it comprehends likewise all the proportions; the arithmetical, which exceeds and is exceeded by an equal number: as in the case of the numbers one, and two, and three; and the geometrical, according to which, as the proportion of the first number is to the second, the same is the ratio of the second to the third, as is the case in the numbers one, two and four; and also in multiplication, which double, or treble, or in short multiply figures to any extent; also in those which are half as much again as the numbers first spoken of, or one third greater, and so on. It also contains the harmonic proportion, in accordance with which that number which is in the middle between two extremities, is exceeded by the one, and exceeds the other by an equal part; as is the case with the numbers three, four, and six.


 Liddell and Scott explain this as meaning such even numbers as become odd when divided, as 2, 6, 10, 14, etc. 


 


 The decade also contains the visible peculiar properties of the triangles, and squares, and other polygonal figures; also the peculiar properties of symphonic ratios, that of the diatessaron in proportion exceeding by one fourth, as is the ratio of four to three; that of fifths exceeding in the ratio of half as much again, as is the case with the proportion of three to two. Also, that of the diapason, where the proportion is precisely twofold, as is the ratio of two to one, or that of the double diapason, where the proportion is fourfold, as in the ratio of eight to two. And it is in reference to this fact that the first philosophers appear to me to have affixed the names to things which they have given them. For they were wise men, and therefore they very speciously called the number ten the decade ( τὴν δεκάδα ), as being that which received every thing ( ὡσανεὶ δεκάδα οὖσαν ), from receiving ( τοῦ δέχεσθαι ) and containing every kind of number, and ratio connected with number, and every proportion, and harmony, and symphony.


 Moreover, at all events, in addition to what has been already said, any one may reasonably admire the decade for the following reason, that it contains within itself a nature which is at the same time devoid of intervals and capable of containing them. Now that nature which has no connection with intervals is beheld in a point alone; but that which is capable of containing intervals is beheld under three appearances, a line, and a superficies, and a solid. For that which is bounded by two points is a line; and that which has two dimensions or intervals is a superficies, the line being extended by the addition of breadth; and that which has three intervals is a solid, length and breadth having taken to themselves the addition of depth. And with these three nature is content; for she has not engendered more intervals or dimensions than these three.