Diophantus

“@GoogleBooks (public domain)”

“History of mathematics”

“With Diophantus the history of Greek arithmetic comes to an end. No original work, that we know of, was done afterwards.”

“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.”

“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.”

“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.”

“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.”

“Diophantos' work is so unique among the Greek treatises which we possess, that he cannot be said to recall the style or subject-matter of any other author, except, indeed, in the fragment on Polygonal Numbers; and even there the reference to Hypsikles is the only indication we can lay hold of.”

“But how much more splendid the methods which reduce the problems which seem to be hardly capable of solution even with the help of surds in such a way that, while the surds, when bidden (so to speak) to plough the arithmetic soil, become true to their name and deaf to entreaty, they are not so much as mentioned in these most ingenious solutions.”

“The Greek mathematician Diophantus of Alexandria... it seems fairly probable... flourished about 250 A.D. ...We gather from the Arithmeticas introduction that it originally consisted of thirteen Books. Of these, only six have survived until now in Greek... The remaining seven were considered irretrievably lost until the recent discovery of four other, hitherto unknown books in Arabic translation, which, since it is attributed to Qustā ibn Lūqā, must have been made around or after the middle of the ninth century.”

“For not more than six books [of Diophantus] are found, though in the proëmium he promises thirteen. If this book, a wonderful and difficult work, could be found entire, I should like to translate it into Latin, for the knowledge of Greek I have lately acquired would suffice for this.”

“No one has yet translated from the Greek into Latin the thirteen books of Diophantus, in which the very flower of the whole of arithmetic lies hid, the ars rei et census [the art of the evaluation of wealth or tax] which to-day they call by the Arabic name of Algebra.”

“Bachet de Méziriac remarked that any number (that is, positive integer) is either a square, or the sum of two, three, or four squares. He did not pretend to possess a proof. He found indications pointing to his statement in certain problems of Diophantus and verified it up to 325. In short, Bachet's statement was just a conjecture, found inductively. ...his main achievement was to put the question: How many squares are needed to represent all integers? Once this question is squarely put, there is not much difficulty in discovering the answer inductively.”

“The Alexandrians used fractions as numbers in their own right, whereas the mathematicians of the classical period spoke only of a ratio of integers, not of parts of a whole and the ratios were used only in proportions. However, even in the classical period genuine fractions... as entities in their own right, were used in commerce. In the Alexandrian period, Archimedes, Heron, Diophantus, and others used fractions freely and performed operations with them. Though... they did not discuss the concept of fractions, apparently these were intuitively sufficiently clear...”

“The first algebraist known to us, the Greek Diophantus (360 A.D.), antedates Aryabhata by a century; but Cajori believes that he took his lead from India.”

“Fermat, commenting about 1637 on Diophantus II, 8 (to solve x2 + y2 = a2), stated that "it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain." This theorem is known as Fermat's last theorem.”

“Diophantus was the first Greek mathematician who frankly recognized fractions as numbers. He was also the first to handle in a systematic way not only simple equations, but quadratics and equations of higher order. In spite of his ineffective symbolism, in spite of the inelegance of his methods, he must be regarded as the precursor of modern algebra. But Diophantus was the last flicker of a dying candle.”

“The "porisms" of Pappus anticipated projective geometry, the problems of Diophantus prepared the ground for the modern theory of equations.”

“Greek algebra before Diophantus was essentially rhetorical.”

“In the solution of simultaneous equations Diophantus adroitly managed with only one symbol for the unknown quantities and arrived at answers, most commonly, by the method of tentative assumption, which consists in assigning to some of the unknown quantities preliminary values, that satisfy only one or two of the conditions. These values lead to expressions palpably wrong, but which generally suggest some stratagem by which values can be secured satisfying all the conditions of the problem.”