280 
dicular to the plane of the triangle, and in magnitude repre- 
sents the double of its area; while the numerator is, as we 
have just seen, in direction tangential to the circle at a, and its 
length represents the product of the lengths of the three sides, 
or the volume of the solid constructed with those sides as 
rectangular edges. We may add, that this tangential line 
aBCa is distinguished from the equally long but opposite tan- 
gent acBA to the same circle aBc at the same point a, by the 
condition that the former is intermediate in direction between 
aB (prolonged through a) and ca, while the latter in like 
manner lies between ac (prolonged) and BA: or we may say 
that the line anca touches, at a, the segment alternate to that 
segment of the circle aBc which has ac for base, and contains 
the point B; while the opposite line acBa touches, at the 
same point, the last mentioned segment itself. The condition 
for the diameter aa, becoming infinite, or for the three points 
aBc being situated on one common straight line, is 
v.aBc = 0, (17) 
This formula (17) is therefore, in this notation, the general 
equation of a straight line in space; (15) is the general 
equation of a circle ; (14) of a plane; and (6) of a sphere.* 
which he inadvertently used as interchangeable in his first communication to the 
Academy: and to make them satisfy the two separate equations, 
exetdd=_aq; 
, 
= 1G — 
He proposes to confine the symbol @’ — q to the signification thus assigned 
for the latter of the two symbols which have been thus defined, and which, on 
aceount of the non-commutative property of multiplication of quaternions, 
ought not to be confounded with each other. 
* The simpler equation of scalar form, s. anc = 0, also represents a spheric 
surface, if B be regarded as the variable point; but a plane, if B be fixed, and 
either A or c alone variable. 
