In fact, directly the essentially anti-Hellenic idea of the irrationals is introduced, the foundations of the idea of number as concrete and definite collapse. Thenceforward, the series of such numbers is no longer a visible row of increasing, discrete, numbers capable of plastic embodiment but a unidimensional _continuum_ in which each “cut” (in Dedekind’s sense) represents a number. Such a number is already difficult to reconcile with Classical number, for the Classical mathematic knows only _one_ number between 1 and 3, whereas for the Western the totality of such numbers is an infinite aggregate. But when we introduce further the imaginary (√-1 or _i_) and finally the complex numbers (general form _a_ + _bi_), the linear continuum is broadened into the highly transcendent form of a number-body, i.e., the content of an aggregate of homogeneous elements in which a “cut” now stands for a number-surface containing an infinite aggregate of numbers of a lower “potency” (for instance, all the real numbers), and there remains not a trace of number in the Classical and popular sense. These number- surfaces, which since Cauchy and Riemann have played an important part in the theory of functions, are _pure thought-pictures_. Even positive irrational number (e.g., √2) could be conceived in a sort of negative fashion by Classical minds; they had, in fact, enough idea of it to ban it as ἄῤῥητος and ἄλογος. But expressions of the form _x_ + _yi_ lie beyond every possibility of comprehension by Classical thought, whereas it is on the extension of the mathematical laws over the whole region of the complex numbers, within which these laws remain operative, that we have built up the function theory which has at last exhibited the Western mathematic in all purity and unity. Not until that point was reached could this mathematic be unreservedly brought to bear in the parallel sphere of our _dynamic_ Western physics; for the Classical mathematic was fitted precisely to its own stereometric world of individual objects and to _static_ mechanics as developed from Leucippus to Archimedes. The brilliant period of the Baroque mathematic—the counterpart of the Ionian—lies substantially in the 18th Century and extends from the decisive discoveries of Newton and Leibniz through Euler, Lagrange, Laplace and D’Alembert to Gauss. Once this immense creation found wings, its rise was miraculous. Men hardly dared believe their senses. The age of refined scepticism witnessed the emergence of one seemingly impossible truth after another.[67] Regarding the theory of the differential coefficient, D’Alembert had to say: “Go forward, and faith will come to you.” Logic itself seemed to raise objections and to prove foundations fallacious. But the goal was reached.