Ancient Greece

"Graecia capta ferum victorem cepit et artes intulit agresti Latio."

"It is the historian's job to draw attention to the personal, social, political and indeed moral issues behind the literary and artistic representations of the Greek world. The historian's job is to present pederasty and all, to make sure that … we come face to face with the way the glory that was Greece was part of a world in which many of our own core values find themselves challenged rather than reinforced."

"The worldliness and naturalness with which the religion of the Greeks is reproached is encountered in their plastic art also. Here too the difference from the oriental is immeasurable. Organic structure takes the place of monstrosity; instead of symbolism and denotation we have what we have learned — through the Greeks — to understand as forms of nature. And yet all of these works breathe a loftiness and nobility which lifts us above the transitory and earthbound world of facts. Before our eyes a miracle takes place: the natural has become one with the spiritual and eternal, without surrendering a whit of its abundance, warmth, and immediacy in the amalgam. Should not the spirit for which exact observance of the natural led to the vision of the eternal and infinite have made the religion of the Greeks the very thing it was?"

"Greek democracy was not, in fact, Greek democracy; it was Athenian, or Corinthian. Although the city-state mentality may seem quaintly parochial today, the same issue is still with us."

"Despite the extraordinary influence of classical Greece on the development of democracy, modern democratic ideas and institutions have also been shaped by many other factors, of which three are particularly important: a republican tradition, the development of representative governments, and certain conclusions that tend to follow from a belief in political equality."

"There is such a thing as fighting the battle of democracy in the front rank too long. It is ever the Aristides experience over again. Everybody remembers Aristides— the sturdy citizen of the Athenian democracy, who was one of the generals at Marathon, one of the victorious captains at Salamis, the conqueror at Plataea, who put through with a punch a very much needed programme of civic reform in Athens, and helped organize the Delian League with the purpose of making Greece a real nation at the time when she was able to be one. He pushed the Athenian democracy to the point of diminishing returns; the people had an attack of fatigue, escorted Aristides to the city gate, and bowed him into the ostracism of silence. That has been the way with democracies. They get over their blue funk after a while. Everbody in Greece is for Aristides now. But he is dead. And it is too late. It is yet a question whether the American democracy has learned its lesson from history so that it knows how to value its Aristides citizens, little and big."

"The other old democracy... the one I teach about... was in Athens. Now Aristotle said democracy means "to rule and to be ruled in turns"... because the system the Athenians set up was designed to ensure that nobody would... cease power. So they took people and... forced them to be political units composed of multiple clans... so different clans had to be together. They couldn't be segregated, and they had a 10 month calendar and every month a different group was in charge. Hence to rule and to be ruled in turns. ...[I]t worked for 200 years. [Much of] our Constitution is based... on the experience of Ancient Athens, of . All of the Founders of this country... were well steeped in the history of the ancient world."

"The Athenian practice had been, even before Plato’s birth, precisely the opposite: the people, the demos, should rule. All important. political decisions—such as war and peace—were made by the assembly of all full citizens. This is now called “direct democracy”; but we must never forget that the citizens formed a minority of the inhabitants—even of the natives. From the point of view here adopted, the important thing is that, in practice, the Athenian democrats regarded their democracy as the alternative to tyranny—to arbitrary rule: in fact, they knew well that a popular leader might be invested with tyrannical powers by a popular vote. So they knew that a popular vote may be wrongheaded, even in the most important matters. (The institution of ostracism recognised this: the ostracised person was banned as a matter of precaution only; he was neither tried nor regarded as guilty.)The Athenians were right: decisions arrived at democratically, and even the powers conveyed upon a government by a democratic vote, may be wrong. It is hard, if not impossible, to construct a constitution that safeguards against mistakes. This is one of the strongest reasons for founding the idea of democracy upon the practical principle of avoiding tyranny rather than upon a divine, or a morally legitimate, right of the people to rule."

"The contrast between the Persian state—and by the same token the late Imperial Roman, Bismarckian, or modern European state—and the Greek polis is far from the only theme that dominates this story. A familiar contrast is between Athenian and Roman notions of freedom and citizenship. The Athenians practiced a form of unfiltered direct democracy that the Romans thought a recipe for chaos; the Romans gave ordinary free and male persons a role in politics, but a carefully structured and controlled one."

"Although the Romans disavowed Athenian democracy, there are many “Roman” arguments for involving the citizenry in political life as deeply as possible. Machiavelli had no taste for Athenian democracy, but preferred citizen armies to mercenary troops, and like Roman writers before him and innumerable writers after him thought that, given the right arrangements, the uncorrupted ordinary people could check the tendency of the rich to subvert republican institutions. That was a commonplace of antiaristocratic republican thinking in eighteenth- and nineteenth-century Europe; it is a standing theme of American populism."

"The key to Athenian democracy was the Assembly, or ecclesia. It was in modern terms legislature, judiciary, and executive, and there was no appeal against its decisions except to a later meeting of itself, or a court that was part of itself. Although its potential membership was 40,000, it operated through many smaller bodies, through courts of 500 members, and in particular through the 500 members of the governing council, or boule, whose members formed the Athenian administration for a year, and the prytany, the 30-strong body whose members formed the managing committee of the boule for a month at a time."

"To the extent that they drew on classical governments for inspiration or illustration, the Founders much preferred republican Rome (or even timocratic Sparta) to Athenian democracy. They used the terms republic and democratic republic, or sometimes representative democracy, to describe early American state governments and the new national system."

"If the early Greeks were cognizant of Babylonian algebra, they made no attempt to develop or even to use it, and thereby they stand convicted of the supreme stupidity in the history of mathematics. ...The ancient Babylonians had a rare capacity for numerical calculation; the majority of Greeks were either mystical or obtuse in their first approach to number. What the Greeks lacked in number, the Babylonians lacked in logic and geometry, and where the Babylonians fell short, the Greeks excelled. Only in the modern mind of the seventeenth and succeeding centuries were number and form first clearly perceived as different aspects of one mathematics."

"Had the early Greek mind been sympathetic to the algebra and arithmetic of the Babylonians, it would have found plenty to exercise its logical acumen, and might easily have produced a masterpiece of the deductive reasoning it worshipped logically sounder than Euclid's greatly overrated Elements. The hypotheses of elementary algebra are fewer and simpler than those of synthetic geometry. ...they could have developed it with any degree of logical rigor they desired. Had they done so, Apollonius would have been Descartes, and Archimedes, Newton. As it was, the very perfection... of Greek geometry retarded progress for centuries."

"The Greeks ordinarily are regarded as the founders of mathematics in the strict sense... for they emphasized the value of abstract generalizations... and the deductive elaboration of these. ...this early intellectual revolution occurred at about the time of a distinct geographical shift in the centers of civilization. The focal points previously had river valleys, such as the Nile, or of the Tigris and Euphrates; but by the middle of the eighth century B.C. these ancient potamic civilizations were confronted with a vigorous young thalassic civilization established about the Mediterranean Sea."

"The Greek search for essences had led the Pythagoreans to picture the universe as a multitude of mathematical points completely subject to the laws of number—a sort of arithmetic geometry... The rival Eleatic philosophy of Parmenides upheld the essential "oneness" of the universe and the impossibility of analyzing it in terms of the "many." Zeno of Elea sought dialectically to defend his master's doctrine by demolishing the Pythagorean association of multiplicity with number and magnitude. ...The paradoxes, as one sees now, involve such notions as infinite sequence, limit, and continuity, concepts for which Zeno nor any of the ancients gave precise definition. ...their influence was profound. The Greeks banned from their mathematics any thought of an arithmetic continuum or of an algebraic variable, ideas which might have led to analytic geometry; and they refused to place any confidence in infinite processes, the methods which would have led to calculus. Whereas the Pythagoreans had envisioned a union of arithmetic and geometry, Greek mathematicians after Zeno saw only the mutual incompatibility of the two fields."

