PHYSICAL AND MATHEMATICAL SCIENCES. 221

same kind were pursued, with his usual ability, by Dr.Barrow.

Roberval devised a method of drawing tangents tocurves, grounded on geometrical principles, but applica-ble to those cases only in which the curve is constructedby the intersection of two lines which go on increasingin a given ratio to each other. The essential principleis, that, in such cases, the increments of the two linesbeing completed into a parallelogram, its diagonal willbe the tangent. But this can only be established by anapplication of the principle of limits.

Barrow originated the idea of what has been calledthe incremental triangle, and is, perhaps, the best adaptedto the purposes of general illustration. The notion thata portion of a curve may be taken, so small that it maywithout error be considered as a straight line, is onewhich the mind readily admits: it is, nevertheless,utterly untrue, and contradictory to the first principles ofgeometry. It is one of those instances of which wehave already noticed some, where language absolutelycontradictory in terms may yet be conventionally em-ployed to avoid circumlocution. The idea really atthe basis of such expressions is that a straight line isthe limit to which a portion of a curve continuallydiminished approaches. But, adopting the incorrectbut more convenient phraseology, the small incrementof the curve, and the corresponding increments of theabscissa and ordinate, form a small triangle. If, fromthe relation of the two latter, we express that of theirinfinitely small increments, we have, upon the principlesof plane trigonometry, the position of the hypotlienuse,or the direction of the tangent to the curve. Nearly thesame view of the subject was taken by Fermat .

John Wallis was one of the most eminent mathema-ticians of the period of which we are treating, and madeimportant advances towards the solution of those greatproblems to which the attention of geometers was thenso powerfully drawn. But a limit seemed yet to beplaced to the success of their efforts, which it wasreserved for the genius of Newton to pass.