was a twofold process, the first inductive and the second deductive. The investigator must begin with what was prior in the order of knowing, that is, with the facts observed through the senses, and he must ascend through induction to generalizations or universal forms or causes which were most remote from sensory experience, yet causing that experience and therefore prior in the order of nature. [Footnote:] The idea that the order of demonstration was the order of nature came from Plato. Aristotle said that the order of discovery was the reverse of the order of demonstration."

"The model of scientific knowledge, in which effects could be shown to follow necessarily from their causes as conclusions from premises, Aristotle held to be mathematics, and where mathematics could be used in the natural sciences their conclusions were also exact and necessary. ... Of the inductive process by which the investigator passed from sensory experience of particular facts or connexions to a grasp of the prior demonstrative principles that explained them, Aristotle gave a clear psychological account. The final stage in the process was the sudden act by which the intuitive reason or νοῦς, after a number of experiences of facts, grasped the universal or theory explaining them, or penetrated to knowledge of the substance causing and connecting them."

"Is the ordinary person incompetent? No judgment is more decisive for one's political philosophy. It was perhaps the single most important difference in judgment between Plato and Aristotle."

"It is difficult to be enthusiastic about Aristotle, because it was difficult for him to be enthusiastic about anything... He realized too completely the Delphic command to avoid excess: he is so anxious to pare away extremes that at last nothing is left. He is so fearful of disorder that he forgets to be fearful of slavery; he.is so timid of uncertain change that he prefers a certain changelessness that near resembles death. He lacks that Heraclitean sense of flux which justifies the conservative in believing that all permanent change is gradual, and justifies the radical in believing that no changelessness is permanent. He forgets that Plato's communism was meant only for the elite, the unselfish and ungreedy few; and he comes deviously to a Platonic result when he says that though property should be private, its use should be as far as possible common. He does not see (and perhaps he could not be expected in his early day to see) that individual control of the means of production was stimulating and salutary only when these means were so simple as to be purchasable by any man; and that their increasing complexity and cost lead to a dangerous centralization of ownership and power, and to an artificial and finally disruptive inequality."

"From quotations I had seen I had a high notion of Aristotle's merits, but I had not the most remote notion what a wonderful man he was. Linnaeus and Cuvier have been my two gods, though in very different ways, but they were mere schoolboys to old Aristotle."

"It is pretty definitely settled, among men competent to form a judgment, that Aristotle was the best educated man that ever walked on the surface of this earth. He is still, as he was in Dante's time, the "master of those that know." It is, therefore, not without reason that we look to him, not only as the best exponent of ancient education, but as one of the worthiest guides and examples in education generally. That we may not lose the advantage of his example, it will be well, before we consider his educational theories, to cast a glance at his life, the process of his development, and his work."

"I sit with Shakespeare and he winces not. Across the color-line I move arm in arm with Balzac and Dumas, where smiling men and welcoming women glide in gilded halls. From out the caves of the evening that swing between the strong-limbed earth and the tracery of the stars, I summon Aristotle and Aurelius and what soul I will, and they come all graciously with no scorn nor condescension. So, wed with Truth, I dwell above the Veil. Is this the life you grudge us, O knightly America? Is this the life you long to change into the dull red hideousness of Georgia? Are you so afraid lest peering from this high Pisgah, between Philistine and Amalekite, we sight the Promised Land?"

"John Philoponus (c. 490-570) of Alexandria... refuted Aristotle's theory that the velocities of falling bodies in a given medium are proportional to their weight, making the observation that "if one lets fall simultaneously from the same height two bodies differing greatly in weight, one will find that the ratio of the times of their motion does not correspond to the ratios of their weights, but the difference in time is a very small one." ...He also criticized Aristotle's antiperistasis theory of projectile motion, which states that the air displaced by the object flows back to push it from behind. Instead Philoponus concluded that "some incorporeal kinetic power is imparted by the thrower to the object thrown" and that "if an arrow or a stone is projected by force in a void, the same will happen much more easily, nothing being necessary except the thrower." This is the famous "impetus theory," which was revived in medieval Islam and again in fourteenth century Europe, giving rise to the beginning of modern dynamics."

"We have in our age new accidents and observations, and such, that I question not in the least, but if Aristotle were now alive, they would make him change his opinion; which may be easily collected from the very manner of his discoursing: For when he writeth that he e­steemeth the Heavens inalterable, &c. because no new thing was seen to be begot therein, or any old to be dissolved, he seems im­plicitely to hint unto us, that when he should see any such accident, he would hold the contrary; and confront, as indeed it is meet, sensible experiments to natural reason: for had he not made any reckoning of the senses, he would not then from the not seeing of any sensible mutation, have argued immutability."

"I do believe for certain, that he [Aristotle] first procured, by the help of the senses, such experiments and observations as he could, to assure him as much as was possible of the conclusion, and that he afterwards sought out the means how to demonstrate it; for this is the usual course in demonstrative sciences. And the reason thereof is, because when the conclusion is true, by the help of the resolutive method, one may hit upon some proposition before demonstrated, or come to some principle known per se; but if the conclusion be false, a man may proceed in infinitum, and never meet with any truth already known."

"The group of philosophical ideas that concerns us has been called essentialism by Popper, who has traced the impact of Plato's metaphysics on political thinking down to modern times. Even before Plato, Greek philosophy began to experience difficulties in dealing with change. If things grew, or passed away, they seem somehow unreal, suggesting that they belonged only to a world of appearances. Heraclitus, in adopting the notion that material things are illusory, maintained that all that really exists is "fire"—that is, process. ...To Plato, true reality exists in the essence, Idea, or eidos. ...In the hands of Aristotle, essentialist metaphysics became somewhat altered. ...[H]e held that [essences] did not exist apart from things. His works embraced the concepts of teleology, empiricism, and natural science... to understand a thing was to know its essence, or to define it. ...A true system of knowledge thus became essentially a classification scheme... Plato and Aristotle... both embraced the notion that ideas or classes are more than just abstractions—that is... both advocated forms of "realism." ...Aristotle ...advocated heirarchical classification... classes were differentiated... by properties held in common... An implication, of enormous historical importance, was that it became very difficult to classify things which change, or... grade into one another, or even to conceive or to discuss them. Indeed, the very attempt to reason in terms of essences almost forces one to ignore everything dynamic or transitory. One could hardly design a philosophy better suited to predispose one toward dogmatic reasoning and static concepts. The Darwinian revolution thus depended upon the collapse of the Western intellectual tradition."

"Aristotle... justly reproves Democritus for saying, that if no medium were interposed, a pismire would be visible in the heavens; asserting, on the contrary, that if vacuity alone intervened, nothing possibly could be seen, because all vision is performed by changes or motions in the organ of sight; and all such changes or motions imply an interposed medium. Between the perceptions of the eye and of the ear there is a striking analogy. Bodies are only visible by their colour; and colour is only perceptible in light; and unless different motions were excited by light in the eye, colour and the distinctions of colour would no more be visible, than, independently of different vibrations communicated to the ear, sound, and the distinctions of sound, would be audible. When the vibrations in a given time are many, the sensation of sharpness or shrillness follows; when the vibrations are, in the same time, comparatively few, the sensation of flatness is the result: but the first sound does not excite many vibrations because it is shrill or sharp, but it is sharp because it excites many vibrations; and the second sound does not excite few vibrations because it is flat or grave, but it is grave because it excites few vibrations."

"On the authority of Aristotle... motion in the planetary world was somehow directed by the more perfect motion in higher spheres, and so on, up to the outermost sphere of fixed stars, indistinguishable from the prime mover. This implied a refined animistic and pantheistic world view, incomparably more rational than the ancient world views of Babylonians and Egyptians, among others, but a world view, nonetheless, hardly compatible with the idea of "inertial motion" which is implied in Buridan's concept of "impetus"... a momentous breaking point... which was to bear fruit... in the hands, first of Copernicus and then of Newton."

"The followers of Aristotle were called peripatetics because the "master of them that know" valued the linkage between cogitation and ambulation (the covered walk in Aristotle's Lyceum was a peripatos)."

"Time for us embraces a whole field of 'before and after', but Aristotle says: 'Before and after are involved in motion, but time is these so far as they are numbered' (Phys. 223a28). Elsewhere he defines time as 'the number of motion in respect of before and after', and he could seriously discuss the question whether there could be time without conscious and thinking beings; 'for if there could be no one to count, there could be nothing counted. ...If nothing can count but soul, and within soul mind, there cannot be time without soul, but only the substratum of time' (ibid. 219b2, 223a22)"

"As we now know, in the evolution of the structure of human activities, profitability works as a signal that guides selection towards what makes man more fruitful; only what is more profitable will, as a rule, nourish more people, for it sacrifices less than it adds. So much was at least sensed by some Greeks prior to Aristotle. Indeed, in the fifth century - that is, before Aristotle - the first truly great historian began his history of the Peloponnesian War by reflecting how early people `without commerce, without freedom of communication either by land or sea, cultivating no more of their territory than the exigencies of life required, could never rise above nomadic life' and consequently `neither built large cities nor attained to any other form of greatness' (Thucydides, Crawly translation, 1,1,2). But Aristotle ignored this insight. Had the Athenians followed Aristotle's counsel - counsel blind both to economics and to evolution - their city would rapidly have shrunk into a village, for his view of human ordering led him to an ethics appropriate only to, if anywhere at all, a stationary state. Nonetheless his doctrines came to dominate philosophical and religious thinking for the next two thousand years - despite the fact that much of that same philosophical and religious thinking took place within a highly dynamic, rapidly extending, order.(...) The anti-commercial attitude of the mediaeval and early modern Church, condemnation of interest as usury, its teaching of the just price, and its contemptuous treatment of gain is Aristotelian through and through. (...) Notwithstanding, and indeed wholly neglecting, the existence of this great advance, a view that is still permeated by Aristotelian thought, a naive and childlike animistic view of the world (Piaget, 1929:359), has come to dominate social theory and is the foundation of socialist thought."

"Current scientific and philosophical usage is so deeply influenced by the Aristotelian tradition, which knows nothing of evolution, that existing dichotomies and contrasts not only usually fail to capture correctly the processes underlying the problems and conflicts discussed in chapter one, but actually hinder understanding of those problems and conflicts themselves."

"[A]s to the spherical shape of the earth... Aristotle begins by answering an objection raised by the partisans of a flat earth... His answer is confused... He has, however, some positive proofs based on observation. (1) In partial eclipses of the moon the line separating the bright from the dark portion is always convex (circular)—unlike the line of demarcation in the phases of the moon, which may be straight or curved in either direction—this proves that the earth, to the interposition of which lunar eclipses are due, must be spherical. ...[H]is explanation shows that he had sufficiently grasped this truth. (2) Certain stars seen above the horizon in Egypt and in Cyprus are not visible further north, and... certain stars set there which in more northern latitudes remain always above the horizon. ...[I]t follows not only that the earth is spherical, but also that it is not a very large sphere. He adds that this makes it not improbable that people are right when they say that the region about the is joined on to India, one sea connecting them. It is here, too, that he quotes the result arrived at by mathematicians of his time, that the circumference of the earth is 400,000 stades. He is clear that the earth is much smaller than some of the stars. On the other hand, the moon is smaller than the earth. Naturally, Aristotle has a priori reasons for the sphericity of the earth. Thus, using once more his theory of heavy bodies tending to the centre, he assumes that, whether the heavy particles forming the earth are supposed to come together from all directions alike and collect in the centre or not, they will arrange themselves uniformly all round, i.e. in the shape of a sphere, since, if there is any greater mass at one part than at another, the greater mass will push the smaller until the even collection of matter all round the centre produces equilibrium."

"If order is to be maintained in existence — and that after all is what God wills, for He is not a God of confusion — first and foremost it must be remembered that every man is an individual man, is himself conscious of being an individual man. If once men are permitted to coalesce into what Aristotle calls "the multitude," a characteristic of beasts, this abstraction (instead of being regarded as less than nothing, as in fact it is, less than the lowliest individual man) will be regarded as something, and no long time will elapse before this abstraction becomes God."

"He penetrated into the whole universe of things, and subjected its scattered wealth to intelligence; and to him the greater number of the philosophical sciences owe their origin and distinction."

"Unfortunately... the philosophy of Aristotle laid it down as a principle, that the celestial motions were regulated by laws proper to themselves, and bearing no affinity to those which prevail on earth. By thus drawing a broad and impassable line of separation between celestial and terrestrial mechanics, it placed the former altogether out of the pale of experimental research, while it at the same time impeded the progress of the latter by the assumption of principles respecting natural and unnatural motions, hastily adopted from the most superficial and cursory and remark, undeserving even the name of observation. Astronomy therefore continued for ages a science of mere record, in which theory had no part, except in so far as it attempted to conciliate the inequalities of the celestial motions with that assumed law of uniform circular revolution which was alone considered consistent with the perfection of the heavenly mechanism."

"In the old philosophy, a curious conjunction of ethical and physical prejudices had led to the notion that there was something ethically bad and physically obstructive about matter. Aristotle attributes all irregularities and apparent dysteleologies in nature to the disobedience, or sluggish yielding, of matter to the shaping and guiding influence of those reasons and causes which were hypostatised in his ideal 'Forms.'"

"Aristotle was the first accurate critic and truest judge — nay, the greatest philosopher the world ever had; for he noted the vices of all knowledges, in all creatures, and out of many men's perfections in a science he formed still one Art."

"Aristotle had not been popular in the ancient world, but his ideas were picked up by the materialistically-minded Arabs as they were developing their culture, and from there his works were introduced into Western Europe. They became the rage, stimulating a whole intellectual revival. It soon became necessary for the church to deal with this point of view, and through the genius of Thomas Aquinas all of the church ideas were rewritten within the framework of Aristotle's ideas with their mythological character reduced to a bare minimum."

"[A] comparison was undertaken between this book—or the related work On the Harmonies...—and Aristotle's books On the Heavens and Metaphysics... I have nothing to worry about in the case of Aristotle... His Most Serene Highness cannot dislike whatever is the more convincing, whether it be that the world was first made at a fixed beginning in time as was my work On the Harmonies, or will be destroyed at some time, or is merely liable to destruction, like the alterations of the ether and the celestial atmosphere; nor will he ever prefer the Master Aristotle to the truth of which Aristotle was ignorant."

"Aristotle wrote a ten-book History of Animals without ever considering the possibility that animals actually had a history."

"Most expositions of Aristotle's doctrines, when they have not been dictated by a spirit of virulent detraction, or unsympathetic indifference, have carefully suppressed all, or nearly all, the absurdities, and only retained what seemed plausible and consistent. But in this procedure their historical significance disappears."

"Aristotle... seems utterly destitute of any sense of the Ineffable. There is no quality more noticeable in him than his unhesitating confidence in the adequacy of the human mind to comprehend the universe... He never seems to be visited by misgivings as to the compass of human faculty, because his unhesitating mind is destitute of awe. He has no abiding consciousness of the fact deeply impressed on other minds, that the circle of the Knowable is extremely limited; and that beyond it lies a vast mystery... impenetrable. Hence the existence of Evil is no perplexity to his soul; it is accepted as a simple fact. Instead of being troubled by it, saddened by it, he quietly explains it as the consequence of Nature not having correctly written her meaning. This mystery which has darkened so many sensitive meditative minds with anguish he considered to be only bad orthography."

"Roger Bacon expressed a feeling which afterwards moved many minds, when he said that if he had the power he would burn all the works of the Stagirite, since the study of them was not simply loss of time, but multiplication of ignorance. Yet in spite of this outbreak every page is studded with citations from Aristotle, of whom he everywhere speaks in the highest admiration."

"Aristotle forever, but Truth even for longer than that."

"Aristotle, that histrionic mountebank, who from behind a Greek mask has so long bewitched the Church of Christ, that most cunning juggler of souls, who, if he had not been accredited as human blood and bone, we should have been justified in maintaining to be the veritable devil."

"Aristotle sees no difference between the falling of a leaf or a stone and the death of the good and noble people in the ship; nor does he distinguish between the destruction of a multitude of ants by an ox depositing on them his excrement and the death of worshippers killed by the fall of the house when its foundations give way. In short, the opinion of Aristotle is this: Everything is the result of management which is constant, which does not come to an end and does not change any of its properties, as e.g., the heavenly beings, and everything which continues according to a certain rule... But that which is not constant, and does not follow a certain rule... is due to chance and not to management; it is in no relation to Divine Providence. Aristotle holds that it is even impossible to ascribe to Providence that management of these things. ...It is the belief of those who turned away from our Law and said: "God hath forsaken the earth." (Ezek. ix. 9)"

"When I saw that Moses' version of the Genesis of the world did not fit sufficiently in many ways with Aristotle and the rest of the philosophers, I began to have doubts about the truth of all philosophers and started to investigate the secrets of nature."

"In his discussion on slavery Aristotle said that when the shuttle wove by itself and the plectrum played by itself chief workmen would not need helpers nor masters slaves. At the time he wrote, he believed that he was establishing the eternal validity of slavery; but for us today he was in reality justifying the existence of the machine. Work, it is true, is the constant form of man's interaction with his environment, if by work one means the sum total of exertions necessary to maintain life; and the lack of work usually means an impairment of function and a breakdown in organic relationship that leads to substitute forms of work, such as invalidism and neurosis. But work in the form of unwilling drudgery or of that sedentary routine which... the Athenians so properly despised—work in these forms is the true province of machines. Instead of reducing human beings to work-mechanisms, we can now transfer the main part of burden to automatic machines. This potentially... is perhaps the largest justification of the mechanical developments of the last thousand years."

"The first clear expression of the idea of an element occurs in the teachings of the Greek philosophers. ... Aristotle ... who summarized the theories of earlier thinkers, developed the view that all substances were made of a primary matter... On this, different forms could be impressed... so the idea of the transmutation of the elements arose. Aristotle's elements are really fundamental properties of matter.... hotness, coldness, moistness, and dryness. By combining these in pairs, he obtained what are called the four elements, fire, air, earth and water... a fifth, immaterial, one was added, which appears in later writings as the quintessence. This corresponds with the ether. The elements were supposed to settle out naturally into the earth (below), water (the oceans), air (the atmosphere), fire and ether (the sky and heavenly bodies)."

"The victory of orthodox Christian doctrine over classical thought was to some extent a , for the theology that triumphed over Greek philosophy has continued to be shaped ever since by the language and the thought of classical metaphysics. For example, the Fourth Lateran Council in 1215 decreed that "in the sacrament of the altar... the bread is transubstantiated into the body [of Christ]." ...Most of the theological expositions of the term "" have interpreted "substance" [according] to the meaning given this term ...in the fifth book of Aristotle's Metaphysics; transubstantiation, then, would appear to be tied to the acceptance of Aristotelian metaphysics or even of Aristotelian physics. ...Transubstantiation is an individual instance of what has been called the problem of "the hellenization of Christianity.""

"All the things that Aristotle has said are inconsistent because they are poorly systematized and can be called to mind only by the use of arbitrary mnemonic devices."

"It appears to me that there can be no question, that Aristotle stands forth, not only as the greatest figure in antiquity, but as the greatest intellect that has ever appeared upon the face of this earth."

"Aristotle, as a philosopher, is in many ways very different from all his predecessors. He is the first to write like a professor: his treatises are systematic, his discussions are divided into heads, he is a professional teacher, not an inspired prophet. His work is critical, careful, pedestrian, without any trace of Bacchic enthusiasm. The Orphic elements in Plato are watered down in Aristotle, and mixed with a strong dose of common sense; where he is Platonic, one feels that his natural temperament has been overpowered by the teaching to which he has been subjected. He is not passionate, or in any profound sense religious. The errors of his predecessors were the glorious errors of youth attempting the impossible; his errors are those of age which cannot free itself of habitual prejudices. He is best in detail and in criticism; he fails in large construction, for lack of fundamental clarity and Titanic fire."

"I do not agree with Plato, but if anything could make me do so, it would be Aristotle's arguments against him."

"I conclude that the Aristotelian doctrines are wholly false, with the exception of the formal theory of the syllogism, which is unimportant. Any person in the present day who wishes to learn logic will be wasting his time if he reads Aristotle or any of his disciples. Nonetheless, Aristotle's logical writings show great ability, and would have been useful to mankind if they had appeared at a time when intellectual originality was still active. Unfortunately, they appeared at the very end of the creative period of Greek thought, and therefore came to be accepted as authoritative. By the time that logical originality revived, a reign of two thousand years had made Aristotle very difficult to dethrone. Throughout modern times, practically every advance in science, in logic, or in philosophy has had to be made in the teeth of opposition from Aristotle's disciples."

"Aristotle is the last Greek philosopher who faces the world cheerfully; after him, all have, in one form or another, a philosophy of retreat."

"Aristotle, so far as I know, was the first man to proclaim explicitly that man is a rational animal. His reason for this view was one which does not now seem very impressive; it was, that some people can do sums."

"Aristotle could have avoided the mistake of thinking that women have fewer teeth than men, by the simple device of asking Mrs Aristotle to keep her mouth open while he counted."

"To modern educated people, it seems obvious that matters of fact are to be ascertained by observation, not by consulting ancient authorities. But this is an entirely modern conception, which hardly existed before the seventeenth century. Aristotle maintained that women have fewer teeth than men; although he was twice married, it never occurred to him to verify this statement by examining his wives' mouths. He said also that children would be healthier if conceived when the wind is in the north. One gathers that the two Mrs. Aristotles both had to run out and look at the weathercock every evening before going to bed. He states that a man bitten by a mad dog will not go mad, but any other animal will (Hiss. Am., 704a); that the bite of the shrewmouse is dangerous to horses, especially if the mouse is pregnant (ibid., 604b); that elephants suffering from insomnia can be cured by rubbing their shoulders with salt, olive oil, and warm water (ibid., 605a); and so on and so on. Nevertheless, classical dons, who have never observed any animal except the cat and the dog, continue to praise Aristotle for his fidelity to observation."

"Socrates and Plato had no time for Athenian democracy, and wanted a revived aristocratic government for their city. But both were moral radicals; they thought ordinary morality was radically misguided, and that public opinion should be ignored when it was at odds with one's conscience or reason. Things are very different in Aristotle. Plato's concern for the balance of the soul was shared by Aristotle, but not his ethical radicalism."

"Justice is of two kinds, justice in distribution and justice in rectification. ... Aristotle thinks primarily of setting things straight, and denies that rectificatory justice contains an element of 'tit for tat'."

"Aristotle's genius was for showing the ways in which we might construct the "best practicable state." This was not mere practicality; the goals of political life are not wholly mundane. The polity comes into existence for the sake of mere life, but it continues to exist for the sake of the good life. The good life is richly characterized, involving as it does the pursuit of justice, the expansion of the human capacities used in political debate, and the development of all the public and private virtues that a successful state can shelter—military courage, marital fidelity, devotion to the physical and psychological welfare of our children, and so on indefinitely."

"Aristotle, who foresaw so many things, never dreamed of the social truth. Cuvier, whose sagacity is so highly lauded, was constrained to yield homage to the genius of Aristotle in Natural History; for myself, who am at this date in full possession of social truth, in politics Aristotle only inspires me with profound pity."

"The old Greek philosophy, which in Europe in the later middle ages was synonymous with the works of Aristotle, considered motion as a thing for which a cause must be found: a velocity required a force to produce and to maintain it. The great discovery of Galileo was that not velocity, but acceleration requires a force. This is the law of inertia of which the real content is: the natural phenomena are described by differential equations of the second order."

"Men often speak of virtue without using the word but saying instead "the quality of life" or "the great society" or "ethical" or even "square." But do we know what virtue is? Socrates arrived at the conclusion that it is the greatest good for a human being to make everyday speeches about virtue-apparently without ever finding a completely satisfactory definition of it. However, if we seek the most elaborate and least ambiguous answer to this truly vital question, we shall turn to Aristotle's Ethics. There we read among other things that there is a virtue of the first order called magnanimity—the habit of claiming high honors for oneself with the understanding that one is worthy of them. We also read there that sense of shame is not a virtue: sense of shame is becoming for the young who, due to their immaturity, cannot help making mistakes, but not for mature and well-bred men who simply always do the right and proper thing. Wonderful as all this is-we have received a very different message from a very different quarter."

"Aristotle...is the first man [Greek], so far as I know, to have collected books."

"Aristotle... distinguished four sorts of explanatory factor... and in later centuries these came to be known as his 'four causes'. The name is unfortunate, since nowadays we usually restrict the term 'cause' to one of his four types... they would have been better called his 'four becauses'—since he was concerned to distinguish, not the different varieties of cause and effect, but rather the different senses in which the question 'Why?' can be asked... [W]e could give four different answers, whose relevance would depend on our precise interpretation of the question. We could refer to: (i) The material constitution... or 'From what?'... (ii) The form, essence, or 'What was it?'... (iii) The precipitating cause or 'By What?'... (iv) The end [destination or purpose], or 'In aid of what?'... These four types of explanation are not necessarily rivals. ...all four types can frequently be cited without inconsistency. Indeed, apart from a few phenomena... which have no function and so 'just happen', Aristotle thought all natural events called for explanation in all four ways."

"The metaphysical doctrine of 'permanent essences' drew empirical support from the success of Aristotle's zoological theory of fixed species, which was its most convincing application to our actual experience of the world. ...[T]he doctrine of fixed organic species simply exemplified, in the special sphere of biology, the permanent character of all 'rationally intelligible' entities. Conversely, Darwin demonstrated that Aristotle's most favored examples failed to support... the metaphysical assumption on which orthodox Greek natural philosophy had been based. Species were not... permanent entities; the earlier 'typological' or 'essentialist' approach to taxonomy inherited from Aristotle misrepresented the long term history of living things. ...However irrelevant the empirical details of Darwin's work may be to general philosophy, the abstract form of his explanatory schema has a much broader significance. So, when Darwin and his successors showed that the whole zoological concepts of 'species' must be reanalysed in populational terms, their demonstration knocked away [a] prop from the traditional metaphysical debate."

"In matter-theory, as in astronomy, the Church's commitment to Aristotle was in due course to prove an embarassment. In both branches of science his speculative distinction between terrestrial and celestial matter was insecure from the very beginning. His own most loyal commentator, ... had already dreamt of a theory unifying all things, and John Philoponos... had rejected the distinction between terrestrial and celestial matter outright. Nevertheless, it was still an axiom of almost a thousand years later."

"Thus to the plain man there may be no metaphor in Aristotle's "substance", Descartes' "machine of nature," Newtonian "force" and "attraction," Thomas Young's "kinetic energy" and Michelangelo's figure of Leda. Placed in their customary contexts these present nothing to him but the face of literal truth. To the initiated, however, who are aware of the "gross original" senses as well as the now literal senses , they may become metaphors. There are no metaphors per se...."

"Aristotle's works are full of platitudes in much the same way that Shakespeare's Hamlet is full of quotations."

"It is an age of Intellectual slaveries; If they meet any thing extraordinary, they prune it with distinctions, or dawb it with false Glosses, til it looks like the Traditions of Aristotle. His followers are so confident of his principles they seek not to understand what others speak, but to make others speak what they understand... Their Aristotle is a Poet in text, his principles are but Fancies, and they stand more on our Concessions, then his Bottom. Hence it is that his followers, notwithstanding the Assistance of so many Ages, can fetch nothing out of him but Notions: And these indeed they use, as He sayeth Lycophron did his Epithets, Non ut Condimentis, sed ut Cibis, Their Compositions are a meer Tympanie of Terms... It is better then a Fight in Quixot, to observe what Duels, and Digladiations they have about Him. One will make him speak Sense another Non-sense and a third both, Aquinas palps him gently, Scotus makes him winch and he is taught like an ape to shew severall tricks. If we look on his adversaries the least amongst them hath foyld him, but Telesius knocked him in the head, and Campanella hath quite discomposed him... Aristotle thrives by scuffles and the world cryes him up, when trueth cryes him down."

"We've got to purge Aristotle from our system." "I've never read him so why do I have to purge him from my system?" "It's proof of his grip on Western Man that he dominates the thinking of people who have never heard of him."

"Aristotle especially, both by speculation and observation... reached something like the modern idea of a succession of higher organizations from lower, and made the fruitful suggestion of "a perfecting principle" in Nature. With the coming in of Christian theology this tendency toward a yet truer theory of evolution was mainly stopped, but the old crude view remained..."

"As man loses touch with his 'inner being', his instinctive depths, he finds himself trapped in the world of consciousness, that is to say, in the world of other people. Any poet knows this truth; when other people sicken him, he turns to hidden resources of power inside himself, and he knows then that other people don't matter a damn. He knows the 'secret life' inside him is the reality; other people are mere shadows in comparison. but the 'shadows' themselves cling to one another. 'Man is a political animal', said Aristotle, telling one of the greatest lies in human history. Man has more in common with the hills, or with the stars, than with other men."

"[Professor] Jin [Canrong] suggested that Aristotle’s works may have been written some seventeen centuries later. He concluded that if the West has lied for so long about Aristotle, Chinese are authorized to believe they lie about pretty much everything. …He claims that Aristotle’s works are mentioned only in sources from the 13th century and later. This is false, as Aristotle’s theories started being debated mentioning his name and his school shortly after his death, whose date is traditionally fixed at 322 BCE, and even before. Aristotle seems to be much better documented than Confucius… …The theory that Aristotle did not exist may be ridiculed abroad, but if the [ Chinese Communist Party ] wanted to test just how much fake news about Western history and supposed Western conspiracies Chinese are prepared to swallow, the answer it got was—quite a lot."

"εὕρηκα [heúrēka]"

"δῶς μοι πᾶ στῶ καὶ τὰν γᾶν κινάσω. [Dôs moi pâ stô, kaì tàn gân kinásō.]"

