OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
181 
a tracing-line possesses ; I mean, direction and two-sidedness (if I may so speak). An 
arrow lying on* a plane, not only points in a particular direction, but has two distinct 
sides, right and left (see art, 34). Hence when we speak of lines described the same 
way by their tracing-points, and parallelograms described the same way by their 
tracing-lines, the expression same way'' includes much more in the latter than in 
the former case. For example, the parallelograms generated by the translation of 
AB along AD, and of DC along DA, though coincident, are described opposite ways. 
Again, the parallelograms generated by the translation AB along AD, and of AD along 
AB, though coincident, are described opposite ways ; for the former is described by 
a right-side motion, and the latter by a left-side (see art. 34). 
(60.) Definition of Surf ace hy reference to Lateral Translation. — It appears to me 
that the simplest way of including the considerations just* alluded to in the general 
conception of surface requisite in Symbolical Geometry, is to define Surface by refer- 
ence to the Lateral Effect of the Translation of one line along another. The Funda- 
mental Assumption (article 29) is justified in this case, as appears from the remarks 
just made, and the peculiar relation u.v=—v.u is naturally interpreted (see art. 34). 
I shall therefore define Surface in Symbolic Geometry to be the Lateral Effect of the 
Translation of one line along another. By '^‘Effect" here I mean simple Geometrical 
effect, i. e. change of position in space. Also I only speak of lateral effect ; because 
all notion of longitudinal effect is excluded by our ordinary conception of surface, 
and we may assume that no shifting which a line undergoes in its own direction can 
generate surface. 
(61.) Longitudinal Effect considered geometrically. — But though the shifting of a 
line in its own direction generates no surface, it produces alteration of position ; and 
hence it constitutes an important conception. The only difficulty, in considering the 
longitudinal effect geometrically, consists in this — How is it that all units of longi- 
tudinal translation are equivalent (see arts. 36, 37 )? while those of lateral translation 
are not, and on what principle can this difference be interpreted ? The answer ap- 
pears to be this; that a. (3 denotes the effect of translating the unit (3 along the per- 
pendicular unit C4 ; that this operation conveys the conception of a particular plane, 
and we must think of C4.(3, |3.y as different operations because they are performed in 
different planes. On the contrary, the translation of a along a, or aXa, conveys no 
conception of a particular plane ; in fact a X « and l3x(3 may be regarded as performed 
in the same plane. Thus, that which before made the difference does not exist in 
this case. 
Generally, u .v — ii .d , when the two parallelograms gene- Fig. 28 . 
rated are e(]ual in magnitude, and lie in the parallel planes; 
but uxu and u'xa! may be always considered as lying in the 
u 
same plane; and consequently difference of magnitude only 
remains to constitute a difference between uxv and u'xv'. 
* Not in but on a plane, that is, on one particular side of the plane ; e. g. on the upper side of this sh§et of 
paper. 
