As to the first, we shall relinquish the opinion of those who affirm that it is enough, in proportions, to consider the nature of the intervals, and of the medieties which fill up their vacancies; and that the demonstration can be made out for any numbers whatsoever that have spaces sufficient to receive the aforesaid proportions. For this being granted, it makes the demonstration obscure, without the help of schemes, and drives us from another theory, which carries with it a delight not unbecoming philosophy. Beginning therefore from the unit, let us place the duples and triples apart; and there will be on the one side, 2, 4, 8; on the other 3, 9, 27; —seven numbers in all, proceeding forward by multiplication four steps from the unit, which is assumed as the common base. For not only here, but upon other occasions, the sympathy of the quaternary number with the septenary is apparent. There is this peculiar to that tetractys or quaternary number thirty six, so much celebrated by the Pythagoreans, which is more particularly worthy admiration,—that it is composed of the first four even numbers and the first four odd numbers; and it is the fourth connection made of numbers put together in order. The first connection is of one and two; the second of odd numbers. For placing the unit, which is common to both, before, he first takes eight and then twenty-seven, as it were pointing out with the finger where to place each particular sort. [These places are so depraved in the original, that the sense is lost.] But it belongs to others to explain these things more accurately and distinctly; while we content ourselves with only what remains, as peculiarly proper to the subject in hand.