James Gow (scholar): Other

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“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.”

James Gow (scholar)

A Short History of Greek Mathematics (1884)

“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.”

James Gow (scholar)

p, 125
A Companion to School Classics (1888)

“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.”

James Gow (scholar)

p, 125
A Short History of Greek Mathematics (1884)

“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.”

James Gow (scholar)

p, 125
A Short History of Greek Mathematics (1884)

“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.”

James Gow (scholar)

A Short History of Greek Mathematics (1884)

“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.”

James Gow (scholar)

p, 125
A Short History of Greek Mathematics (1884)

“The population of Athens 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.”

James Gow (scholar)

p, 125
A Companion to School Classics (1888)