"We can not forbear pointing out an important application of these results to the Critical Philosophy. Kant bases each of his famous four antinomies on the demand of pure reason for unconditioned totality in a regressive series of conditions. This, he says, must be realized either in an absolute first of the series, conditioning all the other members, but itself unconditioned, or else in the absolute infinity of the series without a first; but reason is utterly unable, on account of mutual contradiction, to decide in which of the two alternatives the unconditioned is found. By the principles we have laid down, however, the problem is solved. The absolute infinity of a series is a contradiction _in adjecto_. As every number, although immeasurably and inconceivably great, is impossible unless _unity_ is given as its basis, so every series, being itself a number, is impossible unless a _first term_ is given as a commencement. Through a first term alone is the unconditioned possible; that is, if it does not exist in a first term, it can not exist at all; of the two alternatives, therefore, one altogether disappears, and reason is freed from the dilemma of a compulsory yet impossible decision. Even if it should be allowed that the series has no first term, but has originated _ab œterno_, it must always at each instant have a _last term_; the series, as a whole, can not be infinite, and hence can not, as Kant claims it can, realize in its wholeness unconditioned totality. Since countless terms forever remain unreached, the series is forever limited by them. Kant himself admits that it _can never be completed_, and is only potentially infinite; actually, therefore, by his own admission, it is finite. But a last term implies a first, as absolutely as one end of a string implies the other; the only possibility of an unconditioned lies in Kant's first alternative, and if, as he maintains Reason must demand it, she can not hesitate in her decisions. That _number is a limitation_ is no new truth, and that every series involves number is self-evident; and it is surprising that so radical a criticism on Kant's system should never have suggested itself to his opponents. Even the so-called _moments_ of time can not be regarded as constituting a real series, for a series can not be real except through its divisibility into members whereas time is indivisible, and its partition into moments is a conventional fiction. Exterior limitability and interior divisibility result equally from the possibility of discontinuity. Exterior illimitability and interior indivisibility are simple phases of the same attribute of _necessary continuity_ contemplated under different aspects. From this principle flows another upon which it is impossible to lay too much stress, namely; _illimitability and indivisibility, infinity and unity, reciprocally necessitate each other_. Hence the Quantitative Infinites must be also Units, and the division of space and time, implying