"What Cavalieri and Roberval did for the integral calculus, Descartes (1596-1650) accomplished for the differential branch by his work on the application of algebra to geometry. Cartesian coordinates made possible the extension of investigations on the drawing of tangents to special curves to the more general problem for curves of any kind. To this must be added the fact that he habitually used the index notation; for this had a very great deal to do with the possibility of Newton's discovery of the general binomial expansion and of many other infinite series. Descartes failed, however, to make any very great progress on his own account in the matter of the drawing of tangents, owing to what I cannot help but call an unfortunate choice of a definition for a tangent to a curve in general. Euclid's circle-tangent definition being more or less hopeless in the general case, Descartes had the choice of three:—"