Editing Diophantus (section)

==''Arithmetica'' ==
[[File:Diophantus-cover.png|thumb|Title page of the Latin translation of Diophantus' ''Arithmetica'' by [[Claude Gaspard Bachet de Méziriac|Bachet]] (1621).|328x328px]]
''Arithmetica'' is the major work of Diophantus and the most prominent work on premodern [[algebra]] in Greek mathematics. It is a collection of 290 [[algebra]]ic problems giving numerical solutions of determinate [[equations]] (those with a unique solution) and [[indeterminate equation]]s. ''Arithmetica'' was originally written in thirteen books, but only six of them survive in Greek,<ref>{{cite book |year= 1998 |editor-last= Magill |editor-first= Frank N. |title= Dictionary of World Biography |url= https://books.google.com/books?id=_CMl8ziTbKYC&pg=PA362 |volume= 1 |publisher= Salem Press |page= 362 |isbn= 9781135457396 }}</ref> while another four books survive in Arabic, which were discovered in 1968.{{sfn|Sesiano|2012}} The books in Arabic correspond to books 4 to 7 of the original treatise, while the Greek books correspond to books 1 to 3 and 8 to 10.{{sfn|Sesiano|2012}}

''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.{{sfn|Christianidis|Oaks|2023|p=80}} Algebra was practiced and diffused orally by practitioners, with Diophantus picking up technique to solve problems in arithmetic.{{sfn|Christianidis|Oaks|2013|pp=158-160}}

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

===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:{{sfn|Christianidis|Oaks|2023|pp=53–66}}
# An unknown is named and an equation is set up
# An equation is simplified to a standard form (''al-jabr'' and ''al-muqābala'' in Arabic)
# 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.{{sfn|Christianidis|Oaks|2013|pp=158-160}}

The main difference between Diophantine notation and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.{{sfn|Cooke|1997|pp=167-168}}{{sfn|Derbyshire|2006|pp=35-36}} So for example, what would be written in modern notation as
<math display="block">x^3 - 2x^2 + 10x -1 = 5,</math>
which can be rewritten as
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5,</math>
would be written in Diophantus's notation as

:<math>\Kappa^{\upsilon} \overline{\alpha} \; \zeta \overline{\iota} \;\, \pitchfork \;\, \Delta^{\upsilon} \overline{\beta} \; \Mu \overline{\alpha} \,\;</math>{{lang|grc|ἴ}}<math>\sigma\;\, \Mu \overline{\varepsilon}</math>

{| class="wikitable"
|-
! Symbol
! What it represents
|-
| &nbsp;<math>\overline{\alpha}</math>
| 1 ([[Alpha]] is the 1st letter of the [[Greek alphabet]])
|-
| &nbsp;<math>\overline{\beta}</math>
| 2 ([[Beta]] is the 2nd letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\varepsilon}</math>
| 5 ([[Epsilon]] is the 5th letter of the Greek alphabet)
|-
| &nbsp;<math>\overline{\iota}</math>
| 10 ([[Iota]] is the 9th letter of the [[History of the Greek alphabet|{{em|modern}} Greek alphabet]] but it was the 10th letter of an [[Archaic Greek alphabets|ancient archaic Greek alphabet]] that had the letter [[digamma]] (uppercase: Ϝ, lowercase: ϝ) in the 6th position between [[epsilon]] ε and [[zeta]] ζ.)
|-
| {{lang|grc|ἴσ}}
| "equals" (short for {{lang|grc|[[wiktionary:ἴσος|ἴσος]]}})
|-
| <math>\pitchfork</math>
| represents the subtraction of everything that follows <math>\pitchfork</math> up to {{lang|grc|ἴσ}}
|-
| <math>\Mu</math>
| the zeroth power (that is, a constant term)
|-
| <math>\zeta</math>
| the unknown quantity (because a number <math>x</math> raised to the first power is just <math>x,</math> this may be thought of as "the first power")
|-
| <math>\Delta^{\upsilon}</math>
| the second power, from Greek {{lang|grc|δύναμις}}, meaning strength or power
|-
| <math>\Kappa^{\upsilon}</math>
| the third power, from Greek {{lang|grc|κύβος}}, meaning a cube
|-
| <math>\Delta^{\upsilon}\Delta</math>
| the fourth power
|-
| <math>\Delta\Kappa^{\upsilon}</math>
| the fifth power
|-
| <math>\Kappa^{\upsilon}\Kappa</math>
| 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:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>{x^3}1 {x}10 - {x^2}2 {x^0}1 = {x^0}5</math>
where to clarify, if the modern parentheses and plus are used then the above equation can be rewritten as:{{sfn|Derbyshire|2006|pp=35-36}}
<math display=block>\left({x^3}1 + {x}10\right) - \left({x^2}2 + {x^0}1\right) = {x^0}5</math>

===Contents===
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 <math>4n + 3</math> cannot be the sum of two squares. Diophantus also appears to know that [[Lagrange's four-square theorem|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]].