Diophantus

For the general, see Diophantus (general). For the sophist, see Diophantus the Arab. For the intersex soldier, see Diophantus of Abae.

Diophantus of Alexandria(Ancient Greek: Διόφαντος, romanized: Diophantos) (/daɪoʊˈfæntəs/; fl. 250 AD) was a Greek mathematician who has been referred to as "the father of algebra".

Diophantus was the author of the Arithmetica in thirteen books, ten of which are still extant, made up of arithmetical problems that are solved through algebraic equations. His work was translated into Arabic in the 9th century AD and had influence in the development of later algebra: Diophantus' method of solution matches medieval Arabic algebra in its concepts and overall procedure. The 1621 edition of Arithmetica by Bachet gained fame after Pierre de Fermat wrote his famous "Last Theorem" in the margins of his copy.

In modern use, Diophantine equations are algebraic equations with integer coefficients for which integer solutions are sought. Diophantine geometry and Diophantine approximations are two other subareas of number theory that are named after him. Some problems from the Arithmetica have inspired modern work in both abstract algebra and number theory.

Biography

Diophantus was born into a Greek family and is known to have lived in Alexandria, Egypt, during the Roman era, circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century.

Much of our knowledge of the life of Diophantus is derived from a 5th-century Greek anthology of number games and puzzles created by Metrodorus. One of the problems (sometimes called his epitaph) 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' age x can be expressed as

x = ⁠x/6⁠ + ⁠x/12⁠ + ⁠x/7⁠ + 5 + ⁠x/2⁠ + 4

which gives x a value of 84 years. However, the accuracy of the information cannot be confirmed.

Arithmetica

Title page of the Latin translation of Diophantus' Arithmetica by Bachet (1621).

See also: Arithmetica

Arithmetica is the major work of Diophantus and the most prominent work on premodern algebra in Greek mathematics. It is a collection of 290 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations. Arithmetica was originally written in thirteen books, but only six of them survive in Greek, while another four books survive in Arabic, which were discovered in 1968.

Arithmetica is the earliest extant work present that solve arithmetic problems by algebra. Diophantus however did not invent the method of algebra, which existed before him. Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.

Equations in the book are presently called Diophantine equations. The method for solving these equations is known as Diophantine analysis. Most of the Arithmetica problems lead to quadratic equations. 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.

In Book 3, Diophantus solves problems of finding values which make two linear expressions simultaneously into squares or cubes. In book 4, he finds rational powers between given numbers. He also noticed that numbers of the form $4n+3$ cannot be the sum of two squares. Diophantus also appears to know that every number can be written as the sum of four squares. If he did know this result (in the sense of having proved it as opposed to merely conjectured it), his doing so would be truly remarkable: even Fermat, who stated the result, failed to provide a proof of it and it was not settled until Joseph-Louis Lagrange proved it using results due to Leonhard Euler.

Other works

Diophantus wrote several other books besides Arithmetica, but other than a few fragments, they have not 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.

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

a³ − b³ = c³ + d³.

Polygonal numbers and geometric elements

Diophantus is also known to have written on polygonal numbers, a topic of great interest to Pythagoras and Pythagoreans. Fragments of a book dealing with polygonal numbers are extant.

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.

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 17th and 18th centuries. Diophantus and his works 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. How much he affected India is a matter of debate.

Diophantus has been considered "the father of algebra" because of his contributions to number theory, mathematical notations and the earliest known use of syncopated notation in his book series Arithmetica. However this is usually debated, because Al-Khwarizmi was also given the title as "the father of algebra", nevertheless both mathematicians were responsible for paving the way for algebra today.

Medieval era

Like many other Greek mathematical treatises, Diophantus was forgotten in Western Europe during the Dark Ages, since the study of ancient Greek, and literacy in general, had greatly declined. The portion of the Greek Arithmetica that survived, however, was, like all ancient Greek texts transmitted to the early modern world, copied by, and thus known to, medieval Byzantine scholars. Scholia on Diophantus by the Byzantine Greek scholar John Chortasmenos (1370–1437) are preserved together with a comprehensive commentary written by the earlier Greek scholar Maximos Planudes (1260 – 1305), who produced an edition of Diophantus within the library of the Chora Monastery in Byzantine Constantinople.

Arithmetica became known to mathematicians in the Islamic world in the tenth century when Abu'l-Wefa translated it into Arabic.

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.

Arithmetica was first translated from Greek 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.

Fermat

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

The Latin translation of Arithmetica by Bachet in 1621 became the first Latin edition that was widely available. Pierre de Fermat owned a copy, studied it and made notes in the margins. The 1621 edition of Arithmetica by Bachet 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 aⁿ + bⁿ = cⁿ has no solutions in non-zero integers a, b, and c. 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 the "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 scholar John Chortasmenos (1370–1437) had written "Thy soul, Diophantus, be with Satan because of the difficulty of your other theorems and particularly of the present theorem" next to the same problem.

