Thus, finally, the whole content of Western number-thought centres itself upon the historic _limit-problem_ of the Faustian mathematic, the key which opens the way to the Infinite, that _Faustian infinite_ which is so different from the infinity of Arabian and Indian world-ideas. Whatever the guise—infinite series, curves or functions—in which number appears in the particular case, the _essence_ of it is the _theory of the limit_.[73] This limit is the absolute opposite of the limit which (without being so called) figures in the Classical problem of the quadrature of the circle. Right into the 18th Century, Euclidean popular prepossessions obscured the real meaning of the differential principle. The idea of infinitely small quantities lay, so to say, ready to hand, and however skilfully they were handled, there was bound to remain a trace of the Classical constancy, the _semblance of magnitude_, about them, though Euclid would never have known them or admitted them as such. Thus, zero is a constant, a whole number in the linear continuum between +1 and -1; and it was a great hindrance to Euler in his analytical researches that, like many after him, he treated the differentials as zero. Only in the 19th Century was this relic of Classical number-feeling finally removed and the Infinitesimal Calculus made logically secure by Cauchy’s definitive elucidation of the _limit- idea_; only the intellectual step from the “infinitely small quantity” to the “lower limit of _every possible_ finite magnitude” brought out the conception of a variable number which oscillates beneath any assignable number that is not zero. A number of this sort has ceased to possess any character of magnitude whatever: the limit, as thus finally presented by theory, is no longer that which is approximated to, but _the approximation, the process, the operation itself. It is not a state, but a relation._ And so in this decisive problem of our mathematic, we are suddenly made to see how _historical_ is the constitution of the Western soul.[74] ----- Footnote 73: The technical difference (in German usage) between _Grenz_ and _Grenzwert_ is in most cases ignored in this translation as it is only the underlying conception of “number” common to both that concerns us. _Grenz_ is the “limit” strictly speaking, i.e., the number _a_ to which the terms _a__{1}₁, _a__{2}₂, _a_₃ ... of a particular _series_ approximate more and more closely, till nearer to _a_ than any assignable number whatever. The _Grenzwert_ of a _function_, on the other hand, is the “limit” of the value which the function takes for a given value _a_ of the variable _x_. These methods of reasoning and their derivatives enable solutions to be obtained for series such as (1⁄_m_¹,) (1⁄_m_²,) (1⁄_m_³,) ... (1⁄_m_^{_x_}) or functions such as _x_(2_x_ - 1) _y_ = ——————————————— (_x_+ 2)(_x_- 3)