"Noli turbare circulos meos. or Noli tangere circulos meos."

"How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!"

"Those who claim to discover everything but produce no proofs of the same may be confuted as having actually pretended to discover the impossible."

"Equal weights at equal distances are in equilibrium and equal weights at unequal distances are not in equilibrium but incline towards the weight which is at the greater distance."

"If two equal weights have not the same centre of gravity, the centre of gravity of both taken together is at the middle point of the line joining their centres of gravity."

"Two magnitudes whether commensurable or incommensurable, balance at distances reciprocally proportional to the magnitudes."

"The centre of gravity of any parallelogram lies on the straight line joining the middle points of opposite sides."

"The centre of gravity of a parallelogram is the point of intersection of its diagonals."

"In any triangle the centre of gravity lies on the straight line joining any angle to the middle point of the opposite side."

"It follows at once from the last proposition that the centre of gravity of any triangle is at the intersection of the lines drawn from any two angles to the middle points of the opposite sides respectively."

"I thought fit to... explain in detail in the same book the peculiarity of a certain method, by which it will be possible... to investigate some of the problems in mathematics by means of mechanics. This procedure is... no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards... But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge."

"I am persuaded that it [The Method of Mechanical Theorems] will be of no little service to mathematics; for I apprehend that some, either of my contemporaries or of my successors, will, by means of the method when once established, be able to discover other theorems in addition, which have not yet occurred to me."

"First then I will set out the very first theorem which became known to me by means of mechanics, namely that Any segment of a section of a right angled cone (i.e., a parabola) is four-thirds of the triangle which has the same base and equal height, and after this I will give each of the other theorems investigated by the same method. Then at the end of the book I will give the geometrical [proofs of the propositions]..."

"The centre of gravity of any cylinder is the point of bisection of the axis."

"The centre of gravity of any cone is [the point which divides its axis so that] the portion [adjacent to the vertex is] triple [of the portion adjacent to the base]."

"Any segment of a right-angled conoid (i.e., a paraboloid of revolution) cut off by a plane at right angles to the axis is 1½ times the cone which has the same base and the same axis as the segment"

"The centre of gravity of any hemisphere [is on the straight line which] is its axis, and divides the said straight line in such a way that the portion of it adjacent to the surface of the hemisphere has to the remaining portion the ratio which 5 has to 3."

"Shall we not make an end to this fighting against this geometrical Briareus who uses our ships like cups to ladle water from the sea, drives off our sambuca ignominiously with cudgel-blows, and by the multitude of missiles that he hurls at us all at once outdoes the hundred-handed giants of mythology?"

"When... the Romans assaulted the walls in two places at once, fear and consternation stupefied the Syracusans.... But when Archimedes began to ply his engines, he at once shot against the land forces all sorts of missile weapons... that came down with incredible noise and violence... they knocked down those upon whom they fell in heaps, breaking all their ranks and files. ...huge poles thrust out from the walls, over the ships, sunk some by the great weights... from on high... others they lifted up into the air by an iron hand or beak like a crane's... and... plunged them to the bottom of the sea; or else the ships, drawn by engines within, and whirled about, were dashed against steep rocks... under the walls, with great destruction of the soldiers... aboard them. A ship was frequently lifted up to a great height in the air... and was rolled to and fro... until the mariners were all thrown out, when at length it was dashed against the rocks, or let fall. At the engine [called Sambuca] that Marcellus brought upon the bridge of ships... while it was as yet approaching the wall, there was discharged a... rock of ten talents [600-700 lb. total] weight, then a second and a third, which, striking upon it with immense force and a noise like thunder, broke all its foundation to pieces... and completely dislodged it from the bridge. So Marcellus... drew off his ships to a safer distance, and sounded a retreat... They then took a resolution of coming up under the walls... in the night; thinking that as Archimedes used ropes stretched at length in playing his engines, the soldiers would now be under the shot, and the darts would... fly over their heads... But he... had... framed... engines accommodated to any distance, and shorter weapons; and... with engines of a shorter range, unexpected blows were inflicted on the assailants. Thus... instantly a shower of darts and other missile weapons was again cast upon them. And when stones came tumbling down... upon their heads, and... the whole wall shot out arrows at them, they retired. ...as they were going off, arrows and darts of a longer range inflicted a great slaughter among them, and their ships were driven one against another; while they themselves were not able to retaliate... For Archimedes had provided and fixed most of his engines immediately under the wall; whence the Romans, seeing that indefinite mischief overwhelmed them from no visible means, began to think they were fighting with the gods."

"When Jove looked down and saw the heavens figured in a sphere of glass he laughed and said to the other gods: "Has the power of mortal effort gone so far? Is my handiwork now mimicked in a fragile globe? An old man of Syracuse has imitated on earth the laws of the heavens, the order of nature, and the ordinances of the gods. Some hidden influence within the sphere directs the various courses of the stars and actuates the lifelike mass with definite motions. A false zodiac runs through a year of its own, and a toy moon waxes and wanes month by month. Now bold invention rejoices to make its own heaven revolve and sets the stars in motion by human wit. Why should I take umbrage at harmless and his mock thunder? Here the feeble hand of man has proved Nature's rival.""

"Archimedes said, “Give to me a fulcrum on which to plant my lever, and I will move the world.” And I say, give to woman the ballot, the political fulcrum, on which to plant her moral lever, and she will lift the world into a nobler purer atmosphere."

"Abstract enquiries into the most puzzling problems did not arise in the brain of Archimedes as a spontaneous and hitherto untouched subject, but rather as a reflection of prior enquiries in the same direction and by men separated from his days by as long a period — and far longer — than the one which separates you from the great Syracusian."

"Archimedes originally solved the problem of finding the solid content of a sphere before that of finding its surface, and he inferred the result of the latter problem from that of the former. ...another illustration of the fact that the order of propositions in the treatises of the Greek geometers as finally elaborated does not necessarily follow the order of discovery."

"Some of the later Greeks, such as Archimedes, had just views on the elementary phenomena of and optics. Indeed, Archimedes, who combined a genius for mathematics with physical insight, must rank with Newton, who lived nearly two thousand years later, as one of the founders of mathematical physics."

"In these days an infinite number of chemical tests would be available. But then Archimedes had to think... afresh. The solution flashed upon him as he lay in his bath. He jumped up and ran through the streets to the palace, shouting Eureka! Eureka! (I have found it! ...) This day... ought to be celebrated as the birthday of mathematical physics; the science came of age when Newton sat in his orchard. Archimedes... had made a great discovery. He saw that a body when immersed in water is pressed upwards by the surrounding water with a resultant force equal to the weight of the water it displaces. ...Hence if W lb. be the [known] weight of the crown, as weighed in air, and w lb. be the [unknown] weight of the water which it displaces when completely immersed, W - w [from which (knowing W) the weight w of the equal volume of water can be derived,] would be the extra upward force necessary to sustain the crown as it hung in the water. [Alternatively, the weight of water, equaling the volume of the crown, and overflowing a tub, could be weighed directly.] Now, this upward force can easily be obtained by weighing the body as it hangs in the water [Fig. 3]...But \frac{w}{W} ...is the same for any lump of metal of the same material: it is now called the ... Archimedes had only to take a lump of indisputably pure gold and find its specific gravity by the same process. ...[N]ot only is it the first precise example of the application of mathematical ideas to physics, but also... a perfect and simple example of what must be the method and spirit of the science for all time. The discovery of the theory of specific gravity marks a genius of the first rank."

"The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader."

"There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results.""

"Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton."

"To conceive of a parabolic segment or of a triangle as the sum of infinitely many line segments, is closely akin to the idea of Leibniz, who thought of the integral \int y~dx as the sum of infinitely many terms y~dx. But, in contrast to Leibniz, Archimedes is fully aware that this conception is... incorrect and that the derivation should be supplemented by a rigorous proof."

"The estimations, which occur in the summing of infinite series and in limiting operations, the "epsilontics", as the calculation with an arbitrarily small ε is sometimes called, were for Archimedes an open book. In this respect, his thinking is entirely modern."

"In Euclidean geometry the infinitely small was rejected and in the classical treatises of Archimedes we have the finest example of mathematical rigour in antiquity. Notwithstanding, in the discovery method we find him manipulating line and surface indivisibles skilfully, imaginatively and non-rigorously"

"Almost all modern translations of Archimedes’ works stem from a single Greek manuscript that was copied from an earlier original at Constantinople in the ninth or tenth century, was translated into Latin in the thirteenth century, and eventually disappeared without a trace in the sixteenth century."

"Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer...he found it much easier to prove geometrically... Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs. ...[He] essentially introduced the concept of a thought experiment into rigorous research. ...[He] freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. ...He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids and flat objects) to advance his mathematics."

"Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first."

"Archimedes was a brilliant inventor and a mathematician. He says to the people around him, "Don't just live in the lap of the gods. Don't be dominated by Mother Nature. You, as a man, can take control of your own destiny.""

"According to legend, nothing could get between him [Archimedes] and his work, and sometimes he would even forget to eat. Ideas would come to him at any moment, and he would scribble them on any available surface. Famously, he was in the bath when he discovered the laws of buoyancy, leading him to run naked through the streets shouting "Eureka!" … Eureka means "I have found it," and it could be argued that Archimedes found out more than anyone else before or since."

"Tragically for all of us, he [Archimedes] was cut down by a Roman soldier because he refused to stop working. … If Archimedes hadn't been killed before his time, what could have he achieved? The industrial revolution could have happened two thousand years earlier. He might have kick-started the modern age."

"I was Euphorbus at the siege of Troy."

"By the air which I breathe, and by the water which I drink, I will not endure to be blamed on account of this discourse."

"Dear youths, I warn you cherish peace divine, And in your hearts lay deep these words of mine."

"Τὴν δ' ἀνθρώπου ψυχὴν διῃρῆσθαι τριχῆ, εἴς τε νοῦν καὶ φρένας καὶ θυμόν. νοῦν μὲν οὖν καὶ θυμὸν εἶναι καὶ ἐν τοῖς ἄλλοις ζῴοις, φρένας δὲ μόνον ἐν ἀνθρώπῳ."

"ἀλλήλοις θ᾽ ὁμιλεῖν, ὡς τοὺς μὲν φίλους ἐχθροὺς μὴ ποιῆσαι, τοὺς δ᾽ ἐχθροὺς φίλους ἐργάσασθαι. ἴδιόν τε μηδὲν ἡγεῖσθαι."

"ἐν ὀργῇ μήτε τι λέγειν μήτε πράσσειν"

"Reason is immortal, all else mortal."

"The most momentous thing in human life is the art of winning the soul to good or to evil."

"κοινὰ τὰ φίλων εἶναι καὶ φιλίαν ἰσότητα."

"Power is the near neighbour of necessity."

"Number is the ruler of forms and ideas, and the cause of gods and daemons."

"Sobriety is the strength of the soul, for it preserves its reason unclouded by passion."

"None but God is wise."

"λόγον περὶ θεοῦ σιγᾶν ἄμεινον ἢ προπετῶς διαλέγεσθαι. (In talk about God, silence is better than reckless words.) (Sextus 366)"

"If there be light, then there is darkness; if cold, heat; if height, depth; if solid, fluid; if hard, soft; if rough, smooth; if calm, tempest; if prosperity, adversity; if life, death."

"Rest satisfied with doing well, and leave others to talk of you as they please."

"It is only necessary to make war with five things; with the maladies of the body, the ignorances of the mind, with the passions of the body, with the seditions of the city and the discords of families."

"As soon as laws are necessary for men, they are no longer fit for freedom."

"Friends are as companions on a journey, who ought to aid each other to persevere in the road to a happier life."

"Anger begins in folly, and ends in repentance."

"Choose always the way that seems the best, however rough it may be; custom will soon render it easy and agreeable."

"It is better wither to be silent, or to say things of more value than silence. Sooner throw a pearl at hazard than an idle or useless word; and do not say a little in many words, but a great deal in a few."

"Truth is so great a perfection, that if God would render himself visible to men, he would choose light for his body and truth for his soul."

"There are men and gods, and beings like Pythagoras."

"There is no word or action but has its echo in Eternity. Thought is an Idea in transit, which when once released, never can be lured back, nor the spoken word recalled. Nor ever can the overt act be erased All that thou thinkest, sayest, or doest bears perpetual record of itself, enduring for Eternity."

"There is geometry in the humming of the strings, there is music in the spacing of the spheres."

"As long as Man continues to be the ruthless destroyer of lower living beings, he will never know health or peace. For as long as men massacre animals, they will kill each other. Indeed, he who sows the seed of murder and pain cannot reap joy and love."

"Time is the soul of this world."

"Most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge."

"Man know thyself; then thou shalt know the Universe and God."

"A blow from your friend is better than a kiss from your enemy."

"Write in the sand the flaws of your friend."

"Educate the children and it won't be necessary to punish the men."

"When going to the temple to adore Divinity neither say nor do any thing in the interim pertaining to the common affairs of life."

"Sacrifice and adore unshod."

"Disbelieve nothing wonderful concerning the gods, nor concerning divine dogmas."

"Declining from the public ways, walk in unfrequented paths."

"Govern your tongue before all other things, following the gods."

"The wind is blowing, adore the wind."

"Cut not fire with a sword."

"Assist a man in raising a burden; but do not assist him in laying it down."

"Step not beyond the beam of the balance."

"Having departed from your house, turn not back; for the furies will be your attendants."

"Eat not the heart."

"Eat not the brain."

"Κυάμων ἀπέχεσθαι"

"Abstain from animals."

"Above and before all things, worship GOD!"

"Above all things reverence thy Self."

"Work at these things, practice them, these are the things you ought to desire; they are what will put you on the path of divine virtue — yes, by the one who entrusted our soul with the tetraktys, source of ever-flowing nature. Pray to the gods for success and get to work."

"Practice justice in word and deed, and do not get in the habit of acting thoughtlessly about anything."

"Know that death comes to everyone, and that wealth will sometimes be acquired, sometimes lost. Whatever griefs mortals suffer by divine chance, whatever destiny you have, endure it and do not complain. But it is right to improve it as much as you can, and remember this: Fate does not give very many of these griefs to good people."

"Many words befall men, mean and noble alike; do not be astonished by them, nor allow yourself to be constrained. If a lie is told, bear with it gently. But whatever I tell you, let it be done completely. Let no one persuade you by word or deed to do or say whatever is not best for you."

"Let not sleep fall upon thy eyes till thou has thrice reviewed the transactions of the past day. Where have I turned aside from rectitude? What have I been doing? What have I left undone, which I ought to have done? Begin thus from the first act, and proceed; and, in conclusion, at the ill which thou hast done, be troubled, and rejoice for the good."

"Meditate upon my counsels; love them; follow them; To the divine virtues will they know how to lead thee. I swear it by the One who in our hearts engraved The sacred Tetrad, symbol immense and pure, Source of Nature and model of the Gods."

"Holding fast to these things, you will know the worlds of gods and mortals which permeates and governs everything. And you will know, as is right, nature similar in all respects, so that you will neither entertain unreasonable hopes nor be neglectful of anything."

"You will know that wretched men are the cause of their own suffering, who neither see nor hear the good that is near them, and few are the ones who know how to secure release from their troubles. Such is the fate that harms their minds; like pebbles they are tossed about from one thing to another with cares unceasing. For the dread companion Strife harms them unawares, whom one must not walk behind, but withdraw from and flee."

"Do not even think of doing what ought not to be done."

"Choose rather to be strong in soul than in body."

"It is difficult to walk at one and the same time many paths of life."

"It is requisite to choose the most excellent life; for custom will make it pleasant. Wealth is an infirm anchor, glory is still more infirm; and in a similar manner, the body, dominion, and honour. For all these are imbecile and powerless. What then are powerful anchors. Prudence, magnanimity, fortitude. These no tempest can shake. This is the Law of God, that virtue is the only thing that is strong; and that every thing else is a trifle."

"It is requisite to defend those who are unjustly accused of having acted injuriously, but to praise those who excel in a certain good."

"Neither will the horse be adjudged to be generous, that is sumptuously adorned, but the horse whose nature is illustrious; nor is the man worthy who possesses great wealth, but he whose soul is generous."

"When the wise man opens his mouth, the beauties of his soul present themselves to the view, like the statues in a temple"

"Remind yourself that all men assert that wisdom is the greatest good, but that there are few who strenuously seek out that greatest good."

"Despise all those things which when liberated from the body you will not want; invoke the Gods to become your helpers."

"Wind indeed increases fire, but custom love."

"Those alone are dear to Divinity who are hostile to injustice."

"None can be free who is a slave to, and ruled by, his passions."

"It is not proper either to have a blunt sword or to use freedom of speech ineffectually. Neither is the sun to be taken from the world, nor freedom of speech from erudition."

"Not frequently man from man."

"When a reasonable Soul forsaketh his divine nature, and becometh beast-like, it dieth. For though the substance of the Soul be incorruptible: yet, lacking the use of Reason, it is reputed dead; for it loseth the Intellective Life."

"A good Soul hath neither too great joy, nor too great sorrow: for it rejoiceth in goodness; and it sorroweth in wickedness. By the means whereof, when it beholdeth all things, and seeth the good and bad so mingled together, it can neither rejoice greatly; nor be grieved with over much sorrow."

"Order thyself so, that thy Soul may always be in good estate; whatsoever become of thy body."

"Dispose thy Soul to all good and necessary things!"

"Patience cometh by the grace of the Soul."

"True and perfect Friendship is, to make one heart and mind of many hearts and bodies."

"He is not rich, that enjoyeth not his own goods."

"By Silence, the discretion of a man is known: and a fool, keeping Silence, seemeth to be wise."

"A fool is known by his Speech; and a wise man by Silence."

"The King that followeth Truth, and ruleth according to Justice, shall reign quietly: but he that doth the contrary, seeketh another to reign for him."

"Tell not abroad what thou intendest to do; for if thou speed not, thou shalt be mocked!"

"If thy fellows hurt thee in small things, suffer it! and be as bold with them!"

"Take not thine enemy for thy friend; nor thy friend for thine enemy!"

"Rejoice not in another man's misfortune!"

"Let thy mind rule thy tongue!"

"Hear gladly!"

"Attempt nothing above thy strength!"

"Be not hasty to speak; nor slow to hear!"

"Wish not the thing, which thou mayest not obtain!"

"If thou intend to do any good; tarry not till to-morrow! for thou knowest not what may chance thee this night."

"Use examples; that such as thou teachest may understand thee the better!"

"Reason not with him, that will deny the principal truths!"

"Honor Wisdom; and deny it not to them that would learn; and shew it unto them that dispraise it! Sow not the sea fields!"

"Wisdom thoroughly learned, will never be forgotten. Science is got by diligence; but Discretion and Wisdom cometh of GOD."

"Without Justice, no realm may prosper."

"Happy is that City that hath a wise man to govern it."

"To use Virtue is perfect blessedness."

"Envy has been, is, and shall be, the destruction of many. What is there, that Envy hath not defamed, or Malice left undefiled? Truly, no good thing."

"A solitary man is a God, or a beast."

"None but a Craftsman can judge of a craft."

"Repentance deserveth Pardon."

"The best and greatest winning is a true friend; and the greatest loss is the loss of time."

"It is better to suffer, than to do, wrong."

"He is worst of all, that is malicious against his friends."

"Evil destroyeth itself."

"Better be mute, than dispute with the Ignorant."

"Virtue is harmony."

"The oldest, shortest words— "yes" and "no"— are those which require the most thought."

"There is nothing so easy but that it becomes difficult when you do it reluctantly."

"Concern should drive us into action and not into a depression."

"In this theater of man's life it is reserved only for God and angels to be lookers on."

"Ten is the very nature of number. All Greeks and all barbarians alike count up to ten, and having reached ten revert again to the unity. And again, Pythagoras maintains, the power of the number 10 lies in the number 4, the tetrad. This is the reason: if one starts at the unit (1) and adds the successive number up to 4, one will make up the number 10 (1+2+3+4 = 10). And if one exceeds the tetrad, one will exceed 10 too.... So that the number by the unit resides in the number 10, but potentially in the number 4. And so the Pythagoreans used to invoke the Tetrad as their most binding oath: "By him that gave to our generation the Tetractys, which contains the fount and root of eternal nature...""

"Pythagoras... assumed as first principles the numbers and symmetries existing among them, which he calls harmonies, and the elements compounded of both, that are called geometrical. ...he says that the nature of Number is the Decad."

"Let us not link ourselves with the vilifiers of Plato and the persecutors of Confucius. They were oppressed by citizens who were considered the pride of the country. Thus has the world raised its hand against the great Servitors. Be assured that the Brotherhood formed by Pythagoras appeared dangerous in the eyes of the city guard. Paracelsus was a target for mockery and malignance. Thomas Vaughan seemed to be an outcast, and few wished to meet with him. Thus was the reign of darkness manifested."

"If one examines the reasons for the persecution of the best minds of different nations, and compares the reasons for the persecution and banishment of Pythagoras, Anaxagoras, Socrates, Plato, and others, one can observe that in each case the accusations and reasons for banishment were almost identical and unfounded. But in the following centuries full exoneration came, as if there had never been any defamation. It would be correct to conclude that such workers were too exalted for the consciousness of their contemporaries, and the sword of the executioner was ever ready to cut off a head held high. Pericles was recognized in his time only after people had reduced him to a sorry state. Only in that state could his fellow citizens accept him as an equal! A book should be written about the causes of the persecution of great individuals. By comparing the causes is it possible to trace the evil will."

"Pythagoras, it seems, did not only call the supreme Deity a monad, but also a tetrad, or tetractys... It is, in the golden verses, said to be the fountain of the eternal nature; and by Hierocles, the maker of all things, the intelligent god, the cause of the heavenly and sensible god, that is, of the animated world or heaven. The later Pythogoreans endeavour to give reasons why God should be called Tetractys, from certain mysteries in the number four; but... much more probable... this name was really nothing else but the tetragrammaton, or that proper name of the supreme God amongst the Hebrews, consisting of four letters; nor is it strange Pythagoras should be so well acquainted with the name Jehovah, since, besides travelling into other parts of the East, he is by Josephus, Porphyry, and others, to have conversed with the Hebrews also."

"These thinkers seem to consider that number is the principle both as matter for things and as constituting their attributes and permanent states."

"They thought they found in numbers, more than in fire, earth, or water, many resemblances to things which are and become; thus such and such an attribute of numbers is justice, another is soul and mind, another is opportunity, and so on; and again they saw in numbers the attributes and ratios of the musical scales. Since, then, all other things seemed in their whole nature to be assimilated to numbers, while numbers seemed to be the first things in the whole of nature, they supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number."

"Whenever he heard a person who was making use of his symbols, he immediately took him into his circle, and made him a friend."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"But it is, however, in the teachings of Pythagoras that we find the closest and most frequent identities of teachings and argumentation [about as to the nature and position of the Gods], explained as due to Pythagoras himself having visited India and learned his philosophy there, as tradition asserts. In later centuries we find some peculiarly Sânkhyan and Buddhist ideas playing a prominent part in Gnostic thought."

"The school of the Pythagoras and those of the Neo-Platonists kept up the tradition for Greece, and we know that Pythagoras gained some of his learning in India, while Plato studied, and was initiated in the schools of Egypt. More precise information has been published of the Grecian schools than of others; the Pythagorean had pledged disciples as well as an outer discipline, the inner circle passing through three degrees during five years of probation."

"It is further interesting to remark that the finest characters among women with which ancient Greece presents us were formed in the school of Pythagoras, and the same is true of the men."

"The world is always ungrateful to its great men. Florence has built a statue to Galileo, but hardly even mentions Pythagoras. The former had a ready guide in the treatises of Copernicus, who had been obliged to contend against the universally established Ptolemaic system. But neither Galileo nor modern astronomy discovered the emplacement of the planetary bodies. Thousands of ages before, it was taught by the sages of Middle Asia, and brought thence by Pythagoras, not as a speculation, but as a demonstrated science. "The numerals of Pythagoras," says Porphyry, "were hieroglyphical symbols, by means whereof he explained all ideas concerning the nature of all things." Verily, then, to antiquity alone have we to look for the origin of all things. p. 31"

"Pythagoras called his Gnosis “the knowledge of things that are,” or ἡ γνῶσις τῶν ὄντων, and preserved that knowledge for his pledged disciples only: for those who could digest such mental food and feel satisfied; and he pledged them to silence and secrecy. Occult alphabets and secret ciphers are the development of the old Egyptian hieratic writings, the secret of which was, in the days of old, in the possession only of the Hierogrammatists, or initiated Egyptian priests."

"Pythagoras repeats our archaic doctrine when stating that the Ego (Nous) is eternal with Deity; that the soul only passed through various stages to arrive at divine excellence; while thumos returned to the earth, and even the phren, the lower Manas, was eliminated."

"I have often admired the mystical way of Pythagoras, and the secret Magic of numbers."

"Pythagoras was said to have been the first man to call himself philosopher; in fact, the world is indebted to him for the word philosopher. Before that time the wise men called themselves sages, which was interpreted to mean those who know. Pythagoras was more modest. He coined the word philosopher, which he defined as one who is attempting to find out."

"There remains no firm basis for the belief that Pythagoras was a geometer and in any case no attestation of his having written anything."

"The apparently ancient reports of the importance of Pythagoras and his pupils in laying the foundations of mathematics crumble on touch, and what we can get hold of is not authentic testimony by the efforts latecomers to paper over a crack, which they obviously found surprising, by the use of various kinds of reconstruction and reinterpretation. On the other hand, there are ancient and unassailable indications of a Greek mathematics antedating Pythagoras and quite outside his sphere of influence."

"I wished to show that Pythagoras, the first founder of the vegetable regimen, was at once a very great physicist and a very great physician; that there has been no one of a more cultured and discriminating humanity; that he was a man of wisdom and of experience; that his motive in commending and introducing the new mode of living was derived not from any extravagant superstition, but from the desire to improve the health and the manners of men."

"Koyré's exaltation of the "Platonic and Pythagorean" elements of the Scientific Revolution... was based on a demonstrably false understanding of how Galileo reached his conclusions. Koyré asserted that Galileo merely used experiments as a check on the theories he devised by mathematical reasoning. But later research has definitively established that Galileo's experiments preceded his attempts to give a mathematical account of their results."

"Pythagoras could not have been the discoverer of the relation, because... this property was known and used by scholars and artisans of Oriental lands thousands of years before Pythagoras... While deductive geometry is barely more than twenty-five hundred years old, empirical geometry is probably as old as civilization itself."

"Pythagoras did not possess a proof of the theorem which bears his name... he was temperamentally uninterested in proofs of this nature, as may be gleaned from... his numerological deductions. ...the Pythagorean theorem was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles. On the other hand, there is no doubt that Pythagoras fully appreciated the metaphysical implications. ...this relation ...was to Pythagoras and the Pythagoreans a basic law of nature, and... a brilliant confirmation of their number philosophy."

"Let us suppose that we have set the problem of finding a solution to the equation \scriptstyle x^2=2. \, This is a problem for which the Babylonians around 1700 BC found the excellent approximation \scriptstyle \sqrt{2}. ...This is the identical problem which Pythagoras asserted had no fractional solution and in whose honor he was supposed to have sacrificed a hecatomb of oxen—the problem which caused the existentialist crisis in ancient Greek mathematics. The \scriptstyle \sqrt{2} exists (as the diagonal of the unit square); yet it does not exist (as a fraction)!"

"A foolish consistency is the hobgoblin of little minds, adored by little statesmen and philosophers and divines. With consistency a great soul has simply nothing to do. He may as well concern himself with his shadow on the wall. Speak what you think now in hard words, and to-morrow speak what to-morrow thinks in hard words again, though it contradict every thing you said to-day. — "Ah, so you shall be sure to be misunderstood." — Is it so bad, then, to be misunderstood? Pythagoras was misunderstood, and Socrates, and Jesus, and Luther, and Copernicus, and Galileo, and Newton, and every pure and wise spirit that ever took flesh. To be great is to be misunderstood."

"But there was among them a man of prodigious knowledge who acquired the profoundest wealth of understanding and was the greatest master of skilled arts of every kind; for, whenever he willed with his whole heart, he could with ease discern each and every truth in his ten—nay, twenty—men's lives."

"Pythagoras is the founder of European culture in the Western Mediterranean sphere."

"The games which can be built up from the simple idea of dots and lines... can be a productive source of teaching material. After all, s provided the Pythagoreans and neo-Pythagoreans with important theorems about the summing of series."

"Nor need you question but that Pythagoras a long time be­fore he found the demonstration for which he offered the Heca­tomb, had been certain, that the square of the side subtending the right angle in a rectangle triangle, was equal to the square of the other two sides: and the certainty of the conclusion condu­ced not a little to the investigating of the demonstration, un­derstanding me alwayes to mean in demonstrative Sciences."

"Some fundamental unity was surely to be discerned either in the matter or the structure of things. The Ionic philosophers chose the former field: Pythagoras took the latter. ...The geometry which he had learnt in Egypt was merely practical. ...It was natural to nascent philosophy to draw, by false analogies, and the use of a brief and deceptive vocabulary,2 enormous conclusions from a very few observed facts: and it is not surprising if Pythagoras, having learnt in Egypt that number was essential to the exact description of forms and of the relations of forms, concluded that number was the cause of form and so of every other quality. Number, he inferred, is quantity and quantity is form and form is quality. Footnote2 Primitive men, on seeing a new thing, look out especially for some resemblance in it to a known thing, so that they may call both by the same name. This developes a habit of pressing small and partial analogies. It also causes many meanings to be at attached to the same word. Hasty and confused theories are the inevitable result."

