164 QUATERNIONS AND THE AUSDEHNUNGSLEHRE. 



We have next the addition of plane surfaces (Plangrossen). The 

 equation ABC + DEF + Gm = JRL 



signifies that the plane JKL passes through the point common to the 

 planes ABC, DEF, and GHI, and that the projection by parallel lines 

 of the triangle JKL on any plane is equal to the sum of the pro- 

 jections of ABC, DEF, and GHI on the same plane, the areas being 

 taken positively or negatively according to the cyclic order of the 

 projected points. This makes the equation equivalent to four ordinary 

 equations. 



Finally, we have the addition of volumes, as in the equation 



ABCD+EFGH = LTKL, 



where there is nothing peculiar, except that each term represents the 

 six-fold volume of the tetrahedron, and is to be taken positively or 

 negatively according to the relative position of the points. 



We have also multiplications as follows: The line (Liniengrosse) 

 AB is regarded as the product of the points A and B. The Plangrosse 

 ABC, which represents the double area of the triangle, is regarded as 

 the product of the three points A, B, and C, or as the product of the 

 line AB and the point C, or of BC and A, or indeed of BA and C. 

 The volume ABCD, which represents six times the tetrahedron, is 

 regarded as the product of the points A, B, C, and D, or as the 

 product of the point A and the Plangrosse BCD, or as the product of 

 the lines AB and BC, etc., etc. 



This does not exhaust the wealth of multiplicative relations which 

 Grassmann has found in the very elements of geometry. The 

 following products are called regressive, as distinguished from the 

 progressive, which have been described. The product of the Plan- 

 grossen ABC and DEF is a part of the line in which the planes ABC 

 and DEF intersect, which is equal in numerical value to the product 

 of the double areas of the triangles ABC and DEF multiplied by the 

 sine of the angle made by the planes. The product of the Linien- 

 grosse AB and the Plangrosse CDE is the point of intersection of the 

 line and the plane with a numerical coefficient representing the 

 product of the length of the line and the double area of the triangle 

 multiplied by the sine of the angle made by the line and the plane. 

 The product of three Plangrossen is consequently the point common 

 to the three planes with a certain numerical coefficient. In plane 

 geometry we have a regressive product of two Liniengrossen, which 

 gives the point of intersection of the lines with a certain numerical 

 coefficient. 



The fundamental operations relating to the point, line, and plane 

 are thus translated into analysis by multiplications. The immense 

 flexibility and power of such an analysis will be appreciated by 



