Al-Hawārī completed his Essential Commentary in Marrakesh in 1305. This modest book was written with the goal of supplying numerical examples to the rules of calculation explained in the famous Condensed Book on the Operations of Arithmetic of his teacher Ibn al-Bannāʾ. It is partly because of its modesty that al-Hawārī’s book is a good vehicle for presenting the various facets of practical Arabic arithmetic, which in al-Hawārī’s time included not just computational techniques but also algebra and other problem-solving methods. In this book we give an edition, English translation, and running commentary of al-Hawārī’s commentary, together with a general introduction to Arabic arithmetic to situate it in context. We cover not just the rules for operating on known and unknown numbers, but also their conceptual and historical background. Numbers in Arabic arithmetic were conceived as amounts or measures of something, and thus could be any positive quantity, including fractions and irrational roots. Thus they were neither founded in the Greek notion of multitudes of an indivisible unit, nor were they defined axiomatically as they are today. This way of understanding numbers informed both their ways of expressing numbers and algebraic expressions and how they performed their calculations. On the historical front, we witness the ways that different local arithmetical traditions interacted, and how Greek arithmetic, even if fundamentally incompatible with its Arabic counterparts, was partially absorbed into practical books like al-Hawārī’s.

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## A

#### Abū Kāmil

#### Abū Yaʿqūb

#### Abū l-Wafāʾ

#### ʿAlī al-Sulamī

#### Apollonius

#### Archimedes

#### Aristotle

## B

#### al-Baghdādī

#### al-Bīrūnī

#### Brahmagupta

## D

#### Descartes, René

#### Diophantus

## F

#### al-Fārābī

#### al-Fārisī

#### Fibonacci

## H

#### Ḥabīb ibn Bihrīz

#### al-Ḥajjāj

#### al-Ḥanbalī

#### Hārūn al-Rashīd

#### al-Ḥaṣṣār

#### Hero

#### Hilbert, David

#### Hipparchus

## I

#### Ibn al-Hāʾim

#### Ibn al-Haytham

#### Ibn al-Khawwām

#### Ibn al-Maghribī

#### Ibn al-Nadīm

#### Ibn al-Samḥ

#### Ibn al-Yāsamīn

#### Ibn Badr

#### Ibn Fallūs

#### Ibn Ghāzī

#### Ibn Haydūr

#### Ibn Munʿim

#### Ibn Qunfudh

#### Ibn Sīnā

#### Ibn Turk

#### Ikhwān al-Ṣafāʾ

#### Isḥāq ibn Ḥunayn

## K

#### al-Karajī

#### al-Kāshī

#### al-Khayyām

#### al-Khwārazmī

#### al-Khwārazmī (lexicographer)

#### Kūshyār ibn Labbān

## M

#### Madyan

#### al-Māhānī

#### al-Maʾmūn

#### al-Mawāḥidī

## N

#### Naṣīr al-Dīn al-Ṭūsī

#### Nesselmann, G.H.F.

#### Nicomachus

#### Nine Chapters

## Q

#### al-Qalaṣādī

#### al-Qurashī

#### Qusṭā ibn Lūqā

## S

#### Saidan, A.S.

#### al-Samawʾal

#### Sebokht, Severus

#### Şeker Zāde

#### Sibṭ al-Māridīnī

#### Stifel, Michael

## U

#### Ulugh Beg

#### al-ʿUqbānī

#### al-Uqlīdisī