1. All number is considered either with reference to itself or in relation to something. The former is divided as follows: some are equal, as for example, two; others are unequal, as for example, three.[248] The latter is divided as follows: some are greater, some are less. The greater are divided as follows: into _multiplices_ (multiple), _superparticulares_, _superpartientes_, _multiplices superparticulares_, _multiplices superpartientes_. The less are divided as follows: _Sub-multiplices_ (sub-multiple), _sub-superparticulares_, _sub-superpartientes_, _sub-multiplices sub-superparticulares_, _sub-multiplices sub-superpartientes_. [248] The examples are found in Du Breul. They do not appear in Arevalo. 6. ... The _superparticularis numerus_ is when a greater number contains in itself a lesser number with which it is compared, and at the same time one part of it. 7. For example; III when compared with II contains in itself two and also one, which is the half of two. IV when compared with III, contains three and also one, which is the third of three. Likewise V, when compared with IV, contains the number four and also one, which is the fourth part of the said number four, and so on. 8. The _superpartiens numerus_ is that which contains the whole of a lesser number and in addition two parts of it, either thirds or fifths or other parts. For example, when V is compared with III, the number five contains three and in addition to this two parts of it. Chapter 7. On the third division of all number. 1. Numbers are abstract or concrete. The latter are divided as follows: first, lineal; second, superficial; third, solid. Abstract number is that which is made up of abstract units. For example, III, IV, V, VI, and so on. 2. Concrete number is that which is made up of units that are not abstract, as for example, the number three, if it is understood of magnitude, whether line, superficies, or solid, is called concrete. 4. The number of superficies is that which is constituted not only by length but also by breadth, as triangular, square, pentangular, or circular numbers, and the rest that are contained in a plane surface or superficies. 5. The circular number, when it is multiplied by itself, beginning with itself, ends with itself. For example, _Quinquies quini vicies quinque_. 6. ... The spherical number is that which being multiplied by the circular number begins with itself and ends with itself; for example, five times five are twenty-five, and this circle being multiplied by itself makes a sphere, that is, five times XXV make CXXV. Chapter 8. On the distinction between arithmetic, geometry, and music.