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Matematik Seçkileri
Sayılar Teorisi
Ziyaretçi defteri

Diophantus of Alexandria

Born: about 200
Died: about 284

Diophantus, often known as the 'father of algebra', is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers. However, essentially nothing is known of his life and there has been much debate regarding the date at which he lived.

There are a few limits which can be put on the dates of Diophantus's life. On the one hand Diophantus quotes the definition of a polygonal number from the work of Hypsicles so he must have written this later than 150 BC. On the other hand Theon of Alexandria, the father of Hypatia, quotes one of Diophantus's definitions so this means that Diophantus wrote no later than 350 AD. However this leaves a span of 500 years, so we have not narrowed down Diophantus's dates a great deal by these pieces of information.

There is another piece of information which was accepted for many years as giving fairly accurate dates. Heath [3] quotes from a letter by Michael Psellus who lived in the last half of the 11th century. Psellus wrote (Heath's translation in [3]):-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the very learned Anatolius collected the most essential parts of the doctrine as stated by Diophantus in a different way and in the most succinct form, dedicating his work to Diophantus.

Psellus also describes in this letter the fact that Diophantus gave different names to powers of the unknown to those given by the Egyptians. This letter was first published by Paul Tannery in [7] and in that work he comments that he believes that Psellus is quoting from a commentary on Diophantus which is now lost and was probably written by Hypatia. However, the quote given above has been used to date Diophantus using the theory that the Anatolius referred to here is the bishop of Laodicea who was a writer and teacher of mathematics and lived in the third century. From this it was deduced that Diophantus wrote around 250 AD and the dates we have given for him are based on this argument.

Knorr in [16] criticises this interpretation, however:-

But one immediately suspects something is amiss: it seems peculiar that someone would compile an abridgement of another man's work and then dedicate it to him, while the qualification "in a different way", in itself vacuous, ought to be redundant, in view of the terms "most essential" and "most succinct".

Knorr gives a different translation of the same passage (showing how difficult the study of Greek mathematics is for anyone who is not an expert in classical Greek) which has a remarkably different meaning:-

Diophantus dealt with [Egyptian arithmetic] more accurately, but the very learned Anatolius, having collected the most essential parts of that man's doctrine, to a different Diophantus most succinctly addressed it.

The conclusion of Knorr as to Diophantus's dates is [16]:-

... we must entertain the possibility that Diophantus lived earlier than the third century, possibly even earlier that Heron in the first century.

The most details we have of Diophantus's life (and these may be totally fictitious) come from the Greek Anthology, compiled by Metrodorus around 500 AD. This collection of puzzles contain one about Diophantus which says:-

... his boyhood lasted 1/6th of his life; he married after 1/7th more; his beard grew after 1/12th more, and his son was born 5 years later; the son lived to half his father's age, and the father died 4 years after the son.

So he married at the age of 26 and had a son who died at the age of 42, four years before Diophantus himself died aged 84. Based on this information we have given him a life span of 84 years.

The Arithmetica is a collection of 130 problems giving numerical solutions of determinate equations (those with a unique solution), and indeterminate equations. The method for solving the latter is now known as Diophantine analysis. Only six of the original 13 books were thought to have survived and it was also thought that the others must have been lost quite soon after they were written. There are many Arabic translations, for example by Abu'l-Wafa, but only material from these six books appeared. Heath writes in [4] in 1920:-

The missing books were evidently lost at a very early date. Paul Tannery suggests that Hypatia's commentary extended only to the first six books, and that she left untouched the remaining seven, which, partly as a consequence, were first forgotten and then lost.

However, an Arabic manuscript in the library Astan-i Quds (The Holy Shrine library) in Meshed, Iran has a title claiming it is a translation by Qusta ibn Luqa, who died in 912, of Books IV to VII of Arithmetica by Diophantus of Alexandria. F Sezgin made this remarkable discovery in 1968. In [19] and [20] Rashed compares the four books in this Arabic translation with the known six Greek books and claims that this text is a translation of the lost books of Diophantus. Rozenfeld, in reviewing these two articles is, however, not completely convinced:-

The reviewer, familiar with the Arabic text of this manuscript, does not doubt that this manuscript is the translation from the Greek text written in Alexandria but the great difference between the Greek books of Diophantus's Arithmetic combining questions of algebra with deep questions of the theory of numbers and these books containing only algebraic material make it very probable that this text was written not by Diophantus but by some one of his commentators (perhaps Hypatia?).

It is time to take a look at this most outstanding work on algebra in Greek mathematics. The work considers the solution of many problems concerning linear and quadratic equations, but considers only positive rational solutions to these problems. Equations which would lead to solutions which are negative or irrational square roots, Diophantus considers as useless. To give one specific example, he calls the equation 4 = 4x + 20 'absurd' because it would lead to a meaningless answer. In other words how could a problem lead to the solution -4 books? There is no evidence to suggest that Diophantus realised that a quadratic equation could have two solutions. However, the fact that he was always satisfied with a rational solution and did not require a whole number is more sophisticated than we might realise today.

