1. The first of the figures on a plane surface is the circle, a figure that is plane, and has a circumference, in the middle of which is a point upon which everything converges (_cuncta convergunt_) which geometers call the center, and the Latins call the point of the circle. 2. A quadrilateral figure is one on a plane surface, and it is contained by four straight lines.... 3. A sphere is a figure of rounded form equal in all its parts. A cube is a solid figure which is contained by length, breadth, and thickness. 5. A cone (_conon_) is a solid figure which narrows from a broad base like the right-angled triangle. 6. A pyramid is a solid figure which narrows to a point from a broad base like fire. For fire in Greek is called πῦρ. 7. Just as all number is contained within ten so the outline of every figure is contained within the circle. 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]