hing 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 spiral, first fully discussed by James Bernouilli. This
curve possesses the property of reproducing itself in a variety of
curious and interesting ways; for which reason Bernouilli wished it
inscribed upon his tomb, with the motto,--_Eadem mutata resurgo_. Shall
we wisely shake our heads at all this, as unavaili