RECORDS OF MEETINGS OF 1907 321 



give God as the world ground; expanding the emotional tendency would 

 give God as universal love; and expanding the active tendency would give 

 Gcd as universal action. 



Professor Keyset discussed certain questions connected with the bases 

 of geometry. In Hilbert's 'Foundations of Geometry' culminate the efforts 

 of western thinkers from pre-Euclidean times so to analyze the space 

 intuition as to provide a simple and complete set of independent axioms 

 for the science of space, in particular for what is now distinguished as 

 Euclidean geometry. For Hilbert as for Euclid, the elements of space fall 

 into three classes or systems, points, planes, lines. Hilbert's axioms are 

 statements of certain relations that shall be the fundamental relations 

 satisfied by the elements, but — and this is very important — the elements 

 are not further defined. Accordingly, if one asks, ^Yhat are points, planes 

 and lines ? the answer is. They are any three systems of entities that satisfy 

 the axioms and require no others. Are there other entity systems than 

 those of points, planes and lines, that satisfy the Hilbert axioms ? There 

 are infinitely many such other triplets of systems of entities. One of the 

 most obvious of such triplets consists of the system of points of space ex- 

 cept a single specified point, say P, the system of all the spheres containing 

 P regarded as bereft of P and so called pseudo-spheres, and the system of 

 all the circles containing P regarded as bereft of P and so called pseudo 

 or pathological circles. These three systems perfectly satisfy the Hilbert 

 axioms, and the description of the 'space' composed of these systems of 

 entities regarded as elements is logically identical with Euclidean geometry, 

 though psychologically the geometries are as different as fire and water or 

 as red and bitter. Euclidean geometry is, therefore, one; psychologically 

 many — infinitely many. 



The Section then adjourned. 



R. S. WOODWORTH, 



Secretary. 



BUSINESS IMEETING. 

 March 4, 1907. 



The Academy met at 8: 15 P. INI. at the American Museum of Natural 

 History, Vice-President Grabau presiding. 



The minutes of the last meeting were read and approved. 



Council reported that arrangements were being made for suitable com- 