"That the discovery of incommensurability of lines made a strong impression on Greek thought is indicated by the story of Hippasus... It is demonstrated more reliably by the prominence given to the theory of irrationals by Plato and his school [e.g., Eudoxus of Cnidus]. ...the Greeks were led by Zeno and Hippasus to abandon the pursuit of a full arithmetization of geometry... there was no such thing as algebraic analysis. Geometry was the domain of continuous magnitude, arithmetic was concerned with the discrete set of integers; and the two fields were irreconcilable."

"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."

"In the Greek world mathematics was more closely related to philosophy than to practical affairs, and this kinship has persisted to the present day."

"These three problems—the , the duplication of the cube, and the trisection of the angle—have since been known as the "three famous (or classical) problems" of antiquity. More than 2200 years later it was proved that all three... were unsolvable by means of straightedge and compass alone. ...the better part of Greek mathematics, and much of later mathematical thought, was suggested by efforts to achieve the impossible—or failing this, to modify the rules."

"Comparatively few of the propositions and proofs in the Elements are his [Euclid's] own discoveries. In fact, the proof of the "Theorem of Pythagoras" is the only one directly ascribed to him."

"Euclid, Archimedes, and Apollonius brought geometry to as high a state of perfection as it perhaps could be brought without first introducing some more general and more powerful method than the old method of exhaustion. A briefer symbolism, a Cartesian geometry, an infinitesimal calculus, were needed. The Greek mind was not adapted to the invention of general methods. Instead of a climb to still loftier heights we observe, therefore, on the part of later Greek geometers, a descent during which they paused here and there to look around for details which had been passed by in the hasty ascent."

"Thus, it is again a conclusion to be assumed in advance, only waiting for a confirmation, that Greek mathematics had brought its traces into those Indian works."

"The discovery of incommensurable quantities threw an awful wrench in the machinery of geometry... The difficulty was finally overcome by Eudoxus' theory of proportion. But there was an indirect scare... In Euclid the theory of proportion and similar figures is postponed until the last possible moment, quite contrary to our present practice. Meanwhile, theorems which we prove by proportion were handled by the method of Application of Areas... The credit for discovering this seems to belong to the Pythagoreans"

"The inspiration of Fermat's discussion of the conic sections, and that is practically the whole of his analytic geometry, comes direct from Apollonius. The same had been true of Pappus, fourteen centuries before. His point of departure is the famous four-line problem... This question seems to have stumped both Euclid and Aristaeus, and to have been first solved by Apollonius. In Apollonius's own work we find what is rather the converse of this problem. Almost the first piece of geometrical writing which Fermat did was to prove the three-line case."

"It is important to remember that the ancient Greeks did not have an abstract system of number symbols, and used the letters of the alphabet as number symbols. They also commonly manipulated pebbles to learn arithmetic and used small stones on calculating boards. In this case, number patterns were their common experience of arithmetic. From this use of pebbles, we have inherited the word 'calculation,' from the Latin calculus, which means 'pebble.'"

"How is it that the nation which gave us geometry and carried this science so far, did not create even a rudimentary algebra?"

"The existence of incommensurable geometric magnitudes... necessitated a thorough reexamination and recasting of the foundations of mathematics, a task that occupied much of the fourth century B.C. During this period Greek algebra and geometry assumed the highly organized and rigorously deductive form that is set forth the the 13 books of the Elements that Euclid wrote about in 300 B.C. This systematic exposition of the Greek mathematical accomplishments of the preceding three centuries is the earliest major Greek mathematical text that is now available...(due perhaps to the extent to which the Elements subsumed previous expositions)."

"The Greeks... were well aware of geometric magnitudes that we call "irrational," but simply did not think of them as numbers."

"Logistic is the theory which deals with numerable objects and not with numbers; it does not, indeed, consider number in the proper sense of the term, but assumes 1 to be unity, and anything which can be numbered to be number (thus in place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic. ... It treats, then, on the one hand, that which Archimedes called 'The Cattle Problem,' and on the other hand 'melite' and 'phialite' numbers, the one discussing vials (measures, containters) and the other flocks; and when dealing with other kinds of problems it has regard to the number of sensible bodies and makes its pronouncements as though it were for absolute objects. ... It has for material all numerable objects, and as subdivisions the so-called Greek and Egyptian methods for multiplication and division, as well as the summation and decomposition of fractions, whereby it investigates the secrets lurking in the subject-matter of the problems by means of the procedure that employs triangles and polygons. ... It has for its aim that which is useful in the relations of life in business, although it seems to pronounce upon sensible objects as if they were absolute."

"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."

"The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable... showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established. ...The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes."

"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.""

"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."

"Arithmetic is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic was primarily a philosophical study, having no necessary connection with practical affairs. Indeed the Greeks gave a separate name to the arithmetic of business, λογιστική [accounting or practical logistic]... In general the philosophers and mathematicians of Greece undoubtedly considered it beneath their dignity to treat of this branch, which probably formed a part of the elementary instruction of children."

"And do you not know also that although they make use of the visible forms and reason about them, they are thinking not of these, but of the ideals which they resemble; not of the figures which they draw, but of the absolute square and the absolute diameter, and so on --the forms which they draw or make, and which have shadows and reflections in water of their own, are converted by them into images, but they are really seeking to behold the things themselves, which can only be seen with the eye of the mind?"

"The ancient Geometry had no symbols, nor any notation beyond ordinary language and the specific terms of the science."

"Arithmetic... teaches all the various operations of numbers and demonstrates their properties. ...The Greeks are said have received it from the Phoenicians. The ancients, who have treated arithmetic most exactness, are Euclid, Nicomachus of Alexandria, and . It was difficult either for the Greeks or the Romans to succeed much in arithmetic, as both used only letters of the alphabet for numbers, the multiplication of which, in great calculations, necessarily occasioned abundance of trouble. The Arabic ciphers... are infinitely more commodious, and have contributed very much to the improvement of arithmetic."

"More than any other of his predecessors Plato appreciated the scientific possibilities of geometry... By his teaching he laid the foundation of the science, insisting upon accurate definitions, and logical proof. His opposition to the materialists, who saw in geometry only what was immediately useful to the artisan and the mechanic, is made clear by Plutarch in his Life of Marcellus... "Plato's indignation at it and his invections against it as the mere corruption and annihilation of the one good geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence.""

"Each of these sciences has a subject which is different from the science. I can show you that the art of computation has to do with odd and even numbers in their numerical relations to themselves and to each other. ...And the odd and even numbers are not the same with the art of computation? ..The art of weighing, again, has to do with lighter and heavier; but the art of weighing is one thing, and the heavy and the light another. ...what is that which is not wisdom, and of which wisdom is the science? ...wisdom is the only science which is the science of itself as well as of the other sciences. ...But the science of science... will also be the science of the absence of science."

"The theory of proportions is credited to Eudoxus... and is expounded in Book V of Euclid's Elements. The purpose of the theory is to enable lengths (and other geometric quantities) to be treated as precisely as numbers, while only admitting the use of rational numbers. ...To simplify ...let us call lengths rational if they are rational multiples of a fixed length. Eudoxus' idea was to say that a length \lambda is determined by those rational lengths less than it and those greater than it. ...he says \lambda_1 = \lambda_2...if any rational length < \lambda_1 is also < \lambda_2, and vice versa [any rational length > \lambda_2 is also > \lambda_1]. Likewise \lambda_1 < \lambda_2 if there is a rational length > \lambda_1 but < \lambda_2 [between \lambda_1 and \lambda_2]. This definition uses the rationals to give an infinitely sharp notion of length while avoiding any overt use of infinity. ... The theory of proportions was so successful that it delayed the development of a theory of real numbers for 2000 years. This was ironic, because the theory of proportion can be used to define irrational numbers just as well as lengths. It was understandable though, because the common irrational lengths... arise from constructions that are intuitively clear and finite from the geometric point of view. Any arithmetic approach to the \sqrt2, whether by sequences, decimals, or continued fractions, is infinite and therefore less intuitive. Until the nineteenth century this seemed a good reason... Then the problems of geometry came to a head, and mathematicians began to fear geometric intuition as much as they had previously feared infinity."

