We cannot close our discussion of the curriculum of Irish monastic schools without a brief reference to its limitations which are especially noteworthy in the case of mathematics and other secular sciences. While not venturing into details some writers manage to convey the impression that the scientific knowledge of the Irish monks was as profound as their knowledge of the classics. We have not found any evidence to support this view. The most that can be said in favour of the teachers of this early period is that they willingly imparted all the mathematical and scientific knowledge of their time. But the actual amount of such knowledge possessed by Western Europe prior to the tenth century was relatively small as compared with even the later Middle Ages not to speak of modern times. ARITHMETIC: Prior to the tenth century arithmetic was essentially the art of computation. It was largely devoted to computing Easter, so the terms “Computus” and “Arithmetic” became synonymous.[460] Apart from this practical side it is possible that a theoretical treatment of numbers was not absolutely wanting. The method of reckoning was necessarily crude and little progress was possible while the cumbersome Roman system of notation made computation with large numbers well nigh impossible. It remains for further investigation to show whether the arithmetical knowledge of the Irish monks extended beyond the limits of Bede’s _De Tempora Ratione_,[461] or equalled that of the _Liber de Computo_[462] of Rabanus Maurus. Our acquaintance with source material in this field is too limited to warrant any sweeping statement, but it is significant of the nature of arithmetical knowledge of this period and of the lack of creative ability to find a scholar of such ability as Marianus Scottus in the following century basing his work on Computus[463] on that of Rabanus. Indeed it was not until the introduction of the Arabic system of notation and Hindoo methods through Arabic influence that there was much possibility of progress in Arithmetic. The introduction of the Arabic system is attributed by some to Gerbert[464] (d. 1003 A.D.) but it was not until the end of the twelfth century that the Hindoo-Arabic system became generally established among mathematicians. A century later Arithmetic began to be applied to commerce.[465] _Algebra_ was apparently unknown to the Irish monks during the period under investigation and was probably not introduced into Europe until the twelfth century.[466] GEOMETRY: