PHYSICAL AND MATHEMATICAL SCIENCES. 223

others, of introducing the phraseology of infinite quan-tity, and estimating the values which algebraical expres-sions assume in different cases, when we suppose a terminvolved in them to become infinitely great or infinitelysmall. In his investigations we may trace the germs ofthose methods which, very soon after, in the hands ofNewton, received an expansion fitting them for the pur-poses of analysing the most complex laws of physicalphenomena. But the principles which Wallis hadadopted, were not yet so far developed as to be under-stood in the full extent, or to exhibit the power whichthey really possessed. He was, however, enabled topursue the subject of quadrature upon much moregeneral grounds than aqy of his predecessors. Hitherto,as we have seen, geometers had succeeded in assigningin finite terms the value of the area of the spaces boundedby curve lines, only in a few limited and very simplecases. Des Cartes had generalised the mode of con-ceiving the construction of all curves ; and it hencefollowed, that by certain algebraical processes, thoseapplications of the method of limits which had beendevised by Kepler and Cavalieri for expressing the areas,might be extended in a manner equally general, by vir-tue of the relation subsisting between the abscissa andordinate,—the two variables, which enter Des Cartes equation.

Wallis found, that in all cases where the value of oneof these could be expressed in terms of the other, withoutinvolving fractional or negative indices, he could exhibitthe value of the area, or quadrature of the curve, in finiteterms.

A partial extension was given to this method byNicholas Kauffman, (more commonly known by hisLatinised name, Mercator, ) who devised a method ofreducing some of the expressions into a continued seriesof terms. In this way he obtained the quadrature ofthe hyperbola, in 1667.

Wallis, however, was anxious to extend his solutionsbeyond this limited portion of the subject. He saw a