176 G. PLARR ON THE ESTABLISHMENT OF THE 
without the help of any other auxiliary than the most elementary propositions 
of algebra or geometry ; and then division can be treated of. 
§ 1. Definitions. 
We call vector the “ compound” of, 1st, the mutual distance of two points, and 
of, 2d, the direction of the straight line drawn from one of the points, as origin 
of the vector, to the other point, as extremity of the vector. 
A vector is usually represented by a Greek letter, p suppose, or by the 
letters designating its origin and extremity respectively, AA’ suppose, the 
first being always written to the left of the second, so 
4’ that 
gee p or AA’ 
p 
A are meant to represent the same. 
We designate by Tp, pronounced tensor of p, ‘the 
_ absolute length of the vector ; and by Up, pronounced versor of p, the symbol, 
sut generis, which represents the direction of p. 
In order to express the vector p by means of Tp and U2} we agree by 
definition, (1), that 
Up x Tp, or simply Up Top, 
expresses the construction of the length Tp into the direction belonging to p, 
and designated by Up, the extremities of Tp having to coincide respectively 
with the two points given as origin and extremity of p. 
We may conceive a vector whose length is equal to the unit of length. In 
this respect 
Up x 1, or simply Up, 
is called also the wnit-vector of p; it is the unit of length directed in the 
direction belonging to p. 
We may conceive the wnit-vector added to itself, end to end, on the same 
straight lme a number of times expressed by the same number as Tp; the 
vector comprised between the origin of the first of the unit-vectors and the 
extremity of the last (the last may be fractionary), will have for its expression 
Tp x (Up x 1), or simply Tp Up; 
so that practically the expressions Up Tp and cp Up are representing the same 
vector p. 
We agree by definition that the symbol Up is to be independent of position. 
The same value of Up is to represent directions which are parallel, and drawn 
towards the same region of space. Such directions will be designated by the 
appellation of “ the same direction.” We apply the appellation of “opposed 

