International congress of byzantine studies belgrade, 22 27 august 2016

Illya Bey
Hussite Theological Faculty, Charles University, Prague, Czech Republic; 
[e-mail]
Византийская философия: в поисках определения
Статья посвящена поиску определения понятия «византийская философия». Поднима-
ются такие вопросы, как правомерность существования и употребления этого термина, про-
водится разделение между понятиями «византийская философия» и «философия в Византии». 
Рассматриваются также определения философии, использованные визан тийскими авторами, 
возможность отделения философии от теологии, philosophia theologiae ancilla. Затрагиваются 
вопросы влияния аристотелизма и платонизма на развитие фило софии в Византии.
Ilias Nesseris
Österreichische Akademie der Wissenschaften, Institut für Mittelalterforschung, 
Abteilung Byzanzforschung, Vienna, Austria; 
[e-mail]
Mathematics in the Twelfth-Century Byzantium
It is widely accepted that the twelfth century was the golden era of rhetoric in Byzantium and 
a time which many other literary genres thrived as well. But above else, it may be argued, it was the 
golden era of higher education, since in the many schools that functioned in Constantinople during 
the twelfth century taught many intellectuals of a great calibour, such as Theodore Prodromos or 
Eustathios of Thessalonica to name but a few. The works of Homer, the classics and other authors 
that had entered the school curriculum continued to be taught uninterruptedly, while new works 
and handbooks were composed by these scholars in order to cover their needs in the classroom 
in the fields of poetry, rhetoric and philosophy. Alongside these subjects the Quadrivium, which 
consisted of mathematics –i.e. arithmetic and geometry– together with harmonics (the ancient 
music theory) and astronomy, continued to be taught. 

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Nevertheless the study of mathematics in this period has received very little scholarly attention 
so far, so that quite often it is admitted that there is a gap in our knowledge between the 11
th
and the 
13
th
century, or more precisely between Michael Psellos on the one hand, and Nikephoros Blemmydes 
and the scholars of the early Palaeologan period, Maximos Planudes and George Pachymeres, on 
the other. 
The aim of this paper is to shed some light in this breach, first by referring to the ways in which 
the twelfth century scholars exhibited their knowledge of ancient Greek mathematical works, for 
instance by mentioning or quoting passages from authors like Nikomachos of Gerasa or Euclid. 
Furthermore by focusing more closely on their teaching activities, since in some instances we can 
trace not only what subjects exactly they taught, but also who were their students; such was the case 
of Manuel Sarantenos, the future Patriarch in Nicaea (1217-1222), who was initiated in the study of 
Diophantos by his teacher Nikolaos Kaloethes, an expert on this very difficult subject. It is no matter 
of fortune then that another student of Kaloethes, the monk Prodromos, was to become the instructor 
of Blemmydes in higher mathematics in the beginning of the second decade of the thirteenth century. 
Of even greater importance is of course the location of works that were composed ad hoc for the 
study or teaching of the mathematical works of Antiquity. Thus a central position in our paper occupies 
the identification and analysis of some hitherto unknown scholia of Eustratios of Nicaea, Michael 
Ephesios and a certain Nikolaos Dishypatos in specific passages of the Nikomachos’ Introduction to 
Arithmetic, which serves to illustrate the point that this work continued to be commented upon (earlier 
commentaries existed from the early byzantine period by Iamblichos of Chalkida, John Philoponos 
and Asklepios of Tralleis) and most probably taught in the Comnenian period. 
Finally, another necessary step is the tracing of all the manuscripts with mathematical content 
that were copied at that time, but also of previous centuries that continued to be in use and circulation 
in the scholarly circles of the capital during this period.
In conclusion our detailed analysis aims at showing that the twelfth century clearly constitutes 
one of the missing links in the long chain of the dissemination and the study of mathematics in 
Byzantium bridging thus this gap in our knowledge.