first breadth and the first depth; (for in common the term _first_ is adapted to all of them), viz. to the triad itself, and the tetrad itself. “Again, the Pythagoreans and Plato did not denominate idea from one thing, and ideal number from another. But since the assertion is eminently true, that all things are similar to number, it is evident that number, and especially every ideal number, was denominated on account of its paradigmatic peculiarity. If any one, however, wishes to apprehend this from the appellation itself, it is easy to infer that idea was so called, from rendering as it were its participants similar to itself, and imparting to them _form_, _order_, _beauty_, and _unity_; and this in consequence of always preserving the same form, expanding its own power to the infinity of particulars, and investing with the same species its eternal participants. _Number_ also, since it imparts proportion and elegant arrangement to all things, was allotted this appellation. For the ancients, says Syrianus,[104] call to _adapt_ or _compose_ αρσαι _arsai_, whence is derived αριθμος _arithmos number_. Hence αναρσιον anarsion among the Greeks signifies _incomposite_. Hence too, those Grecian sayings, _you will adapt the balance_, _they placed number together with them_, and also _number and friendship_. From all which number was called by the Greeks _arithmos_, as that which measures and orderly arranges all things, and unites them in amicable league. “Farther still, some of the Pythagoreans discoursed about inseparable numbers alone, i. e. numbers which are inseparable from mundane natures, but others about such as have a subsistence separate from the universe, in which as paradigms they saw those numbers are contained, which are perfected by nature. But others, making a distinction between the two, unfolded their doctrine in a more clear and perfect manner. If it be requisite, however, to speak concerning the difference of these monads, and their privation of difference, we must say that the monads which subsist in quantity, are by no means to be extended to essential numbers; but when we call essential numbers monads, we must assert that all of them mutually differ from each other by _difference_ itself, and that they possess a privation of difference from _sameness_. It is evident also, that those which are in the same order, are contained through mutual comparison, in _sameness_ rather than in difference, but that those which are in different orders are conversant with much diversity, through the dominion of _difference_.