"Pythagoras was indeed the first man to call himself a philosopher. Others before had called themselves wise (sophos), but Pythagoras was the first to call himself a philosopher, literally a lover of wisdom. More importantly, for Pythagoras and his followers philosophy was not merely an intellectual pursuit, but a way of life, the aim of which was the assimilation to God."

"Pythagoras, the son of Mnesarchus, was the most learned of all men of history; and having selected from these writings, he thus formed his own wisdom and extensive learning, and mischievous art."

"Much learning does not teach wisdom; otherwise it would have taught Hesiod and Pythagoras, and again Xenophanes and Hecataeus."

"No one will deny that the soul of Pythagoras was sent to mankind from Apollo's domain, having either been one of his attendants, or more intimate associates, which may be inferred both from his birth, and his versatile wisdom."

"After his father's death, though he was still but a youth, his aspect was so venerable, and his habits so temperate that he was honored and even reverenced by elderly men, attracting the attention of all who saw and heard him speak, creating the most profound impression. That is the reason that many plausibly asserted that he was a child of the divinity. Enjoying the privilege of such a renown, of an education so thorough from infancy, and of so impressive a natural appearance he showed that he deserved all these advantages by deserving them, by the adornment of piety and discipline, by exquisite habits, by firmness of soul, and by a body duly subjected to the mandates of reason. An inimitable quiet and serenity marked all his words and actions, soaring above all laughter, emulation, contention, or any other irregularity or eccentricity; his influence at Samos was that of some beneficent divinity. His great renown, while yet a youth, reached not only men as illustrious for their wisdom as Thales at Miletus, and Bias at Prione, but also extended to the neighboring cities. He was celebrated everywhere as the "long-haired Samian," and by the multitude was given credit for being under divine inspiration."

"Pythagoras conceived that the first attention that should be given to men should be addressed to the senses, as when one perceives beautiful figures and forms, or hears beautiful rhythms and melodies. Consequently he laid down that the first erudition was that which subsists through music's melodies and rhythms, and from these he obtained remedies of human manners and passions, and restored the pristine harmony of the faculties of the soul."

"Pythagoras taught, accordingly, that he had himself been originally Euphorbus, and then Callides, thirdly Hermotimus, fourthly Pyrrhus, and lastly Pythagoras; and that those things which had existed, after certain revolutions of time, came into being again; so that nothing in the world should be thought of as new. He said that true philosophy was a meditation on death; that its daily struggle was to draw forth the soul from the prison of the body into liberty: that our learning was recollection, and many other things which Plato works out in his dialogues, especially in the Phaedo and Timæus. For Plato, after having formed the Academy and gained innumerable disciples, felt that his philosophy was deficient on many points, and therefore went to Magna Græcia, and there learned the doctrines of Pythagoras from Archytas of Tarentum and Timæus of Locris: and this system he embodied in the elegant form and style which he had learned from Socrates. The whole of this, as we can prove, Origen carried over into his book Περὶ ᾿Αρχῶν, only changing the name."

"Nicomachus concludes his first book with a theorem that indicates that mathematics was not yet free from ethical and æsthetic mixture. From Pythagoras onward two ideas were widespread in Greek, especially Platonic, philosophy. These are that the beautiful and the definite are prior to the ugly and the indefinite, and that from them are formed all the parts and classes of the infinite and indefinite. Nicomachus aims to show that in mathematics the same principle holds good in that from equality may be derived all the species of inequality."

"The Ionians were optimistic, heathenly materialists... Every philosopher of the period seems to have had his own theory regarding the nature of the universe around him. ...The sixth century scene evokes the image of an orchestra expectantly tuning up, each player absorbed in his own instrument only, deaf to the caterwaulings of the others. Then there is a dramatic silence, the conductor enters the stage, raps three times with his baton, and harmony emerges from the chaos. The maestro is Pythagoras of Samos, whose influence on the ideas, and thereby on the destiny, of the human race was probably greater than that of any single man before or after him."

"It is impossible to decide whether a particular detail of the Pythagorean universe was the work of the master, or filled in by a pupil—a remark which equally applies to Leonardo or Michelangelo. But there can be no doubt that the basic features were conceived by a single mind; that Pythagoras of Samos was both the founder of a new religious philosophy, and the founder of Science, as the word is understood today."

"It may perhaps help us to realize the human side of our Masters if we remember that many of Them in comparatively recent times have been known as historical characters. The Master K.H. for example, appeared in Europe as the philosopher Pythagoras. Before that He was the Egyptian priest Sarthon, and on yet another occasion chief-priest of a temple at Agade, in Asia Minor, where He was killed in a general massacre of the inhabitants by a host of invading barbarians who swooped down upon them from the hills"

"Ah, Pythagoras' metempsychosis, were that true, This soul should fly from me, and I be changed Unto some brutish beast! All beasts are happy, for when they die, Their souls are soon dissolved in elements; But mine must live still to be plagued in hell."

"Pythagoras and Plato and Boehme and Paracelsus and Thomas Vaughan were men who bore their lamps amidst their fellowmen in life under a hail of nonunderstanding and abuse. Anyone could approach them, but only a few were able to discern the superearthly radiance behind the earthly face. It is possible to name great Servitors of East and West, North and South. It is possible to peruse their biographies; yet everywhere we feel that the superearthly radiance appears rarely in the course of centuries. One should learn from reality. Let us not link ourselves with the vilifiers of Plato and the persecutors of Confucius. They were oppressed by citizens who were considered the pride of the country. Thus has the world raised its hand against the great Servitors. Be assured that the Brotherhood formed by Pythagoras appeared dangerous in the eyes of the city guard. (175)"

"What appeared here, at the center of the Pythagorean tradition in philosophy, is another view of psyche that seems to owe little or nothing to the pan-vitalism or pan-deism (see theion) that is the legacy of the Milesians."

"The votaries of Pythagoras of Samos have this story to tell of him, that he was not an Ionian at all, but that, once on a time in Troy, he had been Euphorbus, and that he had come to life after death, but had died as the songs of Homer relate. And they say that he declined to wear apparel made from dead animal products and, to guard his purity, abstained from all flesh diet, and from the offering of animals in sacrifice. For that he would not stain the altars with blood; nay, rather the honey-cake and frankincense and the hymn of praise, these they say were the offerings made to the Gods by this man, who realized that they welcome such tribute more than they do the hecatombs note and the knife laid upon the sacrificial basket. For they say that he had of a certainty social intercourse with the gods, and learnt from them the conditions under which they take pleasure in men or are disgusted, and on this intercourse he based his account of nature."

"It was through philosophy, he said, that he had come to be surprised at nothing."

"The following became universally known: first, that he maintains that the soul is immortal; second, that it changes into other kinds of living things; third, that events recur in certain cycles and that nothing is ever absolutely new; and fourth, that all living things should be regarded as akin. Pythagoras seems to have been the first to bring these beliefs into Greece."

"He ordained that his disciples should speak well and think reverently of the Gods, muses and heroes, and likewise of parents and benefactors; that they should obey the laws; that they should not relegate the worship of the Gods to a secondary position, performing it eagerly, even at home; that to the celestial divinities they should sacrifice uncommon offerings; and ordinary ones to the inferior deities. (The world he Divided into) opposite powers; the "one" was a better monad, light, right, equal, stable and straight; while the "other" was an inferior duad, darkness, left, unequal, unstable and movable."

"Such things taught he, though advising above all things to speak the truth, for this alone deifies men. For as he had learned from the Magi, who call God Oremasdes, God's body is light, and his soul is truth. He taught much else, which he claimed to have learned from Aristoclea at Delphi."

"According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles. ...The traditionary account, that Pythagoras was the founder of scientific mathematics, is in some degree, supported by the statement of Diogenes Laertius, that he was chiefly occupied with the consideration of the properties of number, weight, and extension, besides music and astronomy. The passage of Cicero (De Nat. Deor. III. 36) may be referred to as evidence that later writers were unable to give any precise account of the mathematical discoveries of Pythagoras. To Pythagoras, however, is attributed the discovery of some of the most important elementary properties contained in the first book of Euclid's Elements. The very important truth contained in Prop. 47, Book I. is also ascribed to Pythagoras. ...Proclus attributes to him the discovery of that right-angled triangle, the three sides of which are respectively 3, 4, and 5 units. To Pythagoras also belongs the discovery, that there are only three kinds of regular polygons which can be placed so as to fill up the space round a point; namely, six equilateral triangles, four squares, and three regular hexagons. Proclus attributes to him the doctrine of incommensurables, and the discovery of the five regular solids, which, if not due to Pythagoras, originated in his school. In Astronomy he is reputed to have held, that the Sun is the centre of the system, and that the planets revolve round it. This has been called, from his name, the Pythagorean System, which was revived by Copernicus, A.D.1541, and proved by Newton."

"Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrationals [or 'proportions'] and the construction of the cosmic figures."

"Pythagoras, as everyone knows, said that "all things are numbers." This statement, interpreted in a modern way, is logical nonsense, but what he meant was not exactly nonsense. He discovered the importance of numbers in music and the connection which he established between music and arithmetic survives in the mathematical terms "harmonic mean" and "harmonic progression." He thought of numbers as shapes, as they appear on dice or playing cards. We still speak of squares or cubes of numbers, which are terms that we owe to him. He also spoke of oblong numbers, triangular numbers, pyramidal numbers, and so on. These were the numbers of pebbles [or calculi] (or as we would more naturally say, shot) required to make the shapes in question."

"Personal religion is derived from ecstasy, theology from mathematics, and both are to be found in Pythagoras."

"The combination of mathematics and theology, which began with Pythagoras, characterized religious philosophy in Greece, in the Middle Ages, and in modern times down to Kant. Orphism before Pythagoras was analogous to Asiatic mystery religions. But in Plato, Saint Augustine, Thomas Aquinas, Descartes, Spinoza, and Kant there is an intimate blending of religion and reasoning, of moral aspiration with logical admiration of what is timeless, which comes from Pythagoras, and distinguishes the intellectualized theology of Europe from the more straightforward mysticism of Asia. It is only in quite recent times that it has been possible to say clearly that Pythagoras was wrong. I do not know of any other man who has been as influential as he was in the sphere of thought. I say this because what appears as Platonism is, when analyzed, found to be in essence Pythagoreanism. The whole conception of an eternal world, revealed to the intellect but not to the senses, is derived from him. But for him, Christians would not have thought of Christ as the Word; but for him, theologians would not have sought logical proofs of God and immortality."

"Inasmuch as I have begun to explain to you how much greater was my impulse to approach philosophy in my youth than to continue it in my old age, I shall not be ashamed to tell you what ardent zeal Pythagoras inspired in me. Sotion used to tell me why Pythagoras abstained from animal food, and why, in later times, Sextius did also. In each case, the reason was different, but it was in each case a noble reason. … Pythagoras … held that all beings were inter-related, and that there was a system of exchange between souls which transmigrated from one bodily shape into another. If one may believe him, no soul perishes or ceases from its functions at all, except for a tiny interval – when it is being poured from one body into another."

"What seems certain is that Pythagoras developed the idea of mathematical logic... He realized that numbers exist independently of the tangible world and therefore their study was untainted by inaccuracies of perception. This meant he could discover truths which were independent of opinion of prejudice and which were more absolute then any previous knowledge."

"Number, its kinds; the first kind, intellectual in the divine mind. Number is of two kinds, the Intellectual (or immateriall) and the Scientiall. The intellectuall is that eternal substance of number, which Pythagoras in his discourse concerning the Gods asserted to be the principle most providentiall of all Heaven and Earth, and the nature that is betwixt them. Moreover, it is the root of divine Beings, and of gods, & of Dæmons. This is that which he termed the principle, fountain,and root of all things, and defined it to be that which before all things exists in the divine mind; from which and out of which all things are digested into order, and remain numbred by an indissolube series. For all things which are ordered in the world by nature according to an artificiall course in part and in whole appear to be distinguished and adorn'd by Providence and the All-creating Mind, according to Number; the exemplar being established by applying (as the reason of the principle before the impression of things) the number præxistent in the Intellect of God, maker of the world. This only in intellectual, & wholly immaterial, really a substance according to which as being the most exact artificiall reason, all things are perfected, Time, Heaven, Motion, the Stars and their various revolutions. ...The other kind of number, Scientiall; its principles. Scientiall Number is that which Pythagoras defines the extension and production into act of the seminall reasons which are in the Monad, or a heap of Monads, or a progressian of multitude beginning from Monad, and a regression ending in Monad."

"In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world.... It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion."

"Pythagoras was a teacher of the purest system of morals ever propounded to man."

"Pythagoras was a man; and with all his imperfections on his head, we shall look among the race of men, for his better, in yain, yea, for his equal, or his second, but in vain. Pythagoras was entirely a Deist, a steady maintainer of the unity of God, and of the eternal obligations of moral virtue. No Christian writings, even to this day, can compete in sublimity and grandeur with what this illustrious philosopher has laid down concerning God, and the end of all our actions; and it is likely, says Bayle, that he would have carried his orthodoxy much farther, had he had the courage to expose himself to martyrdom."

"It was Pythagoras who first called heaven kosmos, because it is perfect, and "adorned" with infinite beauty and living beings."

"It is very important to note that some 2,500 years ago at the least Pythagoras went from Samos to the Ganges to learn geometry…But he would certainly not have undertaken such a strange journey had the reputation of the Brahmins' science not been long established in Europe…"

"From of old, amid the rage of robbery and blood-lust, it came to wise men's consciousness that the human race was suffering from a malady which necessarily kept it in progressive deterioration. Many a hint from observation of the natural man, as also dim half-legendary memories, had made them guess the primal nature of this man, and that his present state is therefore a degeneration. A mystery enwrapped Pythagoras, the preacher of vegetarianism; no philosopher since him has pondered on the essence of the world, without recurring to his teaching. Silent fellowships were founded, remote from turmoil of the world, to carry out this doctrine as a sanctification from sin and misery."

"It was a maxim of Pythagoras that the two most excellent things for man were to speak the truth, and to render benefits to each other."

"The association between religion and mathematically based science has its origins in the mists of history. ...the very dawn of Western culture in sixth-century B.C. Greece. ...[W]hen the Greeks were turning away from the mythological picture immortalized by Homer and Hesiod, the Ionian philosopher Pythagoras of Samos pioneered a worldview in which mathematics was seen as the key to reality. In place of the mythological gods, Pythagoras painted a picture in which the universe was conceived as a great musical instrument resonating with divine mathematical harmonies. ...[inspiring] mystics, theologians, and physicists ever since. ...But to Pythagoras and his followers, mathematics was the key not simply to the physical world, but more importantly to the spiritual world—for they believed that numbers were literally gods. By contemplating numbers and their relationships, the Pythagoreans sought union with the "divine." For them, mathematics was first and foremost a religious activity."

"Around 600 BCE, Pythagoras observed that the tones of a lyre sound most harmonious when the ratio of string lengths forms a simple whole-number fraction. Inspired by such hints, Pythagoras and his followers made a remarkable intuitive leap. They foresaw the possibility of a different kind of world-model, less dependent on the accident of our senses but more in tune with Nature's hidden harmonies, and ultimately more faithful to reality. This is the meaning of... "All things are number.""

"Pythagoras was a man of the most various accomplishments, and appears to have penetrated in different directions into the depths of human knowledge. He sought wisdom in its retreats of fairest promise, in Egypt and other distant countries. In this investigation he employed the earlier period of his life, probably till he was forty, and devoted the remainder to such modes of proceeding, as appeared to him the most likely to secure the advantage of what he had acquired to a late posterity. He founded a school, and delivered his acquisitions by oral communication to a numerous body of followers. He divided his pupils into two classes, the one neophytes, to whom was explained only the most obvious and general truths, the other who were admitted into the entire confidence of the master. These last he caused to throw their property into a common stock, and to live together in the same place of resort. He appears to have spent the latter half of his life in that part of Italy, called Magna Graecia, so denominated in some degree from the numerous colonies of Grecians by whom it was planted, and partly perhaps from the memory of the illustrious things which Pythagoras achieved there."

"But this marvellous man in some way, whether from the knowlege he received, or from his own proper discoveries, has secured to his species benefits of a more permanent nature, and which shall outlive the revolutions of ages, and the instability of political institutions. He was a profound geometrician. The two theorems, that the internal angles of every right-line triangle are equal to two right angles, and that the square of the hypothenuse of every right angled triangle is equal to the sum of the squares of the other two sides, are ascribed to him. In memory of the latter of these discoveries he is said to have offered a public sacrifice to the Gods; and the theorem is still known by the name of the Pythagorean theorem."

"To inculcate a pure and a simple mode of subsistence was also an express object of pursuit to Pythagoras. He taught a total abstinence from every thing having had the property of animal life. … He taught temperance in all its branches, and a resolute subjection of the appetites of the body to contemplation and the exercises of the mind; and, by the unremitted discipline and authority he exerted over his followers, he caused his lessons to be constantly observed. There was therefore an edifying and an exemplary simplicity that prevailed as far as the influence of Pythagoras extended, that won golden opinions to his adherents at all times that they appeared, and in all places."

"One revolution that Pythagoras worked, was that, whereas, immediately before, those who were most conspicuous among the Greeks as instructors of mankind in understanding and virtue, styled themselves sophists, professors of wisdom, this illustrious man desired to be known only by the appellation of a philosopher, a lover of wisdom. The sophists had previously brought their denomination into discredit and reproach, by the arrogance of their pretensions, and the imperious way in which they attempted to lay down the law to the world. The modesty of this appellation however did not altogether suit with the deep designs of Pythagoras, the ascendancy he resolved to acquire, and the oracular subjection in which he deemed it necessary to hold those who placed themselves under his instruction. This wonderful man set out with making himself a model of the passive and unscrupulous docility which he afterwards required from others. He did not begin to teach till he was forty years of age, and from eighteen to that period he studied in foreign countries, with the resolution to submit to all his teachers enjoined, and to make himself master of their least communicated and most secret wisdom.<!-- In Egypt in particular, we are told that, though he brought a letter of recommendation from Polycrates, his native sovereign, to Amasis, king of that country, who fully concurred with the views of the writer, the priests, jealous of admitting a foreigner into their secrets, baffled him as long as they could, referring him from one college to another, and prescribing to him the most rigorous preparatives, not excluding the rite of circumcision. But Pythagoras endured and underwent every thing, till at length their unwillingness was conquered, and his perseverance received its suitable reward."

"When in the end Pythagoras thought himself fully qualified for the task he had all along had in view, he was no less strict in prescribing ample preliminaries to his own scholars. At the time that a pupil was proposed to him, the master, we are told, examined him with multiplied questions as to his principles, his habits and intentions, observed minutely his voice and manner of speaking, his walk and his gestures, the lines of his countenance, and the expression and management of his eye, and, when he was satisfied with these, then and not till then admitted him as a probationer. It is to be supposed that all this must have been personal. As soon however as this was over, the master was withdrawn from the sight of the pupil; and a noviciate of three and five, in all eight years, was prescribed to the scholar, during which time he was only to hear his instructor from behind a curtain, and the strictest silence was enjoined him through the whole period. As the instructions Pythagoras received in Egypt and the East admitted of no dispute, so in his turn he required an unreserved submission from those who heard him: autos iphae "the master has said it," was deemed a sufficient solution to all doubt and uncertainty."

"To give the greater authority and effect to his communications Pythagoras hid himself during the day at least from the great body of his pupils, and was only seen by them at night. Indeed there is no reason to suppose that any one was admitted into his entire familiarity. When he came forth, he appeared in a long garment of the purest white, with a flowing beard, and a garland upon his head. He is said to have been of the finest symmetrical form, with a majestic carriage, and a grave and awful countenance. He suffered his followers to believe that he was one of the Gods, the Hyperborean Apollo, and is said to have told Abaris that he assumed the human form, that he might the better invite men to an easiness of approach and to confidence in him. -->"

"What however seems to be agreed in by all his biographers, is that he professed to have already in different ages appeared in the likeness of man: first as Aethalides, the son of Mercury; and, when his father expressed himself ready to invest him with any gift short of immortality, he prayed that, as the human soul is destined successively to dwell in various forms, he might have the privilege in each to remember his former state of being, which was granted him. From Aethalides he became Euphorbus, who slew Patroclus at the siege of Troy. He then appeared as Hermotimus, then Pyrrhus, a fisherman of Delos, and finally Pythagoras. He said that a period of time was interposed between each transmigration, during which he visited the seat of departed souls; and he professed to relate a part of the wonders he had seen. He is said to have eaten sparingly and in secret, and in all respects to have given himself out for a being not subject to the ordinary laws of nature. Pythagoras therefore pretended to miraculous endowments. Happening to be on the sea-shore when certain fishermen drew to land an enormous multitude of fishes, he desired them to allow him to dispose of the capture, which they consented to, provided he would name the precise number they had caught. He did so, and required that they should throw their prize into the sea again, at the same time paying them the value of the fish."

"The close of the life of Pythagoras was, according to every statement, in the midst of misfortune and violence. Some particulars are related by Iamblichus, which, though he is not an authority beyond all exception, are so characteristic as seem to entitle them to the being transcribed. … Cylon, the richest man, or, as he is in one place styled, the prince, of Crotona, had manifested the greatest partiality to Pythagoras. He was at the same time a man of rude, impatient and boisterous character. He, together with Perialus of Thurium, submitted to all the severities of the Pythagorean school. They passed the three years of probation, and the five years of silence. They were received into the familiarity of the master. They were then initiated, and delivered all their wealth into the common stock. They were however ultimately pronounced deficient in intellectual power, or for some other reason were not judged worthy to continue among the confidential pupils of Pythagoras. They were expelled. The double of the property they had contributed was paid back to them. A monument was set up in memory of what they had been; and they were pronounced dead to the school. … Cylon, from feelings of the deepest reverence and awe for Pythagoras, which he had cherished for years, was filled even to bursting with inextinguishable hatred and revenge. The unparalleled merits, the venerable age of the master whom he had so long followed, had no power to control his violence. His paramount influence in the city insured him the command of a great body of followers. He excited them to a frame of turbulence and riot. He represented to them how intolerable was the despotism of this pretended philosopher. They surrounded the school in which the pupils were accustomed to assemble, and set it on fire. Forty persons perished in the flames. According to some accounts Pythagoras was absent at the time. According to others he and two of his pupils escaped. He retired from Crotona to Metapontum. But the hostility which had broken out in the former city, followed him there. He took refuge in the Temple of the Muses. But he was held so closely besieged that no provisions could be conveyed to him; and he finally perished with hunger, after, according to Laertius, forty days' abstinence."

"It is difficult to imagine any thing more instructive, and more pregnant with matter for salutary reflection, than the contrast presented to us by the character and system of action of Pythagoras on the one hand, and those of the great enquirers of the last two centuries, for example, Bacon, Newton and Locke, on the other. Pythagoras probably does not yield to any one of these in the evidences of true intellectual greatness. In his school, in the followers he trained resembling himself, and in the salutary effects he produced on the institutions of the various republics of Magna Graecia and Sicily, he must be allowed greatly to have excelled them. His discoveries of various propositions in geometry, of the earth as a planet, and of the solar system as now universally recognised, clearly stamp him a genius of the highest order. Yet this man, thus enlightened and philanthropical, established his system of proceeding upon narrow and exclusive principles, and conducted it by methods of artifice, quackery and delusion. One of his leading maxims was, that the great and fundamental truths to the establishment of which he devoted himself, were studiously to be concealed from the vulgar, and only to be imparted to a select few, and after years of the severest noviciate and trial. He learned his earliest lessons of wisdom in Egypt after this method, and he conformed through life to the example which had thus been delivered to him. The severe examination that he made of the candidates previously to their being admitted into his school, and the years of silence that were then prescribed to them, testify this. He instructed them by symbols, obscure and enigmatical propositions, which they were first to exercise their ingenuity to expound. The authority and dogmatical assertions of the master were to remain unquestioned; and the pupils were to fashion themselves to obsequious and implicit submission, and were the furthest in the world from being encouraged to the independent exercise of their own understandings. There was nothing that Pythagoras was more fixed to discountenance, than the communication of the truths upon which he placed the highest value, to the uninitiated. It is not probable therefore that he wrote any thing: all was communicated orally, by such gradations, and with such discretion, as he might think fit to adopt and to exercise."

"All was oracular and dogmatic in the school of Pythagoras. He prized and justly prized the greatness of his attainments and discoveries, and had no conception that any thing could go beyond them. He did not encourage, nay, he resolutely opposed, all true independence of mind, and that undaunted spirit of enterprise which is the atmosphere in which the sublimest thoughts are most naturally generated. He therefore did not throw open the gates of science and wisdom, and invite every comer; but on the contrary narrowed the entrance, and carefully reduced the number of aspirants. He thought not of the most likely methods to give strength and permanence and an extensive sphere to the progress of the human mind. For these reasons he wrote nothing; but consigned all to the frail and uncertain custody of tradition. And distant posterity has amply avenged itself upon the narrowness of his policy; and the name of Pythagoras, which would otherwise have been ranked with the first luminaries of mankind, and consigned to everlasting gratitude, has in consequence of a few radical and fatal mistakes, been often loaded with obloquy, and the hero who bore it been indiscriminately classed among the votaries of imposture and artifice."

"It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle 'of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that “ numbers are the first principles of all entities.” Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct. (p. XV Before the Veil)"

"Aristotle was no trustworthy witness. He misrepresented Plato, and he almost caricatured the doctrines of Pythagoras. There is a canon of interpretation, which should guide us in our examinations of every philosophical opinion: The human mind has, under the necessary operation of its own laws, been compelled to entertain the same fundamental ideas, and the human heart to cherish the same feelings in all ages. It is certain that Pythagoras awakened the deepest intellectual sympathy of his age, and that his doctrines exerted a powerful influence upon the mind of Plato. His cardinal idea was that there existed a permanent principle of unity beneath the forms, changes, and other phenomena of the universe. Aristotle asserted that he taught that numbers are the first principles of all entities. Ritter has expressed the opinion that the formula of Pythagoras should be taken symbolically, which is doubtless correct."

"If the Pythagorean metempsychosis should be thoroughly explained and compared with the modern theory of evolution, it would be found to supply every "missing link" in the chain of the latter. But who of our scientists would consent to lose his precious time over the vagaries of the ancients. Notwithstanding proofs to the contrary, they not only deny that the nations of the archaic periods, but even the ancient philosophers had any positive knowledge of the Heliocentric system. The "Venerable Bedes," the Augustines and Lactantii appear to have smothered, with their dogmatic ignorance, all faith in the more ancient theologists of the pre-Christian centuries. But now philology and a closer acquaintance with Sanskrit literature have partially enabled us to vindicate them from these unmerited imputations. In the Vedas, for instance, we find positive proof that so long ago as 2000 B.C., the Hindu sages and scholars must have been acquainted with the rotundity of our globe and the Heliocentric system. Hence, Pythagoras and Plato knew well this astronomical truth; for Pythagoras obtained his knowledge in India, or from men who had been there, and Plato faithfully echoed his teachings."

"If we carefully trace the terms nazar, and nazaret, throughout the best known works of ancient writers, we will meet them in connection with “Pagan” as well as Jewish adepts. Thus, Alexander Polyhistor says of Pythagoras that he was a disciple of the Assyrian Nazarrt, whom some suppose to be Ezekiel. Diogenes Laertius states most positively that Pythagoras, after being initiated into all the Mysteries of the Greeks and barbarians, “went into Egypt and afterward visited the Chaldeans and Magi;” and Apuleius maintains that it was Zoroaster who instructed Pythagoras. (p. 140)"

"Zoroaster, Pythagoras, Epicharmus, Empedocles, Kebes, Euripides, Plato, Euclid, Philo, Boethius, Virgil, Marcus Cicero, Plotinus, Iamblichus, Proclus, Psellus, Synesius, Origen, and, finally, Aristotle himself, far 'from denying our immortality, support it most emphatically. (p. 251)"

"The laws of Manu are the doctrines of Plato, Philo, Zoroaster, Pythagoras, and of the Kabala. The esoterism of every religion may be solved by the latter. The kabalistic doctrine of the allegorical Father and Son, or Xlarrjp and Aoyos is identical with the groundwork of Buddhism. Moses could not reveal to the multitude the sublime secrets of religious speculation, nor the cosmogony of the universe ; the whole resting upon the Hindu Illusion, a clever mask veiling the Sanctum Sanctorum, and which has misled so many theological commentators. (p. 271)"

"Scholastic skeptics, as well as ignorant materialists, have greatly amused themselves for the last two centuries over the absurdities attributed to Pythagoras by his biographer, Iamblichus. The Samian philosopher is said to have persuaded a she-bear to give up eating human flesh; to have forced a white eagle to descend to him from the clouds, and to have subdued him by stroking him gently with the hand, and by talking to him. On another occasion, Pythagoras actually persuaded an ox to renounce eating beans, by merely whispering in the animal's ear! (Iamblichus: "De Vita Pythag.") Oh, ignorance and superstition of our forefathers, how ridiculous they appear in the eyes of our enlightened generations! Let us, however, analyze this absurdity. Every day we see unlettered men, proprietors of strolling menageries, taming and completely subduing the most ferocious animals, merely by the power of their irresistible will... Every one has either witnessed or heard of the seemingly magical power of some mesmerizers and psychologists. They are able to subjugate their patients for any length of time. Regazzoni, the mesmerist who excited such wonder in France and London, has achieved far more extraordinary feats than what is above attributed to Pythagoras. Why, then, accuse the ancient biographers of such men as Pythagoras and Apollonius of Tyana of either wilful misrepresentation or absurd superstition?"

