Concerning the procreation of the soul as discoursed in Timaeus (11-15)

The first pair of these numbers consists of one and two, the second of three and four, the third of five and six; neither of which pairs make a tetragonal number, either by themselves or joined with any other figures. The fourth consists of seven and eight, which, being added all together, produce a tetragonal number of thirty-six. But the quaternary of numbers set down by Plato have a more perfect generation, of even numbers multiplied by even distances, and of odd by uneven distances. This quaternary contains the unit, the common original of all even and odd numbers. Subsequent to which are two and three, the first plane numbers; then four and nine, the first squares; and next eight and twenty-seven, the first cubical numbers (not counting the unit). Whence it is apparent, that his intention was not that the numbers should be placed in a direct line one above another, but apart and oppositely one against the other, the even by themselves, and the odd by themselves, according to the scheme here given. In this manner similar numbers will be joined together, which will produce other remarkable numbers, as well by addition as by multiplication.

By addition thus: two and three make five, four and nine make thirteen, eight and twenty-seven make thirty-five. Of all which numbers the Pythagoreans called five the nourisher, that is to say, the breeding or fostering sound, believing a fifth to be the first of all the intervals of tones which could be sounded. But as for thirteen,

Admit a right-angled parallelogram, A B C D, the lesser side of which A B consists of five, the longer side A C contains seven squares. Let the lesser division be unequally divided into two and three squares, marked by E; and the larger division in two unequal divisions more of three and four squares, marked by F. Thus A E F G comprehends six, E B G I nine, F G C H eight, and G I H D twelve. By this means the whole parallelogram, containing thirty-five little square areas, comprehends all the proportions of the first concords of music in the number of these little squares. For six is exceeded by eight in a sesquiterce proportion (3:4), wherein the diatessaron is comprehended. And six is exceeded by nine in a sesquialter proportion (2:3), wherein also is included the fifth. Six is exceeded by twelve in duple proportion (1:2), containing the octave; and then lastly, there is the sesquioctave proportion of a tone in eight to nine. And therefore they call that number which comprehends all these proportions harmony. This number is 35, which being multiplied by 6, the product is 210, which is the number of days, they say, which brings those infants to perfection that are born at the seventh month’s end.

To proceed by way of multiplication,—twice 3

Now these numbers aforesaid being endued with all these properties, the last of them, which is 27, has this peculiar to itself, that it is equal to all those that precede together; besides, that it is the periodical number of the days wherein the moon finishes her monthly course; the Pythagoreans make it to be the tone of all the harmonical intervals. On the other side, they call thirteen the remainder, in regard it misses a unit to be half of twenty-seven.

But since the numbers proposed did not afford space