OP THE TRANSLATION OF A DIRECTED MAGNITUDE. 
165 
forms for denoting the same thing. Instead, therefore, of inventing new symbols for 
representing distributive functions, I shall venture to appropriate the almost obsolete 
forms X .y and x'X.y to the purpose. At the same time it must be borne in mind, 
that, according to the views of many eminent mathematicians, the product of x and y 
in Symbolical Algebra may be defined to be any distributive function of x and y, and 
thus the appropriation of x .y and xy.y here proposed is nothing more than a legiti- 
mate application of these forms. 
( 8 .) I shall therefore assume x ,y and xv^y to be symbolical forms for denoting 
distributive functions of x and 3 / ; in other words, I shall consider x .y and x'X.y to 
be completely defined by the equations 
x.y-\-x' .y=.{x-\-j^').y x'Ky-]^^^ y.y=.{x-\-x'') Xy 
x.y-{-x.y'—x\y-\-y') xXy-^xXy'—xX {y-^y'), 
just as the symbolic form a’" is completely defined by the equation 
(9.) Whether x.y and xXy are ^'Commutative''' functions of x and 3 /, i.e. whether 
x.y=y.x, and xXy—yXx, does not appear from these defining equations, and there- 
fore it must be decided by the particular nature of the quantity or operation which 
each of these forms is assumed to represent. 
( 10 .) Signification of the sign +. In Symbolical Algebra the sign -{- may be re- 
garded as simply an abbreviation for the words “ together with," and thus u^v means 
simply u “ together with ” v, or u and v “put together." Now these words “ together 
with" may be taken in a great variety of senses, as the following examples taken 
from ordinary algebra show, viz. 
3-f5 = 8 , 3£-l-55.=780rf., 3-^4y/^\ = {2+J^iy, 
5 miles east-1-5 miles west=0, &c. &c. 
In the first example - 1 - means a “putting together" by simple numerical addition; in 
the second, a “putting together" of certain pieces of gold and silver, with reference 
to a certain conventional value set on them ; in the third, a mere symbolical “ putting 
together;” and so on. Hence it is clear that in using the sign -f- as an abbreviation 
of the words “ together with," the precise nature of the “putting together" is supposed 
to be understood in each case. I shall therefore define the notation, u-\-v, to mean, 
u and ^-^ut together in a sense supposed to he understood. 
Now in some cases it is very important that the precise nature of the “putting to- 
gether" denoted by the sign -f should be clearly understood, and therefore distinctly 
specified. This it will be necessary for me to do here with reference to two remark- 
able significations which have been given to the sign +. 
( 11 .) The first is that signification given to + in Symbolical Geometry. If u and v 
denote two lines of certain magnitudes and drawn in certain direc- Fig. 6. 
tions, then u-\-v is assumed to denote u and v put together as in the 
figure 6 ; that is, the heginning-point or origin of v coinciding with the 
end-point (if I may so use the words) of u. The second signification — H. 
