the system of lines which pass through their corresponding points.  The point-row and pencil of the second order may be real or imaginary, but his theorems still apply.  An illustration of a correspondence of this sort, where the conic is imaginary, is given in § 15 of the first chapter.  In defining conjugate imaginary points on a line, Von Staudt made use of an involution of points having no double points.  His methods, while elegant and powerful, are hardly adapted to an elementary course, but Reye(22) and others have done much toward simplifying his presentation.

186.  Recent developments. It would be only confusing to the student to attempt to trace here the later developments of the science of protective geometry.  It is concerned for the most part with curves and surfaces of a higher degree than the second.  Purely synthetic methods have been used with marked success in the study of the straight line in space.  The struggle between analysis and pure geometry has long since come to an end.  Each has its distinct advantages, and the mathematician who cultivates one at the expense of the other will never attain the results that he would attain if both methods were equally ready to his hand.  Pure geometry has to its credit some of the finest discoveries in mathematics, and need not apologize for having been born.  The day of its usefulness has not passed with the invention of abridged notation and of short methods in analysis.  While we may be certain that any geometrical problem may always be stated in analytic form, it does not follow that that statement will be simple or easily interpreted.  For many mathematicians the geometric intuitions are weak, and for such the method will have little attraction.  On the other hand, there will always be those for whom the subject will have a peculiar glamor—­who will follow with delight the curious and unexpected relations between the forms of space.  There is a corresponding pleasure, doubtless, for the analyst in tracing the marvelous connections between the various fields in which he wanders, and it is as absurd to shut one’s eyes to the beauties in one as it is to ignore those in the other.  “Let us cultivate geometry, then,” says Darboux,(23) “without wishing in all points to equal it to its rival.  Besides, if we were tempted to neglect it, it would not be long in finding in the applications of mathematics, as once it has already done, the means of renewing its life and of developing itself anew.  It is like the Giant Antaeus, who renewed, his strength by touching the earth.”

INDEX

(The numbers refer to the paragraphs)

Abel (1802-1829), 179

Analogy, 24

Analytic geometry, 21, 118, 119, 120, 146, 176, 180

Anharmonic ratio, 46, 161, 184, 185

Apollonius (second half of third century B.C.), 70

Archimedes (287-212 B.C.), 176

Aristotle (384-322 B.C.), 169