Diophantus was among the first to recognized positive rational numbers as numbers, by allowing fractions for coefficients and solutions. He coined the term παρισότης (parisotēs) to refer to an approximate equality. This term was rendered as adaequalitas in Latin, and became the technique of adequality developed by Pierre de Fermat to find maxima for functions and tangent lines to curves.

Diophantine analysis

See also: Diophantine equation

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: 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. 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 4 = 4x + 20 'absurd' because it would lead to a negative value for x. 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.

Rediscovery of books IV-VII

In 1968, Fuat Sezgin found four previously unknown books of Arithmetica at the shrine of Imam Rezā in the holy Islamic city of Mashhad in northeastern Iran. The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa (820–912). Norbert Schappacher has written:

[The four missing books] resurfaced around 1971 in the Astan Quds Library in Meshed (Iran) in a copy from 1198 AD. It was not catalogued under the name of Diophantus (but under that of Qusta ibn Luqa) because the librarian was apparently not able to read the main line of the cover page where Diophantus’s name appears in geometric Kufi calligraphy.

Notation

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.

Similar to medieval Arabic algebra, Diophantus uses three stages to solution of a problem by algebra: 1) An unknown is named and an equation is set up 2) An equation is simplified to a standard form (al-jabr and al-muqābala in Arabic) 3) Simplified equation is solved

Although Diophantus made important advances in symbolism, he still lacked the necessary notation to express more general methods; 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 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".

Diophantus does not give classification of equations in six types like Al-Khwarizmi in extant parts of Arithmetica. He does says that he would give solution to three terms equations later, so this part of work is possibly just lost.

The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. So for example, what would be written in modern notation as $$x^{3}-2x^{2}+10x-1=5,$$ which can be rewritten as $$\left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5,$$ would be written in Diophantus's notation as

$\mathrm {K} ^{\upsilon }{\overline {\alpha }}\;\zeta {\overline {\iota }}\;\,\pitchfork \;\,\Delta ^{\upsilon }{\overline {\beta }}\;\mathrm {M} {\overline {\alpha }}\,\;$ἴ$\sigma \;\,\mathrm {M} {\overline {\varepsilon }}$

Symbol	What it represents
${\overline {\alpha }}$	1 (Alpha is the 1st letter of the Greek alphabet)
${\overline {\beta }}$	2 (Beta is the 2nd letter of the Greek alphabet)
${\overline {\varepsilon }}$	5 (Epsilon is the 5th letter of the Greek alphabet)
${\overline {\iota }}$	10 (Iota is the 9th letter of the modern Greek alphabet but it was the 10th letter of an ancient archaic Greek alphabet that had the letter digamma (uppercase: Ϝ, lowercase: ϝ) in the 6th position between epsilon ε and zeta ζ.)
ἴσ	"equals" (short for ἴσος)
$\pitchfork$	represents the subtraction of everything that follows $\pitchfork$ up to ἴσ
$\mathrm {M}$	the zeroth power (that is, a constant term)
$\zeta$	the unknown quantity (because a number $x$ raised to the first power is just $x,$ this may be thought of as "the first power")
$\Delta ^{\upsilon }$	the second power, from Greek δύναμις, meaning strength or power
$\mathrm {K} ^{\upsilon }$	the third power, from Greek κύβος, meaning a cube
$\Delta ^{\upsilon }\Delta$	the fourth power
$\Delta \mathrm {K} ^{\upsilon }$	the fifth power
$\mathrm {K} ^{\upsilon }\mathrm {K}$	the sixth power

Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's equation into a modern equation would be the following: $${x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5$$ where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as: $$\left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5$$

References

• Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
• Corry, Leo (2015). A Brief History of Numbers (1st ed.). Oxford University Press. p. 71. ISBN 978-0-19-870259-7. OCLC 907194512.
• Christianidis, Jean; Oaks, Jeffrey A. (2023). The Arithmetica of Diophantus: a complete translation and commentary. Abingdon, Oxon New York, NY: Routledge. ISBN 1138046353.
• Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0-471-18082-3.
• Derbyshire, John (2006). Unknown Quantity: A Real And Imaginary History of Algebra. Joseph Henry Press. ISBN 0-309-09657-X.
• Heath, Sir Thomas, Diophantos of Alexandria: A Study in the History of Greek Algebra, Cambridge: Cambridge University Press, 1885, 1910.
• Katz, Victor J.; Parshall, Karen Hunger (2014). Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press. ISBN 978-0-691-14905-9.
• Oaks, Jeffrey; Christianidis, Jean (2013). "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria". Historia Mathematica. 40 (2): 158–160. doi:10.1016/j.hm.2012.09.001.
• Robinson, D. C. and Luke Hodgkin. History of Mathematics, King's College London, 2003.
• Sesiano, Jacques (2011). Books IV to VII of Diophantus' Arithmetica in the Arabic translation attributed to Qusṭā ibn Lūqā. New York Heidelberg Berlin: Springer-Verlag. ISBN 1461381762.