"The towns which arose along the coast of Asia Minor and on the Greek mainland were no longer administration centers of an irrigation society. They were trading towns in which the old-time feudal landlords had to fight a losing battle with an independent, politically conscious merchant class. ...The merchant trader had never enjoyed so much independence, but he knew that this independence was the result of a constant and bitter struggle. The static outlook of the Orient could never be his. He lived in a period of geographical discovery comparable only to those of sixteenth-century Western Europe; he recognized no absolute monarch or power supposedly vested in a static deity. ...he could enjoy a certain amount of leisure, the result of wealth and of slave labor. He could philosophize...The absence of any well-established religion led many... into mysticism, but also stimulated its opposite, the growth of rationalism and the scientific outlook."

"The "exhaustion method"... was the Platonic school's answer to Zeno. It avoided the pitfalls of the infinitesimals by simply discarding them... It had the disadvantage that the result... must be known in advance ...a letter from Archimedes to Eratosthenes... described a nonrigorous but fertile way of finding results ...known as the "Method." It has been suggested... that it represented a school of mathematical reasoning competing with Eudoxus... In Democritus' school, according to the theory of Luria, the notion of a "geometrical atom" was introduced. ...The advantage of the "atom method" over the "exhaustion method" was that it facilitated the finding of new results. Antiquity had thus the choice between a rigorous but relatively sterile, and a loosely-founded but far more fertile method. ...in practically all classical texts the first [the exhaustion] method was used. This... may be connected with the fact that mathematics had become a hobby of the leisure class which was based on slavery, indifferent to invention, and interested in contemplation. It may also be a reflection of the victory of Platonic idealism over Democritian materialism in the realm of mathematical philosophy."

"It is the purpose of this paper to show what is historically wrong with the traditional way the history of ancient Greek mathematics has been written and to call to the new generation of historians of Greek mathematics to rewrite the history on a new and historically sane basis."

"One of the central concepts for the understanding of ancient Greek mathematics has customarily been, at least since the time of and , the concept of 'geometric algebra'. What it amounts to is that Greek mathematics, especially after the discovery of the 'irrational'... is algebra dressed up, primarily for the sake of rigor, in geometrical garb. The reasoning... the line of attack... the solutions... etc. all are essentially algebraic... attired in geometrical accouterments. We... look for the algebraic 'subtext'... of any geometrical proof... always to transcribe... any proposition in[to] the symbolic language of modern algebra... [making] the logical structure of the proof clear and convincing, without thereby losing anything, not only in generality but also in any possible sui generis features of the ancient way of doing things. ...[i.e., that] there is nothing unique and (ontologically) idiosyncratic concerning the way... ancient Greek mathematicians went about their proofs, which might be lost... I cannot find any historically gratifying basis for this generally accepted view... those who have been writing the history of mathematics... have typically been mathematicians... largely unable to relinquish and discard their laboriously acquired mathematical competence when dealing with periods in history during which such competence is historically irrelevant and... anachronistic. Such... stems from the unstated assumption that mathematics is a scientia universalis, an algebra of thought containing universal ways of inference, everlasting structures, and timeless, ideal patterns of investigation which can be identified throughout the history of civilized man and which are completely independent of the form in which they happen to appear at a particular junction of time."

"Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any things or about some things, without specification of definite particular things. These propositions were first enunciated by the Greeks for geometry; and, accordingly, geometry was the great Greek mathematical science. After the rise of geometry centuries passed away before algebra made a really effective start, despite some faint anticipations by the later Greek mathematicians."

"The Greeks would have said... we know a much better way of taking a square root. ...the ancient Greeks thought entirely geometrically, not arithmetically. And they would... do the following. If you want to solve x^2 = N, you should first... think of whether N is bigger than or equal to one. Suppose that case 1) N < 1. ...Draw a [horizontal] line segment of length one and then [within and from the end of that segment]... make a segment of size N. And then with the center of the [length one] segment you draw a circle so this is a [unit length] diameter. And you... [draw a vertical line from the end of the N segment inside the circle] up here [to intersect the circle] and then... look at this quantity x... this [top angle of the largest triangle circumscribed by the circle] is a right angle by Thales theorem, so we have some similar triangles. So [side x, side 1 from the large circumscribed triangle] \frac{x}{1} = \frac{N}{x} [side N, side x from the small left triangle] by similar \triangle's . And so x^2 = N. So Geometrically finding a square root is... a relatively simple... rule or construction, but arithmetically much more difficult. What happens if N is bigger than one? Well then you just interchange the roles of the N and the one. Case 2) N \ge 1. So you start by having a diameter of size N and then you make [a line segment of length] 1 here [from the end of the segment of length N to within that segment] and then otherwise do exactly the same thing [as in the above, case 1]. ...x will be square root, x^2 = N, by the same argument."

"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 extraordinary ability of Diophantus appears rather in... the ingenuity with which he reduces every problem to an equation which he is competent to solve."

"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..."

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

"The oldest definition of Analysis as opposed to Synthesis is that appended to Euclid XIII. 5. It was possibly framed by Eudoxus. It states that "Analysis is the obtaining of the thing sought by assuming it and so reasoning up to an admitted truth: synthesis is the obtaining of the thing sought by reasoning up to the inference and proof of it." In other words, the synthetic proof proceeds by shewing that certain admitted truths involve the proposed new truth: the analytic proof proceeds by shewing that the proposed new truth involves certain admitted truths."

"The history of the Athenian school begins with the teaching of Hippocrates about 420 B.C.; the school was established on a permanent basis by the labours of Plato and Eudoxus; and, together with the neighboring school of Cyzicus, continued to extend on the lines laid down by these three geometricians until the foundation (about 300 B.C.) of the university at Alexandria drew thither most of the talent of Greece."

"Eudoxus... is also reckoned as the founder of the school at Cyzicus. The connection [with the school] of Athens was very close, and it is impossible to disentangle their histories. It is said that Hippocrates, Plato, and Theaetetus belonged to the Athenian school; while Eudoxus, , and Aristaeus belonged to that of Cyzicus. There was always constant intercourse between the two schools, the earliest members of both had been under the influence either of Archytas or of his pupil ..."

"The geometricians of these schools... were especially interested in three problems: namely (i), the duplication of the cube... (ii) the trisection of an angle; and (iii) the squaring of a circle... Now the first two... (considered analytically) require the solution of a quadratic equation; and, since a construction by means of circles (whose equations are of the form x^2 + y^2 + ax + by + c = 0 and straight lines (whose equations are of the form \alpha x + \beta y + \gamma = 0) cannot be equivalent to the solution of a cubic equation, the problems are insoluble if in our constructions we restrict ourselves to the use of circles and straight lines, that is, to Euclidean geometry. If the use of s be permitted, both of these questions can be solved in many ways. The third problem is equivalent to finding a rectangle whose sides are equal respectively to the radius and to the semiperimeter of the circle. These lines have long been known to be incommensurable, but it is only recently that it has been shown by Lindemann that their ratio cannot be the root of a rational algebraical equation. Hence the problem also is insoluble by Euclidean geometry. The Athenians and Cyzicians were thus destined to fail in all three problems, but the attempts to solve them led to the discovery of many new theorems and processes."

"The sum of the three angles of every plane triangle is equal to two right angles. The mathematical truth enunciated in the above theorem is not new It has been known for more than two thousand years. ...Our reason for believing that Thales was not ignorant of the theorem under consideration is found in the beautiful demonstration by which he proved that every angle in a semicircle is a right angle. This appears to have been regarded as the most remarkable of the geometrical achievements of Thales, and it is stated that on inscribing a right angled triangle in a circle he sacrificed an ox to the immortal gods."

"Before giving the proof by which Thales probably established the truth... it will be well to consider the geometrical capital which this Grecian mathematician had at his command. ...i. The angles at the base of an are equal. ...ii. If two straight lines cut one another the vertically opposite angles are equal."

