782 PROCEEDINGS OF THE AMERICAN ACADEMY. 



There is one difference, however, for in Case I this invariant was ex- 

 pressible as a ratio of distances, and consequently was independent of the 

 unit of length assumed. In this case 8 is not so expressible. It is not, 

 therefore, an absolute invariant, but depends upon the units. 



39. Area. The line area again will be defined by the property that 

 triangles having the same vertex and bases on the same line have 

 areas proportional to their bases. By the same argument as used in 

 Case I it is seen that 



Area ABC _ ahC 

 Area A'B'C ~ a'b'C" 



and the same argument will show that the factor of proportionality is 

 the same for the whole plane and hence can be assumed to be unity, 

 and we write 



Area ABC = ahC. (27) 



It is to be kept in mind that here the area is not equal to the perim- 

 eter, for the perimeter is zero. 



By duality the angle area of the triangle ABC is 



Angle area ABC = aBG. (28) 



§ 4. Products. 



40. We have represented a line by a segment joining two of its 

 points A and B and have used the notation 



AB 



to represent this segment. We shall now consider this expression as 

 the product of two unit points and seek to determine its laws of com- 

 bination. When the points are not unit points we shall take the 

 product to be the segment "^ AB multiplied by the product of the mag- 

 nitudes of the points. 



The operation of multiplication will be denoted by writing the quan- 

 tities together as is done in ordinary algebra. A dot between quanti- 

 ties will indicate how the quantities are to be associated, e. g. AB • CD 

 will be used to indicate that the two products AB and CD are to 

 be multiplied together. 



' It should here be kept in mind that segment is not to be confused with 

 its length, which measures only the magnitude of the segment. The length 

 of a segment can be zero without the segment being zero, which is the case 

 of a segment of a line passing through F. 



