See study and English translation by T. L. Heath (2d ed. 1910, repr. 1964).
Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. between 200 and 214, d. between 284 and 298 AD), sometimes called "the father of algebra", a title some claim should be shared by a Persian mathematician al-Khwārizmī, born some 500 years later. Diophantus was an Alexandrian mathematician and the author of a series of books called Arithmetica that deal with solving algebraic equations, many of which are now lost. Pierre de Fermat studied Arithmetica and made a fateful note in the margin of his copy of the book that a certain equation similar to the Pythagorean equation considered by Diophantus has no solutions and he found "a truly marvelous proof of this proposition", the celebrated Fermat's Last Theorem. This led to tremendous advances in number theory, and the study of diophantine equations ("diophantine geometry") and of diophantine approximations remain important areas of mathematical research. Diophantus was the first Greek mathematician who recognized fractions as numbers; thus he allowed positive rational numbers for the coefficients and solutions. In modern use, diophantine equations are usually algebraic equations with integer coefficients, for which integer solutions are sought. Diophantus also made advances in mathematical notation.
Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between 200 and 214 to 284 or 298 AD, although Klaus Barner states that he may have been active as early as 150 BC, thus the life span of about 400 years. While most scholars consider Diophantus to have been a Greek, others speculate him to have been a non-Greek, possibly either a Hellenized Babylonian, an Egyptian, a Jew, or a Chaldean. Much of our knowledge the life of Diophantus is derived from a 5th century Greek anthology of number games and strategy puzzles. One of the problems states:
'Here lies Diophantus,' the wonder behold. Through art algebraic, the stone tells how old: 'God gave him his boyhood one-sixth of his life, One twelfth more as youth while whiskers grew rife; And then yet one-seventh ere marriage begun; In five years there came a bouncing new son. Alas, the dear child of master and sage After attaining half the measure of his father's life chill fate took him. After consoling his fate by the science of numbers for four years, he ended his life.'
This puzzle implies that Diophantus lived to be about 84 years old. However, the accuracy of the information cannot be independently confirmed. This puzzle was the Puzzle No.142 in Professor Layton and Pandora's Box as one of the hardest solving puzzles in the game, which needed to be unlocked by solving other puzzles first.
The Arithmetica is the major work of Diophantus and the most prominent work on algebra in Greek mathematics. It is a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of which Arithmetica consisted only six have survived, though there are some who believe that four Arab books discovered in 1968 are also by Diophantus. Some Diophantine problems from Arithmetica have been found in Arabic sources.
It should be mentioned here that Diophantus never used general methods in his solutions. Hermann Hankel, renowned German mathematician made the following remark regarding Diophantus.
“Our author (Diophantos) not the slightest trace of a general, comprehensive method is discernible; each problem calls for some special method which refuses to work even for the most closely related problems. For this reason it is difficult for the modern scholar to solve the 101st problem even after having studied 100 of Diophantos’s solutions”
During the Dark Ages, Diophantus was forgotten and like many other mathematical treatises from the classical period, Arithmetica survived through the Arab tradition. In 1463 German mathematician Regiomontanus wrote:
Arithmetica was first translated into Latin by Bombelli in 1570, but the translation was never published. However, Bombelli borrowed many of the problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by Xylander. The best known Latin translation of Arithmetica was made by Bachet in 1621 and became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it, and made notes in the margins.
Fermat's proof was never found, and the problem of finding a proof for the theorem went unsolved for centuries. A proof was finally found in 1994 by Andrew Wiles after working on it for seven years. It is believed that Fermat did not actually have the proof he claimed to have. Although the original copy in which Fermat wrote this is lost today, Fermat's son edited the next edition of Diophantus, published in 1670. Even though the text is otherwise inferior to the 1621 edition, Fermat's annotations --- including his famous "Last Theorem" --- were printed in this version.
Fermat was not the first mathematician so moved to write in his own marginal notes to Diophantus; the Byzantine mathematician Maximus Planudes had written "Thy soul, Diophantus, be with Satan because of the difficulty of your theorems" next to the same problem.
Diophantus wrote several other books besides Arithmetica, but very few of them have survived.
Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but this book is entirely lost. Some scholars think that The porisms may have actually been a section of Arithmetica that is now lost.
Although The Porisms is lost, we know three lemmas contained there since Diophantus refers to them in the Arithmetica. One lemma states that the difference of the cubes of two rational numbers is equal to the sum of the cubes of two other rational numbers, i.e. given any a and b, with a > b, there exist c and d, all positive and rational, such that
A book called Preliminaries to the Geometric Elements has been traditionally attributed to Hero of Alexandria. It has been studied recently by Wilbur Knorr, who suggested that the attribution to Hero is incorrect, and that the true author is Diophantus .
Today Diophantine analysis is the area of study where integer (whole number) solutions are sought for equations, and Diophantine equations are polynomial equations with integer coefficients to which only integer solutions are sought. It is usually rather difficult to tell whether a given Diophantine equation is solvable. Most of the problems in Arithmetica lead to quadratic equations. Diophantus looked at 3 different types of quadratic equations: , , and . The reason why there were three cases to Diophantus, while today we have only one case, is that he did not have any notion for zero and he avoided negative coefficients by considering the given numbers to all be positive in each of the three cases above. Diophantus was always satisfied with a rational solution and did not require a whole number which means he accepted fractions as solutions to his problems. Diophantus considered negative or irrational square root solutions "useless", "meaningless", and even "absurd". To give one specific example, he calls the equation 'absurd' because it would lead to a negative value for . One solution was all he looked for in a quadratic equation. There is no evidence that suggests Diophantus even realized that there could be two solutions to a quadratic equation. He also considered simultaneous quadratic equations.
“The symbolism that Diophantus introduced for the first time, and undoubtedly devised himself, provided a short and readily comprehensible means of expressing an equation... Since an abbreviation is also employed for the word ‘equals’, Diophantus took a fundamental step from verbal algebra towards symbolic algebra.”
Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods. This caused his work to be more concerned with particular problems rather than general situations. Some of the limitations of Diophantus' notation are that he only had notation for one unknown and, when problems involved more than a single unknown, Diophantus was reduced to expressing "first unknown", "second unknown", etc. in words. He also lacked a symbol for a general number n. Where we would write , Diophantus has to resort to constructions like : ... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.
Algebra still had a long way to go before very general problems could be written down and solved succinctly.