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HISTORICAL V1KW OF THE

solve certain geometrical questions respecting curves,their tangents and normals, their rectification andquadrature. The incremental triangle conceived byBarrow was nothing else than a sort of geometricalpicture of the limiting ratio of the increments, or thefluxions of the curve or tangent, its ordinate and ab-scissa. The general method, therefore, supplied atonce the means of drawing tangents and normals tocurves, the ratio which determined the tangent beingsimply the differential coefficient of the equation to thecurve. This idea was soon generalised, and the contactof curves with other than rectilinear tangents wasfound to be expressed by an extension of the methodsof the same calculus. The notion of circles whichosculated with curves, or had the same curvature withthem at different points in their periphery, was alreadyfamiliar to the mind of the geometer : the new systemafforded a more (general mode of viewing the subject,and enabled the analyst to compare in this way succes-sive orders of contact. Similar conclusions were de-duced respecting curved surfaces and solid problems.But though the calculus was thus in its first designlimited to such subjects, and borrowed much of its toneand character from those geometrical purposes to whichit was subservient, yet it soon took a wider range, andwas seen to apply to functions of the most comprehen-sive class. Des Cartes had provided the means ofbringing geometry within the dominion of the calculus,by reducing it to algebraic equations. But to bring therelations of quantity in general under its power, and tofurnish the means of considering functions of all kindsin such a way as to investigate their forms and modi-fications by means of the calculus, the further labours ofanalysts were yet demanded, and more general viewsrequired to be taken.

Some of the simpler and more important of theserelations were, however, perceived and investigated atthe earliest stage of the progress of the calculus. Suchwas the consideration of maxima and minima, to which,