Etymologiae, Book III, chapters 7-8: De tertia divisione totius numeri to De differentia arithmeticae, geometriae et musicae
Chapter VII. On the third division of the whole of number Numbers are either (1) discrete or (2) continuous. This is divided thus: (1) linear, (2) superficial, (3) solid. A discrete number is one that is contained by discrete monads, as for example III, IV, V, VI and the rest. A continuous number is one that is contained by joined monads; as for example the number three may be understood in magnitude, that is, in a line, or a space or a solid is called continuous; likewise the numbers four and five. A linear number is one that, beginning from the monad, is written linearly up to infinity. Hence alpha is put for the designation of lines, since this letter signifies one among the Greeks. A superficial number is one that is contained not only by length but also by breadth, as the triangular, the square, the pentagonal or circular numbers, and the rest, which are always contained in the plane foot, that is, the surface. ‸The circular number is that which, when multiplied in like manner, beginning from itself is turned back to itself, as for example five times five, 25. A solid number is one that is contained by length and breadth or height, such as pyramids, which rise up in the manner of a flame. The cube, such as dice. Spheres, which have an equal roundness on every side. A spherical number is one that, multiplied from a circular number, begins from itself and is turned back into itself. Five times five, 25. This circle, when multiplied into itself, makes a sphere, that is, five times 25, 125.
Chapter VIII. On the difference between arithmetic, geometry and music Between Arithmetic and Geometry and Music there is this difference, in finding the means. In Arithmetic you first seek thus. You join the extremes, and divide, and make the mean: suppose you make the extremes VI and XII; you join them together and they make 18; you halve and make 9, which is the arithmetical analogy, that the mean exceeds the first by as many monads as it is exceeded by the extreme. For 9 exceeds 6 by three monads, and is exceeded by 12 by the same. According to geometry you seek thus. The extremes multiplied make as much as the means doubled, suppose 6 and 12 multiplied will make seventy-two, the means 8 and 9 multiplied make the same. According to music thus: By what part the mean exceeds the first, by the same part the mean is exceeded by the extreme. Suppose 6 and 8; they exceed by two parts, which two parts of the third; the mean, 8, is exceeded by the last, 9.