Dictionary of Scientific Biography Dictionary of Scientific Biography

 geometrical or mechanical) by which a figure is generated and rendered easy to conceive.”62 The “written record of [Newton's] first researches in the interlocking structures of Cartesian coordinate geometry and infinitesimal analysis”63 shows him to have been establishing “the foundations of his mature work in mathematics” and reveals “for the first time the true magnitude of his genius.”64 And in fact Newton did contribute significantly to analytic geometry. In his 1671 Methodis fluxionum, he devoted “Prob. 4: To draw tangents to curves” to a study of the different ways in which tangents may be drawn “according to the various relationships of curves to straight lines,” that is, according to the “modes” or coordinate systems in which the curve is specified.65 Newton proceeded “by considering the ratios of limit-increments of the co-ordinate variables (which are those of their fluxions).”66 His “Mode 3” consists of using what are now known as standard bipolar coordinates, which Newton applied to Cartesian ovals as follows: Let x, y be the distances from a pair of fixed points (two “poles”); the equation a ? (e/d)x - y = 0 for Descartes's “second-order ovals” will then yield the fluxional relation ?(e/d)? - ? = 0 (in dot notation) or ?em/d - n = 0 (in the notation of the original manuscript, in which m, n are used for the fluxions ?, ? of x, y). When d = e, “the curve turns out to be a conic.” In “Mode 7,” Newton introduced polar coordinates for the construction of spirals; “the equation of an Archimedean spiral” in these coordinates becomes (a/b)x = y, where y is the radius vector (now usually designated r or p) and x the angle (θ or φ). Newton constructed equations for the transformation of coordinates (as, for example, from polar to Cartesian), and found formulas in both polar and rectangular coordinates for the curvature of a variety of curves, including conics and spirals. On the basis of these results Boyer has quite properly referred to Newton as “an originator of polar coordinates.”67 Further geometrical results may be found in Enumeratio linearum tertii ordinis, first written in 1667 or 1668, and then redone and published, together with De quadratura, as an appendix to the Opticks (1704).68 Newton devoted the bulk of the tract to classifying cubic curves into seventy-two “Classes, Genders, or Orders, according to the Number of the Dimensions of an Equation, expressing the relation between the Ordinates and the Abscissae; or which is much at one [that is, the same thing], according to the Number of Points in which they may be cut by a Right Line.” In a brief fifth section, Newton dealt with “The Generation of Curves by Shadows,” or the theory of projections, by which he considered the shadows produced “by a luminous point” as projections “on an infinite plane.” He showed that the “shadows” (or projections) of conic sections are themselves conic sections, while “those of curves of the second genus will always be curves of the second genus; those of the third genus will always be curves of the third genus; and so on ad infinitum.” Furthermore, “in the same manner as the circle, projecting its shadow, generates all the conic sections, so the five divergent parabolae, by their shadows, generate all the other curves of the second genus.” As C. R. M. Talbot observed, this presentation is “substantially the same as that which is discussed at greater length in the twenty-second lemma [book III, section 5] of the Principia, in which it is proposed to ‘transmute’ any rectilinear or curvilinear figure into another of the same analytical order by means of the method of projections.”69 The work ends with a brief supplement on “The Organical Description of Curves,” leading to the “Description of the Conick-Section by Five Given Points” and including the clear statement, “The Use of Curves in Geometry is, that by their Intersections Problems may be solved” (with an example of an equation of the ninth degree). Newton in this tract laid “the foundation for the study of Higher Plane Curves, bringing out the importance of asymptotes, nodes, cusps,” according to Turnbull, while Boyer has asserted that it “is the earliest instance of a work devoted solely to graphs of higher plane curves in algebra,” and has called attention to the systematic use of two axes and the lack of “hesitation about negative coordinates.”70 Newton's major mathematical activity had come to a halt by 1696, when he left Cambridge for London. The Principia, composed in the 1680's, marked the last great exertion of his mathematical genius, although in the early 1690's he worked on porisms and began a “Liber geometriae,” never completed, of which David Gregory gave a good description of the planned whole.71 For the most part, Newton spent the rest of his mathematical life revising earlier works. Newton's other chief mathematical activity during the London years lay in furthering his own position against Leibniz in the dispute over priority and originality in the invention of the calculus. But he did respond elegantly to a pair of challenge problems set by Johann [I] Bernoulli in June 1696. The first of these problems was “mechanico-geometrical,” to find the curve of swiftest descent. Newton's answer was brief: the “brachistochrone” is a cycloid. The second problem was to find a curve with the following property, “that the two segments [of a right line drawn from a given point through the curve], being Image Size: 240x320 480x640 960x1280 1440x1920 1920x2560 