This book seeks to resemble such a convenient аlpine railway up into the world of mathematics. It can give nothing to the man in whom no chord vibrates rcsponsively to the peculiar harmonies of this realm of thought, who accepts with inward indifference the technological and scientific acquisitions which he uses cvciy day of his life, and which were made possible only by the fruitful contributions of mathematics. But we venture to hope that there will also be readers who will not remain entirely untouched by the aura which breathes in this realm that affects most people as being so unapproachable. It is possible, if fate so wills it, that there may even be some young man with unspent energies and with the necessary ability, who will feel the urge to pursue the subject more seriously, spurred by the distant vision offered to him here. For I confess that what is offered here will not be able to satisfy him, nor should it. From the railway route along which we here guide the reader are excluded, because of their difficulty of access, many of the most important mountain heights and lines of advance.