ONE:Returning to our more immediate subject, we must observe that the Pythagoreans did not maintain, in anticipation of modern quantitative science, that all things are determined by number, but that all things are numbers, or are made out of numbers, two propositions not easily distinguished by unpractised thinkers. Numbers, in a word, were to them precisely what water had been to Thales, what air was to Anaximenes, the absolute principle of existence; only with them the idea of a limit, the leading inspiration of Greek thought, had reached a higher degree of abstraction. Number was, as it were, the exterior limit of the finite, and the interior limit of the infinite. Add to this that mathematical studies, cultivated in Egypt and Phoenicia for their practical utility alone, were being pursued in Hellas with ever-increasing ardour for the sake of their own delightfulness, for the intellectual discipline that they supplieda discipline even12 more valuable then than now, and for the insight which they bestowed, or were believed to bestow, into the secret constitution of Nature; and that the more complicated arithmetical operations were habitually conducted with the aid of geometrical diagrams, thus suggesting the possibility of applying a similar treatment to every order of relations. Consider the lively emotions excited among an intelligent people at a time when multiplication and division, squaring and cubing, the rule of three, the construction and equivalence of figures, with all their manifold applications to industry, commerce, fine art, and tactics, were just as strange and wonderful as electrical phenomena are to us; consider also the magical influence still commonly attributed to particular numbers, and the intense eagerness to obtain exact numerical statements, even when they are of no practical value, exhibited by all who are thrown back on primitive ways of living, as, for example, in Alpine travelling, or on board an Atlantic steamer, and we shall cease to wonder that a mere form of thought, a lifeless abstraction, should once have been regarded as the solution of every problem, the cause of all existence; or that these speculations were more than once revived in after ages, and perished only with Greek philosophy itself.After the explanations required of us they allowed us to go on. The incessant roar of the guns made the girl tremble for fear, and the stinging smoke made her cough. After much trouble we got at last as far as Herstal, where I had promised her a short rest.