Arithmetica

''Arithmetica'' ( grc-gre, Ἀριθμητικά) is an Ancient Greek text on mathematics written by the mathematician Diophantus () in the 3rd century AD. It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate equations.

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 $4n + 3$ 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.

''Arithmetica'' was originally written in thirteen books, but the Greek manuscripts that survived to the present contain no more than six books. 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:

he four missing booksresurfaced 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.

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

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. "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." ''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. "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 $\zeta$ (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."

In ''Arithmetica'', Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations; thus he used what is now known as . The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials. "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." So for example, what would be written in modern notation as
$x^3 - 2x^2 + 10x -1 = 5,$
which can be rewritten as
$\left(1 + 10\right) - \left(2 + 1\right) = 5,$
would be written in Diophantus's syncopated notation as

:$\Kappa^ \overline \; \zeta \overline \;\, \pitchfork \;\, \Delta^ \overline \; \Mu \overline \,\;$$\sigma\;\, \Mu \overline$

where the symbols represent the following:

Unlike in modern notation, 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:
$1 10 - 2 1 = 5$
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:
$\left(1 + 10\right) - \left(2 + 1\right) = 5$

''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. ''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. ''Arithmetica'' also makes use of the identities: "The book makes frequent use of the identities ..which had appeared in Diophantus and had been widely used by the Arabs."
$\begin \left(a^2 + b^2\right) \left(c^2 + d^2\right) &= (ac + db)^2 + (bc - ad)^2 \\ &= (ad + bc)^2 + (ac - bd)^2 \\ \end$

See also

* Diophantus II.VIII
* Muhammad ibn Mūsā al-Khwārizmī

Citations

References

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External links

* 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, .

{{Use dmy dates, date=April 2017

3rd-century books
Ancient Greek mathematical works
History of algebra