OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
163 
ward to meet a perpendicular AC let fall from A upon it, the effect produced, that is, 
the work accumulated, is represented by the product of the translated magnitude 
(the force, namely,) into the amount of translation which takes place, not laterally, 
as before, but longitudinally, that is, along the direction of the force, which longitu- 
dinal translation is manifestly CB. 
(3.) From these three instances the idea naturally arises of some necessary connec- 
tion between the translation of a directed magnitude and the product of the two 
factors, the magnitude translated and the amount of translation ; or, to say the least, 
there appears to be some ground for conceiving that the product in question may be 
the proper form of notation for representing the translation. And, secondly, the 
necessity of distinguishing between lateral and longitudinal translation is clearly 
indicated, inasmuch as the longitudinal effect is zero in the first two cases, while the 
lateral effect is zero in the third case. Taking my clue from these suggestions, I 
shall now proceed to explain my proposed method of notation ; observing, that my 
object is to make it as general as possible consistently with definiteness and utility, 
and that, for this reason, I shall employ all the generalizations of the elementary 
algebraical signs which are now admitted by mathematicians. 
I shall also adopt, to a certain extent, the views of Symbolical Algebra taken by 
the late Mr. Gregory, and published by him in several papers, but especially in one 
read before the Royal Society of Edinburgh on the Foundations of Algebra. I may 
observe, however, that my proposed method of notation does not assume the correct- 
ness of these views, and might be enunciated independently of them ; but they 
appear to my mind to form the most satisfactory theory of Symbolical Algebra. 
I. Preliminary Definitions, Statements, etc, 
(4.) Directed Magnitude. — I use this term to denote any of those magnitudes 
which we represent graphically by arrows ; remarking that an arrow represents three 
things, viz. an origin oy point of application, marked by its feather-extremity a par- 
ticular magnitude represented by its length ; and a particular direction shown by the 
harh. 
(5.) Translation, Lateral and Longitudinal. — “Transla- 
tion” is the term employed to denote that peculiar and 
simplest change of position of a rigid body which consists 
in the parallel and equal motion of all its component 
particles. I shall use the same term to denote a change 
of position of a directed magnitude without change of 
direction, as is shown in fig. 4. The translation therefore 
of a directed magnitude consists in simple alteration of 
^'origin," as from A to B in the figure. 
But this alteration of origin is of a twofold nature, 
being partly lateral and partly longitudinal. If CD be the indefinite line of direction 
Y 2 
