The text provides some information about the mathematician Ibn Turk, a contemporary of al-Khwarizmi. The latter is considered as the founder of algebra; but the former wrote a book on algebra, too. The problem of priority about algebra goes back at least to the beginning of the Xth century.

1. 1
AL-KHWARIZMI, IBN TURK AND ALGEBRA: REVISITING A
CONTROVERSY OVER A PRIORITY OF DISCOVERY.
Driss Lamrabet
(Rabat, Morocco)
Keywords: History of mathematics - al-Khwārizmī- – Ibn Turk - Abū Kāmil – Abū Barza
– Algebraic equations – Central Asia
INTRODUCTION
Here we briefly recall the origin of the controversy concerning the authorship of algebra
and report some pieces of information concerning Ibn Turk (and his grandson Abū Barza),
pieces gathered from various sources others than the unique two sources (namely Ibn al-
Nadīm’s al-Fihrist, Xth century and al-Qifṭī’s Ikhbār al-Ulama’, XIIIth century)
mentioned by the historians of science who were interested in Ibn Turk.
We have indeed found a mention of this scholar in a historian who had personal contact
with him and who reports in his book events according to him: it is al-Baladhurī (d.
279H/892). Ibn Turk’s grandson, on the other hand, is quoted even more frequently by
many historians as will be seen further.
In the light of these elements which appear to be new, we will try to somewhat rehabilitate
Abū Barza, to do him justice and to clear him of the slanders raised against him (and his
ancestor) by Abū Kāmil, this illustrious mathematician whose great reputation has certainly
given his judgments the status of truth and maybe helped to overshadow Ibn Turk’s
mathematical contributions in favour of al-Khwārizmī’s.
Testimonies supporting Abū Barza have intentionally been multiplied below in order to
better situate his place in the community of scholars and thus to better judge the credibility
of what he might claim, and finally to provide information on the period during which Ibn
Turk lived and on his intellectual concerns. All this may, if not confirm, at least corroborate
certain assumptions about this mathematician and his work.
I BIOGRAPHICAL PRECISIONS
The origin of the controversy of priority in question is the statement made by the well-
known Egyptian mathematician Abū Kāmil Shujā‘ bin Aslam (around 340H/951) in the
introduction of his book of algebra1
and in another of his works (still lost) as reported by
Hajjī Khalifa in Kash azh-zhunūn2
([24]). The latter relates that according to Abū Kāmil,
1
In this introduction, Abū Kāmil praises al-Khwārizmī at length and urges mathematicians to recognize him
as the creator of algebra and the one thanks to whom this discipline became clear and accessible.
2
"‫كامل‬ ‫أبو‬ ‫قال‬‫بن‬ ‫شجاع‬‫(كتاب‬ ‫في‬ ‫أسلم‬)‫والمقابلة‬ ‫بالجبر‬ ‫الوصايا‬:‫معروفا‬ ‫كتابا‬ ‫ألفت‬:()‫أصوله‬ ‫في‬ ‫والزيادة‬ ‫وتمامه‬ ‫الجبر‬ ‫بكمال‬‫وأقمت‬
‫كتابي‬ ‫في‬ ‫الحجة‬‫والمقابلة‬ ‫الجبر‬ ‫في‬ ‫والسبق‬ ‫بالتقدمة‬ ‫الثاني‬‫موسى‬ ‫بن‬ ‫لمحمد‬‫على‬ ‫والرد‬‫المحترق‬[]‫المتخرق‬‫المعروف‬‫ينسب‬ ‫مما‬ ‫برزة‬ ‫بأبي‬

2. 2
Abū Barza (d. 298H/910)3
allegedly claimed that the creation of algebra was not the work
of al-Khwārizmī, but of his grandfather, a contemporary of al-Khwārizmī called Abū al-
Fadl ‘Abd-ul-Ḥamīd bin Wāsi‘ Ibn Turk al-Khuttalī4
. Abū Kāmil took a stand in favour of
al-Khwārizmī and severely denigrated Abū Barza's allegations. Apart from Abū Kāmil, the
question is raised by none of the usual sources, except for the rallying of Ibn Khaldūn to
Abū Kāmil's opinion.
The oldest source that provides some information about Ibn Turk is Ibn an-Nādīm's Al-
Fihrist (10th century). This was then reproduced by al-Qiftī (13th century), Hajji Khalifa
(17th century), and Ismā‘il Bāchā al-Baghdādī (20th century). It only quotes the
mathematical contributions and does not mention any other field of activity or date. Further
investigation has lead us to glean additional bits of information. We have indeed found a
direct reference to Ibn Turk made by one of his contemporaries who seems to have been
one of his disciples. This is the historian Al-Balādhurī who, according to the sources, sent
a panegyric to Caliph al-Māmūn in his youth in 218H/833, and lived until around
279H/892.5
Despite the inconvenience that can result, below are reproduced - as footnotes - some of
the few passages that quote Ibn Turk or his grandson Abū Barza. The footnotes include the
reproduction of Arabic passages followed by approximate English translations.
Ibn Turk seems to have been interested in history, since he is quoted by al-Balādhurī in
three chains of transmission. According to this historian, Ibn Turk had been a disciple of
Yahyā b. Ādam6
(d. 203H/818). A late biobibliography (Ismā'īl Bāchā's Hadiyyat al-'arifīn
‫الذي‬ ‫الحميد‬ ‫عبد‬ ‫إلى‬‫جده‬ ‫أنه‬ ‫ذكر‬‫جده‬ ‫إلى‬ ‫نسب‬ ‫فيما‬ ‫معرفته‬ ‫وقلة‬ ‫تقصيره‬ ‫وبينت‬”(‫خليفة‬ ‫حجي‬:""‫الظنون‬ ‫كشف‬[24],‫ج‬2‫ص‬1408)
.
"Abū Kāmil Shujā b. Aslam said in Kitāb al-wasāyā bi al-jabr wa al-muqābalh : I wrote a book known as:
Kamal al-Jabr wa tamāmuh wa al-ziyāda fī usūlih and in the latter, I provided the proof of Muhammad bin
Musā's priority and primacy in algebra, and the answer to the insane [al-mutakharriq] known as Abū Barza
regarding what he attributed to ‘Abd-ul-Ḥamīd, whom he claimed that he was his grandfather; I gave
arguments proving his shortcomings and his lack of knowledge about what he attributed to his grandfather".
(Hajjī Khalifa in Kash azh-zhunūn [24], vol 2, p. 1407).
