James Gow (scholar): Condition

James Gow (scholar) was scholar. Explore interesting quotes on condition.
James Gow (scholar): 44 quotes0 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)

“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)

“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)

“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.”

James Gow (scholar)

A Short History of Greek Mathematics (1884)