"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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 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."

"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 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?"

"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."

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