2:^0 KANSAS ACADEMY OP SCIENCE, 



ON CERTAIN METHODS OF THE GEOMETRY OF POSITION.* 



By Arnold Emch, Boulder, Colo. 

 Read before the Academy, by title, at lola, December 31, 1901. 



I. INTRODUCTION. 



'T^HE present tendency of scientific specialization is generally justi- 

 ■^ tied ; it is in the interest of pure science. This is especially true 

 of the devolopment of mathematical branches, where everything for- 

 eign to the fundamental axioms is carefully discarded. A science in 

 which this spirit is applied intentionally and systematically becomes 

 a branch of philosophy; for instance, Grassmann's Linende Ans- 

 dehnmgslehre, Lobatschesky's Geometry, v. Standt's Geometrie der 

 Lage, Weierstrasse's Theory of Functions, etc. As such they are of 

 the greatest importance for the rigorous development of mathemat- 

 ical thought, and their value cannot be overestimated. It is a ques- 

 tion, however, whether so-called pure methods are always what they 

 pretend to be, and whether they are always to be recommended for 

 pedagogical purposes. To take an example : Is it well to consider 

 certain configurations in space in order to simplify the demonstration 

 of propositions in plane geometry; or is it necessary, in order to 

 be consistent, to apply only previously known propositions of plane 

 geometry ? 



There is no doubt in my mind that the first can be done in a suc- 

 cessful and consistent manner. In this paper I shall attempt to show 

 the value of such methods for the teaching of the geometry of posi- 

 tion. At this point I desire to say that descrij)tive geometry, as well 

 as projective geometry, or the geometry of position, ought to be made 

 regular courses in the mathematical departments of real universities. 

 A knowledge of elementary descriptive geometry gives the student an 

 invaluable power for the mastery of the more diflScult problems of 

 projective geometry and of higher geometry ingeneral.f In what fol- 

 lows I shall assume the knowledge of ordinary descriptive geometry. 



II. ADVANCED PLANE GEOMETRY. 



There are a great number of propositions in plane geometry which 

 appear in their natural light when considered as projections of figures 

 in space. As such they are independent of metrical relations and it 

 is unnatural to prove them by equations. As an example, I may men- 

 tion the homology of triangles. 



*A paper read before the American Association for the Advancement of Science, at Denver, 

 Colo., August, 1901. 



t Regular courses in descriptive and projective geometry are now offered at nearly all uni- 

 versities of continental Europe. 



