Historians From England

"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."

"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."

"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."

"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..."

"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."

"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."

"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."

"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 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."

"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."

"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."

"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."

"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..."

"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."

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

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

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

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

"Historians now debate whether Britain had a ‘managed economy’ by 1939, the very term—die gelenkte wirtschaft—adopted by German economists in the 1930s to describe the Nazi economy."

"Nazi political hegemony in the end prevented German capitalists from acting as capitalists."

"In the long run the [Nazi] movement was moving to a position in which the economic New Order would be controlled by the Party through a bureaucratic apparatus staffed by technical experts and dominated by political interests, not unlike the system that had already been built up in the Soviet Union."

"Göring industrial empire represented, in this sense, one of the major steps towards restricting private industrial capitalism and substituting a ‘völkisch’, state-run industrial economy."

"In a remark made in 1932 to Otto Wagener, one of the party’s economic experts, Hitler observed that the business bourgeoisie “know nothing except their profit. ‘Fatherland’ is only a word for them.”"

"During the years that followed, Hitler and the Nazi movement were anything but the tools of German big business."

"When the regime wanted additional industrial capacity or resources for the military economy which the private sector could not provide, it was created (or conquered) by the state… during the Third Reich state ownership expanded into the productive sectors, based on the strategic industries, aviation, aluminum, synthetic oil and rubber, chemicals, iron and steel, and army equipment. Government finances for state-owned enterprises rose from RM 4,000m in 1933 to RM 16,000m 10 years later; the capital assets of state-owned industry doubled during the same period; the number of state-owned firms topped 500."

"Indeed, many businessmen seem on the evidence to have been wary of the closet anti-capitalism of the rank-and-file Nazis."

"Hitler was ‘an enemy of free-market economics’ and a ‘reluctant dirigiste.’"

"[T]here could be no guarantee in a free market that the interests of race or state as Hitler defined them would take priority over the interests of businessmen or consumers. After 1933 the role of the state in regulating and directing economic life increased sharply, first in order to encourage economic revival, then to divert economic resources to the growth of German military power… [which] led, in an unplanned, incremental way, to the establishment of a kind of command economy."

"The outcome [of the Nazi economy] was a command economy, governed by military priorities, but run by a coalition of state officials, soldiers, party hacks and industrial technocrats."

"Industry was regarded entirely instrumentally by party, state and military, in terms of its ability to provide the sinews of war… The business community was characterized by a defensive opportunism in the face of state power."

"The [Nazi] regime was also able to use the state-owned multi-nationals as a ‘battering ram’ for entering economies that were not occupied territory."

"Some big businessmen did contribute to the Nazi election funds but German capitalism cannot be regarded, on the evidence, as having collectively brought fascism to power in any direct sense. Fascism in Germany was a mass movement brought to power through collusion with a bankrupt but traditional elite, not as the puppet of big business."

"It has been estimated that about one-fifth of Nazi voters and members were drawn from the manual work-force. After 1933 the number of workers in the Nazi Party went up, a product of opportunism perhaps as much as conviction."

"After his defeat in the 1919 election, Mussolini saw no future in trying to out-socialist the socialists. Without a distinct policy, without friends and backing, he was in serious danger of ending up as a confused and egocentric demagogue with a talent for histrionics."

"Mussolini had been envious of the Bolsheviks and for a while fancied himself as the Lenin of Italy."

"There are not many historians who matter. Not many whose works have changed the way people see themselves. Of that little list, there is an even smaller number whose works have mattered to those in another society. The obscure American naval captain Alfred Thayer Mahan was one. His Strategy of Sea Power, published in 1890, had earned him honorary degrees in Oxford and Cambridge by 1893: Mahan told an anxious British public what it most wanted to hear about its navy. Denis Mack Smith is another. His Cavour and Garibaldi, published in 1954, told many Italians what they did not want to hear, but told them at a special point in their history when they had no choice but to listen. Denis Mack Smith became and has remained one of the most important historians of Italy."