Note: according to the editions, we find ‫المحترف‬ (the professional), ‫المحترق‬ (which burns, which catches on
fire) or ‫.المتخرق‬ The two first terms are visibly unrelated to the context.
3
A date reported by ibn Kathīr, al-Dhahabī and others. Ibn al-Jawzī in al-Muntazham provides 292H/905.
4
Al-Qiftī gives al-Gīlī, whereas in all the edited old sources such as al-Balādhurī’s history (d. 279H/892)
and Ibn al-Nādīm's repertory (Xth century), we find al-Khuttalī (from Khuttal, area of the former province
of Khorāsān bordering the river Amou Daria); the preference for al-Gīlī reported by Sayli in [8] is thus
debatable.
5
He was also a disciple of the mathematician Al-Khwārizmī (whom he calls Muhammad. b. Mūsā al-
Khwārizmī al-Ḥāsib), as he reports information he says he heard from this scholar (twice in Ansāb al-ashrāf,
[17], Volume 4, pp. 274 and 349).
6
For instance:
-"‫البص‬ ‫إلى‬ ‫عائشة‬ ‫مسير‬ ‫ذكروا‬ :‫قال‬ ‫مجاهد‬ ‫عن‬ ،‫رجل‬ ‫عن‬ ،‫سفيان‬ ‫عن‬ ،‫آدم‬ ‫بن‬ ‫يحيى‬ ‫حدثني‬ ،‫الحاسب‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫حدثني‬:‫فقال‬ ،‫رة‬
‫وكا‬ ،‫إليه‬ ‫وسلم‬ ‫عليه‬ ‫هللا‬ ‫صلى‬ ‫النبي‬ ‫نساء‬ ّ‫أحب‬ ‫فإنها‬ ‫هذا‬ ‫ومع‬ ،‫اإلحسان‬ ‫من‬ ‫وتأخر‬ ‫لها‬ ‫تقدم‬ ‫ما‬ ‫مبطل‬ ‫وال‬ ،‫البارع‬ ‫فضلها‬ ‫بمذهب‬ ‫ذلك‬ ‫ليس‬‫نت‬
( ."ّ‫حب‬ ‫من‬ ‫مع‬ ‫وكل‬ ‫له؛‬ ً‫ا‬َّ‫ب‬‫ح‬ ‫ّهن‬‫د‬‫أش‬‫البالذري‬[ "‫األشراف‬ ‫أنساب‬ ‫من‬ ‫مل‬ُ‫ج‬":17‫ج‬ ، )‫أ‬ ]2‫ص‬48)
-" ‘Abd-ul-Ḥamīd bin Wasi‘ the arithmetician related to me: Yaḥya bin Ādam related to me, from Sufiān,
from some man, from Mujāhid that: They talked about ‘Aisha's departure to Baṣṣorah then he said: that will

3. 3
[16]) provides 240H/854 as date of Ibn Turk's death, indicating that its source is al-Fihrist,
but the known editions of al-Fihrist give no date. However, such a date seems acceptable
given the information provided by al-Balādhurī. It is close to that proposed as approximate
(235H/848) by Sayli ([8]). Thus, Ibn Turk and al-Khwārizmī lived at the same epoch and
were intellectually mature under the reign of al-Māmūn. Al-Khwārizmī was one of the
scholars close to the caliphs and was an influential member of Bayt al-Hikma (House of
Wisdom), factors that probably played in favor of the spread of his algebra at the expense
of Ibn Turk's, since nothing is known about the cultural and scientific environment in which
the latter lived, except that he dwelt (as did al-Khwārizmī and Abū Barza) in Baghdad.
Ibn an-Nādīm relates in al-Fihrist that Ibn Turk is called Abū-l-Fadl (or Abū Muhammad)
‘Abd-ul-Ḥamīd ben Wāsi‘ Ibn Turk al-Khuttalī the specialist of calculation / the
arithmetician (al-hāsib) and cites among his books: the Comprehensive Book in
Calculation, which includes six books (chapters), the Book of Transactions. [15]) .7
not remove her excellent merit nor cancel her past and future beneficence. Furthermore she was the most
beloved among the Prophet's (pbuh) women, and she beloved him far more than any of them; everybody
[stays] with whom he cherishes. " Al-Baladhūrī: Jumal min Ansāb al-ashrāf [17] ‫,)أ‬ vol 2 p 48)
‫﴿فقوال‬ :‫قرأ‬ ‫ثم‬ ،‫وضررها‬ ‫ونفعها‬ ‫ودواؤها‬ ‫النفوس‬ ‫شفاء‬ ‫به‬ ‫كالكالم‬ ‫رأيت‬ ‫ما‬ :‫قال‬ ‫الثوري‬ ‫أن‬ ‫آدم‬ ‫بن‬ ‫يحيى‬ ‫عن‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫"وحدثني‬‫له‬
" ﴾‫يخشى‬ ‫أو‬ ‫يتذكر‬ ‫لعله‬ ً‫ا‬‫ّن‬‫ي‬‫ل‬ ً‫ال‬‫قو‬(‫ج‬ ، ‫السابق‬ ‫المصدر‬11‫ص‬321)
" ‘Abd-ul-Ḥamīd bin Wasi‘ related to me from Yaḥya bin Ādam that al-Thawrī had said: I have seen nothing
like talking: it conveys the healing of souls and their medicine, their benefit and harm. Then he recited
[Coran, Taha, verse 44]:﴾ Speak to him with gentle words; perhaps he will ponder or fear﴿. (Ibid., vol.11, p.
321)
-‫حدثنى‬ :‫قال‬ ‫الحاسب‬ ‫الختلى‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫"حدثنى‬،‫المدائن‬ ‫مسجد‬ ‫بالسواد‬ ‫بنى‬ ‫جامع‬ ‫مسجد‬ ‫أول‬ :‫قال‬ ‫صالح‬ ‫بن‬ ‫الحسن‬ ‫عن‬ ،‫آدم‬ ‫بن‬ ‫يحي‬
[ ‫(البالذري‬ ".‫اليمان‬ ‫ابن‬ ‫حذيفة‬ ‫يدي‬ ‫على‬ ‫ذلك‬ ‫وجرى‬ ،‫بناؤه‬ ‫وأحكم‬ ‫بعد‬ ‫وسع‬ ‫ثم‬ ،‫وأصحابه‬ ‫سعد‬ ‫بناه‬17])‫ب‬"‫البلدان‬ ‫"فتوح‬ ،‫الجزء‬ ،2‫ص‬
407)
-" ‘Abd-ul-Ḥamīd bin Wasi‘ al-Khuttalī the Specialist in Calculation related to me: Yaḥya bin Ādam related
to me that Ḥassan bin Ṣaliḥ had said: the first great mosque built in al-Sawād was al-Madā'in's Mosque,
build by Sa‘ad and his companions; it was afterwards broadened and its construction perfected thanks to
Ḥudhayfa ibn al-Yamān". (Al-Baladhūrī:. [17] )‫ب‬ ،Futūḥ al-buldān, vol 2, p. 407).