Diophantus looked at three types of quadratic equations ax2 + bx = c, ax2 = bx + c and ax2 + c = bx. 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 a, b, c to all be positive in each of the three cases above.

There are, however, many other types of problems considered by Diophantus. He solved problems such as pairs of simultaneous quadratic equations.
Consider y + z = 10, yz = 9. Diophantus would solve this by creating a single quadratic equation in x. Put 2x = y - z so, adding y + z = 10 and y - z = 2x, we have y = 5 + x, then subtracting them gives z = 5 - x. Now

9 = yz = (5 + x)(5 - x) = 25 - x2, so x2 = 16, x = 4

leading to y = 9, z = 1.

In Book III, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares. For example he shows how to find x to make 10x + 9 and 5x + 4 both squares (he finds x = 28). Other problems seek a value for x such that particular types of polynomials in x up to degree 6 are squares. For example he solves the problem of finding x such that x3 - 3x2 + 3x + 1 is a square in Book VI. Again in Book VI he solves problems such as finding x such that simultaneously 4x + 2 is a cube and 2x + 1 is a square (for which he easily finds the answer x = 3/2).

Another type of problem which Diophantus studies, this time in Book IV, is to find powers between given limits. For example to find a square between 5/4 and 2 he multiplies both by 64, spots the square 100 between 80 and 128, so obtaining the solution 25/16 to the original problem. In Book V he solves problems such as writing 13 as the sum of two square each greater than 6 (and he gives the solution 66049/10201 and 66564/10201). He also writes 10 as the sum of three squares each greater than 3, finding the three squares

1745041/505521, 1651225/505521, 1658944/505521.

Heath looks at number theory results of which Diophantus was clearly aware, yet it is unclear whether he had a proof. Of course these results may have been proved in other books written by Diophantus or he may have felt they were "obviously" true due to his experimental evidence. Among such results are [4]:-

... no number of the form 4n + 3 or 4n - 1 can be the sum of two squares;

... a number of the form 24n + 7 cannot be the sum of three squares.

Diophantus also appears to know that every number can be written as the sum of four squares. If indeed he did know this result it would be truly remarkable for even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Lagrange proved it using results due to Euler.

Although Diophantus did not use sophisticated algebraic notation, he did introduce an algebraic symbolism that used an abbreviation for the unknown and for the powers of the unknown. As Vogel writes in [1]:-

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.

One thing will be clear from the examples we have quoted and that is that Diophantus is concerned with particular problems more often than with general methods. The reason for this is that although he made important advances in symbolism, he still lacked the necessary notation to express more general methods. For instance 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 (12 + 6n)/(n2 -3), Diophantus has to write in words:-

... a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three.

Despite the improved notation and that Diophantus introduced, algebra had a long way to go before really general problems could be written down and solved succinctly.

Fragments of another of Diophantus's books On polygonal numbers, a topic of great interest to Pythagoras and his followers, has survived. In [1] it is stated that this work contains:-

... little that is original, [and] is immediately differentiated from the Arithmetica by its use of geometric proofs.

Diophantus himself refers to another work which consists of a collection of lemmas called The Porisms but this book is entirely lost. We do know three lemmas contained in The Porisms since Diophantus refers to them in the Arithmetica. One such lemma is 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 numbers a, b then there exist numbers c, d such that a3 - b3= c3 + d3.

Another extant work Preliminaries to the geometric elements, which has been attributed to Heron, has been studied recently in [16] where it is suggested that the attribution to Heron is incorrect and that the work is due to Diophantus. The author of the article [14] thinks that he may have identified yet another work by Diophantus. He writes:-

We conjecture the existence of a lost theoretical treatise of Diophantus, entitled "Teaching of the elements of arithmetic". Our claims are based on a scholium of an anonymous Byzantine commentator.

European mathematicians did not learn of the gems in Diophantus's Arithmetica until Regiomontanus wrote in 1463:-

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...

Bombelli translated much of the work in 1570 but it was never published. Bombelli did borrow many of Diophantus's problems for his own Algebra. The most famous Latin translation of the Diophantus's Arithmetica is due to Bachet in 1621 and it is that edition which Fermat studied. Certainly Fermat was inspired by this work which has become famous in recent years due to its connection with Fermat's Last Theorem.

We began this article with the remark that Diophantus is often regarded as the 'father of algebra' but there is no doubt that many of the methods for solving linear and quadratic equations go back to Babylonian mathematics. For this reason Vogel writes [1]:-

... Diophantus was not, as he has often been called, the father of algebra. Nevertheless, his remarkable, if unsystematic, collection of indeterminate problems is a singular achievement that was not fully appreciated and further developed until much later.


Article by: J J O'Connor and E F Robertson

February 1999

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