University Studies 



Vol. X JANUARY ipio No. i 



THE IMAGINARY IN GEOMETRY 1 



BY ELLERY W. DAVIS 



Let the point P with complex coordinates 



.r = .r' + u-", v = v' + ?Y' 



be represented by a "red vector" joining the ordinary "black 

 point" (x' } y') = P' to the "bine point" 



(x' + x", y' + y")=P". 



We may conceive the blue point as in a plane of imaginaries in 

 which the origin of coordinates is over (x' } y'). Similarly, Max- 

 well has superposed a velocity diagram upon a position diagram, 

 and upon that an acceleration diagram. It will sometimes be con- 

 venient to speak of a " black-vector " from the origin to the black- 

 point from which the red vector begins. Though the vector is 

 understood to be always straight the line which represents it may 

 for convenience be drawn curved. It is essential merely that it 

 go from the proper beginning to the proper end. 



2. The coordinates are merely the projections of the vector 

 upon the axes. We can change to new axes rectangular or 

 oblique, leaving the red vector unchanged. We can make the co- 

 ordinates homogeneous. We can extend the method to space 

 coordinates. All the usual formulae for transformation will hold, 



1 Portions of this paper have heen read as follows : Before the Amer- 

 ican Mathematical Society, Chicago, April, 1907; Ithaca, November, 1907; 

 Columbia, November, 1909; and before the British Association for the 

 Advancement of Science, August, 1909. 



University Sti dies. Vol. X, No. 1, January, l'.tlO. 



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