7
-‫وله‬ .‫محمد‬ ‫ابا‬ ‫يكنى‬ ‫وقيل‬ .‫الحاسب‬ ‫الختلى‬ ‫ترك‬ ‫بن‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫الفضل‬ ‫أبو‬ ‫وهو‬ ‫الحميد‬ ‫عبد‬ "،‫الحساب‬ ‫في‬ ‫الجامع‬ ‫كتاب‬ ،‫الكتب‬ ‫من‬
،‫النديم‬ ‫(ابن‬ ".‫العالمات‬ ‫كتاب‬ .‫كتب‬ ‫ستة‬ ‫على‬ ‫ويحتوى‬[15]‫ص‬391)
- " ‘Abd-ul-Ḥamīd: he is Abū al-Faḍl ‘Abd-ul-Ḥamīd b. Wāsi‘ b. Turk al-Khuttlī al-Ḥasib; some report that
his surname is Abū Muḥammad. He has among his books: Kitāb al-Jāmi‘ fī al-ḥisāb, in six chapters, Kitāb
al-mu‘āmalāt. " (Ibn al-Nadīm, [15], p. 391)
-‫ويكنى‬ ‫الجيلي‬ ‫ترك‬ ‫بابن‬ ‫ويعرف‬ ‫أهلها‬ َ‫ْن‬‫ي‬َ‫ب‬ ‫مذكور‬ ‫ا‬َ‫ه‬‫ِي‬‫ف‬ ‫مقدم‬ ‫الحساب‬ ‫بصناعة‬ ‫عالم‬ ‫حاسب‬ ‫رجل‬ ‫ا‬َ‫ذ‬َ‫ه‬ ‫الفضل‬ ‫أبو‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫"عبد‬‫محمد‬ ‫أبا‬
‫وخواص‬ ‫الحساب‬ ‫نوادر‬ ‫كتاب‬ .‫كتب‬ ‫ستة‬ ‫ى‬َ‫ل‬َ‫ع‬ ‫يحتوي‬ ‫الحساب‬ ‫ِي‬‫ف‬ ‫الجامع‬ ‫كتاب‬ :‫منها‬ ‫مستعملة‬ ‫مشهورة‬ ‫تصانيف‬ ‫الحساب‬ ‫ِي‬‫ف‬ ُ‫ه‬َ‫ل‬ .ً‫ا‬‫أيض‬
.‫األعداد‬‫(القفطي‬ "،[14‫ص‬ ]155.)
" ‘Abd-ul-Ḥamīd b. Wāsi‘, Abū al-Faḍl: This man is an arithmetician, erudite in the art of arithmetic,
advanced in it and renowned among its professionals; he is known as Ibn Turk al-Jīlī and has also Abū
Muḥammad as surname. He is the author of famous and widespread books in mathematics including: Kitāb
al-Jāmi‘ fī al-ḥisāb containing six chapters, Kitāb nawādir al-ḥisāb wa khawāṣ al-a‘dād. (al-Qifṭī, [14] p.
155)
-‫سنة‬ ‫توفي‬ ‫الحاسب‬ ‫البغدادي‬ ‫الختلي‬ ‫ترك‬ ‫بن‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ :‫"الختلي‬240‫كتاب‬ .‫الحساب‬ ‫في‬ ‫الجامع‬ ‫كتاب‬ ‫صنف‬ ‫ومائتين‬ ‫أربعين‬
[ ،‫باشا‬ ‫(إسماعيل‬ " )‫الفهرست‬ ‫(من‬ ,‫المعامالت‬16‫ج‬ ،]1‫ص‬506)

4. 4
Regarding Abū Barza, Ibn an-Nadīm calls him Abū Barza Al-Fadl bin Muhammad bien
‘Abd-ul-Ḥamīd ben Turk ben Wāsi‘ al-Khuttalī, and cites among his books: the Book of
Transactions, the Book of Surveying8
.
. Al-Qiftī ([14]) provides more detail on Ibn Turk, whom he presents as an eminent
arithmetician (hāsib) renowned among the people of this art and author of famous books
on arithmetic to which people resorted, including: the Comprehensive Book in Calculation
involving six books (chapters), the Book of Rare Things in Calculation and the Properties
of Numbers. He points out that he is known as Ibn Turk al-Jīlī. However, the list of works
cited differs from Ibn an-Nadīm's one, since it does not include the Book of Transactions
and includes the Book of Rare Things.
Al-Qiftī ([14]) provides more details on Ibn Turk, whom he presents as an eminent
arithmetician (hāsib) renowned among the people of this art and author of famous books
on arithmetic sought for by people, including: the Comprehensive Book in Calculation
involving six books (chapters), the Book of Rare Things in Calculation and the Properties
of Numbers. He points out that he is known as Ibn Turk al-Jīlī. However, the list of works
cited differs from Ibn an-Nadīm's one, since it does not include the Book of Transactions
and includes the Book of Rare Things.
-"al-Khuttalī: ‘Abd-ul-Ḥamīd b. Wāsi‘ b. Turk al-Khuttlī al-Baghdādī al-ḥāsib; he died in 240 H. He wrote:
Kitāb al-Jāmi‘ fī al-ḥisāb, in six chapters, Kitāb al-mu‘āmalāt (from al-Fihrist) (Ismā‘īl Bāshā, [16] ،vol. 1
p. 506).
8
-‫النديم‬ ‫(ابن‬ ".‫المساحة‬ ‫المعامالت،كتاب‬ ‫كتاب‬ ،‫الكتب‬ ‫من‬ ‫وله‬ .‫الختلى‬ ‫واسع‬ ‫بن‬ ‫ترك‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫بن‬ ‫محمد‬ ‫بن‬ ‫الفضل‬ ‫برزة‬ ‫"أبو‬[15]‫ص‬
391)
- " Abū Barza al-Faḍl bin Muḥammad bin ‘Abd-ul-Ḥamīd bin Turk bin Wāsi‘ al-Khuttalī. He has as books:
Kitāb al-mu‘āmalāt, kitāb al-misāḥa" (Ibn al-Nadīm, [15] p. 391)
-،‫مفيدة‬ ‫كتبا‬ ‫ذلك‬ ‫في‬ ‫مصنف‬ ،‫ألجلها‬ ‫مقصود‬ ‫فيها‬ ‫مقدم‬ ‫الحساب‬ ‫بصناعة‬ ‫عالم‬ :،‫الجيلي‬ ‫برزة‬ ‫،أبو‬ ‫واسع‬ ‫بن‬ ‫الحميد‬ ‫عبد‬ ‫بن‬ ‫محمد‬ ‫بن‬ ‫"الفضل‬
( "‫المساحة‬ ‫كتاب‬ ،‫المعامالت‬ ‫كتاب‬ :‫منها‬[، ‫القفطي‬14]‫ص‬ ،168).
