Page:A History of Mathematics (1893).djvu/229

 Newton exemplifies this last assertion by the problem of tangency: Let AB be the abscissa, BC the ordinate, VCH the tangent, Ec the increment of the ordinate, which produced meets VH at T, and Cc the increment of the curve. The right line Cc being produced to K, there are formed three small triangles, the rectilinear CEc, the mixtilinear CEc, and the rectilinear CET, Of these, the first is evidently the smallest, and the last the greatest. Now suppose the ordinate bc to move into the place BC, so that the point c exactly coincides with the point C; CK, and therefore the curve Cc, is coincident with the tangent CH, Ec is absolutely equal to ET, and the mixtilinear evanescent triangle CEc is, in the last form, similar to the triangle CET; and its evanescent sides CE, Ec, Cc, will be proportional to CE, ET, and CT, the sides of the triangle CET. Hence it follows that the fluxions of the lines AB, BC, AC, being in the last ratio of their evanescent increments, are proportional to the sides of the triangle CET, or, which is all one, of the triangle VBC similar thereunto. As long as the points C and c are distant from each other by an interval, however small, the line CK will stand apart by a small angle from the tangent CH, But when CK coincides with CH, and the lines CE, Ec, cC reach their ultimate ratios, then the points C and c accurately coincide and are one and the same. Newton then adds that "in mathematics the minutest errors are not to be neglected." This is plainly a rejection of the postulates of Leibniz. The doctrine of infinitely small