"vi. The angle in a semicircle is a right angle. It is believed that Thales proved this proposition in the following manner: Let ABCH be a circle of which the diameter is BC, and the centre E. ...Draw AE and produce BA to F. Because BE is equal to EA [both being radii of the circle], the angle EAB is equal to EBA; also, because AE is equal to EC, the angle EAC is equal to ECA [being angles at the base of an isosceles triangle]; wherefore, the whole angle BAC is equal to the two angles ABC, ACB. But FAC, the exterior angle of the triangle ABC, is also equal to the two angles ABC, ACB [since the sum of the three angles of the triangle is equal to two right angles, i.e., a straight line]; therefore the angle BAC is equal to the angle FAC, and each of them is therefore a right angle; wherefore the angle BAC in a semicircle is a right angle. Thales's demonstration, if we may call this his, is quite different from the one given in modern text-books; but it is certainly neither less rigid nor less beautiful. The demonstration is the one given in Euclid, but his work, we must remember, is to a large extent compiled from the works of previous writers. It will be seen, however, that this demonstration implies a knowledge of a seventh proposition,—"If one side of a triangle be produced, the exterior angle is equal to the sum of the two interior and opposite angles." Thales must have been familiar with this truth."

"For the mathematician the important consideration is that the foundations of mathematics and a great portion of its content are Greek. The Greeks laid down the first principles, invented the methods ab initio, and fixed the terminology. Mathematics in short is a Greek science, whatever new developments modern analysis has brought or may bring."

"Greek mathematics reveals an important aspect of the Greek genius of which the student of Greek culture is apt to lose sight."

"Dr. James Gow did a great service by the publication in 1884 of his Short History of Greek Mathematics, a scholarly and useful work which has held its own and has been quoted with respect and appreciation by authorities on the history of mathematics in all parts of the world. At the date when he wrote, however, Dr. Gow had necessarily to rely upon the works of the pioneers Bretschneider, Hankel, Allman, and (first edition). Since then the subject has been very greatly advanced... scholars and mathematicians... have thrown light on many obscure points. It is therefore high time for the complete story to be rewritten."

"Euclid, the author of the incomparable Elements, wrote on almost all the other branches of mathematics known in his day. Archimedes's work, all original and set forth in treatises which are models of scientific exposition, perfect in form and style, was even wider in its range of subjects. The imperishable and unique monuments of the genius of these two men must be detached from their surroundings and seen as a whole if we would appreciate to the full the pre-eminent place which they occupy, and will hold for all time, in the history of science."

"It is a defect in the existing histories that, while they state generally the contents of, and the main propositions proved in, the great treatises of Archimedes and Apollonius, they make little attempt to describe the procedure by which the results are obtained. I have therefore taken pains, in the most significant cases, to show the course of the argument in sufficient detail to enable a competent mathematician to grasp the method used and to apply it, if he will, to other similar investigations."

"was the author of a book purporting to be a manual of mathematical subjects such as a student would require to enable him to understand Plato."

"Thales and Pythagoras took their start from Babylonian mathematics but gave it a very different... specifically Greek character... in the Pythagorean school and outside, mathematics was brought to... ever higher development and began gradually to satisfy the demands of stricter logic... through the work of Plato's friends Theaetetus and Eudoxus, mathematics was brought to a state of perfection, beauty and exactness, which we admire in the elements of Euclid. ...the mathematical method of proof served as a prototype for Plato's dialectics and for Aristotle's logic."

"Plato is very fond of appealing to mathematics to show that exact reasoning is possible, not about things which are seen and heard, but about ideal objects which exist in thought only. ... Plato is apparently convinced that the mathematicians will agree with him. And indeed, when we deal with line segments which one sees and which one measures empirically, the question as to the existence of a common measure has no sense; a hair's breadth will measure integrally every line that is drawn. The question of commensurability makes sense only for line segments as objects of thought. It is therefore clear that Theodorus could not appeal to intuition to prove the incommensurability of the sides of his squares."

"After Apollonius Greek mathematics came to a dead stop. It is true that there were some epigones, such as Diocles and Zenodorus... But apart from trigonometry, nothing great nothing new appeared. The geometry of the conics remained in the form Apollonius gave it, until Descartes. ...The "Method" of Archimedes was lost sight of, and the problem of integration remained where it was, until it was attacked anew in the 17th century... Germs of projective geometry were present, but it remained for Desargues and Pascal to bring these to fruition. ...Higher plane curves were studied only sporadically... Geometric algebra and the theory of proportions were carried over into modern times as inert traditions, of which the inner meaning was no longer understood. The Arabs started algebra anew, from a much more primitive point of view... Greek geometry had run into a blind alley."

"Any one can use our algebraic notation, but only a gifted mathematician can deal with the Greek theory of proportions and with geometric algebra."

"An oral tradition makes it possible to indicate the line segments with the fingers; one can emphasize essentials and point out how the proof was found. All of this disappears in the written formulation... as soon as some external cause brought about an interruption in the oral tradition, and only books remained, it became very difficult to assimilate the work of the great predursors, and next to impossible to pass beyond it."

"Much of the narrative heritage of India and Greece goes back to shared ancestral narratives told in early IE times – to ‘protonarratives’. (…) the Greek tradition quite often fuses or amalgamates traditions that were separate in the protonarrative and remain separate in the Sanskrit."

"In parts of their careers, Arjuna and Odysseus show similarities so numerous and detailed that they must be cognate figures, sharing an origin in the proto-hero of an oral proto-narrative."

"Lord Mountstuart Elphinstone (1779-1859) in comparing the ancient Greeks with the ancient Hindus, says: "Their (Hindus) general learning was more considerable; and in the knowledge of the being and nature of God, they were already in possession of a light which was but faintly perceived even by the loftiest intellects in the best days of Athens.""

"Journeys to India and indebtedness to Brahminical wisdom are now ascribed to numerous founders and leaders in Greek thought, such as Plato, Democritus, Pherecydes of Cyrus and, quite often, Pythagoras. We hear that Pythagoras studied astronomy andastrology with the Chaldeans, and psychological as well as soteriological doctrines with the Indian gymnosophists."

"The mythology, as well as the cosmogony of the Egyptians, Greeks and Romans, were borrowed from the doctrines of the Brahmins."

"There are reports by writers of the Hellenistic and Roman periods that Greeks had visited India in much earlier times... In fact Plutarch, Diodoros Sikeliotes and Diogenes Laertios manage between them to send just about every Greek sage into the East (including Pythagoras and Democritos, but notably not Socrates and Aristotle)."

"Plato, through the Pythagoreans and also the Orphics, was subjected to the influence of Hindu thought but he may not have been aware of it as coming from India."

"It will no longer remain to be doubted that the priests of Egypt and the sages of Greece have drawn directly from the original well of India; that only Brahmanism can provide those fragments of their teaching which have come down to us with the clarity which they do not possess."

"‘These coincidences of thought,’ says A.N. Marlow who documents many more, ‘each small in itself, amount to quite a formidable total. As to the…way by which Indian influence reached Greece, I have no new solution to offer and fall back with others on Persia as the intermediary. The problem is, however, that the influences can be seen centuries earlier’"

"Everything is narrow in the Occident. Greece is small — I stifle. Judea is dry — I pant. Let me look a little towards lofty Asia, towards the deep Orient. There I find my immense poem, vast as India’s seas, blessed and made golden by the sun, a book of divine harmony in which nothing jars. There reigns a lovable peace, and even in the midst of battle, an infinite softness, an unbounded fraternity extending to all that lives, a bottomless and shoreless ocean of love, piety, clemency. I have found what I was looking for: the bible of kindness. Great poem, receive me!… Let me plunge into it! It is the sea of milk."

"The West spoke fairly enough, talking of honor, the sanctity of the given word, and of promises; of freedom and enlightenment. It vaunted its poets, its philosophers, Its scientists, Its classical inheritance from that beautiful, far off Greece, whose greatest philosophers, it forgot to mention, had been inspired through Egypt and Persia, by India."

"So, in practice, Western history has used two standards of evidence for transmission: one ultra-lax standard of evidence for transmission from "Greeks", and another ultra-strict standard for transmission to the West. For cases of alleged transmission from the Greeks, mere speculations—a speculative chronology combined with speculative attribution—are regarded as ample evidence of transmission. In the other direction, similarity with a real earlier work, by a real author, together with a clear channel of transmission, do not prove anything, for there is always the possibility of repeated miracles by which any number of people in the West may independently reinvent things just when they could be transmitted."