- " al-Faḍl bin Muḥammad bin ‘Abd-ul-Ḥamīd bin Wāsi, Abū Barza al-Jīlī: an erudite in the art of calculation
in which he was excellent and for which he was requested and on which he wrote useful books including:
Kitāb al-mu‘āmalāt, kitāb al-misāḥa ". (al-Qifṭī, [14] p. 168).

5. 5
Concerning Abū Barza, Al-Qiftī reports that this scholar is called Al-Fadl bin Muhammad
bin ‘Abd-ul-Ḥamīd bin Wāsi‘, Abū Barza Al-Jīlī (instead of al-Khuttalī, and without
adding Ibn Turk), points out that he is an eminent scholar in calculation and that people
flocked to him for this discipline in which he composed interesting books including: the
Book of Transactions and the Book of Surveying. In the chains of transmissions of the
Hadith, he is called Abū Barza Al-Fadl bin Muhammad al-Ḥāsib 9
.
Abū Barza is consequently Ibn Turk's grandson, and this fact is no longer merely
hypothetical as reported by Abū Kāmil.
He is also mentioned in several transmission chains. He must have died quite old, since
among his teachers there were two well-known narrators of the Hadith: Ahmad bin ‘Abd
Allāh bin Younes at-Tamīmī al-Kūfī, who died in 227H/842 and Abū ‘Abd Allāh
Muhammad ben Samā‘a who died in 233H/847. He was to be around 15 years old in
227H/842 as a student and would therefore have been born around 212H/827 (he died in
298H/910); the date of birth of his grandfather Ibn Turk could be around 170H/786, and
the grandfather may have seen his grandson as an adult.
In addition to being proven specialists of calculation, the only aspect mentioned by Ibn an-
Nadīm and al-Qiftī, what precedes reveals that Abū Barza and his ancestor were also
credible narrators ‫رواة‬ (ruwāt) of the Hadith, especially the former. Far from the
disparaging image that Abū Kāmil wanted to give him, Abū Barza enjoyed excellent
reputation and great respect on the part of renowned historians, including al-Khatīb a-
Baghdādī (d. 463H/1072), As-Sam‘anī (d.562H/1167), Ibn al-Jawzī (d .597H/1201) and
Al-Dhahabī (d.748H/1347), as well as on the part of Hadith's specialists such as al-
Bayhaqī (d.458H/1066); all unanimously regarded him as a reliable and highly respectable
person10
. In addition, the two scholars are cited everywhere as scholars with expertise in
the science of calculation and as authorities in this field. Both are qualified as al-Hāsib
(the arithmetician/mathematician), a qualification that was bestowed only to those who
stood out in mathematics.
In light of this information about Abū Barza, his claims of priority for which Abū Kāmil
attacked and denigrated him should therefore be taken more seriously into consideration
and give him due together with his ancestor Ibn Turk.
II A COMMON SOURCE FOR AL-KHWARIZMI AND IBN TURK?
For a more profound analysis, please see Seyli’s paper cited below.
A fragment of Ibn Turk's algebra book was discovered and edited by Sayli in Istambul in
1962, and translated into English in 2007 ([5], i)). Although short, this fragment contains
enough elements to allow comparison. 11
The two works of algebra, that by al-Khwārizmī and that by Ibn Turk, have similarities and
differences, and both are not without raising questions.
Similarities include:
1) Regarding terminology: use of the same terms: shay', māl, jidhr,...
2) Methodologically: the existence of geometric justifications of algorithms in both works,
the similarity going as far as the use of almost identical geometric patterns for the fifth
case 2
x q px+ =
9
See for example :

6. 6
Abū Na’īm al-Ispahānī (d.430H/1038 ) : Ḥulyat al-‘awliyā’ wa ṭabaqāt al-‘aṣfiyā’, asl-
sa’āda, Le Caire, 1974, 7 : 255
Al-Bayhaqī (d.458H/1066) : Shu’ab al-īmān, maktabat al-rashād, Mumbay, Inde, 2003.
2 : 17 - 6 : 45 - 13 : 100
Al_Khaṭīb al-Baghdādī (d. 463H/1072): al-kifāya fī ‘ilm al-riwāya, al-maktaba al-
‘ilmiya, Medine, p. 347
Ismā’īl al-Ṭalīḥī al-Iṣbahānī (d.537H/1142 ): al-targhīb wa al-tarhīb, Dar al-ḥadīth, Le
Caire, 1993, 3 : 24.
Ibn Ḥajar al-Asqalānī (d.852H/1448): Taghlīq al-Ta’līq ‘alā Ṣaḥiḥ al-Bukhārī, al-
Maktab al-Islāmī, Ammān, 1405H, 2 : 55
10
-‫الرملي‬ ‫سماعه‬ ‫بن‬ ‫ومحمد‬ ‫الكوفيين‬ ‫الحماني‬ ‫ويحيى‬ ‫موسى‬ ‫بن‬ ‫وثابت‬ ‫يونس‬ ‫بن‬ ‫هللا‬ ‫عبد‬ ‫بن‬ ‫أحمد‬ ‫عن‬ ‫حدث‬ :‫الحاسب‬ ‫برزة‬ ‫أبو‬ ‫محمد‬ ‫بن‬ ‫"الفضل‬
‫يوسف‬ ‫بن‬ ‫محمد‬ ‫بن‬ ‫أحمد‬ ‫العباس‬ ‫وأبو‬ ‫ماسي‬ ‫بن‬ ‫محمد‬ ‫وأبو‬ ‫قانع‬ ‫بن‬ ‫الباقي‬ ‫عبد‬ ‫عنه‬ ‫روى‬ ‫القومسي‬ ‫حبيب‬ ‫بن‬ ‫ونوح‬ ‫األلهاني‬ ‫سليمان‬ ‫بن‬ ‫ومالك‬
.‫السقطي‬...‫ثقة‬ ‫وكان‬.‫جليل‬ ‫وهو‬ ‫لعمرى‬ ‫أي‬ :‫فقال‬ ‫ثقة؟‬ ‫أكان‬ :‫قلت‬ ‫الحاسب‬ ‫برزه‬ ‫أبي‬ ‫عن‬ ‫البرقاني‬ ‫بكر‬ ‫أبا‬ ‫سألت‬[، ‫البغدادي‬ ‫(الخطيب‬ ".20،]
‫ج‬ )‫أ‬14‫ص‬346-‫رقم‬ ‫الترجمة‬6770).
