concerning the contents of Polygons and Polyhedrons. 341 

 or 



{abY + {acY+{bcY-2{abf . {acf-2{abf{bcf-2{acY . {bcf ; 



the negative of which is the well-kuown form expressing the 

 square of four times the area of the triangle abc. 



There is another and more general theorem of Staudt for two 

 triangles not in the same plane, which may be obtained with 

 equal facility. In fact, if we start from the determinant 



and add to each column respectively the last column multiplied 

 ^y ^^1^ ^^'2> ^?3' respectively, we arrive at the form 



And considering ^j, t/j ; ^^, ij^ ; ^3, 7/3 as the coordinates of a, yS, 7, 

 the projections upon the plane of abc of a triangle ABC, whose 

 plane intersects the former plane in the axis of y, and makes 

 with that plane an angle whose tangent is (e), it is easily seen 

 that this detei'minant is term for term identical with the deter- 

 minant 



which therefore expresses — 16 times the product of the triangles 

 abc and u^<y, i. e. abc x ABC x cosine of the angle between the two. 

 A similar method, if we ascend from sensible to rational geometry, 

 may be given for expressing in terms of the distances the product 

 of any two pyramids (in a hyperspace) by the cosine of the angle in- 

 cluded between the two infinite sj)aces=i'- in which they respectively 

 lie. To pass from the cases which have been considered of two 

 triangles to two polygons, or of two tetrahedrons to two polyhe- 

 drons, generally presents no difficulty ; and for Professor Staudt's 



• In rational or universal geometry, that which is commonly termed infi- 

 nite space (as if it were something ahsolute and unique, and to which, by 

 the conditions of our being, the representative power of the understanding 

 is limited), is regarded as a single liomaloid related to a jjlanc, preeisclv in 

 the same way as a plane is to a right line. Universal geometry brings home 

 to the mind with an irresistible force of conviction the tnith of the Kantian 

 doctrine of localitv. 