"In the many centuries, since Toledo, that Western historians have been talking of transmission from the Greeks, who ever produced a Sanskrit manuscript of Ptolemy? Who ever proved that Aryabhata had seen such a Sanskrit manuscript? Yet every Western reference work on the subject asserts that Indian astronomy is transmitted from the Greeks. ... So, it is not so much that the standards of evidence have changed, but that there are (even as of today) two simultaneous standards of evidence for transmission. One for transmission to the West, and another for purported transmission from the West. Not only is the judge biased, the very rules of evidence are biased!... So, similarity and precedence do not always establish transmission. Whether or not they establish transmission depends upon the direction of transfer. Thus, in practice, there are two standards of evidence for transmission: an ultra-lax standard for transmission from Greeks, and an ultra-strict standard for transmission to the West."

"A hundred facts testify to how great an extent the East was mingled with Hellenic thought during the second century of our era."

"There is no language in the world, even Greek, which has the clarity and the philosophical precision of Sanskrit. India is not only at the origin of everything, she is superior in everything, intellectually, religiously or politically and even the Greek heritage seems pale in comparison."

"Even the loftiest philosophy of the Europeans, the idealism of reason, as it is set forth by Greek philosophers, appears, in comparison with the abundant light and vigour of Oriental idealism, like a feeble Promethean spark in the full flood of heavenly glory of the noonday sun—faltering and feeble, and ever ready to be extinguished."

"We find among the Indians the vestiges of the most remote antiquity... We know that all peoples came there to draw the elements of their knowledge... India, in her splendour, gave religions and laws to all the other peoples; Egypt and Greece owed to her both their fables and their wisdom."

"The Greek sought political liberty. The Hindu has always sought spiritual liberty. Both are one-sided."

"Two curious nations there have been - sprung of the same race, but placed in different circumstances and environments, working put the problems of life each in its own particular way. I mean the ancient Hindu and the ancient Greek. The Indian Aryan - bounded on the north by the snow-caps of the Himalayas, with fresh-water rivers like rolling oceans surrounding him in the plains, with eternal forests which, to him, seemed to be the end of the world - turned his vision inward; and given the natural instinct, the superfine brain of the Aryan, with this sublime scenery surrounding him, the natural result was that he became introspective. The analysis of his own mind was the great theme of the Indo-Aryan. With the Greek, on the other hand, who arrived at a part of the earth which was more beautiful than sublime, the beautiful islands of the Grecian Archipelago, nature all around him generous yet simple - his mind naturally went outside. It wanted to analyse the external world. And as a result we find that from India have sprung all the analytical sciences, and from Greece all the sciences of generalization. The Hindu mind went on in its own direction and produced the most marvellous results. Even at the present day, the logical capacity of the Hindus, and the tremendous power which the Indian brain still possesses, is beyond compare. ...Today the ancient Greek is meeting the ancient Hindu on the soil of India."

"Two nations of yore, namely the Greek and the Aryan placed in different environments and circumstances - the former, surrounded by all that was beautiful, sweet, and tempting in nature, with an invigorating climate, and the latter, surrounded on every side by all that was sublime, and born and nurtured in a climate which did not allow of much physical exercise - developed two peculiar and different ideals of civilization. The study of the Greeks was the outer infinite, while that of the Aryans was the inner infinite; one studied the macrocosm, and the other the microcosm. Each had its distinct part to play in the civilisation of the world. Not that one was required to borrow from the other, but if they compared notes both would be the gainers. The Aryans were by nature an analytical race. In the sciences of mathematics and grammar wonderful fruits were gained, and by the analysis of mind the full tree was developed. In Pythagoras, Socrates, Plato, and the Egyptian neo-Platonists, we can find traces of Indian thought."

"The Greeks, in their mythology, were merely disciples of India and of Egypt."

"Our nations have mutually destroyed each other on that very soil where we went to collect nothing but money, and where the first Greeks travelled for nothing but knowledge."

"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 Brahmans' science not been been long established in Europe."

"We have already acknowledged that arithmetic, geometry, astronomy were taught among the Brahmans. From time immemorial they have known the precession of the equinoxes and were in their calculation far closer to the real figure than the Greeks who came much later. Mr. Le Gentil (a French astronomer who spent several years in India) has with admiration acknowledged the Brahmans' science, as well as the immensity of time these Indians must have needed to reach a knowledge of which even the Chinese never had any notion, and which was unknown to Egypt and to Chaldea, the teacher of Egypt."

"I like to think that someone will trace how the deepest thinking of India made its way to Greece and from there to the philosophy of our times."

"Joseph Waligore is equally critical but has no solution except that they did borrow ‘from Indian philosophy but only had limited contact with the Indians and could not penetrate the core of Indian thought’."

"Hellenism is the pot on the stove, the scoop for the embers, the jug of milk, it is the furnishings, the crockery, what surrounds the body; Hellenism is the warmth of the domestic hearth, perceived as sacred, it is everything belonging to man that puts him in contact with a part of the outside world [...]. Hellenism is purposely surrounding man with furnishings instead of just any objects, transforming the latter into furniture, humanizing the surrounding world, infusing it with a subtle teleological warmth. Hellenism is the stove by which a man sits and enjoys the warmth it emanates, so akin to the warmth he has inside. (Osip Emilyevich Mandelshtam)"

"The "Hellenes" astonished us because, although open to the spiritual disturbances of their age, they appealed to ancient methods to find a solution to the anxieties of the present. Their placid faith in a tradition stemming from Plato and constantly evolving was perhaps the most reassuring aspect of late antique civilization. In fact, many classical and enlightened societies had collapsed under the weight of their own traditionalism, leaving their immediate successors only with a memory of anxieties and nightmares. If this did not happen in the Roman Empire, it is largely due to the "Hellenic Renaissance" and the dialogue between its proponents and the new Christian aristocratic intellectuals. (Peter Brown)"

"Alexander the Great"

"Ancient Greece"

"Four hundred thousand slaves must be devoted to forty thousand citizens; weak and deformed children must be exposed; morality and humanity, as well as all the comforts, elegancies, and pleasures of life, must be sacrificed to this glaring phantom of vanity, superstition, and ambition. Separated from the rest of mankind, they lived together, destitute of all business, pleasure, and amusement, but war and politics, pride and ambition; and there occupations and passions they transmitted from generation to generation, for seven hundred years; as if fighting and intriguing, and not life and happiness, were the end of man, and society; as as if the love of one's country, and of glory, were amiable passions, when not limited by justice and general benevolence; and as if nations were to be chained together for ever, merely that one family might reign among them… Human nature perished under this frigid system of national and family pride."

"The institution of Lycurgus was well calculated to preserve the independence of his country, but had no regard to its happiness, and very little to its liberty. As the people's consent was necessary to every law, it had so far the appearance of political liberty: but the civil liberty of it was little better than that of a man chained in a dungeon; a liberty to rest as he is. The influence of this boasted legislation on the human character was to produce warriors and politicians, and nothing else. To say that this people were happy, is to contradict every quality in human nature, except ambition. They had no other gratification: science and letters were sacrificed, as well as commerce, to the ruling passion."

"There are, at any rate, two or three things in Spartan life that I think ought to appeal to all sections of this generation. The example that the Spartans made of the drunken Helots would find some to applaud it to-day. The alacrity with which some of our modern womanhood are striving now to overtake the modes and the fashions of those girls who took part in the festivals of Greece shows that their practices in this respect command emulation in some quarters. There is much also in the Spartan regime that would appeal to the modern Fascisti. The Spartans had another rule which one has often heard Britons blamed for following—never to trust the foreigner. I do not think that there would have been much chance for a Soviet Delegation in Sparta."

"I like to think of the earth being stirred over grim Sparta. To all boys—and I am speaking of my own recollections—she had a peculiar appeal. I think most boys prefer her to Athens and prefer Leonidas to Alcibiades in spite of the fact that Alcibiades is a much more successful character in the modern world than Leonidas. But there is something in Spartan discipline and mode of life that appeals to strenuous youth, and it may be the cherished recollections of far distant days that make me feel a thrill at the thought of the investigations of the School in Sparta."