-" al-Faḍl bin Muḥammad, Abū Barza al-ḥāsib: He narrated from Aḥmad bin ‘Abd Allāh bin Yūnus, Thābit
bin Mūsā, Yaḥyā al-Ḥimmānī [all of them ] Kūfan, Muḥammad bin Samā‘a al-Ramlī, Mālik bin Sulaymān
al-Alhānī and Nūḥ bin Ḥabīb al-Qawmasī; narrated from him: ‘Abd al-Bāqī bin Qāni‘,Abū Muḥammad bin
Māsī Abū al-‘Abbās Aḥmad bin Muḥammad bin Yūsuf al-Saqaṭī. He was trustworthy … I asked Abū Bakr
al-Barqānī about Abū Barza al-ḥāsib: Was he trustworthy? He answered: Sure! By my life! He was
honourable, too." (al-Khaṭīb al-Baghdādī, [20] ‫)أ‬ vol. 14, P. 346- Biography No. 6770).
-"‫عبد‬ ‫عنه‬ ‫روى‬ ،‫وغيرهم‬ ‫الحماني‬ ‫بن‬ ‫ويحيى‬ ‫موسى‬ ‫بن‬ ‫وثابت‬ ‫يونس‬ ‫بن‬ ‫هللا‬ ‫عبد‬ ‫بن‬ ‫أحمد‬ ‫عن‬ ‫حدث‬ ،‫الحاسب‬ ‫محمد‬ ‫بن‬ ‫الفضل‬ ‫برزة‬ ‫أبو‬
‫صفر‬ ‫من‬ ‫بقين‬ ‫الربع‬ ‫ومات‬ ،‫صدوقا‬ ‫القدر‬ ‫جليل‬ ‫ثقة‬ ‫وكان‬ ،‫السقطي‬ ‫يوسف‬ ‫بن‬ ‫محمد‬ ‫بن‬ ‫أحمد‬ ‫العباس‬ ‫وأبو‬ ‫ماسي‬ ‫بن‬ ‫محمد‬ ‫وأبو‬ ‫قانع‬ ‫بن‬ ‫الباقي‬
‫و‬ ‫وتسعين‬ ‫ثماني‬ ‫سنة‬".‫مائتين‬[ ،‫(السمعاني‬22‫ج‬ ، ]2‫ص‬154).
-" Abū Barza al-Faḍl bin Muḥammad al-ḥāsib: He narrated from Aḥmad bin ‘Abd Allāh bin Yūnus, Thābit
bin Mūsā, Yaḥyā al-Ḥimmānī and others. Related from him: ‘Abd al-Bāqī bin Nāfi‘, Abū Muḥammad bin
Māsī and Abū al-‘Abbās Aḥmad bin Muḥammad bin Yūsuf al-Saqaṭī. He was trustworthy, highly honourable
and honest.
(Al-Sam‘ānī, [22], vol 2, p. 154).
-‫عن‬ ‫حدث‬ .‫الحاسب‬ ‫برزة‬ ‫أبو‬ ،‫محمد‬ ‫بن‬ ‫الفضل‬ "[( ."‫القدر‬ ‫جليل‬ ‫ثقة‬ ‫وكان‬ ،‫قانع‬ ‫بن‬ ‫الباقي‬ ‫عبد‬ ‫عنه‬ ‫روى‬ ‫الحماني‬ ‫يحيى‬13‫الجوزي‬ ‫ابن‬ ،]
‫ج‬13‫ص‬43.)
"- al-Faḍl bin Muḥammad, Abū Barza al-ḥāsib: He narrated from Yaḥyā al-Ḥimmānī ; ‘Abd al-Bāqī bin
Nāfi‘ narrated from him. He was trustworthy and highly honourable
( [13] Ibn al-Jawzī, vol. 13, p. 43).
-‫ابن‬ :‫وعنه‬ .‫سماعة‬ ‫بن‬ ‫ومحمد‬ ،‫الحماني‬ ‫ويحيى‬ ،‫اليربوعي‬ ‫يونس‬ ‫ابن‬ :‫عن‬ ‫روى‬ .‫بغداد‬ ‫حيسوب‬ ‫كان‬ .‫الحاسب‬ ‫برزة‬ ‫أبو‬ .‫محمد‬ ‫بن‬ ‫"الفضل‬
‫توفي‬ .‫ماسي‬ ‫بن‬ ‫محمد‬ ‫وأبو‬ ،‫السقطي‬ ‫محمد‬ ‫بن‬ ‫وأحمد‬ ،‫قانع‬[ ‫(الذهبي‬ ."‫الخطيب‬ ‫وثقه‬ .‫وتسعين‬ ‫ثمان‬ ‫سنة‬ ‫صفر‬ ‫في‬21‫،ج‬ ]22‫ص‬226).
"- al-Faḍl bin Muḥammad, Abū Barza al-ḥāsib: He was the arithmetician of Bagdād. He narrated from: Ibn
Yūnus al-Yarbū‘ī, Yaḥyā al-Ḥimmānī and Muḥammad bin Samā‘a; narrated from him: Ibn Nāfi‘ (sic) ,
Aḥmad bin Muḥammad al-Saqaṭī and Abū Muḥammad bin Māsī. He died in Ṣafar of the year [2]98. Al-
Khaṭīb established his trustworthiness".
(Al-Dhahabī, [21], vol. 22, p. 226).
11
For a more detailed comparative study of the algebra of these two mathematicians, see [2], [2b] and[5].

7. 7
In terms of differences, some authors agree that Ibn Turk's work is mathematically more
elaborate than al-Khwārizmī's one; in particular, the former considered two models for this
fifth case (one for x < p/2 and the other for x > p/2), while the latter was limited to one.
This superiority, without further evidence, led some authors (see for example [4]) to regard
Ibn Turk's contribution as a mere continuation of12
al-Khwarizmī's algebra.
