11. 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. 1 2 3 4 5 6 7 8 [Illustration: 1 /\ / \ 2 / 5 \ 3 / \ 4 / 13 \ 9 / \ 8 / 35 \ 27 - - - - - - - - ] 12. 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, they called it the remainder, despairing, as Plato himself did, of being ever able to divide a tone into equal parts. Then five and thirty they named harmony, as consisting of the two cubes eight and twenty-seven, the first that rise from an odd and an even number, as also of the four numbers, six, eight, nine, and twelve, comprehending both harmonical and arithmetical proportion. Which nevertheless will be more conspicuous, being made out in a scheme to the eye. [Illustration]