"Here is what you do, friends. Forget country. Forget king. Forget wife and children and freedom. Forget every concept, however noble, that you imagine you fight for here today. Act for this alone: for the man who stands at your shoulder. He is everything, and everything is contained within him. That is all I know. That is all I can tell you."

"The ideal Spartan was plucky, indifferent to hardship and pain, a first-rate athlete. The less he talked or, for that matter, thought, the better. It was for him emphatically not to reason why, but always to do and die. He was a soldier and nothing else. The purpose of the Spartan state was war."

"The idea that underlay the young Spartans' training was their obligation to maintain the power of the state and ignore everything that did not directly contribute to it. All the other possibilities of life — imagination, love of beauty, intellectual interests — were put aside. The goal of human aspiration and achievement was to uphold the fatherland. Only what helped the state was good, only what harmed it was bad. A Spartan was not an individual but a part of a well-functioning machine which assumed all responsibility for him, exacted absolute submission from him, molded his character and his mind, and imbued him with the deep conviction that the chief end of man was to kill and be killed."

"It was natural for [Spartan women] to think and speak as Gorgo, the wife of Leonidas, is said to have done, when some foreign lady, as it would seem, told her that the women of Lacedaemon were the only women of the world who could rule men; 'With good reason,' she said, 'for we are the only women who bring forth men'."

"Their [Lacadaemonian] women, it is said, were bold and masculine, overbearing to their husbands in the first place, absolute mistresses in their houses, giving their opinions about public matters freely, and speaking openly even on the most important subjects."

"The freedom which thus prevailed at that time in marriage relations was aimed at physical and political well-being, and was far removed from the licentiousness which was afterwards attributed to their women, so much so that adultery was wholly unknown among them. And a saying is reported of one Geradas,​ a Spartan of very ancient type, who, on being asked by a stranger what the punishment for adulterers was among them, answered: "Stranger, there is no adulterer among us." "Suppose, then," replied the stranger, "there should be one." "A bull," said Geradas, "would be his forfeit, a bull so large that it could stretch over Mount Taÿgetus and drink from the river Eurotas." Then the stranger was astonished and said: "But how could there be a bull so large?" To which Geradas replied, with a smile: "But how could there be an adulterer in Sparta?" Such, then, are the accounts we find of their marriages."

"The king marched against the enemy in close companion­ship with one who had been crowned victor in the great games. And they tell of a certain Spartan who refused to be bought off from a contest at Olympia by large sums of money, and after a long struggle outwrestled his antagonist. When some one said to him then: "What advantage, O Spartan, hast thou got from thy victory?" he answered, with a smile: "I shall stand in front of my king when I fight our enemies.""

"Through his doctrine of the similarity between Sparta and the perfect state, Plato became one of the most successful propagators of what I should like to call ‘the Great Myth of Sparta’—the perennial and influential myth of the supremacy of the Spartan constitution and way of life."

"Men are not born equal in themselves, so I think it beneath a man to postulate that they are. If I thought myself as good as Sokrates I should be a fool; and if, not really believing it, I asked you to make me happy by assuring me of it, you would rightly despise me. So why should I insult my fellow-citizens by treating them as fools and cowards? A man who thinks himself as good as everyone else will be at no pains to grow better. On the other hand, I might think myself as good as Sokrates, and even persuade other fools to agree with me; but under a democracy, Sokrates is there in the Agora to prove me wrong. I want a city where I can find my equals and respect my betters, whoever they are; and where no one can tell me to swallow a lie because it is expedient, or some other man's will."

"Political writers argue in regard to the love of liberty with the same philosophy that philosophers do in regard to the state of nature; by the things they see they judge of things very different which they have never seen, and they attribute to men a natural inclination to slavery, on account of the patience with which the slaves within their notice carry the yoke; not reflecting that it is with liberty as with innocence and virtue, the value of which is not known but by those who possess them, though the relish for them is lost with the things themselves. I know the charms of your country, said Brasidas to a satrap who was comparing the life of the Spartans with that of the Persepolites; but you can not know the pleasures of mine.”"

"To understand Plato, and indeed many later philosophers, it is necessary to know something of Sparta. Sparta had a double effect on Greek thought: through the reality, and through the myth. Each is important. The reality enabled the Spartans to defeat Athens in war; the myth influenced Plato's political theory, and that of countless subsequent writers. The myth, fully developed, is to be found in Plutarch’s Life of Lycurgus; the ideals that it favours have had a great part in framing the doctrines of Rousseau, Nietzsche, and National Socialism. The myth is of even more importance, historically, than the reality."

"There was not simply one ‘orientalizing’ period, there were several."

"Angesichts dieser Sachlage wäre es durchaus nicht als abwegig zu bezeichnen, daß man fragte, was im archaischen Hellas eigentlich nicht aus dem Orient herstammte."

"In view of this state of affairs it could not be called out of the way to ask what there was in Archaic Greece which did not come from the orient."

"Hushed be on Dryads’ wooded rock the rills, And hushed the bleatings on the meads, Now Pan his pipe with breath melodious fills And kisses with moist lip the reeds; While, treading nimble dances all around, Dryads and Hamadryads beat the ground."

"Within the shady grove we chanced to peep, And caught Cythera’s rosy boy asleep: None of his brave artillery had he, But bow and quiver hung upon a tree; While he on rosebuds smiling lay, in warm Slumber fast bound; and o’er his lips a swarm Of honey bees laid sweets and wrought no harm."

"Thou gazest on the stars: Would I might be, O star of mine, the skies With myriad eyes To gaze on thee."

"Thou wert the morning star among the living, Ere thy fair light had fled;— Now, having died, thou art as Hesperus, giving New splendour to the dead."

"Xenophanes said, "I confess myself the greatest coward in the world, for I dare not do an ill thing.""

"One made the observation of the people of Asia that they were all slaves to one man, merely because they could not pronounce that syllable No."

"Euripides was wont to say, "Silence is an answer to a wise man.""

"Alexander wept when he heard from Anaxarchus that there was an infinite number of worlds; and his friends asking him if any accident had befallen him, he returns this answer: "Do you not think it a matter worthy of lamentation that when there is such a vast multitude of them, we have not yet conquered one?""

"Like the man who threw a stone at a bitch, but hit his step-mother, on which he exclaimed, "Not so bad!""

"Pittacus said, "Every one of you hath his particular plague, and my wife is mine; and he is very happy who hath this only"."

"He was a man, which, as Plato saith, is a very inconstant creature."

"The pilot cannot mitigate the billows or calm the winds."

"I, for my own part, had much rather people should say of me that there neither is nor ever was such a man as Plutarch, than that they should say, "Plutarch is an unsteady, fickle, froward, vindictive, and touchy fellow.""

"Remember what Simonides said,—that he never repented that he had held his tongue, but often that he had spoken."

"Custom is almost a second nature."

"Epaminondas is reported wittily to have said of a good man that died about the time of the battle of Leuctra, "How came he to have so much leisure as to die, when there was so much stirring?""

"Have in readiness this saying of Solon, "But we will not give up our virtue in exchange for their wealth.""

"Anacharsis said a man's felicity consists not in the outward and visible favours and blessings of Fortune, but in the inward and unseen perfections and riches of the mind."

"Said Periander, "Hesiod might as well have kept his breath to cool his pottage.""

"Socrates said, "Bad men live that they may eat and drink, whereas good men eat and drink that they may live.""

"And Archimedes, as he was washing, thought of a manner of computing the proportion of gold in King Hiero's crown by seeing the water flowing over the bathing-stool. He leaped up as one possessed or inspired, crying, "I have found it! Eureka!""

"Said Scopas of Thessaly, "We rich men count our felicity and happiness to lie in these superfluities, and not in those necessary things.""

"That proverbial saying, "Ill news goes quick and far.""

"A traveller at Sparta, standing long upon one leg, said to a Lacedæmonian, "I do not believe you can do as much." "True," said he, "but every goose can.""

