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

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## Abu Kamil:

## 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

### Contents

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## 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).

## Sources

- 1965: T.H. O'Beirne:
*Puzzles and Paradoxes* - 1992: David Wells:
*Curious and Interesting Puzzles*... (previous) ... (next): Sun Tsu Suan Ching: $74$