Further reading

• Allard, A. "Les scolies aux arithmétiques de Diophante d'Alexandrie dans le Matritensis Bibl.Nat.4678 et les Vatican Gr.191 et 304" Byzantion 53. Brussels, 1983: 682–710.
• Bachet de Méziriac, C.G. Diophanti Alexandrini Arithmeticorum libri sex et De numeris multangulis liber unus. Paris: Lutetiae, 1621.
• Bashmakova, Izabella G. Diophantos. Arithmetica and the Book of Polygonal Numbers. Introduction and Commentary Translation by I.N. Veselovsky. Moscow: Nauka [in Russian].
• Bashmakova, Izabella G. "Diophante et Fermat", Revue d'Histoire des Sciences 19 (1966), pp. 289–306
• Bashmakova, Izabella G. Diophantus and Diophantine Equations. Moscow: Nauka 1972 [in Russian]. German translation: Diophant und diophantische Gleichungen. Birkhauser, Basel/ Stuttgart, 1974. English translation: Diophantus and Diophantine Equations. Translated by Abe Shenitzer with the editorial assistance of Hardy Grant and updated by Joseph Silverman. The Dolciani Mathematical Expositions, 20. Mathematical Association of America, Washington, DC. 1997.
• Bashmakova, Izabella G. "Arithmetic of Algebraic Curves from Diophantus to Poincaré", Historia Mathematica 8 (1981), 393–416.
• Bashmakova, Izabella G., Slavutin, E.I. History of Diophantine Analysis from Diophantus to Fermat. Moscow: Nauka 1984 [in Russian].
• Christianidis, J. "Maxime Planude sur le sens du terme diophantien "plasmatikon"", Historia Scientiarum, 6 (1996)37-41.
• Christianidis, J. "Une interpretation byzantine de Diophante", Historia Mathematica, 25 (1998) 22–28.
• Czwalina, Arthur. Arithmetik des Diophantos von Alexandria. Göttingen, 1952.
• Diophantus Alexandrinus, Pierre de Fermat, Claude Gaspard Bachet de Meziriac, Diophanti Alexandrini Arithmeticorum libri 6, et De numeris multangulis liber unus. Cum comm. C(laude) G(aspar) Bacheti et observationibus P(ierre) de Fermat. Acc. doctrinae analyticae inventum novum, coll. ex variis eiu. Tolosae 1670, doi:10.3931/e-rara-9423.
• Heath, Sir Thomas (1981). A history of Greek mathematics. Vol. 2. Cambridge University Press: Cambridge.
• Rashed, Roshdi, Houzel, Christian. Les Arithmétiques de Diophante : Lecture historique et mathématique, Berlin, New York : Walter de Gruyter, 2013.
• Rashed, Roshdi, Histoire de l’analyse diophantienne classique : D’Abū Kāmil à Fermat, Berlin, New York : Walter de Gruyter.
• Rashed, Roshdi. L’Art de l’Algèbre de Diophante. éd. arabe. Le Caire : Bibliothèque Nationale, 1975.
• Rashed, Roshdi. Diophante. Les Arithmétiques. Volume III: Book IV; Volume IV: Books V–VII, app., index. Collection des Universités de France. Paris (Société d’Édition "Les Belles Lettres"), 1984.
• Sesiano, Jacques. The Arabic text of Books IV to VII of Diophantus’ translation and commentary. Thesis. Providence: Brown University, 1975.
• Σταμάτης, Ευάγγελος Σ. Διοφάντου Αριθμητικά. Η άλγεβρα των αρχαίων Ελλήνων. Αρχαίον κείμενον – μετάφρασις – επεξηγήσεις. Αθήναι, Οργανισμός Εκδόσεως Διδακτικών Βιβλίων, 1963.
• Tannery, P. L. Diophanti Alexandrini Opera omnia: cum Graecis commentariis, Lipsiae: In aedibus B.G. Teubneri, 1893-1895 (online: vol. 1, vol. 2)
• Ver Eecke, P. Diophante d’Alexandrie: Les Six Livres Arithmétiques et le Livre des Nombres Polygones, Bruges: Desclée, De Brouwer, 1921.
• Wertheim, G. Die Arithmetik und die Schrift über Polygonalzahlen des Diophantus von Alexandria. Übersetzt und mit Anmerkungen von G. Wertheim. Leipzig, 1890.
• Vogel, Kurt (1970). "Diophantus of Alexandria". Dictionary of Scientific Biography. Vol. 4. New York: Scribner.

External links

• O'Connor, John J.; Robertson, Edmund F., "Diophantus", MacTutor History of Mathematics Archive, University of St Andrews
• Diophantus's Riddle Diophantus' epitaph, by E. Weisstein
• Norbert Schappacher (2005). Diophantus of Alexandria : a Text and its History.
• Review of Sesiano's Diophantus Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica, by Jan P. Hogendijk
• Latin translation from 1575 by Wilhelm Xylander
• Scans of Tannery's edition of Diophantus at wilbourhall.org