ng? 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