The brilliant claims of reason striving to extend its dominion beyond the limits of experience, have been represented above only in dry formulae, which contain merely the grounds of its pretensions. They have, besides, in conformity with the character of a transcendental philosophy, been freed from every empirical element; although the full splendour of the promises they hold out, and the anticipations they excite, manifests itself only when in connection with empirical cognitions. In the application of them, however, and in the advancing enlargement of the employment of reason, while struggling to rise from the region of experience and to soar to those sublime ideas, philosophy discovers a value and a dignity, which, if it could but make good its assertions, would raise it far above all other departments of human knowledge—professing, as it does, to present a sure foundation for our highest hopes and the ultimate aims of all the exertions of reason. The questions: whether the world has a beginning and a limit to its extension in space; whether there exists anywhere, or perhaps, in my own thinking Self, an indivisible and indestructible unity—or whether nothing but what is divisible and transitory exists; whether I am a free agent, or, like other beings, am bound in the chains of nature and fate; whether, finally, there is a supreme cause of the world, or all our thought and speculation must end with nature and the order of external things—are questions for the solution of which the mathematician would willingly exchange his whole science; for in it there is no satisfaction for the highest aspirations and most ardent desires of humanity. Nay, it may even be said that the true value of mathematics—that pride of human reason—consists in this: that she guides reason to the knowledge of nature—in her greater as well as in her less manifestations—in her beautiful order and regularity—guides her, moreover, to an insight into the wonderful unity of the moving forces in the operations of nature, far beyond the expectations of a philosophy building only on experience; and that she thus encourages philosophy to extend the province of reason beyond all experience, and at the same time provides it with the most excellent materials for supporting its investigations, in so far as their nature admits, by adequate and accordant intuitions.