A working vocabulary in Arabic.

Completion, restoration. The operation of moving a subtracted term to the other side of an equation to make it positive, adding the same quantity to both sides so nothing negative remains. One of the two moves in the title of al-Khwārizmī's al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa'l-muqābala, and the ultimate source of the English word algebra.

Balancing, confrontation. The second operation in al-Khwārizmī's title, cancelling like terms that appear on both sides of an equation, or subtracting equal quantities from both sides. Together with al-jabr it defines the reduction procedure that turns any quadratic into one of six standard forms.

Calculation, arithmetic, reckoning. The general word for computational mathematics in Arabic. It appears in the full title of al-Khwārizmī's algebra treatise (fī ḥisāb, "in the calculation of") and in the name of many later works, including al-Karajī's al-Kāfī fī'l-ḥisāb ("The Sufficient on Calculation").

Geometry; engineering. The Arabic word for geometry, borrowed from Middle Persian handāz- ("to measure, allot, reckon"). In classical usage handasa covered both the pure discipline of geometry, as in Ibn al-Haytham's optical proofs, and the practical arts of measurement and construction.

Root. The unknown quantity x in al-Khwārizmī's algebra, literally the botanical root, from which the "root of an equation" gets its name in every European language. Contemporary sources sometimes use shayʾ ("thing") for the same idea; both terms appear side by side in early Arabic algebra.

Wealth, possession, the square of the unknown. Literally "wealth" or "an amount of money", but in algebra the square of the root, i.e. x². Historians of mathematics note the double meaning is deliberate: early Arabic algebra grew out of arithmetical problems about amounts of money and their square roots, and the older meaning survives inside the algebraic one.

Thing, an unknown. The other Arabic word for the unknown quantity in an equation, alongside jidhr. Translated into medieval Latin as res ("thing") and into Italian as cosa, giving rise to the Renaissance term for early symbolic algebra: l'arte della cosa, the "art of the thing".

Cube. The third power of the unknown, x³, and the geometrical cube. Sesiano's edition of a tenth century Arabic algebraic compendium gives the standard sequence: jidhr (root, x), māl (square, x²), then mukaʿʿab, "the product of the last two". The word itself is related to kaʿb ("heel, ankle bone, die"), the shape of ancient dice cut from animal bones.

Number. The general word for a numerical quantity. In al-Khwārizmī's algebra, the equations are built from three kinds of objects: units (ʿadad), roots (jidhr) and squares (māl). Plural aʿdād (أعداد).

Angle; corner. The Arabic word for a geometric angle, from the root z-w-y ("to draw together, gather"). Modern usage keeps the mathematical senses right next to the everyday one, zāwiya ḥādda is an acute angle, zāwiya qāʾima is a right angle, and a zāwiya is also just the corner of a room.

Circle. From the verb dāra ("to revolve, turn around"), literally "that which turns". The word describes a closed circuit in motion, fitting the Greek geometric tradition extended by al-Bīrūnī, Ibn al-Haytham, and others.

Triangle. Built on the root th l th ("three"), the word literally means "a shape with three sides". The pattern extends to murabbaʿ (مربع, quadrilateral, from rubʿ, "four"), musaddas (مسدس, hexagon), and muthamman (مثمن, octagon). Arabic mathematical vocabulary makes the numeric structure of shapes unusually clear.