"Pythagoras stands at the fountainhead of our culture. The ideas he set in motion were, according to Daniel Boorstin, "among the most potent in modern history," resulting directly in many of the pillars upon which the modern world is built. In particular, the very existence of science becomes possible only when it is realized that inner, purely subjective, mathematical forms have a resonance with the form and behavior of the external world — a Pythagorean perception. And a world at peace — that is to say, in a nuclear age, the survival of our planet — is predicated upon ideas of universal brotherhood to which Pythagoras, while not the sole author, made an enormous contribution. Even the seeming remoteness of Pythagorean teaching helps one to realize that the current world view, while it seems destined to dominate the planet, is fleeting and temporary and, like others before it, will pass."

"Pythagoras' teachings have enormous relevance in understanding both the sources of our culture and, perhaps more importantly, where it may be heading or may need to head. But to appreciate this we have to understand him in modern terms."

"At the dawn of our century, scientists were proclaiming that our understanding of the world was almost complete. Only one or two small problems in physics remained to be solved. One of these problems had to do with black body radiation and was solved by Max Planck. His solution, however, formed the foundation for quantum mechanics which was to sweep aside almost the whole edifice of fundamental assumptions in physics, and with it our understanding of the world. A hundred years later we are faced with a similar situation. The mechanistic viewpoint that began to dominate our world view in the seventeenth century has almost completed its hegemony. This paradigm, as historian Hugh Kearney points out, stems from only one of three main systems of thought that flowed from Greek thought into the modern world, each of which has dominated our world view at different points in our history. … In spite of the dominance of mechanistic thought in the contemporary world, a perplexing residue of the magical tradition still survives in the form of several issues, solutions to which do not appear possible within the context of a purely mechanical view of the world."

"It is important to recognize that the materialist, scientific paradigm that dominates the late twentieth century world and provides the basis for its dominant institutions, has its basis in the life and work of Pythagoras, one of the most significant representatives of the perennial philosophy and a founder of the magical tradition. This spirit, which gave rise to our world view, is a spirit that must be recaptured if our civilization is to flourish. The choice is a clear one to many, and was summed up in a book title by the late Pythagorean and futurist Buckminster Fuller, Utopia or Oblivion."

"The concept of a harmonious universe ordered according to "the Great Chain of Being" — a chain that connects the continuum of matter, body, mind, soul and spirit — stands as one of the most fundamental ideas of western thought. … It continues to be a profound influence upon the deepest strata of our thought. And yet a major rift has appeared in the consciousness of our time because the theme of harmonia has not been translated into the realm of human conduct. The challenge of our time may be to revive it, and make divine harmony "the great theme" of the next millennium. Any success we have in accomplishing this will be based, in large part, on the achievements of Pythagoras."

"If someone associates with a true Pythagorean, what will he will get from him, and in what quantity? I would say: statesmanship, geometry, astronomy, arithmetic, harmonics, music, medicine, complete and god-given prophecy, and also the higher rewards — greatness of mind, of soul, and of manner, steadiness, piety, knowledge of the gods and not just supposition, familiarity with blessed spirits and not just faith, friendship with both gods and spirits, self-sufficiency, persistence, frugality, reduction of essential needs, ease of perception, of movement, and of breath, good color, health, cheerfulness, and immortality."

"It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed us down a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"The Pythagorean mathematical concepts, abstracted from sense impressions of nature, were... projected into nature and considered to be the structural elements of the universe. [Pythagoreans] attempted to construct the whole heaven out of numbers, the stars being... material points. ...they identified the regular geometric solids... with the different sorts of substances in nature. ...This confusion of the abstract and the concrete, of rational conception and empirical description, which was characteristic of the whole Pythagorean school and of much later thought, will be found to bear significantly on the development of the concepts of calculus. It has often been inexactly described as mysticism, but such stigmatization appears to be somewhat unfair. Pythagorean deduction a priori having met with remarkable success in its field, an attempt (unwarranted...) was made to apply it to the description of the world of events, in which the Ionian hylozoistic interpretations a posteriori had made very little headway. This attack on the problem was highly rational and not entirely unsuccessful, even though it was an inversion of the scientific procedure, in that it made induction secondary to deduction."

"We know that he (Pythagoras) went to India to be instructed; but the capacity of the learner determines his degree of proficiency, and if Pythagoras on his return had so little knowledge in geometry as to consider the forty-seventh of Euclid as a great discovery, he certainly was entirely incapable of acquiring the Indian method of calculation, through his deficiency of preparatory knowledge… each teacher, or head of sect that drew his knowledge from Indian sources, might conceal his instructions to be reckoned an inventor."

"Ionian philosophers... had sought to identify a first principle for all things. Thales had thought to find this in water, but others preferred to think of air or fire as the basic element. The Pythagoreans had taken a more abstract direction, postulating that number... was the basic stuff behind phenomena; this numerical atomism... had come under attack by the followers of Parmenides of Elea... The fundamental tenet of the was the unity and permanence of being... contrasted with the Pythagorean ideas of multiplicity and change. Of Parmenides' disciples the best known was Zeno the Eleatic... who propounded arguments to prove the inconsistency in the concepts of multiplicity and divisibility."

"We may... go to our... statement from Aristotle's treatise on the Pythagoreans, that according to them the universe draws in from the Unlimited time and breath and the void. The cosmic nucleus starts from the unit-seed, which generates mathematically the number-series and physically the distinct forms of matter. ...it feeds on the Unlimited outside and imposes form or limit on it. Physically speaking this Unlimited is [potential or] unformed matter... mathematically it is extension not yet delimited by number or figure. ...As apeiron in the full sense, it was... duration without beginning, end, or internal division—not time, in Plutarch's words, but only the shapeless and unformed raw material of time... As soon... as it had been drawn or breathed in by the unit, or limiting principle, number is imposed on it and at once it is time in the proper sense. ...the Limit, that is the growing cosmos, breathed in... imposed form on sheer extension, and by developing the heavenly bodies to swing in regular, repetitive circular motion... it took in the raw material of time and turned it into time itself."

"Among the sages of this description, to whose useful labors the world is so much indebted, none held a more deservedly conspicuous rank than Pythagoras […] Quitting the land of his nativity […] his zeal for the acquisition of knowledge led him first to Egypt […] Having at this celebrated fountain of learning exhausted the supply without diminishing his thirst, he sought the further means of slaking it, in the then almost unexplored peninsula of India, whence he returned, bringing back with him the doctrine of Metempsychosis, the prejudices against animal diet, the mysterious notions respecting the powers of numbers, and other visionary and fanciful tenets of the East."

"It is certain that the Theory of Numbers originated in the school of Pythagoras."

"Those who dwelt in the common auditorium adopted this oath: "I swear by the discoverer of the Tetraktys, which is the spring of all our wisdom; The perennial fount and root of Nature.""

"The tetrad was called by the Pythagoreans every number, because it comprehends in itself all the numbers as far as to the decad, and the decad itself; for the sum of 1, 2, 3, and 4, is 10. Hence both the decad and the tetrad were said by them to be every number; the decad indeed in energy, but the tetrad in capacity. The sum likewise of these four numbers was said by them to constitute the tetractys, in which all harmonic ratios are included. For 4 to 1, which is a quadruple ratio, forms the symphony bisdiapason; the ratio of 3 to 2, which is sesquialter forms the symphony diapente; 4 to 3, which is sesquitertian, the symphony diatessaron; and 2 to 1, which is a duple ratio, forms the diapason."

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"The Neo-Pythagoreans treated all the divisions of philosophy. In Metaphysics they held that the Unit and the (indeterminate) Two are the basis of all things. the Unit being the form, and the Two the matter. ...The Unit being the prior principle may be identified with Deity, and, as such, was thought of either as the former [creator] of indefinite matter into individual things, or, as in Neo-Platonism, as the transcendent origin of the derivative Unit and Two. Another mode of conception was to identify the numbers with the Platonic Ideas and then to think of the Unit as comprehending them in the same manner as the mind comprehends its thoughts and gives them form. In Logic the Neo-Pythagoreans were for the most part imitators of Aristotle. Their Physics was Aristotelian and Stoic. Their Anthropology was Platonic. In Ethics and Politics they merely reechoed the Academy and the Lyceum with Stoic additions. In all this Neo-Pythagoreanism has little originality."

"Why was the Tetraktys so revered? Because to the eyes of the sixth century BC Pythagoreans, it seemed to outline the entire nature of the universe. In geometry — the springboard to the Greeks' epochal revolution in thought — the number 1 represented a point... 2 represented a line... 3 represented a surface... and 4 represented a three-dimensional tetrahedral solid... The Tetraktys, therefore appeared to encompass all the perceived dimensions of space."

"On the question whether mathematics was discovered or invented, Pythagoras and the Pythagoreans had no doubt — mathematics was real, immutable, omnipresent, and more sublime than anything that could conceivably emerge from the human mind. The Pythagoreans literally embedded the universe into mathematics. In fact, to the Pythagoreans, God was not a mathematician — mathematics was God! ...By setting the stage, and to some extent the agenda, for the next generation of philosophers — Plato in particular — the Pythagoreans established a commanding position in Western thought."

"As a moral philosopher, many of his precepts relating to the conduct of life will be found in the verses which bear the name of the Golden Verses of Pythagoras. It is probable they were composed by some one of his school, and contain the substance of his moral teaching. The speculations of the early philosophers did not end in the investigation of the properties of number and space. The Pythagoreans attempted to find, and dreamed they had found, in the forms of geometrical figures and in certain numbers, the principles of all science and knowledge, whether physical or moral. The figures of Geometry were regarded as having reference to other truths besides the mere abstract properties of space. They regarded the unit, as the point; the duad, as the line; the triad, as the surface; and the tetractys, as the geometrical volume. They assumed the pentad as the physical body with its physical qualities. They seem to have been the first who reckoned the elements to be five in number, on the supposition of their derivation from the five regular solids. They made the cube, earth; the pyramid, fire; the octohedron, air; the icosahedron, water; and the dodecahedron, aether. The analogy of the five senses and the five elements was another favourite notion of the Pythagoreans."

"If we consider the results obtained together, we will not be able to doubt the conclusion to be drawn from them. The ancient priestly geometry of the Indians not only knew the Pythagorean theorem, but it even played the main role in their calculations; with its help, they constructed elements that the Greeks found in a completely different way; with its help, they also found the irrational quantities. And it was precisely these two things that Pythagoras introduced into the Greek-Italian world; these two things, according to the Greeks, he invented. Indeed, even more! The way in which Pythagoras proved his theorem was also, in all likelihood, the same as that which we find in the Vedic Shulba Sutras. After examining the Shulba Sutras, we could have said: If Pythagoras really was in India, as we previously suggested, and initiated himself into the priestly wisdom of the Brahmins, then he could have brought precisely these theorems of geometric science to Greece; — and history has been telling us for several millennia now that this was indeed the case!"

"While most s emphasized the reality of change — in particular, the Atomists, followers of and Democritus — the Pythagoreans stressed the study of the unchangeable elements in nature and society. In their search for the eternal laws of the universe they studied geometry, arithmetic, astronomy, and music (the '). Their most outstanding leader was Archytas of Tarentum...and to whose school, if we follow... E. [Eva] Frank, much of the Pythagorean brand of mathematics may be ascribed. ...Numbers were divided into classes: odd, even, even-times-even, odd-times-odd, prime and composite, perfect, friendly, triangular, square, pentagonal, etc. ...Of particular importance was the ratio of numbers (logos, Lat. ratio). Equality of ratio formed a proportion. They discriminated between an arithmetical (2b = a + c), geometrical (b^2 = ac), and a harmonical (\frac{2}{b} = \frac{1}{a} + \frac{1}{c}) proportion that they interpreted philosophically and socially."

"The Pythagoreans knew some properties of s... how a plane can be filled by... regular triangles, squares, or regular hexagons, and space by cubes... [They] may also have known the regular oktahedron and dodekahedron—the latter figure because pyrite, found in Italy, crystallizes in dodekahedra, and models... date to Etruscan times."

"[T]he most striking result of the Greeks' faith that the world could be understood in terms of rational principles was the invention of abstract mathematics. The most grandiose ambition they conceived was to explain all the properties of Nature in arithmetical terms alone. This was the aim of the Pythagoreans... [T]hey... knew that the phenomena of the Heavens recurred in a cyclical manner; and... discovered ...that the sound of a vibrating string ...is simply related to the length ...and its 'harmonics' always go with simple fractional lengths. ...[S]ince the Pythagoreans were a religious brotherhood... they thought that this search would lead to more than explanations alone. If one discovered the mathematical harmonies in things, one should... discover how to put oneself in harmony with Nature. ...[T]hey had ...positive grounds for thinking that both astronomy and acoustics were at the bottom arithmetical; and the study of simple fractions was called 'music' right down until the late Middle Ages."

"Not one of the philosophical ideas in Part I of the commentary is peculiarly Neoplatonic. The doctrine of the Threeness of things... is found in Aristotle and goes back to the early Pythagoreans or to Homer even; paragraph 8 is mathematical in content rather than philosophical... although there is an allusion in it to the Monad as the principle of finitudes, again a very early Pythagorean doctrine; and these two paragraphs are the source of [Heinrich] Sitter's suggestion of the authorship of Proclus. As a matter of fact, the philosophical notions in Part I have been borrowed for the most part directly from Plato, with two or three exceptions that are Aristotelian... Plato's Theaetetus, Parmenides, and the Laws, are specifically mentioned. The Timaeus forms the background of much of the thought. And the Platonism of a mathematician of the turn of the third century A. D. need not surprise us, if we but recall Aristotle's accusation that the Academy tended to turn philosophy into mathematics."

"§1. The aim of Book X of Euclid's treatise on the Elements is to investigate the commensurable and incommensurable, the rational and irrational continuous quantities. This science (or knowledge) had its origin in the sect (or school) of Pythagoras, but underwent an important development at the hands of the Athenian, Theaetetus, who had a natural aptitude for this as for other branches of mathematics most worthy of admiration."

"§2. Since this treatise (i. e. Book X of Euclid.) has the aforesaid aim and object, it will not be unprofitable for us to consolidate the good which it contains. Indeed the sect (or school) of Pythagoras was so affected by its reverence for these things that a saying became current in it, namely, that he who first disclosed the knowledge of surds or irrationals and spread it abroad among the common herd, perished by drowning: which is most probably a parable by which they sought to express their conviction that firstly, it is better to conceal (or veil) every surd, or irrational, or inconceivable in the universe, and, secondly, that the soul which by error or heedlessness discovers or reveals anything of this nature which is in it or in this world, wanders [thereafter] hither and thither on the sea of nonidentity (i. e. lacking all similarity of quality or accident), immersed in the stream of the coming-to-be and the passing-away, where there is no standard of measurement. This was the consideration which Pythagoreans and the Athenian Stranger held to be an incentive to particular care and concern for these things and to imply of necessity the grossest foolishness in him who imagined these things to be of no account."

"δοκεῖ δὲ αὐτῶι τάδε· ἀρχὰς εἶναι τῶν ὅλων ἀτόμους καὶ κενόν, τὰ δ'ἀλλα πάντα νενομίσθαι [δοξάζεσθαι]. (Diogenes Laërtius, Democritus, Vol. IX, 44)"

"νόμωι (γάρ φησι) γλυκὺ καὶ νόμωι πικρόν, νόμωι θερμόν, νόμωι ψυχρόν, νόμωι χροιή, ἐτεῆι δὲ ἄτομα καὶ κενόν (Tetralogies of Thrasyllus, 9; Sext. Emp. adv. math. VII 135)"

"We know nothing accurately in reality, but [only] as it changes according to the bodily condition, and the constitution of those things that flow upon [the body] and impinge upon it."

"Medicine heals diseases of the body, wisdom frees the soul from passions."

"Coition is a slight attack of apoplexy. For man gushes forth from man, and is separated by being torn apart with a kind of blow."

"Man is a universe in little [Microcosm]."

"Good breeding in cattle depends on physical health, but in men on a well-formed character."

"Πολλοὶ πολυμαθέες νοῦν οὐκ ἔχουσιν."

"Immoderate desire is the mark of a child, not a man."

"[I would] rather discover one cause than gain the kingdom of Persia."

"Men have fashioned an image of Chance as an excuse for their own stupidity. For Chance rarely conflicts with intelligence, and most things in life can be set in order by an intelligent sharpsightedness."

"In a shared fish, there are no bones."

"Education is an ornament for the prosperous, a refuge for the unfortunate."

"Beautiful objects are wrought by study through effort, but ugly things are reaped automatically without toil."

"The animal needing something knows how much it needs, the man does not."

"Moderation multiplies pleasures, and increases pleasure."

"The brave man is not only he who overcomes the enemy, but he who is stronger than pleasures. Some men are masters of cities, but are enslaved to women."

"It is hard to fight desire; but to control it is the sign of a reasonable man."

"The laws would not prevent each man from living according to his inclination, unless individuals harmed each other; for envy creates the beginning of strife."

"To a wise man, the whole earth is open; for the native land of a good soul is the whole earth."

"The man who is fortunate in his choice of son-in-law gains a son; the man unfortunate in his choice loses his daughter also."

"If your desires are not great, a little will seem much to you; for small appetite makes poverty equivalent to wealth."

"Disease of the home and of the life comes about in the same way as that of the body."

"No power and no treasure can outweigh the extension of our knowledge."

"Strength and beauty are the blessings of youth; temperance, however, is the flower of old age."

"Man should know from this rule that he is cut off from truth."

"This argument too shows that in truth we know nothing about anything, but every man shares the generally prevailing opinion."

"And yet it will be obvious that it is difficult to really know of what sort each thing is."

"Now, that we do not really know of what sort each thing is, or is not, has often been shown."

"Verily we know nothing. Truth is buried deep. (Another translation: "Of truth we know nothing, for truth is in a well." Diogenes Laertius, Lives of Eminent Philosophers R.D. Hicks, Ed.)"

"In fact we do not know anything infallibly, but only that which changes according to the condition of our body and of the [influences] that reach and impinge upon it."

"There are two forms of knowledge, one genuine, one obscure. To the obscure belong all of the following: sight, hearing, smell, taste, feeling. The other form is the genuine, and is quite distinct from this. [And then distinguishing the genuine from the obscure, he continues:] Whenever the obscure [way of knowing] has reached the minimum sensibile of hearing, smell, taste, and touch, and when the investigation must be carried farther into that which is still finer, then arises the genuine way of knowing, which has a finer organ of thought."

"[Democritus says:] By convention sweet is sweet, by convention bitter is bitter, by convention hot is hot, by convention cold is cold, by convention color is color. But in reality there are atoms and the void. That is, the objects of sense are supposed to be real and it is customary to regard them as such, but in truth they are not. Only the atoms and the void are real."

"Of practical wisdom these are the three fruits: to deliberate well, to speak to the point, to do what is right."

"He who intends to enjoy life should not be busy about many things, and in what he does should not undertake what exceeds his natural capacity. On the contrary, he should have himself so in hand that even when fortune comes his way, and is apparently ready to lead him on to higher things, he should put her aside and not o'erreach his powers. For a being of moderate size is safer than one that bulks too big."

"If any one hearken with understanding to these sayings of mine many a deed worthy of a good man shall he perform and many a foolish deed be spared."

"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."

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

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

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

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

"'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."

"Fools learn wisdom through misfortune."

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

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

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

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

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

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

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

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

"Fame and wealth without wisdom are unsafe possessions."

"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."

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

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

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

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

"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."

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

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

"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."

"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."

"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."

"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."

"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."

"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."

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

"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."

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

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"Ὅπερ ἔδει δεῖξαι."

"ὅπερ ἔδει ποιῆσαι."

"Καὶ τὸ ὅλον τοῦ μέρους μεῖζον [ἐστιν]."

"Πρῶτος ἀριθμός ἐστιν ὁ μονάδι μόνῃ μετρούμενος."

"μὴ εἶναι βασιλικὴν ἀτραπὸν ἐπί γεωμετρίαν."

"Non est regia [inquit Euclides] ad Geometriam via."

"Δός αὐτῷ τριώβολον, ἐπειδὴ δεῖ αὐτῷ ἐξ ὧν μανθάνει κερδαίνειν."

"The laws of nature are but the mathematical thoughts of God."

"With the completion of Euclid's Elements... For the first time in history masses of isolated discoveries were unified and correlated by a single guided principle, that of rigid deduction from explicitly stated assumptions. ...If the Pythagorean dream of a mathematized science is to be realized, all of the sciences must eventually submit to the discipline that geometry accepted from Euclid."

"The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences."

"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."

"There is irrefutable evidence that a substantial portion of the material recorded in the Elements was known before Euclid, and there is nothing either in the style or in the plan of the treatise to suggest that it was intended as a collection of original contributions. Thus, on the whole... the chief objective... was to put system and rigour into the work of his predecessors."

"There never has been, and till we see it we never shall believe that there can be, a system of geometry worthy of the name, which has any material departures (we do not speak of corrections or extensions or developments) from the plan laid down by Euclid."

"Euclid... gave his famous definition of a point: "A point is that which has no parts, or which has no magnitude." …A point has no existence by itself. It exists only as a part of the pattern of relationships which constitute the geometry of Euclid. This is what one means when one says that a point is a mathematical abstraction. The question, What is a point? has no satisfactory answer. Euclid's definition certainly does not answer it. The right way to ask the question is: How does the concept of a point fit into the logical structure of Euclid's geometry? ...It cannot be answered by a definition."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"Euclid is said to have written the Elements of Music. Two treatises are attributed to Euclid in... the Musici, the... Sectio canonis (the theory of the intervals) and the... (introduction to harmony). The first, resting on the Pythagorean theory of music, is mathematical and clearly and well written, the style and the form of the propositions agreeing well with what we find in the Elements. Its genuineness is confirmed not only by internal evidence... Euclid is twice mentioned by name, in the commentary on Ptolemy's Harmonica published by Wallis... The second treatise is not Euclid's..."

"Euclid alone has looked on Beauty bare."

"Those who have written the history of geometry have thus far carried the development of this science. Not much later than these is Euclid, who wrote the 'Elements,' arranged much of Eudoxus' work, completed much of Theaetetus's and brought to irrefragable proof propositions which had been less strictly proved by his predecessors."

"Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says."

"Inasmuch as many things, while appearing to rest on truth and to follow from scientific principles, really tend to lead one astray from the principles and deceive the more superficial minds, he has handed down methods for the discriminative understanding of these things as well, by the use of which methods we shall be able to give beginners in this study practice in the discovery of paralogisms and to avoid being misled. This treatise, by which he puts this machinery in our hands, he entitled (the book) of Pseudaria, enumerating in order their various kinds, exercising our intelligence in each case by theorems of all sorts, setting the true side by side with the false, and combining the refutation of error with practical illustration. This book then is by way of cathartic and exercise, while the Elements contain the irrefragable and complete guide to the actual scientific investigation of the subjects of geometry."

"Water is the first principle of everything."

"Hope is the only good that is common to all men; those who have nothing else possess hope still."

"Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick."

"Do not ask who started it. Finish it"

"Nothing is more ancient than God, for He was never created; nothing more beautiful than the world, it is the work of that same God; nothing is more active than thought, for it flies over the whole universe; nothing is stronger than necessity, for all must submit to it."

"Thales asserted Water to be the principle of things. For he saw that matter was principally dispensed in moisture, and moisture in water; and it seemed proper to make that the principle of things, in which the virtues and powers of beings, and especially the elements of their generations and restorations, were chiefly found. He saw that the breeding of animals is in moisture ; that the seeds and kernels of plants (as long as they are productive and fresh), are likewise soft and tender; that metals also melt and become fluid, and are as it were concrete juices of the earth, or rather a kind of mineral waters; that the earth itself is fertilised and revived by showers or irrigation, and that earth and mud seem nothing else than the lees and sediment of water; that air most plainly is but the exhalation and expansion of water; nay, that even fire itself cannot be lighted, nor kept in and fed, except with moisture and by means of moisture. He saw, too, that the fatness which belongs to moisture, and which is the support and life of flame and fire, seems a kind of ripeness and concoction of the water."

"Thales had a motto — sophotaton chronos aneuriskei gar panta — which means time is wisest because it discovers everything. We still live by that motto — we mark the time and aid the discoveries by keeping the soul lines intact."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"It was not Zeno, the founder of the Stoics, alone, who taught that the Universe evolves, and its primary substance is transformed from the state of fire into that of air, then into that of water, etc. Heraclitus of Ephesus maintained that the one principle that underlies all phenomena in Nature is fire. The intelligence that moves the Universe is fire, and fire is intelligence. And while Anaximenes said the same of air, and Thales of Miletus (600 years b.c.) of water, the Esoteric Doctrine reconciles all these philosophers, by showing that though each was right, the system of none was complete."

"With regard to the Pythagorean theorem my conjecture is that... it was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles."

"The more one studies the period of Thales—the more one compares the knowledge he bequeathed to prosterity with the one he had found when he began his work—the more does his mathematical stature grow, until one is impelled to range Thales with such figures as Archimedes, Fermat, Newton, Gauss and Poincaré."

"Thales the teacher produced the first geometers, even as Thales the thinker founded the first geometry worthy of the name."

"[W]ith Thales... Aristotle ascribes the statement: "Water is the material cause of all things." This... expresses, as Nietzsche... pointed out, three fundamental ideas of philosophy. First, ...the material cause of all things; second, ...that this ...be answered in conformity with reason, without ...myths or mysticism; third, ...that ...it must be possible to reduce everything to one principle."

"Thales' statement was the first expression of... a fundamental substance, of which all other things were transient forms. The word "substance"... was... not interpreted in the purely material sense ...Aristotle ascribes to Thales also ...All things are full of gods. ...[We can] imagine ...Thales took his view ...from meteorological considerations. ...[W]ater can take the most various shapes... ice and snow... vapor, and... clouds. It seems to turn into earth where the rivers form their delta, and it can spring from the earth. Water is the condition for life. Therefore... [as] a fundamental substance, it was natural to think of water first."

"Since Alyattes would not give up the Scythians to Cyaxares at his demand, there was war [ Battle of Halys ] between the Lydians and the Medes for five years; each won many victories over the other, and once they fought a battle by night. They were still warring with equal success, when it happened, at an encounter which occurred in the sixth year, that during the battle the day was suddenly turned to night. Thales of Miletus had foretold this loss of daylight to the Ionians, fixing it within the year in which the change did indeed happen."

"It has been asserted that metaphysical speculation is a thing of the past and that physical science has extirpated it. The discussion of the categories of existence, however, does not appear to be in danger of coming to an end in our time, and the exercise of speculation continues as fascinating to every fresh mind as it was in the days of Thales."

"While [Thales] was studying the stars and looking upwards, he fell into a pit, and a neat, witty Thracian servant girl jeered at him, they say, because he was so eager to know the things in the sky that he could not see what was there before him at his very feet."

"[Thales] first went to Egypt and hence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attack being in some cases more general [i.e. more theoretical or scientific], in others more empirical [...more in the nature of simple inspection or observation]."

"In committing himself to a form of materialism, Thales rejects a picture of the universe found in the Homeric poems, one which posits, in addition to the natural world, a supernatural quadrant populated by beings who are not subject to such laws as may govern the interactions of all natural bodies. If all things are composed of matter, then it ought to be possible to explain all there is to explain about the universe in terms of material bodies and their law-governed interactions. This simple thought already stands in sharp contrast to a world supposed to be populated by supernatural immaterial beings whose actions may be capricious or deliberate, rational or irrational, welcome or unwelcome, but which as a matter of basic principle cannot be explicated in terms of the forms of regularity found in the natural world. In Thales’ naturalistic universe, it ought to be possible to uncover patterns and laws and to use such laws as the basis for stable predictions about the direction the universe is to take; to uncover causes and to use that knowledge to find cures for illnesses or to develop strategies for optimizing our well-being; and, less practically, to find broad-based explanations to fundamental questions which crop up in every organized society. Such questions persist: Where did the universe come from? What, ultimately, is its basic stuff?"

"According to tradition, Thales is the first to reveal the study of nature to the Greeks; although he had many predecessors, in Theopharastus' view, he so surpassed them as to eclipse everyone before him."

"Living virtuously is equal to living in accordance with one's experience of the actual course of nature"

"Wise people are in want of nothing, and yet need many things. On the other hand, nothing is needed by fools, for they do not understand how to use anything, but are in want of everything."

"He who is running a race ought to endeavor and strive to the utmost of his ability to come off victor; but it is utterly wrong for him to trip up his competitor, or to push him aside. So in life it is not unfair for one to seek for himself what may accrue to his benefit; but it is not right to take it from another."

"If I had followed the multitude, I should not have studied philosophy."

"If I knew that it was fated for me to be sick, I would even wish for it; for the foot also, if it had intelligence, would volunteer to get muddy."

"The universe itself is God and the universal outpouring of its soul"

"We should infer in the case of a beautiful dwelling-place that it was built for its owners and not for mice; we ought, therefore, in the same way to regard the universe as the dwelling-place of the gods."

"Wrongly do the Greeks suppose that aught begins or ceases to be; for nothing comes into being or is destroyed; but all is an aggregation or secretion of pre-existent things: so that all-becoming might more correctly be called becoming-mixed, and all corruption, becoming-separate."

"All things were together, infinite both in number and in smallness; for the small too was infinite."

"And since these things are so, we must suppose that there are contained many things and of all sorts in the things that are uniting, seeds of all things, with all sorts of shapes and colours and savours"

"Mind is infinite and self-ruled, and is mixed with nothing, but is alone itself by itself."

"Thought is something limitless and independent, and has been mixed with no thing but is alone by itself. ... What was mingled with it would have prevented it from having power over anything in the way in which it does. ... For it is the finest of all things and the purest."

"The Greeks follow a wrong usage in speaking of coming into being and passing away; for nothing comes into being or passes away, but there is mingling and separation of things that are. So they would be right to call coming into being mixture, and passing away separation."

"The sun provides the moon with its brightness."

"If one examines the reasons for the persecution of the best minds of different nations, and compares the reasons for the persecution and banishment of Pythagoras, Anaxagoras, Socrates, Plato, and others, one can observe that in each case the accusations and reasons for banishment were almost identical and unfounded. But in the following centuries full exoneration came, as if there had never been any defamation. It would be correct to conclude that such workers were too exalted for the consciousness of their contemporaries, and the sword of the executioner was ever ready to cut off a head held high.... A book should be written about the causes of the persecution of great individuals. By comparing the causes is it possible to trace the evil will."

"In mathematics... the Greek attitude differed sharply from that of the earlier potamic cultures. The contrast was clear in... Thales and Pythagoras, and it continues to show... in Athens during the Heroic Age. ...while Anaxagoras was in prison he occupied himself with an attempt to square the circle... the first mention of a problem that was to fascinate mathematicians for more than 2000 years. ...Here we see a type of mathematics that is quite unlike that of the Egyptians and Babylonians. It is not the practical application of a science of number... but a theoretical question involving a... distinction between accuracy in approximation and exactitude in thought. ...no more the concern of the technologist than those he raised... concerning the ultimate structure of matter."

"The big bang and the steady state debate in some ways echoed that between the ideas of Anaximander and Anaxagoras from two and a half millennia earlier. Anaxagoras had envisaged that at one time "all things were together" and that the motive force for the universe originated at a single point... Anaximander on the other hand wanted a universe determined by "the infinite" and needed an "eternal motion" to explain the balancing process of things coming into being and passing away in an eternal universe... ancient philosophy was debating the alternatives of a creation event starting the universe from a single point versus a continuous creation in an eternal universe."

"Anaxagoras of Clazomenae postulated another element called the aether, which was in constant rotation and carried with it the celestial bodies. He also believed that there was a directing intelligence in nature that he called Nous which gives order to what otherwise would be a chaotic universe. By Nous he meant literally "the Mind of the Cosmos"… Anaxagoras was the last of the Ionian physicists."

"The endless sequence of explanation is explicit in Anaxagoras. Even the ingredients that go to make up something and account for its behaviour are themselves composed of ingredients which are themselves again composed of ingredients. [...] In every case, behaviour is a consequence of both the predominant features (which make it seem to be such and such) and also the hidden features (which can make it do otherwise inexplicable things). And this dual explanation will apply as much to the hidden ingredients as to the macroscopic items we encounter in daily life.But still it remains true for Anaxagoras that in principle the material composition (if we could know it in detail) would account for the current behaviour of each item in the world. Unless the thing is alive, that is. For living things, it looks as though the explanation must be supplemented by appeal to another principle, what Anaxagoras called ‘Mind’."

"My dear Meletus, do you think you are prosecuting Anaxagoras? Are you so contemptuous of these men and think them so ignorant of letters as not to know that the books of Anaxagoras of Clazomenae are full of those theories, and further, that the young men learn from me what they can buy from time to time for a drachma, at most, in the bookshops, and ridicule Socrates if he pretends that these theories are his own, especially as they are so absurd? Is that, by Zeus, what you think of me, Meletus, that I do not believe that there are any gods?—That is what I say, that you do not believe in the gods at all."

"Anaxagoras held that everything is infinitely divisible, and that even the smallest portion of matter contains some of each element. Things appear to be that of which they contain the most. Thus, for example, everything contains some fire, but we only call it fire if that element preponderates. He argues against the void, saying that the clepsydra or an inflated skin shows that there is air where there seems to be nothing."

"In science [Anaxagoras] had great merit. It was he who first explained that the moon shines by reflected light... Anaxagoras gave the correct theory of eclipses, and knew that the moon was below the sun. The sun and stars, he said, are fiery stones, but we do not feel the heat of the stars because they are too distant. The sun is larger than the Peloponnesus. The moon has mountains, and (he thought) inhabitants."

"Anaxagoras was more inclined to the study of physics than of metaphysics, for which reason he is accused by Plato and by Aristotle of not having conceded enough to final causes, and of having converted God into a machine. Accordingly he explained on physical principles the formation of plants and animals, and even celestial phenomena; which drew upon him the charge of atheism. Nevertheless, he regarded the testimony of the senses as subjectively true; but as insufficient to attain to objective truth, which was the privilege of the reason."

"Anaxagoras says that perception is produced by opposites; for like things cannot be affected by like. ...It is in the same way that touch and discern their objects. That which is just as warm or just as cold as we are neither warms us nor cools us... [I]n the same way, we do not apprehend the sweet and the sour by means of themselves. We know cold by warm, fresh by salt, and sweet by sour, in virtue of our deficiency in each; for all these are in us to begin with. And we smell and hear in the same... And all sensation implies pain... for all unlike things produce pain by their contact. Brilliant colours and excessive s produce pain... The larger animals are the more sensitive, and... sensation is proportionate to the size of the organs of sense. ...Rarefied air has more smell ... when air is heated and rarefied, it smells. ...[S]mell is better perceived when it is near than when it is far by reason of its being more condensed, while when dispersed it is weak."

"[W]e may assume [Anaxagoras] belonged to a family which had won distinction in the State. Nor need we reject the tradition that Anaxagoras neglected his possessions to follow science. ...[I]n the fourth century he was... regarded as the type of the man who leads the "theoretic life.""

"One incident belonging to the early manhood of Anaxagoras is... his observation of the huge ic stone which fell into the Aigospotamos in 468-67 B.c. ...[I]t may have occasioned one of his most striking departures from the earlier cosmology, and led to ...the ...view for which he was condemned at Athens."

"Anaxagoras was the first philosopher to take up his abode at Athens."

"We are not to suppose... he was attracted... by... the character of the Athenians. ...Athens had ...become the political centre of the Hellenic world; but it had not yet produced a single scientific man. ...[T]he temper of the citizen body... remained hostile to free inquiry... Sokrates, Anaxagoras, and Aristotle fell victims... to the bigotry of the democracy, though... their offence was political rather than religious. They were condemned not as heretics, but as innovators in the state religion. ..[A] recent historian observes, "Athens... was far from... a place for free inquiry to thrive unchecked." ...T]his ...has been in the minds of ...writers who ...represented philosophy as ...un-Greek. It was in reality thoroughly Greek, though... thoroughly unAthenian."

"It seems... reasonable... that Perikles... brought Anaxagoras to Athens... Holm has shown... the aim... was... to Ionise his ...citizens, to impart ...flexibility and openness of mind which characterised their kinsmen across the sea. ...The close relation in which Anaxagoras stood to Perikles is placed ...by the testimony of Plato ...In the Phaedrus ...Sokrates say[s]: "For all arts that are great, there is need of talk and discussion on the parts of that deal with things on high; for that seems to be the source which inspires high-mindedness and effectiveness in every direction. Perikles added this very acquirement to his original gifts. He fell in... with Anaxagoras, who was a scientific man; and, satiating himself with the theory of things on high, and having attained to a knowledge of the true nature of intellect and folly, which were... what the discourses of Anaxagoras were mainly about, he drew from that source... to further him in the art of speech.""

"Alexander of Aitolia... referred to Euripides as the "nursling of brave Anaxagoras." ...The famous fragment on the blessedness of the scientific life might just as well refer to any other cosmologist as to Anaxagoras, and ...suggests ...a thinker ...more primitive ...[T]here is one fragment which distinctly expounds the central thought of Anaxagoras ...We may conclude ...that Euripides knew the philosopher and his views ..."

"Shortly before the ... ...enemies of Perikles began ...attacks upon him through his friends. Pheidias was the first to suffer, and Anaxagoras... next. ...[H]e was an object of special hatred to the religious party ...even though the charge ...against him does not suggest ...he went out of his way to hurt their susceptibilities. The details of the trial are somewhat obscure... [F]irst ...was ...a psephism by —the same whom Aristophanes laughs at in The Birds—enacting that an impeachment should be brought against those who did not practise religion, and taught theories about "the things on high." ...[A]t the actual trial ... authorities give ... conflicting accounts. ...[F]rom Plato ...the accusation was ...that Anaxagoras taught the sun was a red-hot stone, and the moon earth; ...[H]e ...did hold these views ...[H]e was got out of prison and sent away by Perikles."

"Anaxagoras... went back to ... settled at Lampsakos, and... founded a school there. Probably he did not live long after his exile. The Lampsakenes erected an altar to his memory in their market-place, dedicated to Mind and Truth; and the anniversary of his death was long kept as a holiday for school-children..."

"Diogenes includes Anaxagoras in... philosophers who left... a single book, and... preserved the... criticism... that it was written "in a lofty and agreeable style." ...[F]rom the ...Apology ...the works of Anaxagoras could be bought at Athens for a single drachma; and ...was of some length ...[as] Plato ...speak[s] of it. ...Simplicius had ...a copy, ...and it is to him we owe the preservation of all our fragments, with ...[a few] doubtful exceptions. Unfortunately his quotations seem... confined to the First Book... dealing with general principles, so... we are... in the dark with regard to... details. This is... unfortunate, as... Anaxagoras... first gave the true theory of the moon’s light and... the true theory of eclipses."

"The system of Anaxagoras, like that of Empedokles, aimed at reconciling the Eleatic doctrine that corporeal substance is unchangeable with... a world which... presents the appearance of coming into being and passing away. The conclusions of Parmenides are... accepted and restated. Nothing can be added to all things; for there cannot be more than all, and all is always equal (fr. 5). Nor can anything pass away. What men commonly call coming into being and passing away is... mixture and separation (fr. 17). This... reads almost like a prose paraphrase of Empedokles (fr. 9); and it is... probable... Anaxagoras derived his theory... from his younger contemporary, whose poem was most likely published before his own treatise."

"Empedokles sought to save the world of appearance by maintaining that the opposites—hot and cold, moist and dry—were things, each...real in the Parmenidean sense. Anaxagoras regarded this as inadequate. ...[T]hings of which the world is made are not "cut off with a hatchet" (fr. 8) ...the true formula must be: There is a portion of everything in everything (fr. 11)."

"The statement... there is a portion of everything in everything, is not... referring simply to the original mixture of things before the formation of the worlds (fr. 1). ...[E]ven now “all things are together,” and everything... has an equal number of “portions” (fr. 6). A smaller particle... could only contain a smaller number of portions... [I]f anything is, in the... Parmenidean sense, it is impossible that mere division should make it cease to be (fr. 3). Matter is infinitely divisible... there is no least thing, any more than there is a greatest. But however great or small.., it contains... the same number of "portions,"... [i.e.,] a portion of everything."

"Aristotle says... if we suppose the first principles to be infinite, they may either be one in kind, as with Demokritos, or opposite. Simpliciuss, following Porphyry and Themistios, refers the latter view to Anaxagoras; and Aristotle... implies that the opposites of Anaxagoras had as much right to be... first principles as the "homoeomeries.""

"It is of those opposites... that everything contains a portion. Every particle... large or... small, contains... [all] of those opposite qualities. ...Even snow, Anaxagoras affirmed, was black ...white contains a ...portion of the opposite quality. It is enough to indicate the connexion... with... Herakleitos..."

"Anaxagoras held that, however far you may divide... things—and they are infinitely divisible—you never come to a part so small that it does not contain portions of all the opposites. The smallest portion of bone is... bone. On the other hand, everything can pass into everything else... because the "seeds"... of each form of matter contain a portion of everything... [i.e.,] of all the opposites, though in different proportions. If we... use the word "element" at all, it is these seeds... [T]he "seeds"... he... substituted for the "roots" of Empedokles, were not the opposites in a state of separation, but each contained a portion of them all."

"Though everything has a portion of everything in it, things appear to be that of which there is most in them (fr. 12 sub fin.). ...Air is that in which there is most cold, Fire that in which there is most heat ... [etc.]"

"[W]hen "all things were together," and when the different seeds... were mixed... in infinitely small particles (fr. 1), the appearance... would be that of one of... the primary substances. ...[T]hey did present the appearance of "air and aether"; for the qualities (things) which belong to these prevail in quantity... and everything is... that of which it has most... (fr. 12 sub fin.). Here... Anaxagoras attaches... to Anaximenes. The primary condition... before... formation of... worlds, is much the same in both; only, with Anaxagoras, the original mass is no longer the primary substance, but a mixture of innumerable seeds divided into infinitely small parts. This mass is infinite, like the air of Anaximenes, and... supports itself, since there is nothing surrounding it. ... [T]he "seeds" of all things which it contains are infinite in number (fr. 1). But... the... seeds may be divided into those in which the portions of cold, moist, dense, and dark prevail, and those which have most... warm, dry, rare, and light... the original mass was a mixture of infinite Air and... Fire. ..."

"[T]here is no void in this mixture, an addition... made necessary by... arguments of Parmenides. Anaxagoras added an experimental proof of this to the purely dialectical one of the . He used the klepsydra experiment ... as Empedokles had done... and... showed the corporeal nature of air by means of inflated skins."

"Like Empedokles, Anaxagoras required some external cause to produce motion in the mixture. Body, Parmenides had shown, would never move itself... Anaxagoras called the cause of motion... . ...[T]his ...made Aristotle say that he "stood out like a sober man from the random talkers that had preceded him," and he has often been credited with the introduction of the spiritual into philosophy. ...[D]isappointment [was] expressed ...by Plato and Aristotle as to the way in which Anaxagoras worked out the theory ... Plato makes Sokrates say: "I once heard a man reading a book... of Anaxagoras... saying it was Mind that ordered the world and was the cause of all things. I was delighted... and... thought he... was right. ...But my extravagant expectations were all dashed... when I... found... the man made no use of Mind... He ascribed no causal power... to it in the ordering of things, but to airs, and aethers, and waters, and a host of other strange things." Aristotle... says: "Anaxagoras uses Mind as a ' to account for the formation of the world ; and whenever he is at a loss to explain why anything necessarily is, he drags it in. But in other cases he makes anything rather than Mind the cause." These utterances... suggest... Nous of Anaxagoras did not... stand on a higher level than... Love and Strife of Empedokles..."

"is unmixed (fr. 12), and does not... contain a portion of everything. This would hardly be worth saying of an immaterial mind... The result of its being unmixed is that it "has power over" everything... [i.e.,] it causes things to move. Herakleitos had said as much of Fire, and Empedokles of Strife. Further, it is the "thinnest" of all things, so that it can penetrate everywhere... Nous also "knows all things"... [P]robably... Anaxagoras substituted Nous for the Love and Strife of Empedokles... to retain the old Ionic doctrine of a substance that "knows" all things, and... identify this with the new... substance that "moves" all things. Perhaps... his increased interest in physiological as distinguished from purely cosmological matters... led him to... Mind rather than Soul. ...[T]he originality of Anaxagoras lies ...more in the theory of matter than ...of Nous."

"The formation of a world starts with a rotatory motion which imparts to a portion of the mixed mass in which "all things are together" (fr. 13), and this... motion gradually extends over... wider space. Its rapidity (fr. 9) produced a separation of the rare and the dense, the cold and the hot, the dark and the light, the moist and the dry (fr. 15). This... produces two great masses, the one... of the rare, hot, light, and dry, called... "Aether"; the other... [of] the opposite qualities... called "Air" (fr. 1). ...the Aether or Fire took the outside while the Air occupied the centre (fr. 15)."

"The next stage is the separation of the air into clouds, water, earth, and stones (fr. 16). In this Anaxagoras follows Anaximenes... In his account of the origin of the heavenly bodies... he... [was] more original. ...[A]t the end of fr. 16... stones "rush outwards more than water," and... from the doxographers... heavenly bodies were... stones torn from the earth by the rapidity of its revolution and made red-hot by the speed... Perhaps the fall of the ic stone at Aigospotamoi had something to do with... this theory. ...[W]hile in the earlier stages of the world-formation we are guided... by the analogy of water rotating with light and heavy bodies... in it, we are here reminded... of a sling."

"Anaxagoras adopted the ordinary Ionian theory of innumerable worlds... fr. 4... The words "that it was not only with us that things were separated off, but elsewhere too" can only mean... Nous... caused a rotatory movement in more parts of the boundless mixture than one."

"The cosmology of Anaxagoras is clearly based upon that of Anaximenes... Theophrastos... [states] that Anaxagoras had belonged to the school of Anaximenes. The floating on the air, the dark bodies below the moon, the explanation of the solstices and the "turnings" of the moon by the resistance of air, the explanations given of wind and of thunder and lightning, are all derived from the earlier inquirer."

""There is a portion of everything in everything except , and there are some things in which there is Nous also" (fr. 11). In these words Anaxagoras laid down the distinction between animate and inanimate things. ...[T]he same Nous ..."has power over," ...[i.e.,] sets in motion, all things that have life ...(fr. 12). ...The Nous was the same, but it had more opportunities in one body than another. Man was the wisest... not because he had... better... Nous, but... because he had hands. ...Parmenides ...had ...made the thought of men depend upon ...their limbs."

"As all Nous is the same... plants were regarded as living creatures. ...Plutarch says... [Anaxagoras] called plants "animals fixed in the earth." ...Plants first arose when the seeds... which the air contained were brought down by the rain-water, and animals originated in a similar way."

"It was a happy thought of Anaxagoras to make sensation depend upon irritation by opposites, and to connect it with pain. Many modern theories are based upon a similar idea."

"That Anaxagoras regarded the senses as incapable of reaching the truth... is shown by... fragments preserved by Sextus. But we must not... turn him into a sceptic. ... He did say (fr. 21) that "the weakness of our senses prevents our discerning the truth," but this meant... we do not see the "portions"... [e.g.,] the portions of black which are in the white. Our senses simply show us the portions that prevail. He also said... things which are seen give us the power of seeing the invisible, which is the... opposite of scepticism (fr. 21a)."

"The woman who goes to bed with a man must put off her modesty with her petticoat, and put it on again with the same."

"That tho' a Man were admitted into Heaven to view the wonderful Fabrick of the World, and the Beauty of the stars, yet what would otherwise be Rapture and Extasie, would be but melancholy Amazement if he had not a Friend to communicate it to."

"The unwritten laws of the gods were promulgated against depraved manners, inflicting a severe destiny and penalty on the disobedient; and these unwritten laws are the fathers and leaders of those that are written, and of the dogmas established by men."

"Simplicius attributes to him a work on Opposites, to which he says that Aristotle was indebted for what he says on this subject in his Categories. His Harmonicon is quoted by Nicomachus in his Arithmetic. ...Fragments of the works attributed to him "On the Good and Happy Man," and "On Wisdom," are... extant. ...A letter of Archytas to Plato and Plato's reply are preserved by Laertius."

"Among the mathematical problems Archytas solved or attempted to solve the duplication of the cube, for which purpose he attempted to find two mean proportionals between the two right lines formed by the section of a cylinder, as Laertius expresses it. ...Among his mechanical inventions is mentioned a wooden pigeon that could fly, of which Gellius speaks particularly. The invention of a rattle, perhaps a child's toy, is also attributed to him."

"The fragments of the works "On the Good and Happy Man," and "On Wisdom," were published by T. Gale in 1670, and were given again with other things in his "Opuscula Mythologica," Cambridge, 1671, 8vo.; Amsterdam, 1688, 8vo. The fragment of the Greek text of the work on the "Nature of the Universe," was published at Venice, 1561, 8vo.; with a Latin version by Dom. Pizimentius, under the title "Architæ Tar. X. Prædicamenta." ...A complete collection of the fragments was published by I. Cn. Orelli, Leipzig, 1821, 8vo. The "Political Fragments of Archytas, Charondas, &c., translated from the Greek by Thomas Taylor, was published at London, 1822, 8vo. There is a work by Nic. T. Reimer intitled "Archytas, eiusque Solutio Problematis Cubi Duplicationis," Göttingen, 1798, 8vo."

"Such was the celebrity of this philosopher, that many illustrious names appear in the train of his disciples, particularly Philolaus, Eudoxus, and Plato. ...He excelled, not only in speculative philosophy, but in geometry and mechanics, He is said to have invented a kind of winged automaton, and several curious hydraulic machines. He was in such high reputation for moral and political wisdom, that, contrary to the usual custom, he was appointed seven different times to the supreme magistracy in Tarentum."

"Of his writings none remain except a metaphysical work, "On the Nature of the Universe," in which he has explained the predicaments; and sundry fragments "On Wisdom," and "On the Good and Happy Man," preserved by Stobaeus and edited from him by Gale."

"His death, which is said to have been occasioned by a shipwreck, is made a subject of poetical description by Horace, who celebrates him as an eminent geographer and astronomer."

"Concerning the philosophical tenets of Archytas... Aristotle, who was an industrious collector from the Pythagoreans, borrowed from him the general arrangements which are usually called his ten categories."

"The sum of his moral doctrine is; that virtue is to be pursued for its own sake in every condition of life; that all excess is inconsistent with virtue; that the mind is more injured by prosperity than by adversity; and that there is no pestilence so destructive to human happiness as pleasure."

"It is probable that Aristotle was indebted to Archytas for many of his moral ideas; particularly for the notion which runs through his ethical pieces, that virtue consists in avoiding extremes."

"Even thee, thou measurer of earth and sea, thou counter of the sands, Archytas, how small a portion of earth contains thee now! It profits thee not to have searched the air and traversed the heavens since thou wert to die. So Tantalus, Tithonus, and Minos have died, and Pythagoras too with all his learning hath gone down once more to the grave. But so it is: all must die alike; some to make sport for Mars, some swallowed up in the deep: old and young go crowding to the grave: none escape: I too have perished in the waters. But grudge me not, thou mariner, a handful of earth: so may the storm spend itself on the woods while thou art safe and thy merchandize increases. Is it a small matter with thee to bring ruin on thy children? Yea, perhaps retribution awaits thyself: my curses will be heard, and then no atonement shall deliver thee. 'Tis but the work of a moment—thrice cast earth upon me and hasten on."

"Te maris et terrae numeroque carentis arenae Mensorem cohibent, Archyta, Pulveris exiqui prope litus parva Matinum Munera, nec quidquam tibi prodest The sea, the earth, the innumerable sand, Archytas, thou coulds't measure; now, alas! A little dust on Matine shore has spann'd That soaring spirit; vain it was to pass The gates of heaven, and send thy soul in quest O'er air's wide realms; for thou hadst yet to die."

"[Hypotheses] 1. That the Moon receives its light from the sun. 2. That the earth is in the relation of a point and centre to the sphere in which the moon moves. 3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye. 4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant. 5. That the breadth of the (earth's) shadow is (that) of two moons. 6. That the moon subtends one fifteenth part of a sign of the zodiac."

"We are now in a position to prove the following propositions : — 1. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon (from the earth); this follows from the hypothesis about the halved moon. 2. The diameter of the sun has the same ratio (as aforesaid) to the diameter of the moon. 3. The diameter of the sun has to the diameter of the earth a ratio greater than that which 19 has to 3, but less than that which 43 has to 6; this follows from the ratio thus discovered between the distances, the hypothesis about the shadow, and the hypothesis that the moon subtends one fifteenth part of a sign of the zodiac."

"Proposition 1. Two equal spheres are comprehended by one and the same cylinder, and two unequal spheres by one and the same cone which has its vertex in the direction of the lesser sphere; and the straight line drawn through the centres of the spheres is at right angles to each of the circles in which the surface of the cylinder, or of the cone, touches the spheres."

"Proposition 2. If a sphere be illuminated by a sphere greater than itself, the illuminated portion of the former sphere will be greater than a hemisphere."

"Proposition 3. The circle in the moon which divides the dark and the bright portions is least when the cone comprehending both the sun and the moon has its vertex at our eye."

"Proposition 4. The circle which divides the dark and the bright portions in the moon is not perceptibly different from a great circle in the moon."

"Proposition 5. When the moon appears to us halved, the great circle parallel to the circle which divides the dark and the bright portions in the moon is then in the direction of our eye; that is to say, the great circle parallel to the dividing circle and our eye are in one plane."

"Proposition 6. The moon moves (in an orbit) lower than (that of) the sun, and, when it is halved, is distant less than a quadrant from the sun."

"Proposition 7. The distance of the sun from the earth is greater than eighteen times, but less than twenty times, the distance of the moon from the earth."

"Proposition 8. When the sun is totally eclipsed, the sun and the moon are then comprehended by one and the same cone which has its vertex at our eye."

"Proposition 9. The diameter of the sun is greater than 18 times, but less than 20 times, the diameter of the moon."

"Proposition 10. The sun has to the moon a [volume] ratio greater than that which 5832 has to 1, but less than that which 8000 has to 1."

"Proposition 11. The diameter of the moon is less than 2/45ths [0.04], but greater than 1/30th [0.03] of the distance of the centre of the moon from our eye."

"Proposition 12. The diameter of the circle which divides the dark and the bright portions in the moon is less than the diameter of the moon, but has to it a ratio greater than [0.99] that which 89 has to 90."

"Proposition 13. The straight line subtending the portion intercepted within the earth's shadow of the circumference of the circle in which the extremities of the diameter of the circle dividing the dark and the bright portions in the moon move is less than double of the diameter of the moon, but has to it a ratio greater than [2.0] that which 88 has to 45; and it is less than 1/9th part of the diameter of the sun, but has to it a ratio greater than [1/10th] that which 22 has to 225. But it has to the straight line drawn from the centre of the sun at right angles to the axis and meeting the sides of the cone a ratio greater than [0.097] that which 979 has to 10125."

"Proposition 14. The straight line joined from the centre of the earth to the centre of the moon has to the straight line cut off from the axis towards the centre of the moon by the straight line subtending the (circumference) within the earth's shadow a ratio greater than that which 675 has to 1."

"Proposition 15. The diameter of the sun has, to the diameter of the earth a ratio greater than [6.3] that which 19 has to 3, but less than [7.2] that which 43 has to 6."

"Proposition 16. The sun has to the earth a [volume] ratio greater than [254] that which 6859 has to 27, but less than [368] that which 79507 has to 216."

"Proposition 17. The diameter of the earth is to the diameter of the moon in a ratio greater than [2.51] that which 108 has to 43, but less than [3.16] that which 60 has to 19."

"Proposition 18. The earth is to the moon in a [volume] ratio greater than [15.8] that which 1259712 has to 79507, but less than [31.5] that which 216000 has to 6859."

"You (King Gelon) are aware the 'universe' is the name given by most astronomers to the sphere the center of which is the center of the Earth, while its radius is equal to the straight line between the center of the Sun and the center of the Earth. This is the common account as you have heard from astronomers. But Aristarchus has brought out a book consisting of certain hypotheses, wherein it appears, as a consequence of the assumptions made, that the universe is many times greater than the 'universe' just mentioned. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun on the circumference of a circle, the Sun lying in the middle of the Floor, and that the sphere of the fixed stars, situated about the same center as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the center of the sphere bears to its surface."

"Copernicus, Kepler and Galileo were ‘revisionists’ in rejecting the geocentric system of Ptolemy (which held sway for some 1500 years) and, against an oppressive and repressive mainstream opinion (and officialdom), reinstated—with improvements—the heliocentric system of Aristarchos of Samos (3rd cent BCE)."

"It was the ancient opinion of not a few in the earliest ages of philosophy, That the fixed stars stood immovable in the highest parts of the world; that under the Fixed Stars the Planets were carried about the Sun; that the Earth, as one of the Planets, described the annual course about the Sun, while a diurnal motion it was in the mean time revolved about its own axe; and that the Sun, as the common fire which served to warm the whole, was fixed in the center of the Universe. This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans. And this was the judgment of Anaximander, more ancient than any of them, and of that wise king of the Romans, Numa Pompilius; who, as a symbol of the figure of the World with the Sun in the center, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it."

"Aristarchus... was a contemporary of Euclid. His fame rests on his heliocentric theory... Perhaps "theory" is too strong a word, for his proofs were weak; yet it was a great idea, an idea redeveloped centuries later by Copernicus. ...an observer on Earth sees a half-moon only when &ang;EMS [the angle between the Earth and Sun, from the persective of the Moon] is a right angle. ...sometimes both Sun and Moon are visible when the phase of the Moon is half-moon. So? Measure &ang;MES [the angle between the Moon and Sun, from the persective of earth]... This is what Aristarchus did. ...the third angle is determined [since the sum of the three angles of any triangle is 180&#186;], so the shape but not the size of &#8420; EMS [triangle formed by Earth-Moon-Sun] is known. Consequently, although the actual length of any side is not determinate, the ratio of any pair is. ...[Moon-Earth distance/Sun-Earth distance] ME/SE = cos &alpha; [where &ang;MES = &alpha;]."

"The theory of Copernicus not only founded modern system of astronomy, but made men to examine other articles of their creed, after were thus convinced that they had taught and believed the earth to be stationary 6000 years. All the opinions of the ancients respecting the motion of the earth were speculative hypotheses, arising from the Pythagorean school, which, as we know, considered fire the centre of the world, round which all was moving. Thus we ought explain the passage of Aristarchus of Samos, mentioned by Aristotle in his Arenario. Aristarchus, a Pythagorean, held the idea that the earth revolves round its axis, and at the same time, in an oblique circle round the sun; and that the distance of the stars is so great, that this circle is but a point in comparison with their orbits, and therefore the motion of the earth produces no apparent motion in them. Every Pythagorean might have entertained this idea, who considered the sun or fire as the centre of the world, and who was, at the same time, so correct a thinker, and so good an astronomer, as Aristarchus of Samos. But this was not the Copernican system of the world. It was the motions the planets, their stations, and their retrogradations, which astronomers could not explain, and which led them to the complicated motions of the epicycles, in which the planets moved in cycloids round the earth."

