Book:Abu Kamil/Kitāb al-ṭarā'if fi'l-ḥisāb

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Abu Kamil: Kitāb al-ṭarā'if fi'l-ḥisāb

Published $\text {c. $900$}$
In English:

Book of Rare Things in the Art of Calculation

Subject Matter

  • Systematic procedures for finding integral solutions for indeterminate equations


Historical Note

Possibly the first work that investigates the number of solutions to a general indeterminate equation.

It arose from Abu Kamil considering the question of purchasing $100$ birds at $100$ drachma, the birds being:

ducks at $2$ drachma,
hens at $1$ drachma,
doves $2$ for a drachma,
ring-doves $3$ for $1$ drachma
and larks $4$ for $1$ drachma.

I went into this problem fully and found that there were $2,696$ valid answers. I marvelled at this, only to discover -- when I spoke of it -- that I was reckoned a simpleton or an incompetent, and strangers looked upon me with suspicion. So I decided to write a book ...

The specific number varies among $2,676$, $2,678$ and $2,698$, depending on which authority you quote. Some of them may indeed have got the number wrong.

He continues the theme in his Kitāb al-ṭair (Book of Birds).