These three studies, constituting the Trivium, based as they were directly on the old Roman learning and schools, contained more that was within the teaching knowledge of the time than did the subjects of the Quadrivium, and also subject-matter which was much more in demand.
II. THE QUADRIVIUM
The trivial studies, in most cases before the thirteenth century, sufficed to prepare for the study of theology, though those few who desired to prepare thoroughly also studied the subjects of the quadrivium. In schools not offering instruction in this advanced group some of the elements of its four studies were often taught from the textbooks in use for the Trivium. Particularly was this the case during the early Middle Ages, when the knowledge of arithmetic, geometry, and astronomy possessed by western Europe was exceedingly small. No regular order in the study of the subjects of this group was followed.
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.
To illustrate the reasoning as to geometrical forms which preceded the finding of Euclid we quote from Maurus, who says that the science of geometry "found realization also at the building of the tabernacle and the temple; and that the same measuring rod, circles, spheres, hemispheres, quadrangles, and other figures were employed. The knowledge of all this brings to him, who is occupied with it, no small gain for his spiritual culture." (R. 74 e). After Gerbert's time some geometry proper and the elements of land surveying were introduced. The real study of geometry in Europe, however, dates from the twelfth century, when Euclid was translated into Latin from the Arabic.