"[Aristarchus] is so obscure that Wallis was obliged to annotate him from one end to the other, in the effort to make him intelligible."

"Of the two mathematicians Aristarchus of Samos and Seleucus of Babylon, whose systems came most nearly to his own, he Copernicus] mentions only the first, making no reference to the second. It has often been asserted that he was not acquainted with the views of Aristarchus of Samos regarding the central sun and the condition of the earth as a planet, because the Arenarius, and all the other works of Archimedes, appeared only one year after his death, and a whole century after the invention of the art of printing; but it is forgotten that Copernicus, in his dedication to Pope Paul III., quotes a long passage on Philolaüs, Ecphantus, and Heraclides of Pontus, from Plutarch's work on The Opinions of Philosophers (III., 13), and therefore that he might have read in the same work (II., 24) that Aristarchus of Samos regards the sun as one of the fixed stars."

"As related by Archimedes in the "sand-counter", Aristarchus advanced the bold hypothesis that the earth rotates in a circle about the sun. Most astronomers rejected this... as Archimedes tells us also. [I]n view of the status of mechanics at the time, there are weighty arguments against the motion of the earth... already found in Aristotle and, developed more fully, in Ptolemy. If the earth had such an enormously rapid motion, says Ptolemy, then everything that was not clinched to and riveted to the earth, would fall behind and would therefore appear to fly off in the opposite direction. Clouds... would be overtaken by the rotation of the earth and would lag behind. ...[T]here is nothing to be said against this since the Greeks did not know the law of inertia and required a force to account for every motion. If the earth does not drag the clouds along, they have to lag behind. We do not know how Aristarchus met these arguments."

"The only book of his which has been preserved is a treatise "On the dimensions and distances of the sun and moon," in which we find the results of the first serious attempt to determine these quantities by observation. He observed the angular distance between the sun and the moon at the time when the latter is half illuminated (when the angle at the moon in the triangle earth-moon-sun is a right angle) and found it equal to twenty-nine thirtieths of a right angle or 87&#176;. From this he deduced the result that the distance of the sun was between eighteen and twenty times as great as the distance of the moon. Although this result is exceedingly erroneous, we see at any rate that Aristarchus was more than a mere speculative philosopher, but that he must have been an observer as well as a mathematician. This treatise does not contain the slightest allusion to any hypothesis on the planetary system..."

"We have to depend on statements of subsequent writers when we endeavor to give Aristarchus his proper place in the history of cosmical systems."

"You know that according to most astronomers the world is the sphere, of which the centre is the centre of the earth, and whose radius is a line from the centre of the earth to the centre of the sun. But Aristarchus of Samos has published in outline certain hypotheses, from which it follows that the world is many times larger than that. For he supposes that the fixed stars and the sun are immovable, but that the earth is carried round the sun in a circle which is in the middle of the course; but the sphere of the fixed stars, lying with the sun round the same centre, is of such a size that the circle, in which he supposes the earth to move, has the same ratio to the distance of the fixed stars as the centre of the sphere has to the surface. But this is evidently impossible, for as the centre of the sphere has no magnitude, it follows that it has no ratio to the surface. It is therefore to be supposed that Aristarchus meant that as we consider the earth as the centre of the world, then the earth has the same ratio to that which we call the world, as the sphere in which is the circle, described by the earth according to him, has to the sphere of the fixed stars."

"Archimedes... does not attempt to argue for or against this hypothesis, he merely objects to the unmathematical idea of there being a certain ratio between a point, which has no magnitude, and the surface of a sphere. Of course the meaning of Aristarchus is clear enough, that if we suppose the earth to move round the sun in a large orbit, the distance of the fixed stars must be immensely great as compared with that of the sun, as our motion round the latter would otherwise produce apparent displacements parallax] among the stars, if they are at different distances from the centre of the world, or at any rate, if they are on the surface of a sphere, make the stars in the neighbourhood of the ecliptic appear to close up or spread out according as the earth is at the part of its orbit farthest from them or nearest to them. This is indeed a very startling hypothesis to meet with so far back as the third century before our era... It would almost seem that there was nothing more to say; that Aristarchus had merely thrown out this suggestion or hypothesis without devoting a book or essay to its discussion, and the fact, that his book on the distance of the sun does not contain anything on the subject, tends to confirm this impression."

"One of the persons in the dialogue, being called to account for turning the world upside down, says that he is quite content so long as he is not accused of impiety, "like as Kleanthes held that Aristarchus of Samos ought to be accused of impiety for moving the hearth of the world as the man in order to save the phenomena supposed that the heavens stand still and the earth moves in an oblique circle at the same time as it turns round its axis.""

"The hypothesis of Aristarchus included the rotation of the earth, as might be expected."

"Archimedes distinctly says in his Psammites or Sand-reckoner that Aristarchus was the first to discover that the apparent diameter of the sun is about 1/720th part of the complete circle described by it in the daily rotation, or, in other words, that the angular diameter is about 1/2&#176;, which is very near the truth."

"Archimedes tells us that Aristarchus wrote a book of hypotheses, one of which was that the sun and the fixed stars remain unmoved and that the earth revolves round the sun in the circumference of a circle. Now Archimedes was a younger contemporary of Aristarchus; he must have seen the book of hypotheses in question, and we could have no better evidence for attributing to Aristarchus the first enunciation of the Copernican hypothesis."

"There is... no reason to doubt the unanimous testimony of antiquity that Aristarchus was the real originator of the Copernican hypothesis."

"Apart from its [On the sizes and distances of the Sun and Moon] astronomical content, it is of the greatest interest for its geometry. Thoroughly classical in form and language, as befits the period between Euclid and Archimedes, it is the first extant specimen of pure geometry used with a trigonometrical object, and in this respect is a sort of forerunner of Archimedes' Measurement of a Circle."

"In his book on sizes and distances Aristarchus lays down these six hypotheses: 1. That the moon receives light from the sun. 2. That the earth is in the relation of a point and centre to the sphere in which the moon moves. 3. That, when the moon appears to us halved, the great circle which divides the dark and the bright portions of the moon is in the direction of our eye. 4. That, when the moon appears to us halved, its distance from the sun is then less than a quadrant by one-thirtieth of a quadrant. 5. That the breadth of the (earth's) shadow is (that) of two moons. 6. That the moon subtends one fifteenth part of a sign of the zodiac. Now the first, third, and fourth of these hypotheses practically agree with the assumptions of Hipparchus and Ptolemy. For the moon is illuminated by the sun at all times except during an eclipse, when it becomes devoid of light through passing into the shadow which results from the interception of the sun's light by the earth, and which is conical in form; next the (circle) dividing the milk-white portion which owes its colour to the sun shining upon it and the portion which has the ashen colour natural to the moon itself is indistinguishable from a great circle (in the moon) when its positions in relation to the sun cause It to appear halved, at which times (a distance of) very nearly a quadrant on the circle of the zodiac is observed (to separate them); and the said dividing circle is in the direction of our eye, for this plane of the circle if produced will in fact pass through our eye in whatever position the moon is when for the first or second time it appears halved. But, as regards the remaining hypotheses, the aforesaid mathematicians have taken a different view. For according to them the earth has the relation of a point and centre, not to the sphere in which the moon moves, but to the sphere of the fixed stars, the breadth of the (earth's) shadow is not (that) of two moons, nor does the moon's diameter subtend one fifteenth part of a sign of the zodiac, that is, 2&#176;. According to Hipparchus, on the one hand, the circle described by the moon is measured 650 times by the diameter of the moon, while the (earth's) shadow is measured by it 2&frac12; times at its mean distance in the conjunctions; in Ptolemy's view, on the other hand, the moon's diameter subtends, when the moon is at its greatest distance, a circumference of 0° 31' 20", and when at its least distance, of 0° 35' 20", while the diameter of the circular section of the shadow is, when the moon is at its greatest distance, 0° 40' 40", and when the moon is at its least distance, 0° 46'. Hence it is that the authors named have come to different conclusions as regards the ratios both of the distances and of the sizes of the sun and moon."

"Ptolemy proved in the fifth book of his Syntaxis that, if the radius of the earth is taken as the unit, the greatest distance of the moon at the conjunctions is 64 10/60 of such units, the greatest distance of the sun 1210, the radius of the moon 17/60 33/602, the radius of the sun 5 30/60. Consequently, if the diameter of the moon is taken as the unit, the earth's diameter is 3 2/5 of such units, and the sun's diameter 18 4/5. That is to say, the diameter of the earth is 3 2/5 times the diameter of the moon, while the diameter of the sun is 18 4/5 times the diameter of the moon and 5 1/2 times the diameter of the earth. From these figures the ratios between the solid contents are manifest, since the cube on 1 is 1 unit, the cube on 3 2/5 is very nearly 39 1/4 of the same units, and the cube on 18 4/5 very nearly 6644 1/2, whence we infer that, if the solid magnitude of the moon is taken as a unit, that of the earth contains 39 1/4 and that of the sun 6644 1/2 of such units; therefore the solid magnitude of the sun is very nearly 170 times greater than that of the earth."

"He Apollonius of Perga] spent a very long time with the pupils of Euclid at Alexandria, and it was thus that he acquired such a scientific habit of thought."

"The so called άναλυόμϵνος ('Treasury of Analysis') is... a special body of doctrine provided for the use of those who, after finishing the ordinary Elements, are desirous of acquiring the power of solving problems which may be set them involving (the construction of) lines, and it is useful for this alone. It is the work of three men, Euclid the author of the Elements, Apollonius of Perga, and Aristaeus the elder, and proceeds by way of analysis and synthesis."

"Analysis... takes that which is sought as if it were admitted and passes from it through its successive consequences to something which is admitted as the result of synthesis: for in analysis we assume that which is sought as if it were (already) done (ɣϵɣονός) and we inquire what it is from which this results, and again what is the antecedent cause of the latter, and so on, until by so retracing our steps we come upon something already known or belonging to the class of first principles, and such a method we call analysis as being solution backwards (άνάπαλɩν λὐσɩν)."

"But in synthesis, reversing the process, we take as already done that which was last arrived at in the analysis and, by arranging in their natural order as consequences what were before antecedents, and successively connecting them one with another, we arrive finally at the construction of what was sought; and this we call synthesis."

"Now analysis is of two kinds, the one directed to searching for the truth and called theoretical, the other directed to finding what we are told to find and called problematical. (1) In the theoretical kind we assume what is sought as if it were existent and true, after which we pass through its successive consequences, as if they too were true and established by virtue of our hypothesis, to something admitted: then (a), if that something admitted is true, that which is sought will also be true and the proof will correspond in the reverse order to the analysis, but (b), if we come upon something admittedly false, that which is sought will also be false. (2) In the problematical kind we assume that which is propounded as if it were known, after which we pass through its successive consequences, taking them as true, up to something admitted: if then (a) what is admitted is possible and obtainable, that is, what mathematicians call given, what was originally proposed will also be possible, and the proof will again correspond in reverse order to the analysis, but if (b) we come upon something admittedly impossible, the problem will also be impossible."

"waives the customary distinction between a circle, and ellipse, a parabola, and a hyperbola; these curves are simply conics, all alike. Although conics were studied by , Euclid, Archimedes and Apollonius, in the fourth and third centuries B.C., the earliest truly projective theorems were discovered by Pappus of Alexandria... and it was J. V. Poncelet... who first proved such theorems by purely projective reasoning."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green. But no Indian or Arab ever studied Pappus or cared in the least for his style or his matter. When geometry came once more up to his level, the invention of analytical methods gave it a sudden push which sent it far beyond him and he was out of date at the very moment when he seemed to be taking a new lease of life."

"Direct solutions by means of conics. Pappus gives two solutions of the trisection problem in which conics are applied directly without any preliminary reduction of the problem to a νϵῡσɩς. ...The passage of Pappus from which this solution is taken is remarkable as being one of three passages in Greek mathematical works still extant (two being in Pappus and one in a fragment of Anthemius on burning mirrors) which refer to the focus-and-directrix property of conics. The second passage in Pappus comes under the heading of Lemmas to the Surface-Loci of Euclid . Pappus there gives a complete proof of the theorem that, if the distance of a point from a fixed point is in a given ratio to its distance from a fixed line, the locus of the point is a conic section which is an ellipse, a parabola, or a hyperbola according as the given ratio is less than, equal to, or greater than, unity. The importance of these passages lies in the fact that the Lemma was required for the understanding of Euclid's treatise. We can hardly avoid the conclusion that the property was used by Euclid in his Surface-Loci, but was assumed as well known. It was, therefore, probably taken from some treatise current in Euclid's time, perhaps from Aristaeus's work on Solid Loci."

"The Porisms. Our only source of information about the nature and contents of the Porisms is Pappus. ...With Pappus's account of Porisms must be compared the passages of Proclus on the same subject. ...Proclus's definition... agees well enough with the first, the 'older', definition of Pappus. A porism occupies a place between a theorem and a problem; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle) and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. ...all the positive information which we have about the nature of a porism and the contents of Euclid's Porisms ...is obscure and leaves great scope for speculation and controversy; naturally, therefore, the problem of restoring the Porisms has had a great fascination for distinguished mathematicians ever since the revival of learning. But it has proved beyond them all."

"In the seventh book of his Collections, Pappus reports about a branch of study he calls analyomenos. We can render this name in English as "Treasury of Analysis," or as "Art of Solving Problems," or even as "Heuristic"... A good English translation of Pappus's report is easily accessible... Pappus's text is important in many ways. ...the procedures described by Pappus are by no means restricted to geometric problems; they are, in fact, not even restricted to mathematical problems. ...the paraphrase... emphasizes certain curious phrases of the original: "assume what is required to be done as already done, what is sought as found, what you have to prove is true." This is paradoxical; it is not mere self-deception..."

"Many elementary textbooks of geometry contain a few remarks about analysis, synthesis, and "assuming the problem as solved." There is little doubt that this almost ineradicable tradition goes back to Pappus, although there is hardly a current textbook whose writer would show any direct acquaintance with Pappus. The circumstance alone that it is restricted to textbooks of geometry shows a current lack of understanding..."

"One of the most celebrated geometrical problems of antiquity was the trisection of an angle. It stands side by side with those other famous problems,—the squaring of the circle and the duplication of the cube. ...modern analysis shows that the trisection of an angle is an insoluble problem if in our constructions we confine ourselves to the use of circles and straight lines. i.e. to Euclidean geometry. ...By the use of the conic sections, however, the problem is readily solved in many ways. Pappus... has given us the following beautiful reduction of the problem... "Since we can trisect a right angle," says Pappus, "it follows that the trisection of any angle can be effected if we can trisect an acute angle." ..While the geometricians quoted by Pappus could not solve the problem, Pappus himself, who lived at a time when the conic sections had been developed to some extent, fixed the position of the point E by means of an hyperbola. Pappus also claims, as his own, a solution by means of the conchoid of Nicomedes."

"Snellius, Vieta, Marinus, Ghetaldus, and Fermat, attempted, from the description of the τόωος άναλυομἐνος in the seventh book, to restore several of the lost books of Apollonius, which made a principal part of that collection. Afterwards Schooten, Dr. Wallis, and subsequently Dr. Halley, pursued the same track; and thus a number of these ancient treatises were restored, with various success indeed, and with different characters of elegance and accuracy..."

"From the defective mode of notation among the Greeks... though there be much ingenuity in some of their methods, we need not be surprised at the great inferiority of their system. Dr. Wallis remarks, that the business of the second book of Pappus appears to be nearly equivalent to what is now considered as a very simple proposition, viz. that the multiplication of any numbers... The first book he with much probability conjectures to have been employed about the simple operations of the addition and subtraction of numbers."

"The superior lines treated of by Pappus, and other ancient writers, were the conchoid, the cissoid, the spiral, and the quadratrix; and a few others are slightly alluded to."

"As a writer, Pappus must have been quite versatile if the following list of works attributed to him is any indication: (1) Description of the World. (2) Comments on the Four Books of the Almagest. (3) Interpretation of Dreams. (4) On the Rivers of Libya. (5) Commentary on the Analemma of Diodorus. (6) Comments on Euclid's Elements. (7) Comments on Ptolemy's Harmonica. (8) Collection. Of all these the only one extant even in part is the Collection, which is a summary in eight books of the principal works of preceding Greek mathematicians with comments and lemmas on the works in question."

"One thing... ought to be emphasized in any discussion relative to the Collection and that is its remarkable suggestiveness. In order to understand this, one has only to turn the works of such men as Chasles, [Siegmund] Günther, Descartes, Newton, and Steiner, for in the writings of these men it furnished the basic ideas for analytic geometry, projective geometry, and other allied theories. And if it had so much offer these men, it ought to furnish some suggestions to careful reader of today."

"I have given these very simple [methods] to show that it is possible to construct all the problems of ordinary geometry by doing no more than the little covered in the four figures that I have explained. This is one thing which I believe the ancient mathematicians did not observe, for otherwise they would not have put so much labor into writing so many books in which the very sequence of the propositions shows that they did not have a sure method of finding all, but rather gathered together those propositions on which they had happened by accident."

"This is also evident from what Pappus has done in the beginning of his seventh book, where... he refers to a question which he says that neither Euclid nor Apollonius nor any one else had been able to solve completely..."

"Pappus proceeds as follows: ...If three straight lines are given in position, and if straight lines be drawn from one and the same point, making given angles with three given lines; and if there be given the ratio of the rectangle contained by two of the lines so drawn to the square of the other, the point lies on a solid locus given in position, namely, one of the three conic sections Again, if lines be drawn making given angles with four straight lines given in position, and if the rectangle of two of the lines so drawn bears a given ratio to the rectangle of the other two; then, in like manner, the point lies on a conic section given in position. It has been shown that to only two lines there corresponds a plane locus. But if there be given four lines, the point generates loci not known up to the present time (that is, impossible to determine by common methods), but merely called 'lines'. It is not clear what they are, or what their properties. One of them, not the first but the most manifest, has been examined, and this has proved to be helpful. These, however, are the propositions concerning them. If from any point straight lines be drawn making given angles with five straight lines given in position, and if the solid rectangular parallelepiped contained by three of the lines so drawn bears a given ratio to the sold rectangular parallelepiped contained by the other two and any given line whatever, the point lies on a 'line' given in position. Again, if there be six lines, and if the solid contained by three of the lines bears a given ratio to the solid contained by the other three lines, the point also lies on a 'line' given in position. But if there be more than six lines, we cannot say whether a ratio of something contained by four lines is given to that which is contained by the rest, since there is no figure of four dimensions."

"The question, then, the solution of which... was completed by no one, is this: Having three, four or more lines given in position, it is first required to find a point from which as many other lines may be drawn, each making a given angle with one of the given lines, so that the rectangle of two of the lines so drawn shall bear a given ratio to the square of the third (if there be only three); or to the rectangle of the other two (if there be four), or again, that the parallelepiped constructed upon three shall bear a given ratio to that upon the other two and any given line (if there be five); or to the parallelepiped upon the other three (if there be six); or (if there be seven) that the product obtained by multiplying four of them together shall bear a given ratio to the product of the other three, or (if there be eight) that the product of four of them shall bear a given ratio to the product of the other four. Thus the question admits of extension of any number of lines."

"Since there is always an infinite number of different points satisfying these requirements, it is also required to discover and trace the curve containing all such points. Pappus says that when there are only three or four lines given, this line is one of the three conic sections, but he does not undertake to determine, describe, or explain the nature of the line required when the question involves a greater number of lines. He only adds that the ancients recognized one of them which they had shown to be useful, and which seemed the simplest, and yet was not the most important. This led me to find out whether, by my own method, I could go as far as they had gone."

"Pappus, probably born about 340 A.D., in Alexandria, was the last great mathematician of the Alexandrian school. His genius was inferior to that of Archimedes, Apollonius, and Euclid, who flourished over 500 years earlier. But living, as he did, at a period when interest in geometry was declining, he towered above his contemporaries "like the peak of Teneriffa above the Atlantic." He is the author of a Commentary on the Almagest, a Commentary on Euclid's Elements, a Commentary on the Analemma of Diodorus,—a writer of whom nothing is known. All these works are lost. Proclus, probably quoting from the Commentary on Euclid, says that Pappus objected to the statement that an angle equal to a right angle is always itself a right angle."

"The only work of Pappus still extant is his Mathematical Collections. This was originally in eight books, but the first and portions of the second are now missing. The Mathematical Collections seems to have been written by Pappus to supply the geometers of his time with a succinct analysis of the most difficult mathematical works and to facilitate the study of them by explanatory lemmas. But these lemmas are selected very freely, and frequently have little or no connection with the subject on hand. However, he gives very accurate summaries of the works of which he treats. The Mathematical Collections is invaluable to us on account of the rich information it gives on various treatises by the foremost Greek mathematicians, which are now lost. Mathematicians of the last century considered it possible to restore lost works from the résumé by Pappus alone."

"We shall now cite the more important of those theorems in the Mathematical Collections which are supposed to be original with Pappus. First of all ranks the elegant theorem re-discovered by Guldin, over 1000 years later, that the volume generated by the revolution of a plane curve which lies wholly on one side of the axis, equals the area of the curve multiplied by the circumference described by its centre of gravity."

"Pappus proved... that the centre of gravity of a triangle is that of another triangle whose vertices lie upon the sides of the first and divide its three sides in the same ratio."

"In the fourth book are new and brilliant propositions on the quadratrix which indicate an intimate acquaintance with curved surfaces. He generates the quadratrix..."

"Pappus considers curves of double curvature still further. He produces a spherical spiral by a point moving uniformly along the circumference of a great circle of a sphere, while the great circle itself revolves uniformly around its diameter. He then finds the area of that portion of the surface of the sphere determined by the spherical spiral..."

"A question which was brought into prominence by Descartes and Newton is the "problem of Pappus." Given several straight lines in a plane, to find the locus of a point such that when perpendiculars (or, more generally, straight lines at given angles) are drawn from it to the given lines, the product of certain ones of them shall be in a given ratio to the product of the remaining ones."

"It was Pappus who first found the focus of the parabola, suggested the use of the directrix, and propounded the theory of the involution of points."

"He solved the problem to draw through three points lying in the same straight line, three straight lines which shall form a triangle inscribed in a given circle."

"He is known in three instances to have copied theorems without giving due credit and that he may have done the same tiling in other cases in which we have no data by which to ascertain the real discoverer"

"In Pascal's wonderful work on conics, written at the age of sixteen and now lost, were given the theorem on the anharmonic ratio, first found in Pappus."

"The first important example solved by Descartes in his geometry is the "problem of Pappus"; viz. "Given several straight lines in a plane, to find the locus of a point such that the perpendiculars, or more generally, straight lines at given angles, drawn from the point to the given lines, shall satisfy the condition that the product of certain of them shall be in a given ratio to the product of the rest. Of this celebrated problem, the Greeks solved only the special case when the number of given lines is four, in which case the locus of the point turns out to be a conic section. By Descartes it was solved completely, and it afforded an excellent example of the use which can be made of his analytical method in the study of loci. Another solution was given later by Newton in the Principia."

"Ptolemy had shewn not only that geometry could be applied to astronomy, but had indicated how new methods of analysis like trigonometry might be thence developed. He found however no successors to take up the work he had commenced so brilliantly, and we must look forward 150 years before we find another geometrician of any eminence. That geometrician was Pappus... We know that he had numerous pupils, and it is probable that he temporarily revived an interest in the study of geometry."

"Pappus wrote several books, but the only one which has come down to us is his Συναɣωɣή, a collection of mathematical papers arranged in eight books of which the first and part of the second have been lost. This collection was intended to be a synopsis of Greek mathematics together with comments and additional propositions... it is trustworthy, and we rely largely on it for our knowledge of other works now lost. ...it is most likely that it gives roughly the order in which the classical authors were read at Alexandria. Probably the first book, which is now lost. was on arithmetic. The next four books deal with geometry exclusive of conic sections; the sixth with astronomy including, as subsidiary subjects, optics and trigonometry; the seventh with analysis, conics, and porisms; and the eighth with mechanics."

"The last two books contain a good deal of original work by Pappus ...it should be remarked that in two or three cases he has been detected in appropriating proofs from earlier authors, and it is possible he may have done this in other cases. Subject to this suspicion we may say that Pappus's best work is in geometry."

"He discovered the directrix in the conic sections, but he investigated only a few isolated properties: the earliest comprehensive account was given by Newton and Boscovich. As an illustration of his power I may mention that he solved [book VII, prop. 107] the problem to inscribe in a given circle a triangle whose sides produced shall pass through three collinear points. This question was in the eighteenth century generalised by Cramer... It was sent in 1742 as a challenge to Castillon, and in 1776 he published a solution. Lagrange, Euler, Lhulier, Fuss, and Lexell also gave solutions in 1780. A few years later the problem was set to a Neapolitan lad A. Giordano, who was only 16 but... he extended it to the case of a polygon of n sides which pass through n given points and gave a solution both simple and elegant. Poncelet extended it to conics of any species and subject to other restrictions."

"In mechanics Pappus shewed that the centre of mass of a triangular lamina is the same as that of an inscribed triangular lamina whose vertices divide each of the sides of the original triangle in the same ratio. He also discovered the two theorems on the surface and volume of a solid of revolution which are still quoted in text-books under his name: these are that the volume generated by the revolution of a curve about an axis is equal to the product of the area of the curve and the length of the path described by its centre of mass; and the surface is equal to the product of the perimeter of the curve and the length of the path described by its centre of mass."

"His work as a whole and his comments shew that he was a geometrician of power; but it was his misfortune to live at a time when but little interest was taken in geometry, and when the subject, as then treated, had been practically exhausted."

"Perhaps the topic [of this book] will appear fairly difficult to you because it is not yet familiar knowledge and the understanding of beginners is easily confused by mistakes; but with your inspiration and my teaching it will be easy for you to master, because clear intelligence supported by good lessons is a fast route to knowledge."

"If we arrive at an equation containing on each side the same term but with different coefficients, we must take equals from equals until we get one term equal to another term. But, if there are on one or on both sides negative terms, the deficiencies must be added on both bides until all the terms on both sides are positive. Then we must take equals from equals until one term is left on each side."

"As a square number is known to be the product of a number multiplied by itself, so every polygonal number, multiplied by one number and added to another, both of which depend upon the number of its angles, produces a square number. I shall prove this, and shall show also how from a given side to find its polygon and conversely. Some auxiliary propositions must first be proved."

"The solution in integers, or in rational numbers, of indeterminate equations belongs to diophantine analysis. The name honors Diophantus, whose treatise of thirteen books, of which only six survive, was the first on the subject. The Latin translation (A.D. 1621) of this suggestive fragment directly inspired Fermat to his creation of the modern higher arithmetic. It also inspired something much less desirable. Diophantus contented himself with special solutions of his problems; the majority of his numerous successors have done likewise, until diophantine analysis today is choked by a jungle of trivialities bearing no resemblance to cultivated mathematics. It is long past time that the standards of Diaphantus be forgotten though he himself be remembered with becoming reverence."

"This work of Diaphantus... was the first Greek mathematics, if indeed it was Greek, to show a genuine talent for algebra. ...He had begun to use symbols operationally. This long stride forward is all the more remarkable because his algebraic notation... was almost as awkward as Greek logistic. That he accomplished what he did with the available techniques places him beyond question among the great algebraists."

"I have decided first to consider the majority of the authors who up to now have written about [algebra], so that I can fill in what they have missed out. They are very many, and among them Mohammed ibn Musa [Al-Khwarizmi], an Arab, is believed to be the first [...] I believe that the word “algebra” came from him, because some years ago, Brother Luca [Pacioli] of Borgo San Sepolcro of the Minorite order, having set himself the task of writing on this science, as much in Latin as in Italian, said that the word “algebra” was Arabic [...] and that the science came from the Arabs. Many who have written after him have believed and said likewise, but in recent years, a Greek work on this discipline has been discovered in the Library of our Lord in the Vatican, composed by a certain Diophantus of Alexandria, a Greek author [...] Antonio Maria Pazzi and I have translated five books (of the seven) [...] In this work we have found that he cites the Indian authors many times, and thus I have been made aware that this discipline belonged to the Indians before the Arabs."

"Admitting the Hindu and Alexandrian authors [such as Diophantus], to be nearly equally ancient, it must be conceded in favor of the Indian algebraist, that he was more advanced in the science […] In the whole science [of algebra], he [Diophantus] is very far behind the Hindu writers […] he is hardly to be considered as the inventor, since he seems to treat the art as already known."

"If his works were not written in Greek, no one would think for a moment that they were the product of Greek mind. There is nothing in his works that reminds us of the classic period of Greek mathematics. His were almost entirely new ideas on a new subject. In the circle of Greek mathematicians he stands alone in his specialty. Except for him, we should be constrained to say that among the Greeks algebra was almost an unknown science."

"In this work [Arithmetica] is introduced the idea of an algebraic equation expressed in algebraic symbols. His treatment is purely analytical and completely divorced from geometrical methods."

"He appears to be the first who could perform such operations as (x - 1) \times (x - 2) without reference to geometry. Such identities as (a + b)^2 = a^2 + 2ab + b^2, which with Euclid appear in the elevated rank of geometric theorems, are with Diophantus the simplest consequences of the algebraic laws of operation."

"In the solution of simultaneous equations Diophantus adroitly managed with only one symbol for the unknown quantities and arrived at answers, most commonly, by the method of tentative assumption, which consists in assigning to some of the unknown quantities preliminary values, that satisfy only one or two of the conditions. These values lead to expressions palpably wrong, but which generally suggest some stratagem by which values can be secured satisfying all the conditions of the problem."

