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 unavailing?  Can we not see the hand of Providence, all through history, leading men wiselier than they knew?  If not, may it not be possible that we have read the wrong book,—­the Universal Gazetteer, perhaps, instead of the true History?  When Plato and Plato’s followers wrought out the theory of those Conic Sections, do we imagine that they saw the great truth, now evident, that every whirling planet in the silent spaces, yes, and every falling body on this earth, describes one of these same curves which furnished to those Athenian philosophers what you, my practical friend, stigmatize as idle amusement?  Comfort yourself, my friend:  there was many a Callicles then who believed that he could better bestow his time upon the politics of the state, neglecting these vain speculations, which to-day are found to be not quite unprofitable, after all, you perceive.

And so in the instance which suggested these reflections, all this eager study of unmeaning curves (if there be anything in the starry universe quite unmeaning) was leading gradually, but directly, to the discovery of the most wonderful of all mathematical instruments, the Calculus preeminently.  In the quadrature of curves, the method of exhaustions was most ancient,—­whereby similar circumscribed and inscribed polygons, by continually increasing the number of their sides, were made to approach the curve until the space contained between them was exhausted, or reduced to an inappreciable quantity.  The sides of the polygons, it was evident, must then be infinitely small.  Yet the polygons and curves were always regarded as distinct lines, differing inappreciably, but different.  The careful study of the period to which we refer led to a new discovery, that every curve may be considered as composed of infinitely small straight lines.  For, by the definition which assigns to a point position without extension, there can be no tangency of points without coincidence.  In the circumference of the circle, then, no two of the points equidistant from the centre can touch each other; and the circumference must be made up of infinite all rectilineal sides joining these points.

A clear conception of this fact led almost immediately to the Method of Tangents of Fermat and Barrow; and this again is the stepping-stone to the Differential Calculus,—­itself a particular application of that instrument.  Dr. Barrow regarded the tangent as merely the prolongation of any one of these infinitely small sides, and demonstrated the relations of these sides to the curve and its ordinates.  His work, entitled “Lectiones Geometricae,” appeared in 1669.  To his high abilities was united a simplicity