classical-scholars

"Achilles—bring back the story, Goddess, of the formidable hero descended through Aeacus from thundering Jupiter, but denied His heaven, his deeds indeed famous through Homeric song, but with much more to celebrate."

"And for yourself, may the gods grant you Your heart's desire, a husband and a home, And the blessing of a harmonious life. For nothing is greater or finer than this, When a man and woman live together With one heart and mind, bringing joy To their friends and grief to their foes."

"I have borne what no man Who has walked this earth has ever yet borne. I have kissed the hand of the man who killed my son."

"Don't try to sell me on death, Odysseus. I'd rather be a hired hand back up on earth, Slaving away for some poor dirt farmer, Than lord it over all these withered dead."

"Shimmering, iridescent, deathless Aphrodite."

"When the defiant boy, whose heart had never trembled, saw this girl at the head of her troop of companions he stiffened, and every bone in his body absorbed liquid fire."

"Rage: Sing, Goddess, Achilles' rage, Black and murderous, that cost the Greeks Incalculable pain."

"There's no way to get around the mind of Zeus."

"Human generations are like leaves in their seasons. The wind blows them to the ground, but the tree Sprouts new ones when spring comes again. Men too. Their generations come and go."

"We know how to tell many believable lies, But also, when we want to, how to speak the plain truth."

"Whoever escapes marriage And women's harm, comes to deadly old age Without any son to support him."

"It was glorious to see—if your heart were iron, And you could keep from grieving at all the pain."

"Ah, my friend, if you and I could only Get out of the war alive and then immortal and ageless all of our days, I would never again fight among the foremost Or send you into battle where men win glory. But as it is, death is everywhere In more shapes that we can count, And since no mortal is immune or can escape, Let's go forward, either to give glory To another man, or get glory from him."

"Don't try to cut any deals with me, Hector. Do lions make peace treaties with men? Do wolves and lambs agree to get along?"

"Crewmates, by now we are hardly strangers to evil and hardship. We've suffered worse. God will grant us an end to these sufferings also. ... Take heart once again and dispel your fears and depression. Maybe the day'll come when even this will be joy to remember."

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

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

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

"Probably Greek logistic, or calculation, extended to more difficult operations... and... probably Greek arithmetic, or theory of numbers, owed much more to induction than is permitted to appear by its first and chief professors."

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

"It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:½ :: (1-½):(2⁄3-½).Iamblichus says that this proportion was called ύπeναντία originally and that Archytas and Hippasus first called it harmonic. Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern άρμονία: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:12:6 :: 12-8:8-6"

"The history of Greek mathematics is, for the most part, only the history of such mathematics as are learnt daily in all our public schools. ...If it was not wanted, as it ought to have been, by our classical professors and our mathematicians, it would have served at any rate to quicken, with some human interest, the melancholy labours of our schoolboys."

"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."

"Apart from the rites and worship peculiar to each family, gens, curia, and tribe, the Romans recognised a vast number of gods and goddesses whose worship was the concern of the whole state. The necessary ceremonies were, in many cases, placed in the charge of sodalicia or clubs... which elected their own members. But the worship of all deities not otherwise provided for was superintended by the pontifices. The College of Pontifices is said to have been founded by Numa, and was in regal times, presided over by the king himself. But when kings were abolished, their religious functions were divided between two officers, the and the or Sacrificulus. The latter, though he was sometimes treated as the chief priest, in reality only offered some of the sacrifices which the king formerly offered... The general supervision of the state religion belonged to the Pontifex Maximus. The Pontifex Maximus lived in the Regia, the ancient palace."

"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."

"Officials could be removed during their year of office by vote of the ecclesia, and periodical opportunities were given for raising complaints..."

"Every official was required to undergo, before assuming office... approval before a law court. This was an inquiry into his conduct, his exactness in paying taxes, etc., and it sometimes happened that he was rejected... Every official was also required to take an oath of allegiance."

"My aim is... to place before a young student a nucleus of well-ordered knowledge, to which he is to add intelligent notes and illustrations from his daily reading."

"It happened fortunately that during this period of turmoil the guidance of the Christian Church, the one powerful and permanent institution, was chiefly in the hands of the splendid order of St. Benedict. This saint... seeing that idleness was the besetting danger of monastic establishments, founded at Monte Cassino... a model abbey, in which industry was the daily rule. Among other employments, reading and writing were approved as powerful agents in distracting the mind from unholy thoughts, and in Benedictine monasteries the mechanical exercise of copying mss. became one of the regular occupations."

"The bones... are given in a heap to a student who has no idea of a skeleton. Here is the defect which I am trying partly to supply..."

