Arithmetica

Arithmetica (Ancient 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. 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.
Author	Diophantus

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 ${\displaystyle 4n+3}$ 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.

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:

[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.

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.

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.

In modern algebra a Laurent polynomial is linear combination of some variables, raised to integer powers, which behaves under multiplication, addition, and subtraction. Algebra of Diophantus, similar to medieval arabic algebra is aggregation of objects of different types with no operations present

For example, the Laurent polynomial written as ${\displaystyle 6{\tfrac {1}{4}}x^{-1}+25x^{2}-9}$ in modern notation is written by Diophantus as "6 4′ inverse Powers, 25 Powers lacking 9 units", or "a collection of ${\displaystyle 6{\tfrac {1}{4}}}$ object of one kind with 25 object of second kind which lack 9 objects of third kind with no operation present".

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

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

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 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. So for example, what would be written in modern notation as $${\displaystyle x^{3}-2x^{2}+10x-1=5,}$$ which can be rewritten as $${\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5,}$$ would be written in Diophantus's syncopated notation as

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

where the symbols represent the following:

Symbol	What it represents
${\displaystyle {\overline {\alpha }}}$	1 (Alpha is the 1st letter of the Greek alphabet)
${\displaystyle {\overline {\beta }}}$	2 (Beta is the 2nd letter of the Greek alphabet)
${\displaystyle {\overline {\varepsilon }}}$	5 (Epsilon is the 5th letter of the Greek alphabet)
${\displaystyle {\overline {\iota }}}$	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)
${\displaystyle \pitchfork }$	represents the subtraction of everything that follows ${\displaystyle \pitchfork }$ up to ἴσ
${\displaystyle \mathrm {M} }$	the zeroth power (that is, a constant term)
${\displaystyle \zeta }$	the unknown quantity (because a number ${\displaystyle x}$ raised to the first power is just ${\displaystyle x,}$ this may be thought of as "the first power")
${\displaystyle \Delta ^{\upsilon }}$	the second power, from Greek δύναμις, meaning strength or power
${\displaystyle \mathrm {K} ^{\upsilon }}$	the third power, from Greek κύβος, meaning a cube
${\displaystyle \Delta ^{\upsilon }\Delta }$	the fourth power
${\displaystyle \Delta \mathrm {K} ^{\upsilon }}$	the fifth power
${\displaystyle \mathrm {K} ^{\upsilon }\mathrm {K} }$	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 syncopated equation into a modern symbolic equation would be the following:$${\displaystyle {x^{3}}1{x}10-{x^{2}}2{x^{0}}1={x^{0}}5}$$ where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:$${\displaystyle \left({x^{3}}1+{x}10\right)-\left({x^{2}}2+{x^{0}}1\right)={x^{0}}5}$$However the distinction between "rhetorical algebra", "syncopated algebra" and "symbolic algebra" is considered outdated by Jeffrey Oaks and Jean Christianidis. The problems were solved on dust-board using some notation, while in books solution were written in "rhetorical style".

Arithmetica also makes use of the identities:$${\displaystyle {\begin{alignedat}{4}\left(a^{2}+b^{2}\right)\left(c^{2}+d^{2}\right)&=(ac+db)^{2}+(bc-ad)^{2}\\&=(ad+bc)^{2}+(ac-bd)^{2}\\\end{alignedat}}}$$

See also

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

Citations

1. "Diophantus of Alexandria (Greek mathematician)". Encyclopædia Britannica. Retrieved 11 April 2013.
2. Magill, Frank N., ed. (1998). Dictionary of World Biography. Vol. 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. Oaks, Jeffrey; Christianidis, Jean. The Arithmetica of Diophantus A Complete Translation and Commentary. p. 80.
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.
10. Oaks, Jeffrey; Christianidis, Jean (2013). "Practicing algebra in late antiquity: The problem-solving of Diophantus of Alexandria". Historia Mathematica. 40: 150.
11. Oaks, Jeffrey; Christianidis, Jean (2023). The Arithmetica of Diophantus A Complete Translation and Commentary. pp. 51–52.
12. Oaks, Jeffrey; Christianidis, Jean (2021). The Arithmetica of Diophantus A Complete Translation and Commentary. pp. 53–66.
13. (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 ${\displaystyle \zeta }$ (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."
14. (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."
15. (Cooke 1997, "Mathematics in the Roman Empire" pp. 167-168)
16. (Derbyshire 2006, "The Father of Algebra" pp. 35-36)
17. Oaks, Jeffrey; Christianidis, Jean (2023). The Arithmetica of Diophantus A Complete Translation and Commentary. pp. 78–79. There are two major flaws with this trichotomy. First, the language written in books is not always the language in which problems were worked out. In Arabic, problems were often solved in notation on a dust-board or some other temporary surface, and then for inclusion in a book a rhetorical version was composed. Also, because of the two-dimensional character of the Arabic notation, it would have been written and read visually, independent of real or imagined speech. It thus fits nicely into Nesselmann's "symbolic" category. The rhetorical version of the same work, on the other hand, was categorized as being "rhetorical". These two ways of writing algebra do not reflect two stages of the development of algebra but are different ways of expressing the same ideas. Second, Nesselmann was unaware of the conceptual differences between premodern and modern algebra, and thus, he could not have appreciated the leap made in the time of Viète and Descartes that included a radical shift in how notation was interpreted.
18. (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.
• 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 L. (2009). Diophantus of Alexandria: A Study in the History of Greek Algebra. Martino Fine Books. ISBN 978-1-57898-754-2.
• 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.
• 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.

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, doi:10.3931/e-rara-9423.