One of the objects of a Congress such as this is to exhibit the connecting links of the different departments of Science, taking that term in its widest acceptation. Hence the organisers of this meeting have taken care that the relations between the different sections shall be exhibited in broad relief. In undertaking a sketch of this kind, which must necessarily be somewhat indefinite in character, we must forget that all is in all; as far as algebra and analysis are concerned, the follower of Pythagoras would be disconcerted by the extent of his task, remembering the celebrated formula of the School: “Things are numbers.” From this point of view my subject would be in-exhaustible. But I shall make no such pretensions, and for the best of reasons. By passing in rapid review the development of our science throughout the ages, and particularly in the last century, I hope to be able to give an adequate indication of the part played by mathematical analysis in its relations to other sciences.