"Greek algebra before Diophantus was essentially rhetorical."

"The "porisms" of Pappus anticipated projective geometry, the problems of Diophantus prepared the ground for the modern theory of equations."

"Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle."

"Fermat, commenting about 1637 on Diophantus II, 8 (to solve x2 + y2 = a2), stated that "it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain." This theorem is known as Fermat's last theorem."

"The first algebraist known to us, the Greek Diophantus (360 A.D.), antedates Aryabhata by a century; but Cajori believes that he took his lead from India."

"Diophantus himself, it is true, gives only the most special solutions of all the questions which he treats, and he is generally content with indicating numbers which furnish one single solution. But it must not be supposed that his method was restricted to these very special solutions. In his time the use of letters to denote undetermined numbers was not yet established, and consequently the more general solutions which we are now enabled to give by means of such notation could not be expected from him. Nevertheless, the actual methods which he uses for solving any of his problems are as general as those which are in use to-day; nay, we are obliged to admit, that there is hardly any method yet invented in this kind of analysis of which there are not sufficiently distinct traces to be discovered in Diophantus."

"There is scarcely any one who states purely arithmetical questions, scarcely any who understands them. Is this not because arithmetic has been treated up to this time geometrically rather than arithmetically? This certainly is indicated by many works ancient and modern. Diophantus himself also indicates this. But he has freed himself from geometry a little more than others have, in that he limits his analysis to rational numbers only; nevertheless the Zetcica of Vieta, in which the methods of Diophantus are extended to continuous magnitude and therefore to geometry, witness the insufficient separation of arithmetic from geometry. Now arithmetic has a special domain of its own, the theory of numbers. This was touched upon but only to a slight degree by Euclid in his Elements, and by those who followed him it has not been sufficiently extended, unless perchance it lies hid in those books of Diophantus which the ravages of time have destroyed. Arithmeticians have now to develop or restore it. To these, that I may lead the way, I propose this theorem to be proved or problem to be solved. If they succeed in discovering the proof or solution, they will acknowledge that questions of this kind are not inferior to the more celebrated ones from geometry either for depth or difficulty or method of proof: Given any number which is not a square, there also exists an infinite number of squares such that when multiplied into the given number and unity is added to the product, the result is a square."

"Many scholars go right through the period of their mathematical education without being introduced to indeterminate (Diophantine) problems. This should seem somewhat strange to us when we reflect on how many real problems of life require solutions which are meaningless unless they are whole numbers. The absence of Diophantine analysis from school curricula is difficult to justify. The methods... are not beyond the capabilities of schoolchildren. It is true that the problems often involve tedious calculations, but this should not be used as an excuse for ignoring the valuable principles of this analysis. Diophantus himself was usually content with finding just one solution... But once one solution has been obtained, the general solution can readily be obtained from it."

"In 130 indeterminate equations, which Diophantus treats, there are more than 50 different classes... It is therefore difficult for a modern, after studying 100 Diophantic equations, to solve the 101st; and if we have made the attempt, and after some vain endeavours read Diophantus' own solution, we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a quick turn reaches the goal, often enough a goal with reaching which we should not be content; we expected to have to climb a toilsome path, but to be rewarded at the end by an extensive view; instead of which, our guide leads by narrow, strange, but smooth ways to a small eminence; he has finished! He lacks the calm and concentrated energy for a deep plunge into a single important problem; and in this way the reader also hurries with inward unrest from problem to problem, as in a game of riddles, without being able to enjoy the individual one. Diophantus dazzles more than he delights. He is in a wonderful measure shrewd, clever, quick-sighted, indefatigable, but does not penetrate thoroughly or deeply into the root of the matter. As his problems seem framed in obedience to no obvious scientific necessity, but often only for the sake of the solution, the solution itself also lacks completeness and deeper signification. He is a brilliant performer in the art of indeterminate analysis invented by him, but the science has nevertheless been indebted, at least directly, to this brilliant genius for few methods, because he was deficient in the speculative thought which sees in the True more than the Correct. That is the general impression which I have derived from a thorough and repeated study of Diophantus' arithmetic."

"The geometrical algebra of the Greeks has been in evidence all through our history from Pythagoras downwards, and no more need be said of it here except that its arithmetical application was no new thing to Diophantus. It is probable, for example, that the solution of the quadratic equation, discovered first by geometry, was applied for the purpose of finding numerical values for the unknown as early as Euclid, if not earlier still. In Heron the numerical solution of equations is well established, so that Diophantus was not the first to treat equations algebraically. What he did was to take a step forward towards an algebraic notation."

"In … a series of lectures at the University of Padua in 1464, he [Regiomontanus] introduced the idea that Arabic algebra descended from Diophantus’s Arithmetica. This heralded the initiation of a myth cultivated by humanists for centuries. Diophantus … became the alleged origin of European algebra. … By overrating the importance of Diophantus … humanist writers created a new mythical identity of European mathematics."

"The creation of the formal language of mathematics is identical with the foundation of modern algebra. ...As far as Greek sources are concerned, the special influence of the Arithmetic of Diophantus on the content, but even more so on the form, of this Arabic science is unmistakable. ...concurrently with the elaboration... of the theory of equations which the Arabs had passed on to the West, the original text of Diophantus began, as early as the fifteenth century, to become well known and influential. But it was not until the last quarter of the sixteenth century that Vieta undertook to modify Diophantus' technique in a really critical way. He thereby became the true founder of modern mathematics."

"The Alexandrians used fractions as numbers in their own right, whereas the mathematicians of the classical period spoke only of a ratio of integers, not of parts of a whole and the ratios were used only in proportions. However, even in the classical period genuine fractions... as entities in their own right, were used in commerce. In the Alexandrian period, Archimedes, Heron, Diophantus, and others used fractions freely and performed operations with them. Though... they did not discuss the concept of fractions, apparently these were intuitively sufficiently clear..."

"Another feature of Alexandrian algebra is the absence of any explicit deductive structure. The various types of numbers... were not defined. Nor was there any axiomatic basis on which a deductive structure could be erected. The work of Heron, Nichomachus, and Diophantus, and of Archimedes as far as his arithmetic is concerned, reads like the procedural texts of the Egyptians and Babylonians... The deductive, orderly proof of Euclid and Apollonius, and of Archimedes' geometry is gone. The problems are inductive in spirit, in that they show methods for concrete problems that presumably apply to general classes whose extent is not specified. In view of the fact that as a consequence of the work of the classical Greeks mathematical results were supposed to be derived deductively from an explicit axiomatic basis, the emergence of an independent arithmetic and algebra with no logical structure of its own raised what became one of the great problems of the history of mathematics. This approach to arithmetic and algebra is the clearest indication of the Egyptian and Babylonian influences... Though the Alexandrian Greek algebraists did not seem to be concerned about this deficiency... it did trouble deeply the European mathematicians."

"Bachet de Méziriac remarked that any number (that is, positive integer) is either a square, or the sum of two, three, or four squares. He did not pretend to possess a proof. He found indications pointing to his statement in certain problems of Diophantus and verified it up to 325. In short, Bachet's statement was just a conjecture, found inductively. ...his main achievement was to put the question: How many squares are needed to represent all integers? Once this question is squarely put, there is not much difficulty in discovering the answer inductively."

"No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra."

"For not more than six books [of Diophantus] are found, though in the proëmium he promises thirteen. If this book, a wonderful and difficult work, could be found entire, I should like to translate it into Latin, for the knowledge of Greek I have lately acquired would suffice for this."

"The Greek mathematician Diophantus of Alexandria... it seems fairly probable... flourished about 250 A.D. ...We gather from the Arithmeticas introduction that it originally consisted of thirteen Books. Of these, only six have survived until now in Greek... The remaining seven were considered irretrievably lost until the recent discovery of four other, hitherto unknown books in Arabic translation, which, since it is attributed to Qustā ibn Lūqā, must have been made around or after the middle of the ninth century."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"But how much more splendid the methods which reduce the problems which seem to be hardly capable of solution even with the help of surds in such a way that, while the surds, when bidden (so to speak) to plough the arithmetic soil, become true to their name and deaf to entreaty, they are not so much as mentioned in these most ingenious solutions."

"Diophantos' work is so unique among the Greek treatises which we possess, that he cannot be said to recall the style or subject-matter of any other author, except, indeed, in the fragment on Polygonal Numbers; and even there the reference to Hypsikles is the only indication we can lay hold of."

"Diophantos lived in a period when the Greek mathematicians of great original power had been succeeded by a number of learned commentators, who confined their investigations within the limits already reached, without attempting to further the development of the science. To this general rule there are two most striking exceptions, in different branches of mathematics, Diophantos and Pappos. These two mathematicians, who would have been an ornament to any age, were destined by fate to live and labour at a time when their work could not check the decay of mathematical learning. There is scarcely a passage in any Greek writer where either of the two is so much as mentioned. The neglect of their works by their countrymen and contemporaries can be explained only by the fact that they were not appreciated or understood."

"The reason why Diophantos was the earliest of the Greek mathematicians to be forgotten is also probably the reason why he was the last to be re-discovered after the Revival of Learning. The oblivion, in fact, into which his writings and methods fell is due to the circumstance that they were not understood. That being so, we are able to understand why there is so much obscurity concerning his personality and the time at which he lived. Indeed, when we consider how little he was understood, and in consequence how little esteemed, we can only congratulate ourselves that so much of his work has survived to the present day."

"Between the time of Regiomontanus and that of Rafael Bombelli Diophantos was once more forgotten, or rather unknown. The Algebra of Bombelli appeared in 1572, and in the preface to this work the author tells us that he had recently discovered a Greek book on Algebra in the Vatican Library, written by a certain Diofantes... Though Bombelli did not carry out his plan of publishing Diophantos in a translation, he has nevertheless taken all the problems of Diophantos' first four Books and some of those of the fifth, and embodied them in his Algebra, interspersing them with his own problems. ...notation in this work of Bombelli's ...shows ...some advance upon that of Diophantos."

"With the help of books only he Wilhelm Xylander] studied the subject of Algebra, as far as was possible from what men like Cardan had written and by his own reflection, with such success that not only did he fall into what Herakleitos called... the conceit of "being somebody" in the field of Arithmetic and "Logistic," but others too who were themselves learned men thought him an arithmetician of exceptional merit. But when he first became acquainted with the problems of Diophantos his pride had a fall so sudden and so humiliating that he might reasonably doubt whether he ought previously to have bewailed, or laughed at himself. He considers it therefore worth while to confess publicly in how disgraceful a condition of ignorance he had previously been content to live, and to do something to make known the work of Diophantos, which had so opened his eyes."

"It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. ...Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4 = 4x + 20 as ᾰτοπος because it would give x = -4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly."

"The history of Alexandrian mathematics begins with the Elements of Euclid and closes with the Algebra of Diophantus, both of which are founded on the discoveries of several preceding centuries."

"The solution of the higher indeterminates depends almost entirely on very favourable numerical conditions and his methods are defective. But the extraordinary ability of Diophantus appears rather in the other department of his art, namely the ingenuity with which he reduces every problem to an equation which he is competent to solve."

"Diophantus shows great Adroitness in selecting the unknown, especially with a view to avoiding an adfected quadratic. ...The most common and characteristic of Diophantus' methods is his use of tentative assumptions which is applied in nearly every problem of the later books. It consists in assigning to the unknown a preliminary value which satisfies one or two only of the necessary conditions, in order that, from its failure to satisfy the remaining conditions, the operator may perceive what exactly is required for that purpose. ...a third characteristic of Diophantus [is] ...the use of the symbol for the unknown in different senses. ...The use of tentative assumptions leads again to another device which may be called... the method of limits. This may best be illustrated by a particular example. If Diophantus wishes to find a square lying between 10 and 11, he multiplies these numbers by successive squares till a square lies between the products. Thus between 40 and 44, 90 and 99 no square lies, but between 160 and 176 there lies the square 169. Hence x^2 = \tfrac{169}{16} will lie between the proposed limits."

"Sometimes... Diophantus solves a problem wholly or in part by synthesis. ...Although ...Diophantus does not treat his problems generally and is usually content with finding any particular numbers which happen to satisfy the conditions of his problems, ...he does occasionally attempt such general solutions as were possible to him. But these solutions are not often exhaustive because he had no symbol for a general coefficient."

"Though the defects in Diophantus' proofs are in general due to the limitation of his symbolism, it is not so always. Very frequently indeed Diophantus introduces into a solution arbitrary conditions and determinations which are not in the problem. Of such "fudged" solutions, as a schoolboy would call them, two particular kinds are very frequent. Sometimes an unknown is assumed at a determinate value... Sometimes a new condition is introduced."

"The Arithmetica... is deficient, sometimes pardonably, sometimes without excuse, in generalization. The book of Porismata, to which Diophantus sometimes refers, seems on the other hand to have been entirely devoted to the discussion of general properties of numbers. It is three times expressly quoted in the Arithmetica... Of all these propositions he says... 'we find it in the Porisms'; but he cites also a great many similar propositions without expressly referring to the Porisms. These latter citations fall into two classes, the first of which contains mere identities, such as the algebraical equivalents of the theorems in Euclid II. ...The other class contains general propositions concerning the resolution of numbers into the sum of two, three or four squares. ...It will be seen that all these propositions are of the general form which ought to have been but is not adopted in the Arithmetica. We are therefore led to the conclusion that the Porismata, like the pamphlet on Polygonal Numbers, was a synthetic and not an analytic treatise. It is open, however, to anyone to maintain the contrary, since no proof of any porism is now extant."

"With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress. Diophantus also represents the outbreak of a movement which probably was not Greek in its origin, and which the Greek genius long resisted, but which was especially adapted to the tastes of the people who, after the extinction of Greek schools, received their heritage and kept their memory green."

"It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantaneously destroyed and shall remain forever exposed to the play of the eternal waves."

"Not much younger than these (sc. Hermotimus of Colophon and Philippus of Mende) is Euclid, who put together the Elements, collecting many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefragable demonstration the things which were only somewhat loosely proved by his predecessors. This man lived in the time of the first Ptolemy. For Archimedes, who came immediately after the first (Ptolemy), makes mention of Euclid: and, further, they say that Ptolemy once asked him if there was in geometry any shorter way than that of the elements, and he answered that there was no royal road to geometry. He is then younger than pupils of Plato but older than Eratosthenes and Archimedes; for the latter were contemporary with one another, as Eratosthenes somewhere says."

"This, therefore, is mathematics: she reminds you of the invisible form of the soul; she gives life to her own discoveries; she awakens the mind and purifies the intellect; she brings light to our intrinsic ideas; she abolishes oblivion and ignorance which are ours by birth."

"If we listen to those who like to record antiquities, we shall find them attributing this theorem to Pythagoras and saying that he sacrificed an ox on its discovery. For my part, though I marvel at those who first noted the truth of this theorem, I admire more the author of the Elements for the very lucid proof by which he made it fast."

"The Platonic doctrine of Ideas has been, in all ages, the derision of the vulgar, and the admiration of the wife. Indeed, if we consider that ideas are the most sublime objects of speculation, and that their nature is no less bright in itself, than difficult to investigate, this opposition in the conduct of mankind will be natural and necessary; for, from our connection with a material nature, our intellectual eye, previous to the irradiations of science, is as ill adapted to objects the most splendid of all, "as the eyes of bats to the light of day.""

"The mathematician speculates the causes of a certain sensible effect, without considering its actual existence; for the contemplation of universals excludes the knowledge of particulars; and he whose intellectual eye is fixed on that which is general and comprehensive, will think but little of that which is sensible and singular."

"Let us now explain the origin of geometry, as existing in the present age of the world. For the demoniacal Aristotle observes, that the same opinions often subsist among men, according to certain orderly revolutions of the world: and that sciences did not receive their first constitution in our times, nor in those periods which are known to us from historical tradition, but have appeared and vanished again in other revolutions of the universe; nor is it possible to say how often this has happened in past ages, and will again take place in the future circulations of time. But, because the origin of arts and sciences is to be considered according to the present revolution of the universe, we must affirm, in conformity with the most general tradition, that geometry was first invented by the Egyptians, deriving its origin from the mensuration of their fields: since this, indeed, was necessary to them, on account of the inundation of the Nile washing away the boundaries of land belonging to each. Nor ought It to seem wonderful, that the invention of this as well as of other sciences, should receive its commencement from convenience and opportunity. Since whatever is carried in the circle of generation proceeds from the imperfect to the perfect."

"A transition, therefore, is not undeservedly made from sense to consideration, and from this to the nobler energies of intellect. Hence, as the certain knowledge of numbers received its origin among the Phœnicians, on account of merchandise and commerce, so geometry was found out among the Egyptians from the distribution of land. When Thales, therefore, first went into Egypt, he transferred this knowledge from thence into Greece: and he invented many things himself, and communicated to his successors the principles of many. Some of which were, indeed, more universal, but others extended to sensibles."

"But after these, Pythagoras changed that philosophy, which is conversant about geometry itself, into the form of a liberal doctrine, considering its principles in a more exalted manner; and investigating its theorems immaterially and intellectually; who likewise invented a treatise of such things as cannot be explained in geometry, and discovered the constitution of the mundane figures."

"After Pythagoras, Anaxagoras the Clazomenian succeeded, who undertook many things pertaining to geometry. And Oenopides the Chian, was somewhat junior to Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained mathematical glory. To these succeeded Hippocrates, the Chian, who invented the quadrature of the lunula, and Theodorus the Cyrenean, both of them eminent in geometrical knowledge. For the first of these, Hippocrates composed geometrical elements: but Plato, who was posterior to these, caused as well geometry itself, as the other mathematical disciplines, to receive a remarkable addition, on account of the great study he bestowed in their investigation. This he himself manifests, and his books, replete with mathematical discourses, evince: to which we may add, that he every where excites whatever in them is wonderful, and extends to philosophy. But in his time also lived Leodamas the Thasian, Architas the Tarentine, and Theætetus the Athenian; by whom theorems were increased, and advanced to a more skilful constitution. But Neoclides was junior to Leodamas, and his disciple was Leon; who added many things to those thought of by former geometricians. So that Leon also constructed elements more accurate, both on account of their multitude, and on account of the use which they exhibit: and besides this, he discovered a method of determining when a problem, whose investigation is sought for, is possible, and when it is impossible."

"But Eudoxus the Cnidian, who was somewhat junior to Leon, and the companion of Plato, first of all rendered the multitude of those theorems which are called universals more abundant; and to three proportions added three others; and things relative to a section, which received their commencement from Plato, he diffused into a richer multitude, employing also resolutions in the prosecution of these."

"Again, Amyclas the Heracleotean, one of Plato's familiars, and Menæchmus, the disciple, indeed, of Eudoxus, but conversant with Plato, and his brother Dinostratus, rendered the whole of geometry as yet more perfect. But Theudius, the Magnian, appears to have excelled, as well in mathematical disciplines, as in the rest of philosophy. For he constructed elements egregiously, and rendered many particulars more universal. Besides, Cyzicinus the Athenian, flourished at the same period, and became illustrious in other mathematical disciplines, but especially in geometry. These, therefore, resorted by turns to the Academy, and employed themselves in proposing common questions."

"But Hermotimus, the Colophonian, rendered more abundant what was formerly published by Eudoxus and Theætetus, and invented a multitude of elements, and wrote concerning some geometrical places. But Philippus the Mendæan, a disciple of Plato, and by him inflamed in the mathematical disciplines, both composed questions, according to the institutions of Plato, and proposed as the object of his enquiry whatever he thought conduced to the Platonic philosophy."

"And thus far historians produce the perfection of this science. But Euclid was not much junior to these, who collected elements, and constructed many of those things which were invented by Eudoxus; and perfected many which were discovered by Theætetus. Besides, he reduced to invincible demonstrations, such things as were exhibited by others with a weaker arm. But he lived in the times of the first Ptolemy: for Archimedes mentions Euclid, in his first book, and also in others. Besides, they relate that Euclid was asked by Ptolomy, whether there was any shorter way to the attainment of geometry than by his elementary institution, and that he answered, there was no other royal path which led to geometry. Euclid, therefore, was junior to the familiars of Plato, but more ancient than Eratosthenes and Archimedes (for these lived at one and the same time, according to the tradition of Eratosthenes) but he was of the Platonic sect, and familiar with its philosophy: and from hence he appointed the constitution of those figures which are called Platonic, as the end of his elementary institutions."

"For this, to draw a right line from every point, to every point, follows the definition, which says, that a line is the flux of a point, and a right line an indeclinable and inflexible flow."

"If two right lines cut one another, they will form the angles at the vertex equal. ... This... is what the present theorem evinces, that when two right lines mutually cut each other, the vertical angles are equal. And it was first invented according to Eudemus by Thales..."

"To a given right line to apply a parallelogram equal to a given triangle in an angle which is equal to a given right lined angle. According to the Familiars of Eudemus, the inventions respecting the application, excess, and defect of spaces, is ancient and belongs to the Pythagoric muse. But junior mathematicians receiving names from these, transferred them to the lines which are called conic, because one of these they denominate a parabola, but the other an hyperbola, and the third an ellipsis; since, indeed these ancient and divine men, in the plane description of spaces on a terminated right line, regarded the things indicated by these appellations. For when a right line being proposed, you adapt a given space to the whole right line, then that space is said to be applied, but when you make the longitude of the space greater than that of the right line, then the space is said to exceed; but when less, so that some part of the right line is external to the described space, then the space is said to be deficient. And after this manner, Euclid, in the sixth book, mentions both excess and defect. But in the present problem he requires application..."

"The person who has found him is unable to tell this to others as he has seen it, for the discovery is not made by the soul who makes a statement, but by the soul who is initiated in and lies outstretched towards the divine light, not moving with its own movement, but keeping its own silence as it were. For if it is by nature not able to grasp the essential nature of other realities either by name or by a defining proposition or by scientific knowledge, but by intuitive thought (noêsis) alone, as he himself says in the Letters, how could it discover the essential nature of the Demiurge in any other way than intuitively (noerôs)? How could the soul, having found him in this way, be able to report what it had seen by means of nouns and verbs and convey this to others? After all, because discursive thought proceeds through combination, it is unable to express the nature that is unified and simple. ... If discovery takes place by the soul who keeps silent, how could the flow of language through the mouth be sufficient to bring to light the essential nature of what has been discovered?"

"He is verbose and dull, but luckily he has preserved for us quotations from other and better authorities."

"The thought of Proclus towered over the entire philosophy of his time as the last great system of Greco-Roman speculation, and offers our thought the dual value of the most elaborate solution to all problems, not only of the Neoplatonic school but of classical philosophy and the form in which it communicated almost immediately to Christian thought in the Middle Ages and the modern age. (Le scuole neoplatoniche, “'The Neoplatonic Schools”', XXXVII, p. 222)"

"Neoplatonic philosophy finds in Proclus the satisfaction of a systematic need of an analytical and deductive nature, while the form of the systematisation proper to Plotinus's Enneads was instead methodical-didactic; therefore, in the history of Neoplatonism and the Alexandrian school, it must be admitted that the full historical awareness of the value and significance of the school came precisely from this philosopher. (Le scuole neoplatoniche, “'The Neoplatonic Schools”', XXXVIII, pp. 228-229)."

"What Science can be more accurate than Geometry? What Science can afford Principles more evident, more certain, yea I will add, more simple than Geometrical Axioms, or exercises a more strictly accurate Logic in drawing its Conclusions? But Aristotle and Proclus affirm that Unity (they had more rightly said Numbers) the Principle of Arithmetic, is more simple than a Point which is the Principle of Geometry, or rather of Magnitude. Because a Point implies Position, but Unity does not. A Point, says Aristotle, and Unity are not to be divided, as Quantity: Unity requires no Position, a Point does. But this Comparison of a Point in Geometry with Unity in Arithmetic is of all the most unsufferable, and derives the worst Consequences upon Mathematical Learning."

"It is well known that the commentary of Proclus on Eucl. Book I is one of the two main sources of information as to the history of Greek geometry which we possess, the other being the Collection of Pappus."

"We shall often... have occasion to quote from the so-called 'Summary' of Proclus... Occupying a few pages of Proclus's Commentary on Euclid, Book I, it reviews, in the briefest possible outline, the course of Greek geometry from the earliest times to Euclid, with special reference to the evolution of the Elements. At one time it was often called the 'Eudemian summary', on the assumption that it was an extract from the great History of Geometry in four Books by Eudemus, the pupil of Aristotle. But a perusal of the summary itself is sufficient to show that it cannot have been written by Eudemus; the most that can be said is that, down to a certain sentence, it was probably based, more or less directly, upon data appearing in Eudemus's History."

"The Porisms. Our only source of information about the nature and contents of the Porisms is Pappus. ...With Pappus's account of Porisms must be compared the passages of Proclus on the same subject. ...Proclus's definition... agees well enough with the first, the 'older', definition of Pappus. A porism occupies a place between a theorem and a problem; it deals with something already existing, as a theorem does, but has to find it (e.g. the centre of a circle) and, as a certain operation is therefore necessary, it partakes to that extent of the nature of a problem, which requires us to construct or produce something not previously existing. ...all the positive information which we have about the nature of a porism and the contents of Euclid's Porisms ...is obscure and leaves great scope for speculation and controversy; naturally, therefore, the problem of restoring the Porisms has had a great fascination for distinguished mathematicians ever since the revival of learning. But it has proved beyond them all."

"Of his surviving works, the Commentary, which treats Book I of Euclid's Elements, is the most valuable. Proclus apparently intended to discuss more of the Elements, but there is no evidence that he ever did so."

"According to the account of Proclus (Book II. c. 4 ), Pythagoras was the first who gave to Geometry the form of a deductive science, by shewing the connexion of the geometrical truths then known, and their dependence on certain first principles."

"It is also problematical whether Proclus could have ever written such a clear, sober, and concise piece of work. His predominant interest in any subject, even mathematics, is always the epistemological aspect of it. He must ever inquire into the how and the why of the knowledge relevant to that subject, and its kind or kinds; and such speculation is apt with him to intrude into the discussion of even a definition or proposition. Moreover Proclus can never forego theologizing in the Pythagorean vein. Mathematical forms are for him but veils concealing from the vulgar gaze divine things. Thus right angles are symbols of virtue, or images of perfection and invariable energy, of limitation, intellectual finitude, and the like, and are ascribed to the Gods which proceed into the universe as the authors of the invariable providence of inferiors, whereas acute and obtuse angles are symbols of vice, or images of unceasing progression, division, partition, and infinity, and are ascribed to the Gods who give progression, motion, and a variety of powers. This epistemological interest and this tendency to symbolism are entirely lacking in our commentary; and another trait peculiar to Proclus is also absent, namely, his inordinate pedantry, his fondness of quoting all kinds of opinions from all sorts of ancient thinkers and of citing these by name with pedagogical finicalness. Obviously the author of our commentary had a philosophical turn of mind, but he was a temperate thinker compared with Proclus."

"A full history of Greek geometry and astronomy during this period written by Eudemus, a pupil of Aristotle, has been lost. It was well known to Proclus, who, in his commentaries on Euclid, gives a brief account of it. This abstract constitutes our most reliable information. We shall quote it frequently under the name of Eudemian Summary."

"About the time of Anaxagoras, but isolated from the Ionic school, flourished Œnopides of Chios. Proclus ascribes to him the solution of the following problems: From a point without, to draw a perpendicular to a given line, and to draw an angle on a line equal to a given angle. That a man could gain a reputation by solving problems so elementary as these, indicates that geometry was still in its infancy, and that the Greeks had not yet gotten far beyond the Egyptian constructions."

"A scholiast on Euclid, thought to be Proclus, says that Eudoxus practically invented the whole of Euclid's fifth book."

"The term 'axiom' was used by Proclus, but not by Euclid. He speaks, instead, of 'common notions'—common either to all men or to all sciences."

"The regular solids were studied so extensively by the Platonists that they received the name of Platonic figures The statement of Proclus that the whole aim of Euclid in writing the Elements was to arrive at the construction of the regular solids is obviously wrong The fourteenth and fifteenth books treating of solid geometry are apocryphal."

"Extracts... made by Proclus indicate that Ptolemy did not regard the parallel-axiom of Euclid as self-evident, and that Ptolemy was the first of the long line of geometers from ancient time down to our own who toiled in the vain attempt to prove it."

"Pappus... is the author of a Commentary on the Almagest, a Commentary on Euclid's Elements, a Commentary on the Analemma of Diodorus,—a writer of whom nothing is known. All these works are lost. Proclus, probably quoting from the Commentary on Euclid, says that Pappus objected to the statement that an angle equal to a right angle is always itself a right angle."

"The ancients, who under the leadership of Pythagoras first made science systematic, defined philosophy as the love of wisdom... [Οἱ παλαιοὶ καὶ πρώτοι μεθοδεύσαντες ἐπιστήμην κατάρξαντος Πυθαγόρου ὡρίζοντο φιλοσοφίαν εἶναι φιλίαν σοφίας...] This 'wisdom' he defined as the knowledge, or science, of the truth in real things, conceiving 'science' to be a steadfast and firm apprehension of the underlying substance. and 'real things' to be those which continue uniformly and the same in the universe and never depart even briefly from their existence; these real things would be things immaterial..."

"Bodily, material things are... continuously involved in continuous flow and change—in imitation of the nature and peculiar quality of that eternal matter and substance which has been from the beginning... The bodiless things, however, of which we conceive in connection with or together with matter, such as qualities, quantities, configurations, largeness, smallness, equality, relations, actualities, dispositions, places, times, all those things... whereby the qualities in each body are comprehended—all these are of themselves immovable and unchangeable, but accidentally they share in and partake of the affections of the body to which they belong. Now it is with such things that 'wisdom' is particularly concerned, but accidentally also with... bodies."

