Hogendijk, J. P.

Revistes Catalanes amb Accés Obert

Reviews 191
k×t (pp. 899-905), an Arabic-French 
glossary of technical terminology (pp. 
907-955), and very complete indices 
of proper names (pp. 957-958), topics 
(pp. 958-965), titles of sources quoted 
(pp. 966-967) and manuscripts (p. 
968), followed by lists of sources and 
bibliography (pp. 969-972). Concern-
ing the bibliography, I wonder why the 
only translation of the Almagest 
quoted is the old French translation by 
N. Halma (Paris, 1813 and 1816) (see 
pp. 697 and 971). Two other sources 
are quoted through manuscripts, in 
spite of the fact that they have been 
edited: al-Fargh×n÷’s al-K×mil f÷ l-
asÐurl×b (p. 654), edited very recently 
by Richard Lorch (2005) and Abý l-
Waf×’s F÷m× ya¬t×ju ilay-hi al-½×niþ 
min þamal [þilm?] al-handasa ed. by 
¼×li¬ A¬mad al-þAl÷ (Baghdad, 1979).  
Leaving aside these small details, 
this volume is a great contribution, 
both in size and importance, to our 
knowledge of Ibn al-Haytham’s 
mathematical astronomy, written by a 
first rate scholar who has dedicated his 
life to research on the history of 
Islamic mathematics. The editions 
have been prepared with the utmost 
care and skill, the translations are 
excellent and the mathematical 
commentaries are detailed and clear. I 
can only thank the author for 
presenting us with this gift. 
 
      Julio Samsó 
 
 
Menso Folkerts, The development 
of mathematics in medieval Europe: 
The Arabs, Euclid, Regiomontanus. 
Aldershot: Variorum collected 
studies series: CS811, 2006. 354 
pages, ISBN 0-86078-957-8, price 
62.50 pounds.  
 
The volume under review contains 
twelve articles by Menso Folkerts on 
the history of medieval as well as 
Renaissance mathematics. Eight arti-
cles had been previously published 
between 1974 and 2002; the articles 
on Arabic mathematics in the West (I), 
Euclid in the Middle Ages (III) and 
Regiomontanus’ approach to Euclid 
(VII) are revised versions, and article 
XII, “Algebra in Germany in the 
Fifteenth Century” is new. Articles I 
and VII appear in English translation 
for the first time.  
 The main emphasis of the book is 
on Euclid’s Elements, its medieval 
transmission, and its influence in the 
works of some Renaissance authors. 
Folkerts’ book is a good introduction 
to this important but complicated 
subject. Several articles in Folkerts’ 
book deal with Renaissance arithmetic 
and algebra on the basis of unpu-
blished manuscripts of Regiomontanus 
(1436-1476).  
 In a review of this size it is 
impossible to do justice to the book as 
a whole. I will therefore restrict myself 
to the question why Folkerts’ book is 
important for the history of medieval 
Islamic science. Folkerts discovered 
the complete version of the arithmetic 
of al-Khwārizmī, see the summary and 
analysis in article II. Article I is a 
concise introduction to the transmis-
sion of Arabic mathematics to the 
West. Folkers arrives at the surprising 
conclusion that the medieval transla-
tions from Arabic to Latin were “the 
Reviews 192 
basis of the mathematics of the 
seventeenth century” and had in the 
seventeenth century “a greater influen-
ce than Renaissance mathematics” (I 
p. 13). In the wonderfully clear article 
no. III, Folkerts details the way in 
which Latin version of Euclid’s Ele-
ments are related to Arabic versions. 
In III, p. 12 footnote 63 add Rüdiger 
Arnzen’s edition of al-Nayrīzī’s 
commentary to Elements book 1 (Köln 
-Essen, 2002, ISBN 3-00-009172-6) 
on the basis of the newly discovered 
manuscript in Qom.  
 Folkerts makes clear that many 
aspects of the transmission of medie-
val Islamic science to Europe are still 
shrouded in mystery. There are 
curious coincidences between Euro-
pean and Arabic writings, such as: 
semi-regular polyhedra in the work of 
Piero della Francesca (died 1492), 
which are the same as in the work of 
Abu’l-Wafā’ (died 998; X, p. 307-
308); approximate rules for the 
determination of sides of regular 
polygons, probably of Arabic origin, 
in a circle in the work of Regiomonta-
nus (V, 283); and mathematical 
problems in the works of Leonardo 
Fibonacci (ca. 1200; IX, 14-15) which 
are also found in the Algebras of al-
Karajī (ca. 1000) and al-Khayyām (ca. 
1100), even though the standard view 
holds that these works were unknown 
in the Western part of the Islamic 
world. In all these examples, the 
question is whether the coincidences 
are the result of transmission from the 
Arabic, and if so, in what way. 
Folkerts also mentions tantalizing 
coincidences between European ma-
thematics and other cultures. Two 
examples from the work of Regiomon-
tanus are: his determination of the di-
agonals from the sides of a concyclic 
quadrilateral, which is related to the 
work of the Indian mathematician 
Brahmagupta (VIII, 425); and his use 
of the Chinese remainder theorem (V, 
p. 200). Regiomontanus is not unique, 
because Leonardo Fibonacci uses a 
result which was also known to Indian 
mathematicians and which can be 
expressed in modern terms by the 
formula (a2
 
+ b2)(c2
 
+ d2)=(ac + bd)2
 
+(ad – bc)2
 
. If knowledge of Indian 
and Chinese mathematics was trans-
mitted to Fibonacci and Regiomonta-
nus, the only plausible channel would 
have been through the Islamic world. 
But no traces of these problems and 
results have yet been found in 
medieval Arabic texts. This may of 
course reflect our own ignorance about 
the medieval Islamic mathematical 
tradition. Thus, the later European 
tradition can provide a wealth of 
fascinating subjects of research which 
may substantially enrich our know-
ledge of the Islamic mathematical tra-
dition as well. 
Together with the previous volume 
on Essays in early Medieval Math-
ematics, (Aldershot 2003), the book 
under review is indispensable for all 
research scholars in the history of 
science in antiquity, the Islamic and 
European medieval period, and the 
Renaissance.  
 
Jan P. Hogendijk 
 


Menso Folkerts, The development of mathematics in medieval Europe: The Arabs,

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