Diophantus

From New World Encyclopedia


Title page of the 1621 edition of Diophantus' Arithmetica, translated into Latin by [[Claude Gaspard Bachet de Méziriac]].

Diophantus of Alexandria (Greek: Διόφαντος ὁ Ἀλεξανδρεύς b. between 200 and 214, d. between 284 and

298 C.E.) was a Hellenistic mathematician. He is sometimes called "the Father of Algebra," a title

he shares with Muhammad ibn Mūsā al-Khwārizmī. He is the author of a series of classical mathematical books called The Arithmetica

and worked with equations which we now call Diophantine equations; the method to solve those problems is now called [[Diophantine

analysis]]. The study of Diophantine equations is one of the central areas of number theory. Diophantus also authored a tract "[[On Polygonal

Numbers]]" and a collection of propositions called "Porisms." The findings and works of Diophantus have influenced mathematics greatly and

caused many other questions to arise. The most famous of these is Fermat's Last Theorem. Diophantus also made advances in mathematical

notation and was the first Hellenistic mathematician who frankly recognized fractions as numbers.

Biography

Little is known about the life of Diophantus. He lived in Alexandria, Egypt, probably from between 200 and 214 to 284 or 298 C.E. There

is still a lot of speculation as to when he lived. In Rafael Bombelli's "Algebra," published in 1572, Diophantus lived under Antoninus Pius (138-

161 C.E.), but there is no proof. Another source, a letter of Psellus (11thc c.), mentions Diophantus and Anatolius as writers on the Egyptian

method of reckoning. It is taken, from that source, that Diophantus most probably flourished around 250 C.E. Most scholars consider Diophantus

to have been a Greek[1][2], though it has

been suggested that he may have been a Hellenized Babylonian."[3] Almost everything we know about Diophantus comes from a single 5th century

Greek anthology, which is a collection of number games and strategy puzzles. Here is one of the puzzles:

“This tomb holds Diophantus. Ah, what a marvel! And the tomb tells scientifically the measure of his life. God vouchsafed that he should be a

boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a

seventh, and in the fifth year after his marriage He granted him a son. Alas! late-begotten and miserable child, when he had reached the

measure of half his father's life, the chill grave took him. After consoling his grief by this science of numbers for four years, he reached the

end of his life.”

The translation and solution of this epigram-problem tells us that Diophantus' boyhood lasted 14 years, acquired a beard at 21, and married at

age 33. He got a son five years later, but he died at age 42 - Diophantus, at this time, was 80 years old. He tried to distract himself from

the grief with the science of numbers, and died 4 years later at 84.

This puzzle reveals that Diophantus lived to be about 84 years old. We cannot be sure if this puzzle is accurate or not.

Arithmetica

See also: Arithmetica

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[citation needed]. Some Diophantine problems from Arithmetica have been found in Arabic sources.

History

After Diophantus's death, the Dark Ages began, spreading a shadow on math and science, and causing knowledge of Diophantus and the

Arithmetica to be lost in Europe for about 1500 years. Sir Heath stated in his "Diophantus of Alexandria," "After the loss of Egypt the work

of Diophantus long remained almost unknown among the Byzantines; perhaps one copy only survived (of theHypatian recension), which was

seen by Michael Psellus and possibly by the scholiast to Iamblichus, but of which no trace can be found after the capture of Constantinople in

1204." Possibly the only reason that some of his work has survived is that many Arab scholars studied his works and preserved this

knowledge for later generations. In 1463 German mathematician Regiomontanus wrote: “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 hidden . . . .”

The first Latin translation of Arithmetica was by Bombelli who translated much of the work in 1570 but it was never published. Bombelli

did however borrow many of Diophantus's problems for his own book Algebra. The editio princeps of Arithmetica was published in 1575 by

Xylander. The most famous Latin translation of Arithmetica was by Bachet in 1621 which was the first translation of Arithmetica available

to the public.

Margin writing by Fermat and Planudes

Problem II.8 in the Arithmetica (edition of 1670), annotated with Fermat's comment which became Fermat's last theorem.

The 1621 edition of Arithmetica by Bombelli gained fame after Pierre de Fermat wrote his famous "Last Theorem" in

the margins of his copy:

“If an integer n is greater than 2, then has no solutions in non-zero integers ,

, and . I have a truly marvelous proof of this proposition which this margin is too narrow to contain.”

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.

Other works

Diophantus did not just write Arithmetica, but very few of his other works have survived.

The Porisms

Diophantus himself refers to a work which consists of a collection of lemmas called The Porisms (or Porismata), but

this book is entirely lost. Many scholars and researchers believe that The Porisms may have actually been a section included inside

Arithmetica or indeed may have been the rest of Arithmetica. [citation needed]

Although the Porisms is 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 , then there exist numbers and such that .

On polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers. Fragments of one of Diophantus' books on polygonal numbers, a topic of

great interest to Pythagoras and his followers, has survived. An extant work called Preliminaries to the Geometric Elements, which has been attributed to Hero of Alexandria, has been studied recently

and it is suggested that the attribution to Hero is incorrect, and that the work is actually by Diophantus [4].

