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SCIENCE 



[N. S. Vol. XLHI. No. 1113 



this is precisely the same as the group of 

 linear substitutions, if points in space are 

 given in rectilinear coordinates. Prom this 

 it was not a long stretch to the conjecture 

 that the properties of objects unchanged by 

 projection must be expressible in some way 

 in terms of the invariants discovered by 

 algebraists. To work out this thought de- 

 manded the ardor and mathematical in- 

 genuity of a race of intellectual giants like 

 Cayley and Sylvester, Aronhold in Berlin, 

 Hermite, Clebsch and Brioschi. Through 

 their toil a special calculus was developed, 

 and some progress made toward answering 

 the central question : What, under the pro- 

 jective group, are the different possible in- 

 variant properties of single algebraic loci, 

 and what the chief invariant relations of 

 two or more loci or systems of loci ? Here 

 then was established a definite standard, by 

 which it could be judged whether geometry 

 was a science, or only the ideal program of 

 a science. The group of operations, the 

 simplest objects to be considered, and the 

 invariant relations of those objects under 

 the group : these covered the content, at 

 least of projective geometry. 



Probably no climax of equal significance 

 for pure mathematics has been reached 

 since Newton and Leibnitz took the scat- 

 tered fragments of a theory of limits and 

 from them created the differential and in- 

 tegral calculus. It was in 1872 that Felix 

 Klein published from the University of 

 Erlangen a brief program, or formal address 

 upon assuming a professorate. The title 

 was: Comparative observations upon mod- 

 ern geometrical investigations, and its cen- 

 tral thesis was in essence the formal defi- 

 nition, just now mentioned, of geometry. 

 There are many sorts of geometry, but all 

 are alike in this, that each studies its own 

 peculiar group of transformations, and 

 seeks to discover and classify the properties 

 of objects which are invariant under all the 



transformations of its group. This was 

 then verified by a survey of all kinds of 

 geometry developed up to that epoch. 



Of especial interest is of course our ele- 

 mentary geometry, the standard Euclidean. 

 We know its objects; what is the group that 

 it studies? Klein answers: The absolute 

 position in space may be changed, for that 

 change no one can distinguish. An ex- 

 change of right for left, as in the space seen 

 in a mirror, does not alter relations that 

 we call geometric. Moreover all size is 

 merely relative, hence uniform expansion 

 or shrinkage in all directions is an opera- 

 tion of the group. Hence rigid motion in- 

 cluding rotations, homogeneous expansions 

 and reflections against a plane, those with 

 their myriaform resultants constitute the 

 group of ordinary geometry. I have men- 

 tioned the group of projective geometry; 

 others are the geometry of circles and 

 spheres, admitting to its group all opera- 

 tions of elementary geometry and in addi- 

 tion all reflections upon spherical mirrors; 

 the two kinds of non-Euclidean geometry, 

 the four-dimensional geometry of lines, in- 

 augurated by Pliicker, and the geometry of 

 contact-transformations, deflned and begun 

 by Sophus Lie, of Norway. Many others 

 can easily be noted and named, all fitting 

 Klein's description in so far as they are 

 developed, by any student of mechanics, 

 hydrodynamics, optics or indeed any per- 

 fected theory in physics. As science tends 

 to become deductive, and as geometry is the 

 most complete type of a deductive science, 

 and now since Klein's program elucidates 

 the ideal or norm of geometry, so it may 

 well arrest the attention and illuminate the 

 procedure of every systematic scientific 

 investigator. 



It must have been this mode of con- 

 ceiving the essence of geometry that was 

 before his mind when Gino Loria, the his- 

 torian of modern geometry, wrote the fol- 



