hold that the limited is the other element, would be equivalent to holding that the whole subject of the analysis was a mere part or element of the analysis. The limited (the book) is what we are analysing, and therefore it would be nonsense to say that the limited was one factor in the analysis, while the limit was the other factor. This would be analysing a total thing into that total thing and something else. But if the limited cannot be the other term of the analysis, that other term must be the unlimited. What else can it be? The limited, then—in this case the book—consists of the limit and of the unlimited, and these are the two elements which go to the constitution of everything. Suppose the limits—for example, the two ends of a line—taken away, and no ends left, that which would remain would be the unlimited. But that cannot be conceived, you will say. Certainly it cannot. But it can be conceived to this extent, that if that part of a line which we call its ends or limits be taken away, and no new limits posited, then the remaining part considered in and by itself, is necessarily the unlimited. This element, which truly cannot be conceived without the other element, is the ἄπειρον of the Pythagoreans; and it cannot be conceived for this reason, that conception is itself constituted by the union or fusion of these two elements, the limit and the unlimited. Such is the Pythagorean doctrine, and it seems to me to be not only perfectly intelligible, but also perfectly true.
