284 .REV. T. p. KIPvKMAN ON LINEAR CONSTRUCTIONS. 



these geometrical results, so easy to remember, and so easy 

 of demonstration, been previously laid down ? 



We return to our paradigm 2 + a?o . y? . ^^| . x^y^ . yiZi.Zf^oik . x^ 

 yT .z^. I9 zr. 0, considering it as a purely geometrical datum. 

 It is necessary that we should both interpret and prove 

 this proposition (2 m 0), without borrowing any aid from 

 arithmetic, and that we should show how a surface of the 

 second order is given thereby, and can be therefrom con- 

 structed. The interpretation is more easy to be given than to 

 be understood. Every one can conceive the reality denoted 

 by xy, a parallelogram having a certain angle, or that re- 

 presented hjxy Zf a. prism whose edges have given inclina- 

 tions. But what geometrical entity is x y z w, or a; x 2 y? 

 It is assuredly a volume or solid of four dimensions ; for 

 the product of four right lines can be only a figure of some 

 kind, which does not straightway become an absurdity 

 because the inhabitants of this planet find it difficult to 

 imagine its existence. In like manner *•? . yl , z\ . Xzy^ . ytZ^ 

 s^^ .Xe.yy. Zs, the product of fifteen right lines, represents 

 a volume of fifteen dimensions, and the proposition before 

 us (2 zn 0) asserts that a given number of such volumes, 

 constructed of course in space of fifteen dimensions, and 

 having edges equal to certain lines given in common space, 

 viz., the co-ordinates of our ten points, 1 2 3... 9 0, have a 

 sum equal to zero. 



If the reader feel distressed with the effort to imagine 

 such transcendental volumes in space of more than three 

 dimensions, that is no affair of mine ; my duty being, not to 

 supply him with additional senses, but with sound argu- 

 ments, of which he is competent to judge with even fewer 

 than five. Had the reader been so unhappy as to enter 

 this world deprived of the sense of touch, he would probably 

 have been as much in the dark about the geometrical im' 

 port o{ x . y . z, as he now is, being endowed with only five 

 senses, concerning the real existence of these solids of fifteen 



