On this account, the idea of irrational numbers—the unending decimal fractions of our notation—was unrealizable within the Greek spirit. Euclid says—and he ought to have been better understood—that incommensurable lines are “_not related to one another like numbers_.” In fact, it is the idea of irrational number that, once achieved, separates the notion of number from that of magnitude, for the magnitude of such a number (π, for example) can never be defined or exactly represented by any straight line. Moreover, it follows from this that in considering the relation, say, between diagonal and side in a square the Greek would be brought up suddenly against a quite other sort of number, which was fundamentally alien to the Classical soul, and was consequently feared as a secret of its proper existence too dangerous to be unveiled. There is a singular and significant late-Greek legend, according to which the man who first published the hidden mystery of the irrational perished by shipwreck, “for the unspeakable and the formless must be left hidden for ever.”[50] The fear that underlies this legend is the selfsame notion that prevented even the ripest Greeks from extending their tiny city-states so as to organize the country-side politically, from laying out their streets to end in prospects and their alleys to give vistas, that made them recoil time and again from the Babylonian astronomy with its penetration of endless starry space,[51] and refuse to venture out of the Mediterranean along sea-paths long before dared by the Phœnicians and the Egyptians. It is the deep metaphysical fear that the sense- comprehensible and present in which the Classical existence had entrenched itself would collapse and precipitate its cosmos (largely created and sustained by art) into unknown primitive abysses. And to understand this fear is to understand the final significance of Classical number—that is, _measure in contrast to the immeasurable_—and to grasp the high ethical significance of its limitation. Goethe too, as a nature-student, felt it—hence his almost terrified aversion to mathematics, which as we can now see was really an involuntary reaction against the _non-Classical_ mathematic, the Infinitesimal Calculus which underlay the natural philosophy of his time.