Could the similarities between the works of the two scholars stem from the existence of an
algebraic tradition in Central Asia, where both originated? Hoyrup [3] distinguishes a sub-
scientific tradition (sub-scientific, represented by calculations related to commercial
transactions and practical geometry such as surveying) and another scientific conveyed in
particular by the works of mathematics and astronomy translated from Greek and Sanskrit.
According to him, the advent of algebra is linked to the first tradition. Hoyrup, [3] carries
out the analysis of algebra writings in several authors including Liber mensurationum, a
Latin translation of a book on surveying written by a certain Abū Bakr who seems to use
a sort of “archaic” algebra which can be related to an Old Babylonian or Seleucid tradition.
After a careful comparative study, Hoyrup conjectures that all of Abū Bakr, al-Khwārizmī
and Ibn Turk had to draw their algebra from the same tradition that was common in their
time among users of al-jabr (ahl al-jabr or asḥāb al-jabr : « al-jabr-people » or
« followers of al-jabr » according to Hoyrup’s terminology).
Among the questions raised by the possible existence of such a tradition in Central Asia is:
In which language was it written? Does the existence of a common Arabic terminology
among the two mathematicians mean that prior Arabization work had already been done?
Was Abū Kāmil, who said he learned algebra through al-Khwārizmī’s book, unaware of
the existence of such a tradition? Having subsequently read Ibn Turk's algebra and found
similarities, would he have cried plagiarism?
He points out the attitude of Abū Kāmil who seems to have made (deliberately?) a clean
sweep of practices other than those initiated by al-Khwārizmī:
“In Abu Kamil’s Algebra, the idea of a special group of al-jabr-people seems to have disappeared.
Instead, the subject, is now understood as the discipline of Al-Khwarizmi’s Kitab fi al-jabr wa’l-
muqabala” (Hoyrup, ii) , p. 263).
Then he concludes:
“So, we are led to the conclusion that both authors supplemented their treatise on the
methods of the “al-jabr-people” with material borrowed from another sub-scientific
tradition. They did so, however, from a conception of mathematics foreign to both sub-
scientific traditions (as far as it can be judged from the indirect evidence at hand), namely
from the idea that mathematics should be supplied with proofs. This, and not only the use
of letters to identify geometric entities and the way to explain the construction of a
geometric figure, was in the scientific mathematical tradition initiated by the Greeks. The
fundamental feat of the two authors was to bring the two levels of mathematical activity
together for mutual fructification. (ibid., p. 266)
12
Including Youschkevitch, ([10], 44), Djebbar ([1], p. 45), Katz et al., [2b], p.144, Hoyrup, [3]

8. 8
Note: Ibn an-Nādīm quotes another mathematician and astronomer, Sanad b. ‘Ali , a Jew
converted to Islam by Caliph al-Māmūn (and thus a contemporary of al-Khwārizmī and
Ibn Turk) and whose al-Fihrist reports that he composed a book of algebra (Kitāb al-jabr
wa-l-muqābalah), still lost.13
Bibliography
[1] -Djebbar; A.: L’algèbre arabe. Genèse d’un art. Vuibert-Adapt, Paris, 2005.
[2] -Hoyrup, J. :
i) “The Formation of Islamic Mathematics”: Sources and Conditions. Preprints og Reprints
1987 Nr 1. (available online)
ii) “Algebraic Traditions behind Ibn Turk and al-Khwārizmī'. In Acts of the International
Symposium on Ibn Turk, Khwarezmī, Farabī, Beyronī and Ibn Sīna
(Ankara 9-12 september 1985). Available online.
[2b] -Katz, Victor J. & Karen Hunger Parshall: Taming the Unknown. A History of
Algebra from Antiquity to the Earliest Twentieth Century. Princeton University Press,
2014.
[3] -Lamrabet, D. :
i) An introduction to the History of Maghrebian Mathematics (2020):
Published by Amazon
https://www.amazon.fr/Introduction-History-Maghrebian-Mathematics/dp/B084DG7LN3
French edition: Introduction à l’histoire des mathématiques maghrébines.
https://www.amazon.fr/INTRODUCTION-LHISTOIRE-MATHEMATIQUES-
MAGHREBINES-Lamrabet/dp/B084DGPNNY
ii) Divers aspects du progrès en mathématiques. Publications de la Faculté des Lettres et
des Sciences Humaines, Rabat ; Série : Colloques et Séminaires. N° 112, 2003.
[4] -Rashed, R.: i) Entre arithmétique et algèbre; Les Belles Lettres, Paris, 1984.
ii) Al-Khwārizmī, Le commencement de l’algèbre; Paris, Librairie Scientifique et Technique Albert
Blanchard ( Sciences dans l’histoire), 2007.
[5] -Sayli, A .:
i) Logical Necessities in Mixed Equations by ‘Abd-ul-Ḥamīd Ibn Turk and the Algebra of
his Time. Foundation for Science, Technology and Civilisation. Editor: Prof. Mohamed El
Gomati. Production: Amar Nazir. Jan. 2007.
13
Some authors (Suter, followed by A. Saidan) dispute the attribution of this book to Sanad by claiming that
it is in fact al-Khwārizmī's book of algebra. Their argument is that Ibn an-Nadīm presents the latter without
mentioning his algebra, just before introducing Sanad b. ‘Ali.

9. 9
https://muslimheritage.com/logical-necessities-in-mixed-equations-abd-al-hamid-ibn-
turk-and-the-algebra-of-his-time/
ii) Al-Khwarizmi, Abdu’l-Hamid Ibn Turk and the Place of Central Asia in the History of
Science and Cuture. Foundation for Science, Technology and Civilisation. Editor: Prof.
Mohamed El Gomati. Production: Amar Nazir. December 2006.
Was available at :
http://www.muslimheritage.com/uploads/Place_of_Central_Asia_in_History_of_Science
_and_Culture.pdf (June 2008).
[6] -Youchkevitch,A.P.: Les mathématiques arabes (VIII-XVe Siècles). Trad. par M.
Caznave et K. Jaouiche. Vrin, Paris, 1976.
https://fr.wikipedia.org/wiki/%27Abd_al-Ham%C4%ABd_ibn_Turk

10. 10
[12]-‫أصبيعة‬ ‫أبي‬ ‫ابن‬:‫األطباء‬ ‫طبقات‬ ‫في‬ ‫األنباء‬ ‫عيون‬،‫بيروت‬ ،‫رضا‬ ‫نزار‬ .1965.
