# Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?

By "algebra", I'm assuming the solving of polynomial equations like every school child does. I accept that many kids loathe algebra, and that solving equations can be dreary and weariful, especially if you have to derive the Cubic Formula! But I don't understand the diction. What are the "broken parts"? What's being reunited?

### algebra [16]

Algebra symbolizes the debt of Western culture to Arab mathematics, but ironically when it first entered the English language it was used as a term for the setting of broken bones, and even sometimes for the fractures themselves (‘The helpes of Algebra and of dislocations’, Robert Copland, Formulary of Guydo in surgery 1541). This reflects the original literal meaning of the Arabic term al jebr, ‘the reuniting of broken parts’, from the verb jabara ‘reunite’. The anatomical connotations of this were adopted when the word was borrowed, as algebra, into Spanish, Italian, and medieval Latin, from one or other of which English acquired it. In Arabic, however, it had long been applied to the solving of algebraic equations (the full Arabic expression was ’ilm aljebr wa’lmuqābalah ‘the science of reunion and equation’, and the mathematician al-Khwarizmi used aljebr as the title of his treatise on algebra – see ALGORITHM), and by the end of the 16th century this was firmly established as the central meaning of algebra in English.

Word Origins (2005 2e) by John Ayto, p 16 Left column.

• The standard interpretation is given in Wikipedia with reference to Boyer and Oaks-Alkhateeb:"The word 'al-jabr' presumably meant something like 'restoration' or 'completion' and seems to refer to the transposition of subtracted terms to the other side of an equation; the word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation." Mar 16 at 9:31
• How come Boyer and Oaks-Alkhateeb version is the "standard" interpretation? It seems as speculative as Gandz. Mar 16 at 14:27
• @M.Farooq At least, it is based on how al-Khwarizmi actually uses the words rather than "Egyptians knew and wrote books on algebra as early as 1600 B.C." and "it would be very strange if the Assyrians... were quite ignorant of this art". Mar 17 at 0:25

See Roshdi Rashed, Classical Mathematics from Al-Khwarizmi to Descartes (Routledge, 2014), page 107 [but I've not seen: R. Rashed (editor), Al-Khwārizmī: The Beginnings of Algebra (2009)]:

The term al-jabr (algebra) is indeed an Arabic term, a name for the action of the verb (a maṣdar, according to the grammarians), the root of which has the general meaning of rectifying or correcting something using some form of constraint, such as setting a broken bone, for example. It is thus an ordinary-language term that can take on multiple meanings. It was devoid of technical meaning before al-Khwārizmī for the first time gave it a two-fold technical signification. When associated with the word almuqābala, it designates both a discipline and an operation. The successors of al-Khwārizmī will quickly give pride of place to the first word, ‘algebra’, to name the discipline, and derive from this single word the name of the professional, ‘algebraist’. This usage already appears in Thābit ibn Qurra (826–901). But this word also designates an operation: that of ‘restoring’ an equation, that is, adding to its two members the subtracted terms. For example, in

$$x^2 + c – bx = d$$ where $$c > d$$,

the operation ‘algebra’ consists in adding $$bx$$ to each side,

$$x^2 + c = bx + d$$,

and the operation al-muqābala, that is, ‘opposition’ or ‘reduction’ amounts to

$$x^2 + (c – d) = bx$$.

The point of the two related operations is to bring the equation back to one of the canonical types that al-Khwārizmī defined a priori.

The word jabr (root jubara) has a number of meanings in ordinary Arabic, two of which are relevant here. The first is “to reduce a fracture”, while the second, which is more general, is “to force, to compel”. The traditional interpretation has connected jabr with the fist meanings for two main reasons. [...] We believe, on the contrary, that the root jabara was employed by the medieval algebraists in its second sense, “to compel”.

Hence the science al-jabr wa’l-muqabalah is the science by which one forces the unknown to acquire certain values and checks the appropriate value by comparing the results with the original conditions of the problem.

• What is the meaning of $almuqābala$? I don't know Arabic but it seems like it means "a comparison." Gandz also cites some German scholars on the meaning of this term. Mar 16 at 15:47
• @M. Farooq According to online dictionaries of unknown quality, the modern meaning of muqabala is "meeting, encounter, interview", but also "opposition", said to derived from a root meaning "placing face to face". Mar 16 at 18:24
• I will favor the meaning "comparison" in this context. The German scholars in Gandz's paper suggested Gegenuberstellung for $almuqabala$, which is (perhaps) placing face to face or placing against each other. Mar 16 at 18:34
• @M. Farooq German "Gegenüberstellung" can mean either confrontation or comparison (as a special case, it can also refer to a police lineup). Mar 17 at 7:49

I have also wondered about the term al-jabr which ultimately became algebra. Re Please see Solomon Gandz (1926) The Origin of the Term “Algebra”, The American Mathematical Monthly, 33:9, 437-440, DOI: 10.1080/00029890.1926.11986615.

