Chapter 13. On the first principles of geometry.
1. ... A point is that which has no part. A line is length without breadth. A straight line is one which lies evenly in respect to its points. A superficies is that which has length and breadth alone.
Chapter 14. On the numbers of geometry.
1. You search into the numbers of geometry as follows: the extremes being multiplied, amount to as much as the means multiplied; as for example, VI and XII being multiplied, make LXXII; the means VIII and IX being multiplied, amount to the same.
ON MUSIC
INTRODUCTION
As an educational subject music is the oldest of those grouped under the heading of the seven liberal arts. In Plato’s time music and gymnastic were the staples of education, and the former term meant chiefly the study of poetry, with music in the proper sense of the word as a mere adjunct. As the different subjects, such as grammar, rhetoric, geometry, arithmetic, appeared in the curriculum, the field of music narrowed and it held a less commanding place. Conflicting points of view in regard to it appear to have arisen. The older educational tradition connected music with grammar and the other literary studies. On the other hand, the influence of the Pythagorean theory of number and of its application to music tended to dissociate grammar and music, and to place the latter in relation to the mathematical sciences. It has been noticed that among the older Roman writers from whom evidence on this matter can be drawn—Cicero, Varro, Seneca, Quintilian, and others—the association of music and grammar appears the natural one, while in the Roman writers of the second, third, and fourth centuries both traditions prevail, with an increasing preference for placing music among the mathematical sciences, where it finally found itself when the canon of the seven liberal arts was formed, and where it remained to the end of the middle ages.[254]
In Isidore little is to be found to justify the mathematical environment of music. It is true that at times he defines it as a mathematical science[255] and he insists on the musical view of the universe as a necessary complement to other views. “Without music,” he says, “there can be no perfect knowledge, for there is nothing without it. For even the universe itself is said to have been formed under the guidance of harmony.”[256] But, with the exception of a paragraph on the musical mean, his treatment is entirely taken up with the non-mathematical aspect of the subject, and the definition “music is the practical knowledge of melody”[257] is the one that more closely fits the occasion.
The treatment[258] of music is of about the same length as that of arithmetic, and is devoted mainly to definitions of musical terms and brief descriptions of wind and stringed instruments. It appears that Isidore knew nothing of music in a technical sense.[259]