[13]-‫الجوزي‬ ‫ابن‬:‫واألمم‬ ‫الملوك‬ ‫تاريخ‬ ‫في‬ ‫المنتظم‬(18‫بن‬ ‫محمد‬ ‫بن‬ ‫علي‬ ‫بن‬ ‫الرحمن‬ ‫عبد‬ ‫الفرج‬ ‫ألبي‬ ،) ‫جزءا‬
‫الجوزي‬‫سنة‬ ‫المتوفى‬597. ‫ه‬‫الجزء‬13،‫العلمية‬ ‫الكتب‬ ‫دار‬ .‫عطا‬ ‫القادر‬ ‫عبد‬ ‫ومصطفى‬ ‫عطا‬ ‫القادر‬ ‫عبد‬ ‫محمد‬ ‫تحقيق‬ .
.‫بيروت‬1412‫ه‬/1992
[14]-‫القفطي‬ )‫(ابن‬:‫الحكماء‬ ‫بأخبار‬ ‫العلماء‬ ‫إخبار‬.)‫تاريخ‬ ‫(بدون‬ ‫القاهرة‬ .‫المتنبي‬ ‫مكتبة‬ .
[15]-‫النديم‬ ‫ابن‬:‫الفهرست‬،‫بيروت‬ ،‫المعرفة‬ ‫دار‬ ،1978
[16]-‫البغدادي‬ ‫باشا‬ ‫إسماعيل‬:‫العارفين‬ ‫هدية‬‫بيروت‬ ،‫العربي‬ ‫التراث‬ ‫إحياء‬ ‫دار‬ .1955
[17]-‫البالذري‬)‫ا‬ :‫جابر‬ ‫بن‬ ‫يحيى‬ ‫بن‬ ‫أحمد‬ ،‫األشراف‬ ‫أنساب‬ ‫من‬ ‫جمل‬ ‫كتاب‬(13‫األجزاء‬ ،)‫جزءا‬2‫و‬4‫و‬11‫تحقيق‬ .
‫بيروت‬ ،‫الفكر‬ ‫دار‬ ،‫زركلي‬ ‫رياض‬ ‫والدكتور‬ ‫زكار‬ ‫سهيل‬ ‫الدكتور‬1996
)‫ب‬.‫البلدان‬ ‫فتوح‬‫الثاني‬ ‫القسم‬،‫بيروت‬ ،‫المعارف‬ ‫مؤسسة‬ .‫الطباع‬ ‫أنيس‬ ‫وعمر‬ ‫الطباع‬ ‫أنيس‬ ‫هللا‬ ‫عبد‬ ‫.تحقيق‬1987
[ .18]-‫البيهقي‬:)‫الشاملة‬ ‫المكتبة‬ ‫(في‬ ‫اإليمان‬ ‫شعب‬
[19]-‫الخوارومي‬: ‫موسى‬ ‫بن‬ ‫محمد‬ ،)‫أ‬‫والمقابلة‬ ‫الجبر‬ ‫كتاب‬‫والدكتور‬ ‫مشرفة‬ ‫مصطفى‬ ‫علي‬ ‫الدكتور‬ ‫وتعليق‬ ‫تقديم‬ .
،‫مصر‬ ‫فرع‬ ،‫والنشر‬ ‫للطباعة‬ ‫العربي‬ ‫الكاتب‬ ‫دار‬ .‫أحمد‬ ‫مرسي‬ ‫محمد‬19
[20]-‫البغدادي‬ ‫الخطيب‬)‫أ‬ :‫بغداد‬ ‫تاريخ‬(17‫الجزء‬ ،)‫جزءا‬14‫الغرب‬ ‫دار‬ .‫عواد‬ ‫معروف‬ ‫بشار‬ ‫الدكتور‬ ‫تحقيق‬ .
‫بيروت‬ ،‫اإلسالمي‬2001.
[21]-‫الذهبي‬:‫اإل‬ ‫تاريخ‬‫سالم‬(52‫الجزء‬ .)‫جزءا‬22‫بيروت‬ ،‫العربي‬ ‫الكتاب‬ ‫دار‬ .‫تدمري‬ ‫السالم‬ ‫عبد‬ ‫عمر‬ ‫تحقيق‬ .
1991.
[22]-‫السمعاني‬:‫األنساب‬(5‫البارودي‬ ‫عمر‬ ‫هللا‬ ‫عبد‬ ‫تحقيق‬ .)‫أجزاء‬.‫الجزء‬،‫الثقافية‬ ‫واألبحاث‬ ‫الخدمات‬ ‫مركز‬ .‫الثاني‬
‫بيروت‬ ،‫الجنان‬ ‫دار‬1408‫ه‬/1988.
[23]-‫المكتبة‬‫الشاملة‬‫الموقع‬ ‫من‬ ‫ل‬َّ‫م‬َ‫ح‬‫:ت‬http//Islamport.com‫سنة‬ ‫خالل‬ ‫زيارات‬ ‫(بضع‬2008)
[24]-‫خليفة‬ ،‫حاجي‬:‫والفنون‬ ‫الكتب‬ ‫أسامي‬ ‫عن‬ ‫الظنون‬ ‫كشف‬‫إ‬ ‫دار‬ ،‫يالتقايا‬ ‫الدين‬ ‫شرف‬ ‫محمد‬ ‫تحقيق‬ .‫التراث‬ ‫حياء‬
‫بيروت‬ ،‫العربي‬

11. 11
ANNEX: RESOLUTION OF AFFINE AND QUADRATIC EQUATIONS BY AL-
KHWĀRIZMī; A BRIEF SUMMARY
(Extract from my book: An introduction to the History of Maghrebian Mathematics (2020):
Published by Amazon:
https://www.amazon.fr/Introduction-History-Maghrebian-Mathematics/dp/B084DG7LN3
The summary here is limited to one example and does not include the geometric
justifications.
To solve a linear equation of degree one or two, Arab-Muslim mathematicians
mobilized four main tools: A) the algebraic way inaugurated by al-Khwārizmī b) the
procedure of the trays of the balance (ṭarīqat al-kaffāt), c) the method of proportions, d)
and the arithmetical tool.
1.1 The algebraic method of al-Khwārizmī14
The resolution of affine and quadratic equations was already known in Mesopotamia
nearly two millennia before Christ. Many tablets contain indeed problems leading to such
equations along with the presentation of solutions. Similarly, there are such problems in
Indian mathematics, and some geometrical constructions in Euclid's Elements can be
interpreted as solutions of quadratic equations; Diophantus (circa 100) - whose
indeterminate algebra seemed to have been known to Arab-Moslem mathematicians circa
10th century - uses ingenious processes to solve problems, but such processes remained
nevertheless limited to the cases dealt with. However, there is no track of a systematic and
general study before al-Khwārizmī on the methods of resolution of algebraic equations
and the expression of roots in terms of coefficients. At best, there was what might be
termed as recipes applied with no reference to a unifying theoretical frame, and one of al-
Khwārizmī's merits and not the least one, is to have introduced order and brought out
general rules.
