Page:Popular Science Monthly Volume 58.djvu/268

260 points and circles associated with it, which has been developed by Brocard, Lemoine, Emmerich, Vigarié and others, was within the reach of the Greeks; but this does not destroy the force of the remark above.

The operations of mathematics are divided fundamentally into two kinds, analytic, which employ the symbolism and methods of algebra (in its broadest sense), and geometric, which consists of the operation of pure reason upon geometric figure. The two are now only partially exclusive, however, for analysis is frequently assisted by geometry, and geometric results are frequently obtained by analytic methods.

With the Greeks, Hindoos and Arabs, the only peoples who concerned themselves to any extent with mathematics until comparatively modern times, the operations of algebra and geometry were entirely distinct. With the Hindoos and Arabs algebra received more attention than geometry and with the Greeks the reverse was true. Many of the theorems of Euclid are capable of an algebraic interpretation, and this fact was probably well known, but nevertheless the theorems themselves are expressed in geometric terms and are proved by purely geometric means; and they do not, therefore, constitute a union of analysis with geometry in the modern sense.

The seventeenth century brought the invention of analytic geometry by Descartes and that of the calculus by Newton and Leibnitz. These methods opened hitherto undreamed-of possibilities in geometric research and led to the systematic study of curves of all descriptions and to a generalization of view in connection with the geometry of the right line, circle and conics, as well as of the higher curves, which has been of the greatest value to the modern mathematician. To point out by a very simple illustration the nature of this work of generalization let us consider the case of a circle and straight line in the same plane, the line being supposed to be of indefinite extent. According to the relative position of this line and circle the Greek geometer would say that the line either meets the circle, or is tangent to the circle, or that the line does not meet the circle at all. We say now, however, that the line always meets the circle in two points, which may be real and distinct, real and coincident or imaginary. Thus a condition of things which the Greek was obliged to consider under three different cases we can deal with now as a single case. This generalized view is a direct consequence of the analytic treatment of the question.

It will be seen from the illustration used above that two very important conceptions are introduced into geometry by the use of the analytic method. One of these is the conception of coincident or consecutive points of intersection, as in the case of a tangent, and the other is that of imaginary elements, as in the case of the imaginary points of