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markable points and circles associated with it, which has been developed 

 by Brocard, Lemoine, Emmerich, Vigarie 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 con- 

 cerned 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 atten- 

 tion 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 geome- 

 try 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 geom- 

 etry of the right line, circle and conies, 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 im- 

 portant conceptions are introduced into geometry by the use of the 

 analytic method. One of these is the conception of coincident or con- 

 secutive 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 