The contribution of this great scholar is a real revolution in the mathematical thought. He
introduced a supplementary level of abstraction in mathematics. If representing a known
number by a symbol (e.g.: twelve by: 12) already constitutes an abstraction, introducing
the concept of unknown constitutes an abstraction of a higher level, because this one can
be any number of any nature: a length, an area, a volume, an amount of money, an age,
etc. With this mathematician, we attend the introduction of new mathematical objects, and
their progressive taming to subject them to the same rules of calculation as on known
numbers.
14
al-Khwārizmī's book will be my reference here, because of its considerable role in the dissemination of
algebra. The reader may refer to my paper on the controversy raised by Abū Barza about the priority of Ibn
Turk, a contemporary of al-Khwārizmī whose part of a book of algebra was discovered and published in
Turkey in 1960; see also papers by A. Sayli cited in the bibliography.

12. 12
More precisely, in al-Khwārizmī's work the following important events stand out:
- Explicit demarcation of the borders of a new mathematical field, baptized ḥisāb al-Jabr
wa al-muqabalah, often abrieged afterwards in (ḥisāb) al-Jabr;
- Definition of new concepts including: numbers, roots, squares;
- Distinction of six canonical cases;
- Introduction of two new operations with their names: al-Jabr and al-Muqābalah;
-Description of the algorithms corresponding to the resolution for each case;
- Expression of solutions in terms of the coefficients of the equation;
- Development of geometrical models and justification of algorithms for the last three
cases of equations.
- Implementation of what precedes in various practical situations to solve diverse
problems, such as those related to commercial transactions, the science of inheritance
shares and surveying.
Here is the meaning of the two terms al-Jabr and al-Muqābala (lit. restoration
and opposition):
- al-Jabr corresponds to the operation of transposing negative terms from a side
in an equation to the other one, so that in both sides there remain only positive terms.
- al-Muqabala is the "simplification" of similar terms in the two sides of an
equation.
Although the author does not mention the term, he makes use of the operation al-ḥaṭṭ
(division of the two sides by a non-zero number). This term appeared later in Ibn al-Bannā'
(no.M134).
al-Khwārizmī considers three sorts of numbers: The simple number (called ‘adad /‘adad
mufrad ‫=عدد‬ number), supposed to be known; the root (jidhr ‫=جذر‬ root; or shay' ‫=شيء‬
thing) which stands usually for the unknown; the square (māl, ‫مال‬ etymologically
possessed goods, amount of a sum) which indicates generally the square of the unknown15
represented for example by x² in today's notation.
He distinguishes on the other hand six canonical forms (abwāb ‫أبواب‬ or ḍurūb ‫)ضروب‬ for
equations, to which any affine or quadratic16
equation can be reduced by means of "al-
jabr" and "al-muqabala":
1- Squares equal to roots : 2
= bxax
2- Squares equal to numbers: c=ax
2
3- Roots equal to numbers: c=bx
4- Roots and squares equal to numbers: 2
a +bx = cx reduced to 2
+ px = qx
5- Squares and numbers equal to roots: 2
a c = bxx + reduced to 2
+q = pxx ;
6- Squares equal to roots and numbers: 2
a = bx +cx reduced to 2
= px +qx
15
His successors added the cube (ka‘
b) , square-square (māl al-māl), square-cube (ka‘
b al-māl), etc. For full details
about the different meanings of māl see Jeffrey A. Oaks & Haitham M. Alkhateeb in Historia Mathematica 32 (2005)
400-425.
16
al-Khwārizmī employs the verb ‘ādala but not the noun mu’ādala. It may be worth noticing that these
six cases do not exhaust the cases of the general equation 2
0ax bx c+ + = ; negative roots were indeed not
considered by al-Khwārizmī (while Indian algebra dealt with them).

13. 13
He avoided in this way the use of negative coefficients. Besides, if he gave the two roots
when they are positive, he did not accept negative or zero roots. This tradition survived
much later after him, negative numbers and zero having reached only late the complete
status of number. Here is an example excerpted from al-Khwārizmī’s book on algebra (my
translation according to the Arabic publishing by M. Musharrafa and M. Mursi):
"I divided ten into two parts, then multiplied each of both by itself; I
obtained fifty eight".
He afterward gives the solution in the following way:
"Take one of the two parts as the root, and the other one ten minus the root;
then multiply ten minus the root by itself, which makes one hundred numbers, one square,
minus twenty roots; multiply the root by itself, which produces one square; make then the
sum; it results from this one hundred and two squares minus twenty roots equal to fifty
eight. By al-jabr, restore to one hundred and two squares the twenty roots subtracted, and
add them to the fifty eight numbers; this gives one hundred and two squares equal to fifty
eight numbers and twenty roots; reduce this to a single square by taking half of what you
have. It results from this fifty and one square equal to twenty nine and ten roots; operate
by al-muqābala by subtracting twenty nine from fifty. It results from it twenty one and
one square equal to ten roots. Take half of the roots, which gives five, that you multiply
by itself. From the result twenty five, subtract the twenty one which are with the square;
it remains four, of which you take the square root, which is two, that you subtract from
half of the roots which is five. It remains three, which is one of the two parts; the other
one is then seven. This problem led you to one of the six cases which is: squares and
numbers equal to roots." [ al-Khwārizmī].
One perceives on this very simple example, the considerable mental effort which
mathematicians had to deploy for lack of symbols. The tradition of al-Khwārizmī was
respected by his successors as well in The Middle East and in The Muslim West as by
European mathematicians until the XVII-th where algebraic notation was not known.
Translated into modern symbolism, what precedes leads to the resolution of the equation
100 +2x² - 20x = 58 . The transformations indicated by al-Khwārizmī are the following
ones:
100 +2x² - 20x = 58
100 + 2 x2
= 58 + 20 x by al-jabr;
50+x² = 29+10x by al-ḥaṭṭ17
;
21+x² =10x by al-muqābala;
The author is thus lead to an equation of the fifth type
2
+q = pxx ; he provides one
solution, x1=3,
according to the formula 2
1 / 2 ( / 2)x p p q= − − then the other solution: x2=10-
x1=7.
17
al-Khwārizmī used the verb ḥaṭṭa in the solution of a problem of heritage shares; the name seems to be
found for the first time in Ibn al-Bannā’ al-Marrākushī (654-721H/1256-1321).