GEOMETRY: ANCIENT AND MODERN. 261 



intersection of a line and circle which are co-planar and non-intersect- 

 ing in the ordinary sense. It is impossible to exaggerate the im- 

 portance of these conceptions. Without them the beautiful fabric of 

 modern geometry would not stand a moment. It will be seen to many 

 readers, no doubt, that a fabric built upon such a foundation will have 

 very much the same stability as a 'castle in Spain.' Such, however, is 

 far from the case. The analysis by which our operations proceed is a 

 thoroughly well founded and trustworthy instrument, and when we 

 give to it the geometric interpretation which we are entirely justified in 

 doing, we find frequently that it reveals to us facts which our senses 

 unaided by its finer powers of interpretation could not have discovered. 

 These facts require for their adequate explanation the recognition of 

 the so-called imaginary elements of the figure. Let us take one more 

 illustration. If from a point outside of, but in the same plane with, a 

 circle we draw two tangents to the circle and connect the points of 

 tangency with a straight line, the original point and the line last men- 

 tioned stand in an important relation to each other and are called re- 

 spectively pole and polar with regard to the circle. Now suppose the 

 point is inside the circle. The whole construction just described be- 

 comes then geometrically impossible, but analytically we can draw from 

 a point within a circle two imaginary tangents to the circle, and simi- 

 larly we can connect the imaginary points of tangency by a straight 

 line, and this straight line is found to be a real line. Moreover, in its 

 relations to the point and circle it exhibits precisely the same properties 

 which are found in the case of the pole and polar first described. Hence 

 this point and line are also included in the general definition of pole 

 and polar. Such examples might be multiplied indefinitely, but they 

 would all go to emphasize the fact of the great power of generalization 

 which resides in the methods of analytic geometry. 



While the power of the analytic method as an instrument of re- 

 search is far greater than that of the older pure geometric method, yet 

 to many minds it lacks somewhat the beauty and elegance of that 

 method as an intellectual exercise. This is due to the fact that its 

 operations, like all algebraic operations, are largely mechanical. Given 

 the equations representing a certain geometric condition, we subject 

 these equations to definite transformations and the results obtained de- 

 note certain new geometric conditions. We have been whisked from 

 the data to the result very much as we are hurried over the country 

 in a railroad train. We may have noted the features of the country as 

 we passed through it or we may not; we arrive at our destination just 

 the same. Pure geometric research, on the other hand, resembles travel 

 on foot or horseback. We must scrutinize the landmarks and keep a 

 careful watch on the direction in which we are traveling, lest we take 

 •-a wrong turn and fail to reach our destination. The result is that we 