"To quote the words of Timaeus, in Plato, "What is that which always is, and has no birth, and what is that which is always becoming but never is? The one is apprehended by the mental processes, with reasoning, and is ever the same; the other can be guessed at by opinion in company with unreasoning sense, a thing which becomes and passes away, but never really is." Therefore, if we crave for the goal which is worthy and fitting for man, namely happiness of life—and this is accomplished by philosophy alone and nothing else, and philosophy means... for us desire for wisdom, and wisdom the science of the truth of things... it is reasonable and most necessary to distinguish and systematize the accidental qualities of things."

"Things... are some of them continuous...which are properly and peculiarly called 'magnitudes'; others are discontinuous, in a side-by-side arrangement, and, as it were, in heaps, which are called 'multitudes,' a flock, for instance, a people, a heap, a chorus, and the like. Wisdom, then, must be considered to be the knowledge of these two forms. Since, however, all multitude and magnitude are by their own nature of necessity infinite—for multitude starts from a definite root and never ceases increasing; and magnitude, when division beginning with a limited whole is carried on, cannot bring the dividing process to an end... and since sciences are always sciences of limited things, and never of infinites, it is accordingly evident that a science dealing with magnitude... or with multitude... could never be formulated.... A science, however, would arise to deal with something separated from each of them, with quantity, set off from multitude, and size, set off from magnitude."

"Since of quantity, one kind is viewed by itself, having no relation to anything else, as 'even,' 'odd,' 'perfect,' and the like, and the other is relative to something else, and is conceived of together with its relationship to another thing, like' double,' , greater,' 'smaller,' 'half,' 'one and one-half times,' 'one and one-third times,' and so forth, it is clear that two scientific methods will lay hold of and deal with the whole investigation of quantity: arithmetic, [with] absolute quantity; and music, [with] relative quantity."

"Two other sciences in the same way will accurately treat of 'size': geometry, the part that abides and is at rest, [and] astronomy, that which moves and revolves."

"Without the aid of these, then, it is not possible to deal accurately with the forms of being nor to discover the truth in things, knowledge of which is wisdom, and evidently not even to philosophize properly, for "just as painting contributes to the menial arts toward correctness of theory, so in truth lines, numbers, harmonic intervals, and the revolutions of circles bear aid to the learning of the doctrines of wisdom," says the Pythagorean Androcydes. Likewise Archytas of Tarentum, at the beginning of... On Harmony, says... in about these words: "It seems to me that they do well to study mathematics, and it is not at all strange that they have correct knowledge about each thing, what it is. For if they knew rightly the nature of the whole, they were also likely to see well what is the nature of the parts. About geometry, indeed, and arithmetic and astronomy, they have handed down to us a clear understanding, and not least also about music. For these seem to be sister sciences; for they deal with sister subjects, the first two forms of being.""

"Plato, too, at the end of the thirteenth book of the Laws, to which some give the title The Philosopher... adds: "Every diagram, system of numbers, every scheme of harmony, and every law of the movement of the stars, ought to appear one to him who studies rightly; and what we say will properly appear if one studies all things looking to one principle, for there will be seen to be one bond for all these things, and if anyone attempts philosophy in any other way he must call on Fortune to assist him. For there is never a path without these... The one who has attained all these things in the way I describe, him I for my part call wisest, and this I maintain through thick and thin." For it is clear that these studies are like ladders and bridges that carry our minds from things apprehended by sense and opinion to those comprehended by the mind and understanding, and from those material, physical things, our foster-brethren known to us from childhood, to the things with which we are unacquainted, foreign to our senses, but in their immateriality and eternity more akin to our souls, and above all to the reason which is in our souls."

"In Plato's Republic, when the interlocutor of Socrates appears to bring certain plausible reasons to bear upon the mathematical sciences, to show that they are useful to human life, arithmetic for reckoning, distributions, contributions, exchanges, and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertakings, revealing beforehand the proper procedure and suitable season, Socrates, reproaching him says: "You amuse me, because you seem to fear that these are useless studies that I recommend; but that is very difficult, nay, impossible. For the eye of the soul, blinded and buried by other pursuits, is rekindled and aroused again by these and these alone, and it is better that this be saved than thousands of bodily eyes, for by it alone is the truth of the universe beheld.""

"If geometry exists, arithmetic must also needs be implied... But on the contrary 3, 4, and the rest might be 5 without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry."

"And once more is this true in the case of music; not only because the absolute is prior to the relative, as 'great' to 'greater' and 'rich' to 'richer' and 'man' to 'father,' but also because the musical harmonies, diatessaron, diapente, and diapason, are named for numbers; similarly all of their harmonic ratios are arithmetical ones, for the diatessaron [] is the ratio of 4 : 3, the diapente [] that of 3 : 2, and the diapason [perfect ] the double ratio [2 : 1]; and the most perfect, the di-diapason [], is the quadruple ratio [4 : 1]."

"More evidently still astronomy attains through arithmetic the investigations that pertain to it, not alone because it is later than geometry in origin—for motion naturally comes after rest—nor because the motions of the stars have a perfectly melodious harmony, but also because risings, settings, progressions, retrogressions, increases, and all sorts of phases are governed by numerical cycles and quantities. So then we have rightly undertaken first the systematic treatment of this, as the science naturally prior, more honorable, and more venerable, and, as it were, mother and nurse of the rest; and here we will take our start for the sake of clearness."

"All that has by nature, with systematic method, been arranged in the universe, seems both in part and as a whole to have been determined and ordered in accordance with number, by the forethought and the mind of him that created all things; for the pattern was fixed, like a preliminary sketch, by the domination of number preëxistent in the mind of the world-creating God, number conceptual only and immaterial in every way, but at the same time the true and the eternal essence, so that with reference to it, as to an artistic plan, should be created all these things: time, motion, the heavens, the stars, all sorts of revolutions."

"Scientific number, being set over such things as these, should be harmoniously constituted, in accordance with itself; not by any other but by itself."

"Everything that is harmoniously constituted is knit together out of opposites..."

"Number is limited multitude or a combination of units or a flow of quantity made up of units; and the first division of number is even and odd."

"The even is that which can be divided into two equal parts without a unit intervening in the middle; and the odd is that which cannot..."

"Every number is at once half the sum of the two on either side of itself..."

"Some... agreeing with , believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion..."

"A Jew, Nicomachus, of Gerasa, published an Arithmetic, which, or rather a Latin translation of it) remained for a thousand years a standard authority on the subject. Geometrical demonstrations are here abandoned, and the work is a mere classification of the results then known, with numerical illustrations: the evidence of the truth of the propositions enunciated, for I cannot call them proofs, being in general an induction from numerical instances. The object of the book is the study of the properties of numbers, and particularly of their ratios. Nicomachus commences with the usual distinctions between even, odd, prime, and perfect numbers; he next discusses fractions in a somewhat clumsy manner; he then turns to polygonal and to solid numbers; and finally treats of ratio, proportion, and progressions. Arithmetic of this kind is usually termed Boethian, and the work of Boethius on it was a recognised text-book in the middle ages."

"The sixth century [BC] was the time, and Greece the place, for human beings to reject once and for all the pernicious number mysticism of the East. Instead, Pythagoras and his followers eagerly accepted it as the celestial revelation of a higher mathematical harmony. Adding vast masses of sheer numerological nonsense of their own to an already enormous bulk, they transmitted this ancient superstition to the golden age of Greek thought, which passed it on to the first century A.D. to the decadent arithmologist Nicomachus. He, enriching his already opulent legacy with a wealth of original rubbish, left it to be sifted by the Roman Boethius, the dim mathematical light of the Middle Ages, thereby darkening the mind of Christian Europe with the venerated nonsense, and encouraging the gemaria of the Talmudists to flourish like a weed."

""Arithmetic Introduction" is the most complete exposition extant of Pythagorean arithmetic. It deals in great part with the same subjects as the arithmetical books of Euclid's "Elements," but where Euclid represents numbers by straight lines, Nicomachus uses arithmetical notation with ordinary language when undetermined numbers are expressed. His treatment of s and s was of influence on medieval arithmetic, especially through Boetius."

"Boetius wrote mathematical texts which were considered authoritative in the Western world for more than a thousand years. ...His "institutio arithmetica," a superficial translation of Nicomachus, did provide some Pythagorean number theory which was absorbed in medieval instruction as part of the ancient and : arithmetic, geometry, astronomy, and music."

"Neo-Pythagoreans, like Nicomachus of Gerasa... and Iamblichus... revel in... number mysticism. ...The purpose of the layman's Introduction to Arithmetic... is to explain in a manner intelligible to every one, the wonderful and divine properties of numbers. In a very entertaining way he tells about s... square, rectangular and s... gnomic numbers and spatial numbers, about ratios and the distinction between multiple and epimoric."

"His native place was Gerasa probably the modern Jerash about 56 miles northeast of Jerusalem. Two treatises bear the name of Nicomachus, the Introductionis arithmeticæ libri duo and the Enchiridion harmonikon. ...A third treatise called Theologoumena arithmetica is anonymous but is probably the work of Nicomachus."

"In mathematical studies he is among the first to attempt a systematic treatment of Arithmetic distinct from Algebra."

"Nicomachus is preparing for the answer, mathematical knowledge, and so he says, the knowledge which distinguishes accurately "the accidents of the Existents." Accident may refer either to a contingent or to a necessary attribute... Here it is the latter. To distinguish such accidents seems at first glance an incredibly difficult task. But Nicomachus, like his descendants, simplifies it by reducing the accidents to two: Magnitude and Multitude."

"Nicomachus... mentions the customary Pythagorean divisions of quantum and the science that deals with each. Quantum is either discrete or continuous. Discrete quantum in itself considered, is the subject of Arithmetic; if in relation, the subject of Music. Continuous quantum, if immovable, is the subject of Geometry; if movable, of Spheric (Astronomy). These four sciences formed the of the Pythagoreans. With the (which Nicomachus does not mention) of Grammar, Logic, and Rhetoric, they composed the seven liberal arts taught in the schools of the Roman Empire."

"Nicomachus gives three definitions of number. The first, "determinate multitude," is by genus and difference, and is ascribed by Iamblichus to Eudoxus. The second, "a group (i.e., organized complex) of units," [Iamblichus] ascribes to Thales and the Egyptians. The third stream of quantity composed of units Philoponus explains as another attempt to distinguish the particular kind of quantum treated in Arithmetic. The Unit was conceived either mystically as an Idea whose "essence" passes in some way into concrete individuals and even into the Ideas to organize them, or spatially and temporally as the boundary of individuals. The former conception gave rise to fantastic speculations on the cosmic meaning of number, examples of which Nicomachus has... given us; the latter gave rise to the s..."

"The philosophical arithmetic of the Greeks, αριθμητική, of which the arithmetic of Nicomachus is a specimen, corresponds in a measure to our number theory; the subject was designed for mature students as a preparation for the study of philosophy, and was not at all intended for children. Arithmetica is... the study of that which is implied in number."

"The Introduction to Arithmetic of Nicomachus is but a restatement of facts which were common property... long before him, and that, except for the few unimportant propositions the discovery of which our author... claims for himself, the book is largely unoriginal. This naturally leads to the inference that the Introduction must be closely connected with other mathematical treatises, which served as the fountains... Because so little remains of this literature, [our inference] is difficult to demonstrate..."

"Iamblichus, when he refers to the Introduction as the... Art of Arithmetic, exactly describes it... The Introduction belongs... among the artes or... concise, practical descriptions and systematic expositions of the principles of various arts and sciences, a type of treatise exceedingly common in ancient times, and one which... made scant claim to originality. Designed for the use of students... they may best be compared to the modern school and college text-book. ...we must consider that the Introduction to Arithmetic differs from the great original treatises of Diophantus and Heron. Because in clearness, conciseness, compendiousness, orderly arrangement and adaptability for scholastic use, it satisfied the demands of seekers after education or general information, it remained the standard work of its class for many centuries."

"Nicomachus planned his Introduction so as to explain the mathematical principles involved in the difficult Platonic passages concerning the world-soul in the Timaeus and the marriage-number in the Republic."

"In geometry we begin with the point, which is indimensional. This is the beginning of the first dimensional form, the line, and by movement the point generates the line. Now Nicomachus had a similar idea of the nature of multitude and number; they form a series, as it were a moving stream, which proceeds out of unity, the monad. Just as the point is not part of the line (for it is indimensional, and the line is defined as that which has one dimension), but is potentially a line, so the monad is not a part of multitude nor of number, though it is the beginning of both, and potentially both. The monad is unity, absence of multitude, potentiality; out of it the dyad first separates itself and 'goes forward' and then in succession follow the other numbers."

"The quadratures of lunes, which were considered to belong to an uncommon class of propositions on account of the close relation (of lunes) to the circle, were first investigated by Hippocrates, and his exposition was thought to be in correct form... He started with, and laid down as the first of the theorems useful for his purpose, the proposition that similar segments of circles have the same ratio to one another as the squares on their bases have... And this he proved by first showing that the squares on the diameters have the same ratio as the circles. For, as the circles are to one another, so also are similar segments of them. For similar segments are those which are the same part of the circles respectively, as for instance a semicircle is similar to a semicircle, and a third part of a circle to a third part... It is for this reason also... that similar segments contain equal angles...'"

"One would suppose that the relation between the pseudo-didactic and the didactic syllogism, was the same as that between the pseudo-dialectic and the dialectic; so that, if the pseudo-dialectic deserved to be called sophistic or , the pseudo-didactic would deserve these appellations also; especially, since the formal conditions of the syllogism are alike for both. This Aristotle does not admit, but draws instead a remarkable distinction. The (he says) is a dishonest man, making it his professional purpose to deceive; the pseudo-graphic man of science is honest always, though sometimes mistaken. So long as the pseudo-graphic syllogism keeps within the limits belonging to its own special science, it may be false, since the geometer may be deceived even in his own science [of] geometry, but it cannot be sophistic or eristic; yet whenever it transgresses those limits, even though it be true and though it solves the problem proposed, it deserves to be called by those two epithets. Thus, there were two distinct methods proposed for the quadrature of the circle—one by Hippokrates, on geometrical principles, the other by Bryson, upon principles extra-geometrical. Both demonstrations were false and unsuccessful; yet that of Hippokrates was not sophistic or eristic, because he kept within the sphere of geometry; while that of Bryson was so, because it travelled out of geometry. Nay more, this last would have been equally sophistic and eristic, and on the same ground, even if it had succeeded in solving the problem. If indeed the pseudo-graphic syllogism be invalid in form, it must be considered as sophistic, even though within the proper scientific limits as to [the] matter; but, if it be correct in form and within these same limits, then however untrue its premisses may be, it is to be regarded as not sophistic or eristic."

"Of original writings... we have only a fragment concerning the lunes of Hippocrates, quoted by Simplicius... and taken from Eudemus's lost History of Geometry..."

"Hippocrates of Chios... the most famous mathematician of his century... is credited with the idea of arranging theorems so that later ones can be proven on the basis of earlier ones, in the manner familiar to us from... Euclid. He is also credited with introducing the indirect method of proof into mathematics. His text on geometry, called the Elements, is lost."

"The circle being after rectilineal figures, the most simple in appearance, geometricians very naturally soon began to seek for its measure. Thus we find that the philosopher Anaxagoras occupied himself with the question in prison. Then Hippocrates of Chios tried the same problem, and it led him to the discovery of what is called the lune, a surface in the shape of a crescent, bounded by two arcs and exactly equal to a given square. He also found two unequal lines which were together equal to a rectilineal figure, so that if their relation could have been found the solution of the problem would have been obtained. But this no one has yet been able to do, nor is it likely ever to be done."

"Hippocrates of Chios... attempted the solution [for squaring the circle] and was the first actually to square a curvilinear figure. He constructed semicircles on the three sides of an isosceles right-angled triangle and showed that the sum of the two lunes thus formed is equal to the area of the triangle itself. ...His proof involves the proposition that the areas of circles are proportional to the squares of their diameters,—a proposition which Eudemus... tells us that Hippocrates proved. To the quadrature problem as such, however, his contribution was not important."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C."

"Hippocrates of Chios... was one of the greatest of the Greek geometricians. He... began life as a merchant. The accounts differ as to whether he was swindled by the Athenian custom-house officials who were stationed at the Chersonese, or whether one of his vessels was captured by an Athenian pirate near ... somewhere about 430 B.C. he came to Athens to try to recover his property in the law courts. ...the Athenians seem only to have laughed at him for his simplicity, first in allowing himself to be cheated, and then in hoping to recover his money. While prosecuting his cause he attended the lectures of various philosophers, and finally (in all probability to earn a living) opened a school of geometry himself. He seems to have been well acquainted with the Pythagorean philosophy, though there is no sufficient authority that he was ever initiated as a Pythagorean."

"[Hippocrates] wrote the first elementary text-book of geometry... on which probably Eudlid's Elements was founded; and therefore he may be said to have sketched out the lines on which geometry is still taught in English schools."

"It is supposed that the use of letters in diagrams to describe a figure was made by him or introduced about this time, as he employs expressions such as "the point on which letter A stands" and "the line on which AB is marked.""

"Hippocrates... denoted the square on a line by the word... power which it still retains in algebra."

"In [his] textbook Hippocrates introduced the method of "reducing" one theorem to another, which being proved, the thing proposed necessarily follows; of this method the ' is an illustration. No doubt the principle had been used occasionally before, but he drew attention to it as a legitimate mode of proof which was capable of numerous applications."

"[Hippocrates] elaborated the geometry of the circle: proving, among other propositions, that similar segments of a circle contain equal angles; that the angle subtended by the chord of a circle is greater than, equal to, or less than a right angle as the segment of the circle containing it is less than, equal to, or greater than a semicircle (Euc. III, 31); and probably several other of the propositions in the third book of Euclid. It is most likely that he also established the propositions that [similar] circles are to one another as the squares of the diameters (Euc. XII, 2), and that similar segments are as the squares of their chords. The proof given in Euclid of the first of these theorems is believed to be due to Hippocrates."

"The most celebrated discoveries of Hippocrates were... in connection with the quadrature of the circle and the duplication of the cube, and owing to his influence these problems played a prominent part in the history of the Athenian school."

"He commenced by finding the area of a lune contained between a semicircle and a quadrilateral arc standing on the same chord... as follows. Let ABC be an isosceles right-angled triangle inscribed in the semicircle ABOC, whose centre is O. On AB and AC as diameters describe semicircles as in the figure. Then, since by Euc. I, 47,sq. on\,BC = sq. on\,AC + sq. on\,AB,therefore, by Euc. XII, 2,area\;\frac{1}{2} \bigodot on\,BC = area\;\frac{1}{2} \bigodot on\,AC + area\;\frac{1}{2} \bigodot on\,ABTake away the common parts.\therefore area\,\triangle ABC = sum\;of\;areas\;of\;lunes\;AECD\;and\;AFBG.Hence the area if the lune AECD is equal to half that of the triangle ABC."

"He next inscribed half a regular hexagon ABCD in a semicircle whose centre was O, and on OA, AB, BC, and CD as diameters described semicircles of which those on OA and AB are drawn in the figure [2]. Then AD [by equilateral triangles within the half-hexagon] is double any of the lines OA, AB, BC, and CD,\therefore\;square\;on\;AD = sum\;of\;sqs.\;on\;OA, AB, BC, and\;CD,\therefore\;area\;\frac{1}{2} \bigodot on\,ABCD = sum\;of\;areas\;of\;\frac{1}{2} \bigodot s\;on\;OA, AB, BC, and\;CD.Take away the common parts\therefore\;area\;trapezium\;ABCD = 3\;lune\;AEBF + \frac{1}{2} \bigodot on\;OA.If therefore the area of this latter lune be known, so is that of the semicircle described on OA as diameter."

"Hippocrates also enunciated various other theorems connected with lunes... of which the theorem last given is a typical example. I believe that they are the earliest examples in which areas bounded by curves were determined by geometry."

"Hippocrates of Chios who lived in Athens in the second half of the fifth century B.C., and wrote the first text book on Geometry, was the first to give examples of curvilinear areas which admit of exact quadrature. These figures are the menisci or lunulae of Hippocrates."

"If on the sides of a right-angled triangle ACB semi-circles are described on the same side, the sum of the areas of the two lunes AEC, BDC is equal to that of the triangle ACB. If the right-angled triangle is isosceles, the two lunes are equal, and each of them is half the area of the triangle. Thus the area of the lunula is found."

"If AC = CD = DB = radius OA (see Fig. 3), the semi-circle ACE is &frac14; of the semi-circle ACDB. We have now&#9680;AB - 3&#9680;AC = [trapezium] ACDB - 3 &middot; meniscus ACE, [where the meniscus is the lunulae, i.e., lune] and each of these expressions is &frac14;&#9680;AB or half the circle on &frac12;AB as diameter. If then the meniscus AEC were quadrable so also would be the circle on &frac12;AB as diameter. Hippocrates recognized the fact that the meniscus is not quadrable, and he made attempts to find other quadrable lunulae in order to make the quadrature of the circle depend on that of such quadrable lunulae."

"The most important name from the point of view of this chapter is Hippocrates of Chios. He is indeed the first person of whom it is recorded that compiled a book of Elements. This is lost, but Simplicius has preserved in his commentary on the Physics of Aristotle a fragment from Eudemus's History of Geometry giving an account of Hippocrates's quadratures of certain lunules or lunes."

"It would appear that Hippocrates was in Athens during a considerable portion of the second half of the fifth century, perhaps from 450 to 430 B.C. We have quoted the story that what brought him there was a suit to recover a large sum which he had lost, in the course of his trading operations, through falling in with pirates; he is said to have remained in Athens on this account a long time, during which he consorted with the philosophers and reached such a degree of proficiency in geometry that he tried to discover a method of squaring the circle. This is of course an allusion to the quadratures of lunes."

"He was the first to observe that the problem of doubling the cube is reducible to that of finding two mean proportionals in continued proportion between two straight lines. The effect of this was, as Proclus says, that thenceforward people addressed themselves (exclusively) to the equivalent problem of finding two mean proportionals between two straight lines."

"[Number is] the commanding and self-begotten container of the eternal duration of mundane concerns."

"The ancient theologists and priests... testify that the soul is united with the body as if for the sake of punishment; and so is buried in body as in a sepulchre."

"There is a fire in the middle at the centre, which is the Vesta of the universe, the house of Jupiter, the mother of the Gods, and the basis coherence and measure of nature."

"Fragment 1. (Stob.21.7; Diog.#.8.85) The world's nature is a harmonious compound of infinite and finite elements; similar is the totality of the world in itself, and of all it contains. b. All beings are necessarily finite or infinite, or simultaneously finite and infinite; but they could not all be infinite only."

"Fragment 2. All things, at least those we know, contain number; for it is evident that nothing whatever can either be thought or known, without number. Number has two distinct kinds: the odd, and the even, and a third, derived from a mingling of the other two kinds, the even-odd. Each of its subspecies is susceptible of many very numerous varieties; which each manifests individually."

"Fragment 3. The harmony is generally the result of contraries; for it is the unity of multiplicity, and the agreement of discordances. (Nicom.Arith.2:509)."

"Fragment 4. This is the state of affairs about nature and harmony. The essence of things is eternal; it is a unique and divine nature, the knowledge of which does not belong to man. Still it would not not be possible that any of the things that are, and are known by us, should arrive to our knowledge, if this essence was not the internal foundation of the principles of which the world was founded, that is, of the finite and infinite elements. Now since these principles are not mutually similar, neither of similar nature, it would be impossible that the order of the world should have been formed by them, unless the harmony had intervened... the dissimilar things, which have neither a similar nature, nor an equivalent function, must be organized by the harmony, if they are to take their place in the connected totality of the world."

"In the old Pythagorean representation of the celestial system, according to Philolaus, the five planets were mentioned... among the ten deified bodies which revolve round the central fire (the focus of the universe έστἱα) "immediately beneath the region of fixed stars;" these were succeeded by the Sun, Moon, Earth, and... the anti-Earth..."

"The first publication of the Pythagorean doctrines is pretty uniformly attributed to Philolaus. He composed a work on the Pythagorean philosophy in three books, which Plato is said to have procured at the cost of 100 minae through , who purchased it from Philolaus, who was at the time in deep poverty. ...Out of the materials which he derived from these books Plato is said to have composed the Timaeus. But in the age of Plato the leading features of the Pythagorean doctrines had long ceased to be secret; and if Philolaus taught the Pythagorean doctrines at Thebes, he was hardly likely to feel much reluctance in publishing them... little more can be regarded as trustworthy, except that Philolaus was the first who published a book on the Pythagorean doctrines, and that Plato read and made use of it."

"According to the Pythagorean Philolaus, "the Dekad, the full and perfect number, was of supreme and universal efficacy as the guide and principle of life, both to the Kosmos and to man. The nature of number was imperative and lawgiving, affording the only solution of all that was perplexing or unknown; without number, all would be indeterminate and unknowable.""

"Philolaus divided the world into three parts—viz. Olympus, which holds within itself the purity of the elements, i. e. probably the central fire and the fire outwardly embracing the world; Kosmus, or the world in a limited sense, i. e. the perfectly ordered world, which comprises all mundane bodies except the earth; and Uranus, i. e. the part of the universe which belongs to the terrestrial sphere."

"Philolaus, in a fragment preserved by (Eclog. Phys. p. 51), says, "that there is a fire in the middle at the centre, which is the Vesta of the universe, the house of Jupiter, the mother of the Gods, and the basis, coherence, and measure of nature." Hence... they are greatly mistaken who suppose the Pythagoreans meant the Sun by the fire at the centre..."

"The importance of the decuple system in relation to the Pythagoreans is much greater. For as they considered numbers over ten to be only the repetition of the first ten numbers, all numbers and all powers of numbers appeared to them to be comprehended in the decad, which is therefore called by Philolaus, great all-powerful and all-producing, the beginning and the guide of the divine and heavenly, as of the terrestrial life."

"In regard to Philolaus, we are told... that he derived geometrical determinations (the point, the line, the surface, the solid) from the first four numbers, so he derived physical qualities from five, the soul from six; reason, health, and light, from seven; love, friendship, prudence, and inventive faculty from eight. Herein (apart from the number schematism) is contained the thought that things represent a graduated scale of increasing perfection; but we hear nothing of any attempt to prove this in detail, or to seek out the characteristics proper to each particular region."

"Plato was instructed in their nature by Philolaus his preceptor, and the disciple of Pythagoras."

"Pythagoras, in the sacred discourse, calls number "the ruler of forms and ideas." But Philolaus, "the commanding and self-begotten container of the eternal duration of mundane concerns." And and all those... under the probation of the quinquennial silence... "the first exemplar of the mundane fabric, and the judiciary instrument of its artificer.""

"Such... being the middle nature of the soul, Plato, with great propriety, in the Phædrus, and in his tenth book of laws, defines it to be number moving itself; which definition he received from Philolaus, and Philolaus from Pythagoras."

"[Eratosthenes] ... is a mathematician among geographers, and yet a geographer among mathematicians; and consequently on both sides he offers his opponents occasions for contradiction."

"Eratosthenes declares that it is no longer necessary to inquire as to the cause of the overflow of the Nile, since we know definitely that men have come to the sources of the Nile and have observed the rains there."

"In comparison with the great size of the earth the protrusion of mountains is not sufficient to deprive it of its spherical shape or to invalidate measurements based on its spherical shape. For Eratosthenes shows that the perpendicular distance from the highest mountain tops to the lowest regions is ten stades [c.5,000-5,500 feet]. This he shows with the help of dioptras which measure magnitudes at a distance."

"Eratosthenes... knew that the Sun was straight overhead in... Syene at noon on the , but that it was 7.2 degrees south of straight overhead in , located 794 kilometers farther north. He concluded... 794 kilometers corresponded to 7.2 degrees out of the 360 degrees... around Earth's circumference, so that the circumference must be 794 km x 360°/7.2°≈39,700 km... remarkably close to the modern value of 40,000 km. Amusingly Christopher Columbus totally bungled this... confusing Arabic miles with Italian miles..."

"Eratosthenes of Cyrene, employing mathematical theories and geometrical methods, discovered from the course of the sun the shadows cast by an equinoctial gnomon, and the inclination of the heaven that the circumference of the earth is two hundred and fifty-two thousand stadia, that is, thirty-one million five hundred thousand paces."

"I have composed a book on the length of the year in which I show that the tropical year contains 365 days plus a fraction of a day which is not exactly 1/4 days as the mathematicians-astronomers suppose, but which is less than 1/4 by about 1/300."

"Hypatia was a University lecturer denounced by Church dignitaries and torn to pieces by Christians. Such will probably be the fate of this book: therefore it bears her name. What I have written here I believe and shall not retract or change for similar episcopal denunciations."

"In speech articulate and logical, in her actions prudent and public-spirited. The city gave her suitable welcome and accorded her special respect."

"Most people, both Christian and non-Christian, saw Hypatia’s killing as a brutal, unprovoked murder that exploded out of a toxic set of circumstances for which Hypatia bore little responsibility."

"that sectors in equal circles are to one another as the angles on which they stand has been proved by me in my edition of the Elements..."

"‘Yet some few of such investigations we have in the five first propositions of Euclid’s thirteenth book … seems to be the work of Theo, […] rather than of Euclid himself.’"

"All our Greek texts of the Elements up to a century ago…purport in their titles to be either ‘from the edition of Theon’…or ‘from the lectures of Theon... [Greek commentaries] commonly speak of the writer of the Elements instead of using his name."