"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 population of Athens and consisted of slaves, resident aliens, and citizens. Slaves were excessively numerous. At a census taken in B.C. 309, the number of slaves was returned at 400,000, and it does not seem likely that there were fewer at any time during the classical period. They were mostly ns, ns, Thracians, and ns, imported from the coasts of the Propontis. ...They were employed for domestic purposes, or were let out for hire in gangs as labourers, or were allowed to work by themselves paying a yearly royalty to their masters. ...hardly any Athenian citizen can have been without two or three. The family of Aeschines (consisting of 6 persons) was considered very poor because it possessed only 7 slaves. On the other hand, Plutarch says that let out 1,000 and Hipponicus 600 slaves to work the gold mines in Thrace. The state possessed some slaves of its own, who were employed chiefly as policemen and clerks. Slaves enjoyed considerable liberties in Athens, and had some rights, even against their masters. They did not serve as soldiers, or sailors, except when the city was in great straits, as at the battle of Arginussae... The worst prospect in store for them was that their masters might be engaged in a lawsuit, for the evidence of a slave (except in a few cases) was not admitted in a court of justice unless he had been put to torture. Slaves were sometimes freed by their masters, with some sort of public ceremony, or (for great services) by the state which paid their value to their masters."

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

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

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

"A student of history, who cares little for Greek or mathematics in particular, but who likes to watch how things grow, will be able to extract from these pages a notion of the whole history of mathematical science down to Newton's time..."

"Each of these smaller corporations... to which an Athenian citizen belonged, had... business of its own—money to spend, officers to appoint, rules to make—very similar to that which the state transacted on a larger scale. And it is not to be supposed that Athenians were at all ashamed to take part in such minor business, as English gentlemen are to sit on a vestry or a town council. On the contrary, a large part of the population left their private affairs for slaves to manage, and devoted themselves entirely to their public duties."

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

"The historian is not only a lover of truth, not only a chronicler of events. These, indeed, he must be at his peril, but how much more! Insight into human nature—and this implies the rarest knowledge and finest sympathy of which man is capable; the power of tracing the delicate relation between deed and motive, and the pressure of action upon circumstance and circumstance upon action; knowledge of the world, in short, in the highest sense of that expression."

"Historical, sociological, literary, linguistic, archeological and other techniques must be brought to bear when they are applicable to the material at hand."

"That both the Gilgamesh Epic and the Odyssey deal with the episodic wanderings of a hero, would not be sufficiently specific to establish a genuine relation between them. But when both epics begin with the declaration that the hero gained experience from his wide wanderings, and end with his homecoming, a relationship dimly appears. ...when we note that whole episodes are in essential agreement, we are on firmer ground. For instance, both Gilgamesh and Odysseus reject a goddess's proposal for marriage; and each of the heroes interviews his dead companion in Hades."

"Out of the Armana Age synthesis emerged the earliest traditions of Israel and Greece."

"The parallels that form the core of this book fit into a historical framework in the wake of the Armana Age during the closing centuries of the second millenium. Prior to the Armana Age (i.e., before 1400 B.C.) Egyptian, Canaanite, Mesopotamian, Anatolian, Aegean and other influences met around the East Mediterranean to form an international order, by which each was in turn affected."

"It would be foolhardy to swell the pages of this book with an exhaustive list of Greco-Hebrew differences. Everyone knows that Homer is very different from the Bible."

"The customs of both the Greeks and Hebrews in that heroic age were often alien to their respective descendants in the classical periods. We shall have to bear in mind that the gulf separating classical Israel (of the great Prophets) from classical Greece (of the scientists and philosophers) must not be read back into the heroic age when both peoples formed part of the same international complex."

"Just as Mycenaean civilisation is a development of the Minoan, so too the Philistines who came to Palestine from Caphtor were an offshoot of the same general Aegean civilisation. ...until David's victories around 1000 B.C., the Philistines dominated the Hebrews, so that, militarily at least, the Davidic monarchy was the Hebrew response to the Philistine stimulus. ...the Achaeans, Trojans, Philistines and Hebrews during the closing centuries of the second millennium belonged to the same international complex of peoples, sharing many conventions and institutions, specifically in military matters."

"The text of Homer about the Mycenaen Age with its memories of the Trojan War, and the Hebrew text covering from the Conquest through David's reign, cover ground with much in common geographically, chronologically and ethnically."

"Manslaughter was requited through blood revenge. Accordingly the offender, to escape the avenger, would be forced to flee, cut off from his land and people, at the mercy of strangers far from home. [Examples are] 2 Samuel (14: 5-7)... Iliad a6: 571-574... Odyssey (15: 271-278)... (Genesis 4: 14)... (Genesis 4: 15)"

"The priestly guilds were highly mobile, with the result that cultic practices crossed ethnic lines over wide areas."

"The Greeks viewed the Mediterranean not as a barrier but as a network of routes connecting people who dwelt along its shores. This is familiar to any student of Greece. ...the Hebrews express themselves similarly in passages like Psalm 8: 9 ("crossing the paths of the seas")."