"Lenin was the contemporary politician whom [Mussolini] most admired and he studied the Russian revolution closely to see what lessons it offered. Lenin seemed to him ‘the very negation of socialism’ because he had not created a dictatorship of the proletariat or of the socialist party, but only of a few intellectuals who had found the secret of winning power. Mussolini was, in truth, envious."

"Mussolini had once belonged to the Bolshevik wing of the Italian Socialist party and still in 1924 confessed admiration for Lenin, while Trotsky was quoted as saying that Mussolini was his best pupil."

"The election [of 1919] indicated that the prevailing sentiment in the country was to the left and Mussolini acknowledge this face by still in 1920 calling himself a socialist, albeit a dissident. He continued to campaign for nationalization of land, workers’ participation in the running of factories and partial expropriation of capital,…"

"[Mussolini] forcibly denounced those socialists who thought religion a matter for individual conscience or had their children baptized. Science had proved that God did not exist and the Jesus of history was an ignorant Jew whose family thought him mad, and who was a pigmy compared to the Buddha. Religion, he said, was a disease of the psyche, an epidemic to be cured by psychiatrists, and Christianity in particular was vitiated by preaching the senseless virtues of resignation and cowardice, whereas the new socialist morality should celebrate violence and rebellion."

"[Mussolini] brought a radical Marxist strand to the Avanti! newspaper, soon doubling its circulation. With a growing audience, Mussolini redoubled the urgency of his utopian propaganda; ‘private property is theft’ and should be abolished as Italy moved through the phase of collectivism forwards to the ultimate goal of communism."

"[In 1938] Mussolini anti-clericalism was thus reassuring itself. Sometimes he now acknowledged that he was an outright disbeliever... [that] the papacy was a malignant tumor in the body of Italy and must 'be rooted out once and for all,’ because there was no room in Rome for both the Pope and himself."

"The Mussolini of 1944 reasserted the socialist beliefs of his youth because he now felt that he had been cheated by the world of finance and industry: after having gained immensely from fascism,… To maintain some intellectual coherence he tried to pretend that, notwithstanding appearances, he had never deserted the socialist programme he had put forward for fascism in 1919; he had allowed certain tactical deviations in the interim, but for the most part, his basic views had never changed. In anonymous articles he now confirmed that he had been right when, in 1910, he called on the proletariat to capture power from the capitalists by a bloody revolution. . . He now decreed that all industrial firms employing more than 100 workers would be nationalized."

"[Mussolini’s] desire for revenge against the bourgeoisie was genuine enough; his intention was to inject the germs of social revolution into Italian society in order to ensure that, if Italy should lose the war, whoever won would have a difficult time of it. In this way, fascism, which had once invented the myth of having saved Italy from bolshevism, ended up deliberately (and more successfully) doing the exact opposite."

"Always an extremist, [Mussolini] inclined to the belief of Babeuf and Blanqui in violent insurrection by a minority in order to establish authoritarian rule. His articles reveal a retreat from belief in class solidarity and a growing attachment to revolution for revolution’s sake, power for the sake of power."

"Bombacci had once been a friend and disciple of Lenin and revived the story that, according to Lenin, Mussolini had been the one serious socialist in Italy. No doubt the Duce was influenced by Bombacci when he called fascism of Salò [Italian Social Republic] the only truly socialist government in existence—with the possible exception of Soviet Russia."

"a) The definition of history? Every historian would agree, I think, that history is a kind of research or inquiry. What kind of inquiry it is I do not yet ask. The point is that generically it belongs to what we call the sciences: that is, the forms of thought whereby we ask questions and try to answer them. Science in general, it is important to realize, does not consist in collecting what we already know and arranging it in this or that kind of pattern. It consists in fastening upon something we do not know, and trying to discover it. Playing patience with things we already know may be a useful means towards this end, but it is not the end itself. It is at best only the means. It is scientifically valuable only in so far as the new arrangement gives us the answer to a question we have already decided to ask. That is why all science begins from the knowledge of our own ignorance: not our ignorance of everything, but our ignorance of some definite thing-the origin of parliament, the cause of cancer, the chemical composition of the sun, the way to make a pump work without muscular exertion on the part of a man or a horse or some other docile animal. Science is finding things out: and in that sense history is a science."