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HISTORICAL VIEW OP THE

scurity of the subject have, been removed, but muchserious misapprehension and bitter controversy wouldhave been avoided. But the inventor of a great prin-ciple, conscious of the value and power of the ideawhich he has conceived, is not disposed to stop andenquire very closely into the exact methods by w r hich it isbest supported and proved, or even rendered intelligible:this was pre-eminently the case with Newton. The viewhe gave of it was just that which a master genius takesof a comprehensive subject: without entering into anyniceties, he expressed the whole at once by a bold meta.phor. The limit of the rate of the function’s increase hecalled the fluxion of the flowing quantity or fluent, theactual value of the quantity, space, or other result gener-ated at a given stage of the variation. He consideredthe velocity with which it flows, the space traversed in agiven time, or which would be if the velocity wereuniform : all these ideas, derived from mechanical con-siderations, could not, strictly speaking, have any placein a purely analytical investigation. Nevertheless, theyafforded by no means an unapt illustration, and wereeven so precisely applicable by analogy, that that ana-logy was preserved even to the details of the proof:moreover, such a view of the matter applied directlyto those particular subjects to which it was in the firstinstance mainly designed to apply : viz. the generationof curvilinear spaces, and, subsequently, the doctrines ofmechanical force and motion.

Leibnitz had recourse to the consideration of theinfinitely small elements simultaneously generated, of thefunction and the variable, of the abscissa and ordinate,&c.; between quantities infinitely small, and even betweenorders of these, each successively infinitely smaller thanthe preceding, he conceived finite ratios to subsist; — amethod involving various metaphysical difficulties whichhave given rise to lengthened and complex discussion,but which is after all not essential to the subject, andbelongs rather to the terms in which it has been couchedthan to the ideas signified by them. Leibnitz called