James Gow (scholar): Number

James Gow (scholar) was scholar. Explore interesting quotes on number.

James Gow (scholar): 44 quotes 0 likes

“Diophantus shows great Adroitness in selecting the unkown, 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.”

James Gow (scholar)

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

Lie , Books , Lying , Failure

“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.”

James Gow (scholar)

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

Problem , Parting

“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)

Books , Quotes , Time

“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)

Men , Nature , Drawing , Falseness

“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.”

James Gow (scholar)

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

Appearance

“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 Pontifex Maximus and the Rex Sacrorum 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.”

James Gow (scholar)

A Companion to School Classics (1888)

God , Family , Religion , Reality

“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 ύπ eναντία originally and that Archytas and Hippasus first called it harmonic.”

James Gow (scholar)

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:<center>12:6 :: 12-8:8-6</center>
Footnote, citing Vide Cantor, Vorles [Vorlesüngen über Geschichte der Mathematik ?] p 152. Nesselmann p. 214 n. Hankel. p. 105 sqq.
A Short History of Greek Mathematics (1884)

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