The author proposes an alternative and exactly notes your concern that al-jabr meant joining the broken parts. Gandz states that this word is not of Arabic origin and most likely it is Assyrian word.

here are still remnants in the mathematical literature suggesting that in olden times the term al-jabr alone was used for the science of equations, and the term al-jabriyyun was taken for the masters of algebra. $${ }^{1}$$ On the other hand the term al-muqabalah alone, according to its real meaning of "putting face to face, confronting, equation,' seems to be the most appropriate name for equations in general. With these difficulties in mind, the writer undertook to search out the real meaning of jabara in the related Semitic languages. Now the Assyrian name gabr\hatu-mahâru means to be equal, to correspond, to confront, or to put two things face to face; see Delitzsch, Assyrisches Handwöterbuch, under gabru and maharu, pp. $$193,401,$$ and Muss-Arnolt, Assyrian Dictionary, under gabru and maxaru, $$^{2}$$ pp. $$210,525 .$$ From the first of these we have the etymology of the Hebrew geber and gibbor. Geber is the mature man leaving the state of boyhood and being equal in rank and ralue to the other men of the assembly or army. Gibbôr is the hero who is strong enough to fight and overcome his equals and rivals in the hostile army. Gabara $$=j$$ abar $$a$$, in its original Assyrian meaning, is therefore the corresponding name for the Arabic qabala (verbal noun muqabalah), and an appropriate name for equations in general. The Egyptians $$^{3}$$ knew and wrote books on algebra as early as 1600 B.C., and it would be very strange if the Assyrians, having the same level of culture as the Egyptians and having close political and economic relations with them, were quite ignorant of this art. Gabr must have been the original Assyrian form of the word. The Arabs received this ancient science, with its original Assyrian name (in Arabic pronunciation al-jabr) from the Aramaeans and Syrians, who lived on Assyrian territory, and added the Arabic name al-muqabalah, which is nothing else than the literal Arabic translation of $$a l$$ jabr. This took place many hundreds of years before Mohammed ibn Musâ al-Kowârizmi. Later on the real meaning of the word was forgotten, and the simple meaning seems not to have been regarded as scholarly enough for good usage. The scholastic method at that time was common in both the philosophical and the theological schools. The scholars tried to find in the Bible, the Koran, and the old philosophical texts everything but the simple, plain meaning. The same method was followed towards these two mathematical terms. The masters simply declared them to signify the first two operations of algebra, namely the removal of the negative and positive quantities, without worrying much about philological reasons. This conception of the two words was already well known at the time of Mohammed ibn Musâ al Khowârizmî, and the latter used the terms in the traditional way without any further explanation, as shown by both Rosen and Ruska. For more than a thousand years this scholastic interpretation prevailed in the Arabic and European worlds. In reality, however, it would seem that the expression 'Ilm al-jabr w'al-muqabalah ought to be rendered simply as Science of equations, al-jabr being the Assyrian and al-muqabalah the Arabic name for equation.

His version seems plausible.

• The "scholastic interpretation" Gandz discusses is no longer extant, and his own etymological speculations are rather outdated as well. Mar 16 at 9:35
• Setting the bones (separate terms of a polynomial) together to order/align for a zero sum. Apr 16 at 20:20

I published an article on the meaning of the terms al-jabr and al-muqabala in Arabic algebra, which you can download from my Academia.edu page: "Simplifying equations in Arabic algebra" Historia Mathematica 34 (2007), 45-61. To be brief, in an equation that we would write as 10 - x = 4x, the "10 - x" (ten dirhams less a thing") was regarded as a deficient 10. One must "restore" it (al-jabr means "to restore") to a full 10, and then also add x ("a thing" to the other side to balance the equation.

Around 820 CE, the polymath Muhammed Ibn Musa al-Khwarizmi published his book, al-Kitab al-Mukhtasar fi Hisab al-Jabr wal Muqabalah in Abbasid Baghdad. This was translated into Latin as Liber Algebrae er Almucabola and was referred to as al-Jabr. It's full title is translated as The Encyclopedic Treatise on Calculation by Balancing and Completion.

The term al-jabr refers to the balancing where terms in an algebraic identity are transposed from one side to the other as is taught in many schools, even now. Presumably the term al-jabr, in Abbasid Baghdadi Arabic meant balancing in some sense and thos was transposed to the mathematical setting of al-jabr's treatise.

• Could you provide a reference that shows al-jabr meant balancing? Mar 18 at 2:48
• @M. Farooq: Have a look at the wikipedia article on al-jabr; this is where the translation of the title was provided and where it said that balancing was the correct translation of the term. Mar 18 at 2:51
• Mozibur Ullah, Wikipedia is not a scholarly source. The etymology section does cite any reference which talks about balancing. en.wikipedia.org/wiki/Algebra Mar 18 at 2:58
• @M. Farooq: Wikipedia has been quoted by responsible sources and so there is nothing wrong and everything right to take it to be a reference source. Mar 18 at 3:02
• @M. Farooq: I said to look at al-Jabr ... (and not algebra). Mar 18 at 3:05