“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)

Adopted from Wikiquote. Last update June 3, 2021. History

Help us to complete the source, original and additional information

Do you have more details about the quote "Sometimes… Diophantus solves a problem wholly or in part by synthesis…. Although… Diophantus does not treat his problem…" by James Gow (scholar)?
James Gow (scholar) photo
James Gow (scholar)22
scholar 1854–1923

Related quotes

“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) (1854–1923) 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) (1854–1923) scholar

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

“Heuristic Search Hypothesis. The solutions to problems are represented as symbol structures. A physical symbol system exercises its intelligence in problem solving by search—that is, by generating and progressively modifying symbol structures until it produces a solution structure.”

Allen Newell (1927–1992) American cognitive scientist

Source: Computer Science as Empirical Inquiry: Symbols and Search (1975), p. 120.

Russell L. Ackoff photo

“Successful problem solving requires finding the right solution to the right problem. We fail more often because we solve the wrong problem than because we get the wrong solution to the right problem.”

Russell L. Ackoff (1919–2009) Scientist

Source: 1970s, Redesigning the future, 1974, p. 8.

Prevale photo

“The secret to solving a problem is to have the strength to find a solution. When the solution does not exist, you invent it.”

Prevale (1983) Italian DJ and producer

Original: Il segreto per risolvere un problema è avere la forza di trovare una soluzione. Quando la soluzione non esiste, la inventi.
Source: prevale.net

Thomas Little Heath photo

“It may be in some measure due to the defects of notation in his time that Diophantos will have in his solutions no numbers whatever except rational numbers, in [the non-numbers of] which, in addition to surds and imaginary quantities, he includes negative quantities. …Such equations then as lead to surd, imaginary, or negative roots he regards as useless for his purpose: the solution is in these cases ὰδοπος, impossible. So we find him describing the equation 4=4x+20 as ᾰτοπος because it would give x=-4. Diophantos makes it throughout his object to obtain solutions in rational numbers, and we find him frequently giving, as a preliminary, conditions which must be satisfied, which are the conditions of a result rational in Diophantos' sense. In the great majority of cases when Diophantos arrives in the course of a solution at an equation which would give an irrational result he retraces his steps and finds out how his equation has arisen, and how he may by altering the previous work substitute for it another which shall give a rational result. This gives rise, in general, to a subsidiary problem the solution of which ensures a rational result for the problem itself. Though, however, Diophantos has no notation for a surd, and does not admit surd results, it is scarcely true to say that he makes no use of quadratic equations which lead to such results. Thus, for example, in v. 33 he solves such an equation so far as to be able to see to what integers the solution would approximate most nearly.”

Thomas Little Heath (1861–1940) British civil servant and academic

Diophantos of Alexandria: A Study in the History of Greek Algebra (1885)

“The problem of thermonuclear reactions is not a usual physics problem. This is a problem that must transform society and the world. Our generation, which gave to mankind atomic energy and thermonuclear energy in the form of explosions, is responsible to humanity for solving the main problem of energy – obtaining energy from water. People are waiting for the solution of this problem. Our duty is to solve it within the lifetime of our generation, and therefore we must set out on this path.”

Gersh Budker (1918–1977) Soviet physicist

as quoted by D. D. Ryutov in [G.I. Budker: reflections & remembrances, by Boris N. Breizman, Springer, 1993, http://books.google.com/books?id=e0bxFrmNtykC&printsec=frontcover#PRA1-PA278,M1, 1-56396-070-2, 278]

“The systems approach to problems does not mean that the most generally formulated problem must be solved in one research project. However desirable this may be, it is seldom possible to realize it in practice. In practice, parts of the total problem are usually solved in sequence. In many cases the total problem cannot be formulated in advance but the solution of one phase of it helps define the next phase. For example, a production control project may require determination of the most economic production quantities of different items. Once these are found it may turn out that these quantities cannot be produced on the available equipment in the available time. This, then, gives rise to a new problem whose solution will affect the solution obtained in the first phase.”

C. West Churchman (1913–2004) American philosopher and systems scientist

Source: 1940s - 1950s, Introduction to Operations Research (1957), p. 7

Milton Friedman photo

“The government solution to a problem is usually as bad as the problem and very often makes the problem worse.”

Milton Friedman (1912–2006) American economist, statistician, and writer

“Interview with Milton Friedman”, Playboy magazine (Feb. 1973)

“A problem is something to solve,” Phillip says. “If there’s no solution, it’s not a problem, so stop treating it like one.”

Jonathan Tropper (1970) American writer

Source: This is Where I Leave You

Related topics