Influence

Diophantus' work has had a large influence in history. Editions of Arithmetica exerted a profound influence on the development of algebra in

Europe in the late sixteenth and through the seventeenth and eighteenth centuries. Diophantus and his works have also influenced [[Arab

mathematics]] and were of great fame among Arab mathematicians. Diophantus' work created a foundation for work on algebra and in fact

much of advanced mathematics is based on algebra. As far as we know Diophantus did not affect the lands of the Orient much and how

much he affected India is a matter of debate.

The Father of Algebra?

Diophantus is often called “the Father of Algebra" because he contributed greatly to number theory, mathematical notation, and because

Arithmetica contains the earliest known use of syncopated notation [5]. However, it seems that many of the methods for solving linear and quadratic equations used by Diophantus go back to

Babylonian mathematics. For this reason mathematical historian Kurt Vogel writes: “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.” According to some historians of mathematics, like Florian Cajori, Diophantus got the first knowledge of algebra from India, [6][7], although other historians disagree.[8]

Diophantine analysis

Today Diophantine analysis is the area of study where integral (whole number) solutions are sought for equations, and Diophantine equations

are polynomial equations with integral coefficients to which only integral 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.

There are no general comprehensive methods of solving used by Diophantus (that is found). Heath stated, "every question requires a quite

special method, which often will not serve even for the most closely allied problems. It is on that account difficult for a modern mathematician

even after studying 100 Diophantine solutions to solve the 101st problem; and if we have made the attempt, and after some vein endeavors

read Diophantus' own solution, we shall be astonished to see how suddenly he leaves the broad high-road, dashes into a side-path and with a

quich turn reaches the goal, often enough a goal with reaching which we should not be content; we expected to have to climb a toilsome

path, but to be rewarded at the end by an extensive view; instead of which out guide leads by narrow, strange, but smooth ways to a small

eminence; he has finished!"

Mathematical notation

Diophantus made important advances in mathematical notation. He was the first person to use algebraic notation and symbolism. Before him

everyone wrote out equations completely. Diophantus introduced an algebraic symbolism that used an abridged notation for frequently occurring

operations, and an abbreviation for the unknown and for the powers of the unknown. Mathematical historian Kurt Vogel states:

“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.”[citation needed]

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. For example, Diophantus lacked symbols for the

operation of multiplication; this probably became as such since his coefficients are all definite numbers or fractions, and the results are recorded

without showing previous work leading to the result. 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.

Heath stated, "Diophantus evidently put down his equations in the ordinary course of writing, i.e. they were written straight on, as are the steps

in the propositions of Euclid, and not put in separate lines for each step in the process of simplification."

Algebra still had a long way to go before very general problems could be written down and solved succinctly.

See also

Notes

  1. Boyer, Carl B. (1991). "Revival and Decline of Greek Mathematics", A History of Mathematics, Second Edition, John Wiley & Sons, Inc., 178. ISBN 0471543977. “At the beginning of this period, also known as the Later Alexandrian Age, we find the leading Greek algebraist, Diophantus of Alexandria, and toward its close there appeared the last significant Greek geometer, Pappus of Alexandria.” 
  2. [[Roger Cooke|Cooke, Roger]] (1997). "The Nature of Mathematics", The History of Mathematics: A Brief Course. Wiley-Interscience, 7. ISBN 0471180823. “Some enlargement in the sphere in which symbols were used occurred in the writings of the third -century Greek mathematician Diophantus of Alexandria, but the same defect was present as in the case of Akkadians.” 
  3. D. M. Burton (1991, 1995). History of Mathematics, Dubuque, IA (Wm.C. Brown Publishers).
  4. Knorr, Wilbur: Arithmêtike stoicheiôsis: On Diophantus and Hero of Alexandria, in: Historia Matematica, New York, 1993, Vol.20, No.2, 180-192
  5. Carl B. Boyer, A History of Mathematics, Second Edition (Wiley, 1991), page 228
  6. Florian Cajori, A History of Elementary Mathematics, 1898
  7. Saradakanta Ganguli, Notes on Indian Mathematics. A Criticism of George Rusby Kaye's Interpretation, Isis, Vol. 12, No. 1 (Feb., 1929), pp. 132-145
  8. [http://logica.ugent.be/albrecht/thesis/Reception.pdf "Heeffer, Albrecht, The Reception of Ancient Indian Mathematics by Western Historians, Ghent University, Belgium."]

References
ISBN links support NWE through referral fees

  • A. Allard, "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibo. Nat. 4678 et les Vaticani gr. 191 et 304,"

Byzantion 53. Brussels, 1983: 682-710.

  • P. Ver Eecke, Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer,

1926.

  • T. L. Heath, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
  • D. C. Robinson and Luke Hodgkin. History of Mathematics, King's College London, 2003.
  • P. L. Tannery, Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895.
  • Jacques Sesiano, Books IV to VII of Diophantus’ Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā, Heidelberg: Springer-

Verlag, 1982. ISBN 0-387-90690-8.

External links

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