702 GEOMETRY modes of transforming one plane curve into another, by making the given curve a peculiar basis for the locus of a new curve. They thus transformed the circle into all the conic sections, without any reference to a cone. The great Newton also invented a means to the same end, so that the consideration of the ellipse and parabola became independent of that of any solid. Thus these methods, es- pecially that of Le Poivre, anticipated descrip- tive geometry, and perhaps prepared the way for it. In 1700 Parent generalized the method of Descartes from representing a line to repre- senting a curve surface by an equation be- tween the distances of a point in the surface from three given planes, at right angles to each other ; but this was not methodically arranged, and it was left for Clairaut, in 1731, to finish this great step. Meanwhile Newton's fluxions and Leibnitz's differential calculus had come into use, and Newton, Maclaurin, and Cotes had made the most exhaustive investigation into curves of the third degree, and many fine dis- coveries in regard to curves in general. The enthusiasm which Newton's example aroused in England and Scotland for pure geometry was followed by a lull of about a century, when Mpnge by his "Descriptive Geometry" gave the whole study new life. The essence of de- scriptive geometry lies in the transmutation of figures, the reduction of* geometry of three dimensions to geometry in a plane. One beau- tiful example of this branch of science may be found in linear perspective, which simply projects the points of a solid upon a plane, by straight lines of light from the eye. Carnot, at the beginning of this century, in his "Ge- ometry of Position" and "Theory of Trans- versals," also introduced valuable methods ; in the first showing how to indicate the direction of lines more exactly by the use of positive and negative signs, and how to use the idea of mo- tion more effectively than before in geometry ; in the second introducing that general form of the theory of transversals, i. e., of the intersec- tions of a system of lines by one not belonging to the system, which Ohasles employs so hap- pily in his Geometrie superieure (1852). This writer develops two principles in the corre- spondence of figures: one, the principle of duality, by which for a given figure a second is found such that points, planes, and straight lines in one correspond to planes, points, and lines in the other ; the second, the principle of homography, by which for any figure a second is drawn such that points, planes, and lines in one correspond to points, planes, and lines in the other ; the utility of each being to trans- fer the demonstrations of truth in one figure to the problems of another figure. "We have alluded to the difficulty of appreciating the value of some of the new methods of treating geometry which have been discovered or in- vented in recent times, more especially the " quaternions" of Sir W. R. Hamilton and the "doctrine of extension" of Dr. H. Grassmann. From a somewhat protracted study of both systems, the present writer is satisfied that any attempt to give a condensed account of them would only serve to perplex the reader. Es- pecially is it difficult to comprehend either system without a more than ordinary acquaint- ance with the history of mathematical science during the present century, and particularly with the efforts to give a geometrical interpre- tation of what are called in algebra imaginary quantities. The beginner in geometry will find many text books, of which none is more popu- lar than the "Elements of Geometry and Trigonometry," by Prof. Charles Davies, from the works of A. M. Legendre (New York, 1858). Much more condensed and suggestive is an "Elementary Treatise on Plane and Solid Geometry," by Prof. Benjamin Peirce (Boston, 1858). An easier treatise than either of these, by Prof. G. R. Perkins, has been published in New York. The true style of Greek ge- ometry may be found in Playfair's " Euclid." For advanced studies the following list of works is recommended : " Modern Geometry," by Mulcahy (London, 1859), giving some idea of the new methods, but not employing analyt- ical geometry ; " Elementary Treatise on Plane and Spherical Trigonometry," and "Element- ary Treatise on Curves, Functions, and Forces," by Benjamin Peirce (Boston, 1858), giving in its most condensed form the necessary intro- ductory knowledge of the notation of trigo- nometry, analytical geometry, and the calculus ; "Analytical Geometry," by Charles Davies (New York, 1855), giving a more popular ex- pression of the same knowledge ; a " Treatise on Conic Sections, containing an Account of some of the most important Modern Algebraic and Geometric Methods," by G. Salmon (Lon- don, 1855) ; a " Treatise on the Higher Plane Curves," by the same author ; Sir Isaac New- ton's Enumeratio Linearum Tertii Ordinis; Sir W. R. Hamilton's "Lectures on Quater- nions" (Dublin, 1853) and "Elements of Qua- ternions " (London, 1866) ; " An Elementary Treatise on Quaternions," by P. G. Tait (Ox- ford, 1867) ; Chasles's Traite de geometric su- perieure (Paris, 1852), Memoire de geometrie sur les proprietes geometriques des coniques spheriques (Brussels, 1831 ; soon after trans- lated into English), and Aper$u historique sur Vorigine et le developpement des metJiodes en geometrie (Brussels, 1837; translated into Ger- man, Halle, 1839; a work which will richly repay a close study) ; Cam of s Geometrie de position (Paris, 1803), De la correlation des figures de geometrie (1801), and Memoire sur la relation qui existe entre les distances respec- tives de cinque points quelconques pris dans Vespace, suivi d"*un essai sur la theorie des trans- versales (1806, and 4to, 1815); Monge's Geo- metrie descriptive (Paris, 7th ed., 1846, inclu- ding Application de Valgebre d la geometrie) ; Systematische Entwiclcelung der Abhangiglceit geometriscTier Gestalten von einander, mit Be- rucknehtigung der Arbeiten alter und ncucr
