OF THE TRANSLATION OF A DIRECTED MAGNITUDE. 
179 
(57.) The lateral translation-product is therefore well adapted to denote rotation. 
Thus (always supposing that v is at right angles to a), the symbol of a line making 
an angle 0 with v, and in a plane at right angles to a, is 
which in fact is the same thing as 
{cos ^-|-(Da.) sin 
for cos 6 and sin 0 represent 
and 
and g®®*- 
fl2 
^ 1.2 ' 
1.2. 3. 4’ 
1.2.3“^ ] 
1.2. 3. 4. 5 
for expressing 
(«D..) 
(flDa.)2 
I 1 0 
whence, since (Da.)’^=— • (Da.)^=+ &c., the symbolic equivalence of the forms 
geD<t. ggg gjg ^ 
is manifest. But the quantity operated upon must necessarily be at right angles to 
a ; for, otherwise, (Da.)^ is not equivalent to — , as appears from art. 54, equation (3.). 
(58.) Concluding Remarks . — I have now said enough, I think, to explain the 
nature of the proposed symbolization, and the general rules which regulate the 
application of the two translation-products. I have based the whole theory simply 
and exclusively on the conception of translation, taking my clue from the three 
suggesting instances, the parallelogram, the couple, and work. As regards the lateral 
effect of translation, the theory is nothing but a general development of oxiv geometrical 
notion of multiplication; for what is a rectangle, ABCD, considered apart from 
arithmetical measure, but the effect or product of the translation of Fig. 24. 
AB along AD ? and this we represent by writing AD before or after ® ^ 
AB. But this method of representation is clearly incomplete when 
we put for AB and AD their numerical representations; and why? a d 
because the special superficial unit then is omitted. For, suppose AB=3, AD=4; 
then, if we say that the rectangle ABCD is the product of 3 and 4, or 12, we mean, 
12 superficial units. Now, by omitting the superficial unit in our representation, we 
leave out all conception of the plane in which the rectangle lies. All that I have 
done above is to restore the superficial unit, and determine its proper representative 
symbol. As regards the longitudinal-effect (suggested by the conception of work), it 
appears that all units are absolutely equivalent, and therefore may be all confounded 
in the common symbol of unity. 
1 now proceed to give Applications of the Symbolic forms* u.v and uXv, &c. 
* I may refer here to an imperfect attempt I made in a paper read before the Cambridge Philosophical 
Society (Nov. 1846) to base the symbolic form T)u.v on the conception of perpendicularity in art. 19 above. 
2 A 2 
