l895-] BAKER — DIRECTED MAGNITUDES. 165 



Making use of the space concepts of symmetry and completeness 

 and carrying out this operation of multiplication in utmost symmetry 

 and completeness, we have as the only possible definite result, 

 (i _|_ _;)" = 2.718.... = (?, one of the pair of numerical values so 



obtrusive in all nature. 



If we bunch the positions which constitute the point and distribute 

 the points into the most symmetrical shape, the number of positions 

 or the weight will be Ictt, a multiple of the other numerical value so 

 omnipresent in the universe, tt = 3. 14 1592.. . 



As might be expected from this investigation, no other numerical 

 operations being indicated, there are no other numerical values com- 

 parable with e and Ti. 



V. 



A vector is a quantity having size, sense and direction of extension, 

 but not location of extension, such as impulse, velocity, distance 

 toward a Centauri, etc. 



A geometrical quantity or posited vector is a magnitude having size, 

 sense, direction and location of extension, such as velocity along a 

 given path, etc. Size of extension is the size in the ordinary accepta- 

 tion of the term. Direction or quality of extension is that property of 

 extension which prevents the quantities from coinciding when brought 

 together. Parallel lines have the same quality of extension. Solids 

 have no distinctive extension and are, therefore, scalars. Geometrical 

 quantities have the same location of extension when the qualities are 

 not only the same but coincident : the extension of the one is the 

 continuation of the extension of the other. 



Geometrical quantities are equal when their properties are the 

 same, viz. , magnitude of extension, direction of extension and location 



of extension. 



VI. 



We now consider the location of points, as well as their weights, 

 first taking the case of unit points. In this case it is almost axiomatic 

 that the sum of two unit points will be the point midway between 

 them (called the mean point) with a weight of two, since this is the 

 only result that combines the properties of the components, taking 

 into consideration both weights and locations, and which gives each 

 the same potency in determining the result. 



An extension of this thought will show that in the case of points 

 with unequal weights the mean point must be in line with the com- 



