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.
[1] It is a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) and indeterminate 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.
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] 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 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.
[8] Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.
[9] 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[10] For example, the Laurent polynomial written as
in modern notation is written by Diophantus as "6 4′ inverse Powers, 25 Powers lacking 9 units", or "a collection of
[11] 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[12] 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[9] In Arithmetica, Diophantus is the first to use symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;[13] 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.
would be written in Diophantus's syncopated notation as where the symbols represent the following: [15][16] Unlike in modern notation, the coefficients come after the variables and addition is represented by the juxtaposition of terms.
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:[16]
The problems were solved on dust-board using some notation, while in books solution were written in "rhetorical style".