What we find about the progression of the other numbers is more indefinite and unsatisfying, and the Notion loses itself in them. Up to five there may certainly be a kind of thought in numbers, but from six onwards they are merely arbitrary determinations. 2. _Application of the System to the Universe_. This simple idea and the simple reality contained therein, must now, however, be further developed in order to come to reality as it is when put together and expanded. The question now meets us as to how, in this relation, the Pythagoreans passed from abstract logical determinations to forms which indicate the concrete use of numbers. In what pertains to space or music, determinations of objects formed by the Pythagoreans through numbers, still bear a somewhat closer relation to the thing, but when they enter the region of the concrete in nature and in mind, numbers become purely formal and empty. a. To show how the Pythagoreans constructed out of numbers the system of the world, Sextus instances (adv. Math. X. 277-283), space relations, and undoubtedly we have in them to do with such ideal principles, for numbers are, in fact, perfect determinations of abstract space. That is to say, if we begin with the point, the first negation of vacuity, “the point corresponds to unity; it is indivisible and the principle of lines, as the unity is that of numbers. While the point exists as the monad or One, the line expresses the duad or Two, for both become comprehensible through transition; the line is the pure relationship of two points and is without breadth. Surface results from the threefold; but the solid figure or body belongs to the fourfold, and in it there are three dimensions present. Others say that body consists of one point” (_i.e._ its essence is one point), “for the flowing point makes the line, the flowing line, however, makes surface, and this surface makes body. They distinguish themselves from the first mentioned, in that the former make numbers primarily proceed from the monad and the undetermined duad, and then points and lines, plane surfaces and solid figures, from numbers, while they construct all from one point.” To the first, distinction is opposition or form set forth as duality; the others have form as activity. “Thus what is corporeal is formed under the directing influence of numbers, but from them also proceed the definite bodies, water, air, fire, and the whole universe generally, which they declare to be harmonious. This harmony is one which again consists of numeral relations only, which constitute the various concords of the absolute harmony.”