=259.= There is an astonishing imagination, even in the science of mathematics.... We repeat, there was far more imagination in the head of Archimedes than in that of Homer.--VOLTAIRE. _A Philosophical Dictionary (Boston, 1881), Vol. 3, p. 40. Article “Imagination.”_ =260.= As the prerogative of Natural Science is to cultivate a taste for observation, so that of Mathematics is, almost from the starting point, to stimulate the faculty of invention. --SYLVESTER, J. J. _A Plea for the Mathematician, Nature, Vol. 1, p. 261; Collected Mathematical Papers, Vol. 2 (Cambridge, 1908), p. 717._ =261.= A marveilous newtrality have these things mathematicall, and also a strange participation between things supernaturall, immortall, intellectuall, simple and indivisible, and things naturall, mortall, sensible, componded and divisible.--DEE, JOHN. _Euclid (1570), Preface._ =262.= Mathematics stands forth as that which unites, mediates between Man and Nature, inner and outer world, thought and perception, as no other subject does.--FROEBEL. _[Herford translation] (London, 1893), Vol. 1, p. 84._ =263.= The intrinsic character of mathematical research and knowledge is based essentially on three properties: first, on its conservative attitude towards the old truths and discoveries of mathematics; secondly, on its progressive mode of development, due to the incessant acquisition of new knowledge on the basis of the old; and thirdly, on its self-sufficiency and its consequent absolute independence.--SCHUBERT, H. _Mathematical Essays and Recreations (Chicago, 1898), p. 27._ =264.= Our science, in contrast with others, is not founded on a single period of human history, but has accompanied the development of culture through all its stages. Mathematics is as much interwoven with Greek culture as with the most modern problems in Engineering. She not only lends a hand to the progressive natural sciences but participates at the same time in the abstract investigations of logicians and philosophers. --KLEIN, F. _Klein und Riecke: Ueber angewandte Mathematik und Physik (1900), p. 228._