Long lethargy and intellectual inanition brooded over Christian Europe.  The darkness of the Middle Ages reached its midnight, and slowly the dawn arose,—­musical with the chirping of innumerable trouveres and minnesingers.  As early as the Tenth Century, Gerbert, afterwards Pope Sylvester II., had passed into Spain and brought thence arithmetic, astronomy, and geometry; and five hundred years after, led by the old tradition of Moorish skill, Camille Leonard of Pisa sailed away over the sea into the distant East, and brought back the forgotten algebra and trigonometry,—­a rich lading, better than gold-dust or many negroes.  Then, in that Fifteenth Century, and in the Sixteenth, followed much that is of interest, not to be mentioned here.  Copernicus, Galileo, Kepler,—­we must pass on, only indicating these names of men whose lives have something of romance in them, so much are they tinged with the characteristics of an age just passing away forever, played out and ended.  The invention of printing, the restoration of classical learning, the discovery of America, the Reformation, followed each other in splendid succession, and the Seventeenth Century dawned upon the world.

The Seventeenth Century!—­forever remarkable alike for intellectual and physical activity, the age of Louis XIV. in France, the revolutionary period of English history, say, rather, the Cromwellian period, indelibly written down in German remembrance by that Thirty-Years’ War,—­these are only the external manifestations of that prodigious activity which prevailed in every direction.  Meanwhile the two sciences of algebra and geometry, thus far single, each depending on its own resources, neither in consequence fully developed, as nothing of human or divine origin can be alone, were united, in the very beginning of this epoch, by Descartes.  This philosopher first applied the algebraic analysis to the solution of geometrical problems; and in this brilliant discovery lay the germ of a sudden growth of interest in the pure mathematics.  The breadth and facility of these solutions added a new charm to the investigation of curves; and passing lightly by the Conic Sections, the mathematicians of that day busied themselves in finding the areas, solids of revolution, tangents, etc., of all imaginable curves,—­some of them remarkable enough.  Such is the cycloid, first conceived by Galileo, and a stumbling-block and cause of contention among geometers long after he had left it, together with his system of the universe, undetermined.  Descartes, Roberval, Pascal, became successively challengers or challenged respecting some new property of this curve.  Thereupon followed the epicycloids, curves which—­as the cycloid is generated by a point upon the circumference of a circle rolled along a straight line—­are generated by a similar point when the path of the circle becomes any curve whatever.  Caustic curves, spirals without number, succeeded, of which but one shall claim our notice,—­the logarithmic