4. ARITHMETIC. Naturally little could be done in this subject as long as the Roman system of notation was in use (see footnote, i, p. 64), and the Arabic notation was not known in western Christian Europe until the beginning of the thirteenth century, and was not much used for two or three centuries later. So far as arithmetic was taught before that time, it was but little in advance of that given to novitiates in the monasteries, except that much attention was devoted to an absurd study of the properties of numbers, [8] and to the uses of arithmetic in determining church days, calculating the date of Easter, and interpreting passages in the Scriptures involving measurements (R. 74 d). The textbook by Rhabanus Maurus _On Reckoning_, issued in 820, is largely in dialogue (catechetical) form, and is devoted to describing the properties of numbers, "odd, even, perfect, imperfect, composite, plane, solid, cardinal, ordinal, adverbial, distributive, multiple, denunciative, etc."; to pointing out the scriptural significance of number; [9] and to an elaborate explanation of finger reckoning, after the old Roman plan (see p. 65). Near the end of the tenth century Gerbert, [10] afterwards Pope Sylvester II, devised a simple abacus-form for expressing numbers, simple enough in itself, but regarded as wonderful in its day. This greatly simplified calculation, and made work with large numbers possible. He also devised an easier form for large divisions. Gerbert's form for expressing numbers may be shown from the following simple sum in addition: _Arabic Form_ _Roman Form_ _Gerbert's Form_ _M C X I_ 1204 MCCIV I II IV 538 DXXXVIII V III VIII 2455 MMCCCCLV II IV V V 619 DCXIX VI I IX ----- --------- ------------------- 4816 MMMMDCCCXVI IV VIII I VI No study of arithmetic of importance was possible, however, until the introduction of Arabic notation and the use of the zero. 5. GEOMETRY. This study consisted almost entirely of geography and reasoning as to geometrical forms until the tenth century, when Boethius' work on _Geometry_, containing some extracts from Euclid, was discovered by Gerbert. The geography of Europe, Asia, and Africa also was studied, as treated in the textbooks of the time, and a little about plants and animals as well was introduced. The nature of the geographic instruction may be inferred from Figure 46, which reproduces one of the best world maps of the day. The main geographical features of the known world can be made out from this, but many of the mediaeval maps are utterly unintelligible.