"Spintharus, speaking in commendation of Epaminondas, says he scarce ever met with any man who knew more and spoke less."

"It is a thing of no great difficulty to raise objections against another man's oration,—nay, it is a very easy matter; but to produce a better in its place is a work extremely troublesome."

"What is bigger than an elephant? But this also is become man's plaything, and a spectacle at public solemnities; and it learns to skip, dance, and kneel."

"No man ever wetted clay and then left it, as if there would be bricks by chance and fortune."

"Alexander was wont to say, "Were I not Alexander, I would be Diogenes.""

"Like watermen, who look astern while they row the boat ahead."

"Socrates said he was not an Athenian or a Greek, but a citizen of the world."

"Anaximander says that men were first produced in fishes, and when they were grown up and able to help themselves were thrown up, and so lived upon the land."

"Athenodorus says hydrophobia, or water-dread, was first discovered in the time of Asclepiades."

"Let us not wonder if something happens which never was before, or if something doth not appear among us with which the ancients were acquainted."

"Why does pouring oil on the sea make it clear and calm? Is it for that the winds, slipping the smooth oil, have no force, nor cause any waves?"

"The great god Pan is dead."

"I am whatever was, or is, or will be; and my veil no mortal ever took up."

"When Hermodotus in his poems described Antigonus as the son of Helios, "My valet-de-chambre," said he, "is not aware of this.""

"There is no debt with so much prejudice put off as that of justice."

"It is a difficult thing for a man to resist the natural necessity of mortal passions."

"He is a fool who lets slip a bird in the hand for a bird in the bush."

"When Demosthenes was asked what was the first part of oratory, he answered, "Action;" and which was the second, he replied, "Action;" and which was the third, he still answered, "Action.""

"Xenophon says that there is no sound more pleasing than one's own praises."

"Lampis, the sea commander, being asked how he got his wealth, answered, "My greatest estate I gained easily enough, but the smaller slowly and with much labour.""

"The general himself ought to be such a one as can at the same time see both forward and backward."

"Statesmen are not only liable to give an account of what they say or do in public, but there is a busy inquiry made into their very meals, beds, marriages, and every other sportive or serious action."

"Leo Byzantius said, "What would you do, if you saw my wife, who scarce reaches up to my knees?… Yet," went he on, "as little as we are, when we fall out with each other, the city of Byzantium is not big enough to hold us.""

"Cato said, "I had rather men should ask why my statue is not set up, than why it is.""

"It was the saying of Bion, that though the boys throw stones at frogs in sport, yet the frogs do not die in sport but in earnest."

"Both Empedocles and Heraclitus held it for a truth that man could not be altogether cleared from injustice in dealing with beasts as he now does."

"Simonides calls painting silent poetry, and poetry speaking painting."

"As Meander says, "For our mind is God;" and as Heraclitus, "Man's genius is a deity.""

"Pythagoras, when he was asked what time was, answered that it was the soul of this world."

"He that first started that doctrine, that knavery is the best defence against a knave, was but an ill teacher, advising us to commit wickedness to secure ourselves."

"Nήπιος, ὃς τὰ ἕτοιμα λιπὼν ἀνέτοιμα διώκει."

"Τοῖς ἐγρηγορόσιν ἕνα καὶ κοινὸν κόσμον εἶναι, τῶν δὲ κοιμωμένων ἕκαστον εἰς ἴδιον ἀποστρέφεσθαι."

"When the candles are out, all women are alike."

"To err in opinion, though it be not the part of wise men, is at least human."

"Οὐ γὰρ ὡς ἀγγεῖον ὁ νοῦς ἀποπληρώσεως ἀλλ' ὑπεκκαύματος μόνον ὥσπερ ὕλη δεῖται ὁρμὴν ἐμποιοῦντος εὑρετικὴν καὶ ὄρεξιν ἐπὶ τὴν ἀλήθειαν. ὥσπερ οὖν εἴ τις ἐκ γειτόνων πυρὸς δεόμενος, εἶτα πολὺ καὶ λαμπρὸν εὑρὼν αὐτοῦ καταμένοι διὰ τέλους θαλπόμενος, οὕτως εἴ τις ἥκων λόγου μεταλαβεῖν πρὸς ἄλλον οὐχ οἴεται δεῖν φῶς οἰκεῖον ἐξάπτειν καὶ νοῦν ἴδιον, ἀλλὰ χαίρων τῇ ἀκροάσει κάθηται θελγόμενος, οἷον ἔρευθος ἕλκει καὶ γάνωμα τὴν δόξαν ἀπὸ τῶν λόγων, τὸν δ᾽ ἐντὸς: εὐρῶτα τῆς ψυχῆς καὶ ζόφον οὐκ ἐκτεθέρμαγκεν οὐδ᾽ ἐξέωκε διὰ φιλοσοφίας."

"By these criteria let Alexander also be judged! For from his words, from his deeds, and from the instruction' which he imparted, it will be seen that he was indeed a philosopher."

"Alexander established more than seventy cities among savage tribes, and sowed all Asia with Greek magistracies."

"If it were not my purpose to combine foreign things with things Greek, to traverse and civilize every continent, to search out the uttermost parts of land and sea, to push the bounds of Macedonia to the farthest Ocean, and to disseminate and shower the blessings of Greek justice and peace over every nation, I should not content to sit quietly in the luxury of idle power, but I should emulate the frugality of Diogenes. But as things are, forgive me Diogenes, that I imitate Heracles, and emulate Perseus, and follow in the footsteps of Dionysus, the divine author and progenitor of my family, and desire that victorious Greeks should dance again in India and revive the memory of the Bacchic revels among the savage mountain tribes beyond the Caucasus."

"What spectator... would not exclaim... that through Fortune the foreign host was prevailing beyond its deserts, but through Virtue the Hellenes were holding out beyond their ability? And if the enemy gains the upper hand, this will be the work of Fortune or of some jealous deity or of divine retribution; but if the the Greeks prevail, it will be Virtue and daring, friendship and fidelity, that will win the guerdon of victory?"

"That remiss and slow-paced justice (as Euripides describes it) that falls upon the wicked by accident, by reason of its uncertainty, ill-timed delay, and disorderly motion, seems rather to resemble chance than providence. So that I cannot conceive what benefit there is in these millstones of the Gods which are said to grind so late, as thereby celestial punishment is obscured, and the awe of evil doing rendered vain and despicable."

"It is a desirable thing to be well descended; but the glory belongs to our ancestors."

"Ἡ ἀνάπαυσις τῶν πόνων ἐστὶν ἄρτυμα."

"It is a true proverb, that if you live with a lame man, you will learn to halt."

"The very spring and root of honesty and virtue lie in good education."

"It is wise to be silent when occasion requires, and better than to speak, though never so well."

"For water continually dropping will wear hard rocks hollow."

"The very spring and root of honesty and virtue lie in the felicity of lighting on good education."

"According to the proverb, the best things are the most difficult."

"To sing the same tune, as the saying is, is in everything cloying and offensive; but men are generally pleased with variety."

"Children are to be won to follow liberal studies by exhortations and rational motives, and on no account to be forced thereto by whipping."

"Nothing made the horse so fat as the king's eye."

"Democritus said, words are but the shadows of actions."

"'Tis a wise saying, Drive on your own track."

"Eat not thy heart; which forbids to afflict our souls, and waste them with vexatious cares."

"Abstain from beans; that is, keep out of public offices, for anciently the choice of the officers of state was made by beans."

"When men are arrived at the goal, they should not turn back."

"The whole life of man is but a point of time; let us enjoy it, therefore, while it lasts, and not spend it to no purpose."

"An old doting fool, with one foot already in the grave."

"They relate of Diogenes of Sinope, when he began to be a philosopher, that the Athenians were celebrating a festival, and there were public banquets and shows and mutual festivities, and drinking and revelling all night, and he, coiled up in a corner of the market-place intending to sleep, fell into a train of thought likely seriously to turn him from his purpose and shake his resolution, for he reflected that he had adopted without any necessity a toilsome and unusual kind of life, and by his own fault sat there debarred of all the good things. At that moment, however, they say a mouse stole up and began to munch some of the crumbs of his barley-cake, and he plucked up his courage and said to himself, in a railing and chiding fashion, "What say you, Diogenes? Do your leavings give this mouse a sumptuous meal, while you, the gentleman, wail and lament because you are not getting drunk yonder and reclining on soft and luxurious couches?" Whenever such depressions of mind are not frequent, and the mind when they take place quickly recovers from them, after having put them to flight as it were, and when such annoyance and distraction is easily got rid of, then one may consider one's progress in virtue as a certainty."

"Whenever we begin so much to love good men that we deem happy, "not only," as Plato says, "the temperate man himself, but also the man who hears the words that flow from his wise lips," and even admire and are pleased with his figure and walk and look and smile, and desire to adapt ourselves to his model and to stick closely to him, then may we think that we are making genuine progress. Still more will this be the case, if we admire the good not only in prosperity, but like lovers who admire even the lispings and paleness of those in their flower, as the tears and dejection of Panthea in her grief and affliction won the affections of Araspes, so we fear neither the exile of Aristides, nor the prison of Anaxagoras, nor the poverty of Socrates, nor the condemnation of Phocion, but think virtue worthy our love even under such trials, and join her, ever chanting that line of Euripides, "Unto the noble everything is good.""

"For the enthusiasm that can go so far as not to be discouraged at the sure prospect of trouble, but admires and emulates what is good even so, could never be turned away from what is noble by anybody. Such men ever, whether they have some business to transact, or have taken upon them some office, or are in some critical conjuncture, put before their eyes the example of noble men, and consider what Plato would have done on the occasion, what Epaminondas would have said, how Lycurgus or Agesilaus would have dealt; that so, adjusting and re-modelling themselves, as it were, at their mirrors, they may correct any ignoble expression, and repress any ignoble passion."

"As those persons who despair of ever being rich make little account of small expenses, thinking that little added to a little will never make any great sum."

"Antiphanes said merrily that in a certain city the cold was so intense that words were congealed as soon as spoken, but that after some time they thawed and became audible; so that the words spoken in winter articulated next summer."

"We are more sensible of what is done against custom than against Nature."

"You ask of me then for what reason it was that Pythagoras abstained from eating of flesh. I for my part do much admire in what humor, with what soul or reason, the first man with his mouth touched slaughter, and reached to his lips the flesh of a dead animal, and having set before people courses of ghastly corpses and ghosts, could give those parts the names of meat and victuals, that but a little before lowed, cried, moved, and saw; how his sight could endure the blood of slaughtered, flayed, and mangled bodies; how his smell could bear their scent; and how the very nastiness happened not to offend the taste, while it chewed the sores of others, and participated of the saps and juices of deadly wounds."

"Are you not ashamed to mix tame fruits with blood and slaughter? You are indeed wont to call serpents, leopards, and lions savage creatures; but yet yourselves are defiled with blood, and come nothing behind them in cruelty. What they kill is their ordinary nourishment, but what you kill is your better fare."

"For the sake of some little mouthful of flesh, we deprive a soul of the sun and light, and of that proportion of life and time it had been born into the world to enjoy. And then we fancy that the voices it utters and screams forth to us are nothing else but certain inarticulate sounds and noises, and not the several deprecations, entreaties, and pleadings of each of them."

"What meal is not expensive? That for which no animal is put to death. … one participating of feeling, of seeing, of hearing, of imagination, and of intellection; which each animal hath received from Nature for the acquiring of what is agreeable to it, and the avoiding what is disagreeable."

"In the beginning, some wild and mischievous beast was killed and eaten, and then some little bird or fish was entrapped. And the love of slaughter, being first experimented and exercised in these, at last passed even to the laboring ox, and the sheep that clothes us, and to the poor cock that keeps the house; until by little and little, unsatiableness being strengthened by use, men came to the slaughter of men, to bloodshed and wars."

"O sons of the Greeks, go on! Free your fatherland, and free your children, your wives, and the shrines of your paternal gods, and the tombs of your ancestors! Now the struggle is for all!"

"September of the year 490 B.C. was to my mind a more cardinal moment of fate for Europe than August 1914. Western civilisation, as we know it with its merits and its faults, was saved in its infancy at Marathon, and ten years later by Leonidas and by the men of Salamis. Rome was then in her in fancy; and had it not been for that decade there would have been nothing to prevent Eastern Europe being orientalised and the ultimate fight for the hegemony of Europe would have been left to the Persians and the Carthaginians. But for the Greeks there would have been no civilisation as we know it, and we should all have been dark-skinned people with long noses."

"The great conflict between Greece and Persia – or, to be more accurate, between a handful of states in mainland Greece and the whole might of the Persian empire at its zenith – must always remain one of the most inspiring episodes in European history. As Aeschylus and Herodotus clearly saw (despite the obfuscations of national pride and propaganda) this had been an ideological struggle, the first of its kind known to us. On one side, the towering, autocratic figure of the Great King; on the other, the voluntary and imperfect discipline of proudly independent citizens. In Herodotus's account, Xerxes' soldiers are driven forward to fight under the lash; the recurrent Persian motif of flogging, mutilation and torture throughout his narrative repays study. The Greeks, on the other hand, fought because they had a personal stake in victory: their struggle was to preserve a hard-won and still precarious heritage of freedom."

"The whole concept of political and intellectual liberty, of the constitutional state – however individually inefficient or corrupt – depended on one thing: that the Greeks, for whatever motive, decided to stand out against the Oriental system of palace absolutism, and did so with remarkable success."

"Common resistance and sacrifice in the face of a profoundly alien invader had begun, however slowly and imperfectly, to forge a sense of what afterwards came to be known as the Panhellenic ideal, of an identifiable and unique Greek spirit which no other race could share. This was perhaps the best and most lasting legacy of the Persian Wars."

"There has never been a war fought for purer motives than the war against Persia. Marathon and Salamis are still words that "send a ringing challenge down through the generations." Their victories still seem a miracle as they seemed to the men who won them. The mighty were put down from their seats and those of low degree exalted, and for fifty years and more Persia could do nothing to Greece. What followed was one of the most triumphant rebirths of the human spirit in all history, when the bitter differences that divide men were far in the background and freedom was in the air — freedom in the great sense, not only equality before the law, but freedom of thought and speech."

"We have no texts explaining the rites and ceremonial of the Dionysiac mysteries in the Greco-Etrusco-Roman world, although there are allusions which can often be clarified with the aid of Indian texts... By studying Shivaite rites [from India], the only ones which have continued down to our own times, the real practices of the Dionysiac rites and “mysteries” may be reconstructed."

"Our history of philosophy can only be the Greco-Roman-Christian one. We know neither the time of formation nor any kind of history about other, Asian philosophemes. Moreover, the simple beginnings of Greek philosophy, which developed from mythology, makes it unimportant for our purpose to ask whether this mythology did or did not have a foreign origin... ."

"It is worth quoting Anthony Grafton’s summation of Scaliger’s assault on the prisca theologia presumptions of his contemporaries here, as Scaliger’s position strongly foreshadows the nineteenth-century philhellenist view of cultural relations in the ancient world: “In astronomy and astrology, it had been the Greeks, not the Babylonians and the Egyptians, who performed most of the observations and, above all, tabulated and systematized the results. The ancient Near East had been not a world of gold, populated by calm sages, but a world of iron, haunted by superstitious fears and only fitfully illuminated by the work of certain science- minded priests — themselves prone to spin out unfounded speculations.”’"

"The first Graeco-Egyptian astrologists did not invent the discipline they claimed to teach the Hellenic world. They used Egyptian sources going up to the Persian period which were themselves at least partially derived from ancient Chaldaean documents. Traces of this primitive substratum still survive in our much later texts, erratic blocks transported on to more recent soil. When we find mentions there of ‘the king of kings’ or ‘satraps’ we are no longer in Egypt but in the ancient Orient … We limit ourselves to noting that in all appearances, the priests who were the authors of Egyptian astrology stayed relatively faithful to the ancient Oriental tradition."