Arithmetica

Arithmetica (Greek: Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus (c. 200/214 AD – c. 284/298 AD) in the 3rd century AD.[1] It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

Cover of the 1621 edition, translated into Latin from Greek by Claude Gaspard Bachet de Méziriac.
AuthorDiophantus

Summary

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.

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 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.

Arithmetica was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books.[2] 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.[3] The four books are thought to have been translated from Greek to Arabic by Qusta ibn Luqa (820–912).[2] 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.[4]

Arithmetica became known to mathematicians in the Islamic world in the tenth century[5] when Abu'l-Wefa translated it into Arabic.[6]

Syncopated algebra

Diophantus was a Hellenistic mathematician who lived circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.[7] Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, and it is different from Babylonian mathematics in that Diophantus is concerned primarily with exact solutions, both determinate and indeterminate, instead of simple approximations.[8]

In Arithmetica, Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;[8] thus he used what is now known as syncopated algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.[9] So, for example, what we would write as

Diophantus would have written as

Κυ ᾱ̄ ζ  ί̄  ⫛ Δυ  β̄ Μ  ᾱ̄ ἴσ Μ  ε̄

where the symbols represent the following:[10][11]

Symbol Representation
 ᾱ̄ represents 1
 β̄ represents 2
 ε̄ represents 5
 ί̄ represents 10
ζ represents the unknown quantity
ἴσ (short for ἴσος) represents "equals"
⫛ represents the subtraction of everything that follows it up to ἴσ
Μ represents the zeroth power (i.e. a constant term)
Δυ represents the second power, from Greek δύναμις, meaning strength or power
Κυ represents the third power, from Greek κύβος, meaning a cube
ΔυΔ represents the fourth power
ΔΚυ represents the fifth power
ΚυΚ represents the sixth power

Note that the coefficients come after the variables and that addition is represented by the juxtaposition of terms. A literal symbol-for-symbol translation of Diophantus's syncopated equation into a modern symbolic equation would be the following:[10]

and, to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:[10]

Arithmetica is a collection of some 150 solved problems with specific numbers and there is no postulational development nor is a general method explicitly explained, although generality of method may have been intended and there is no attempt to find all of the solutions to the equations.[8] Arithmetica does contain solved problems involving several unknown quantities, which are solved, if possible, by expressing the unknown quantities in terms of only one of them.[8] Arithmetica also makes use of the identities:[12]

See also

Citations
  1. "Diophantus of Alexandria (Greek mathematician)". Encyclopædia Britannica. Retrieved 11 April 2013.
  2. Magill, Frank N., ed. (1998). Dictionary of World Biography. 1. Salem Press. p. 362. ISBN 9781135457396.
  3. Hogendijk, Jan P. (1985). "Review of J. Sesiano, Books IV to VII of Diophantus' Arithmetica". Retrieved 6 July 2014. Only six of the thirteen books of the Arithmetica of Diophantus (ca. A.D. 250) are extant in Greek. The remaining books were believed to be lost, until the recent discovery of a medieval Arabic translation of four of the remaining books in a manuscript in the Shrine Library in Meshed in Iran (see the catalogue [Gulchin-i Ma'ani 1971-1972, pp. 235-236]. The manuscript was discovered in 1968 by F. Sezgin).
  4. Schappacher, Norbert (April 2005). "Diophantus of Alexandria : a Text and its History" (PDF). p. 18. Retrieved 9 October 2015.
  5. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 234) "Note the omission of Diophantus and Pappus, authors who evidently were not at first known in Arabia, although the Diophantine Arithmetica became familiar before the end of the tenth century."
  6. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 239) "Abu'l-Wefa was a capable algebraist as well as a trigonometer. He commented on al-Khwarizmi's Algebra and translated from Greek one of the last great classics, the Arithmetica of Diophantus."
  7. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about A.D. 250, but dates a century or more earlier or later are sometimes suggested[...] If this conundrum is historically accurate, Diophantus lived to be eighty-four-years old. [...] The chief Diophantine work known to us is the Arithmetica, a treatise originally in thirteen books, only the first six of which have survived."
  8. (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 180-182) "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent. But whereas Babylonian mathematicians had been concerned primarily with approximate solutions of determinate equations as far as the third degree, the Arithmetica of Diophantus (such as we have it) is almost entirely devoted to the exact solution of equations, both determinate and indeterminate. [...] Throughout the six surviving books of Arithmetica there is a systematic use of abbreviations for powers of numbers and for relationships and operations. An unknown number is represented by a symbol resembling the Greek letter ζ (perhaps for the last letter of arithmos). [...] It is instead a collection of some 150 problems, all worked out in terms of specific numerical examples, although perhaps generality of method was intended. There is no postulation development, nor is an effort made to find all possible solutions. In the case of quadratic equations with two positive roots, only the larger is give, and negative roots are not recognized. No clear-cut distinction is made between determinate and indeterminate problems, and even for the latter for which the number of solutions generally is unlimited, only a single answer is given. Diophantus solved problems involving several unknown numbers by skillfully expressing all unknown quantities, where possible, in terms of only one of them."
  9. (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
  10. (Derbyshire 2006, "The Father of Algebra" pp. 35-36)
  11. (Cooke 1997, "Mathematics in the Roman Empire" pp. 167-168)
  12. (Boyer 1991, "Europe in the Middle Ages" p. 257) "The book makes frequent use of the identities [...] which had appeared in Diophantus and had been widely used by the Arabs."

References
  • Boyer, Carl B. (1991). A History of Mathematics (Second ed.). John Wiley & Sons, Inc. ISBN 0-471-54397-7.
  • Derbyshire, John (2006). Unknown Quantity: A Real And Imaginary History of Algebra. Joseph Henry Press. ISBN 0-309-09657-X.
  • Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0-471-18082-3.

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.

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