302 
A statement in this connection is note- 
worthy : 
But if we have to multiply line segments which 
are not both in the plane passing through the absolute 
unit we cannot apply the preceding rule. For this 
reason I do not consider the multiplication of such 
segments. 
The treatment of division follows in a 
natural manner, and it is proved that indi- 
rect quantities share with direct quantities 
the property that if the dividend is a sum 
we obtain by dividing each term of the sum 
by the divisor several quotients whose sum 
is the quotient sought. 
Then comes a discussion of powers and 
roots establishing the fact that (cos v+ 
1 
e sin v)” has m different values and only m. 
In the next paragraph Wessel shows that 
the m power of a line-segment may be put 
in the form e"*+™”—, where e”” represents 
the length and mb the angle of direction, 
and that thus we have a new method of rep- 
resenting the direction of line-segments in 
the same plane by the aid of natural loga- 
rithms. This last is not again referred to, 
but it is readily seen that Wessel was in 
possession of all three of the present methods 
of representing the complex number, 
a+b/V—1, r(cos ¢+“—1 sin ¢) and ret’. 
At the close of this section the author 
remarks: 
At another time, with the permission of the 
Academy, I will present the complete proofs of these 
theorems. Having given an account of the way in 
which we must, in my judgment, understand the 
sum, the product, the quotient and power of line seg- 
ments, I shall restrict myself to a few applications of 
the method. 
The first application is to a demonstration 
of Cotes’s theorem in which the fundamen- 
tal theorem of algebraic equations is as- 
sumed as previously established. The 
second is to the resolution of plane poly- 
gons. In this certain characteristic nota- 
tions occur. The first side of the quadri- 
lateral considered is taken equal to the 
SCIENCE. 
(N.S. Von. VI. No. 139- 
absolute unit ; the sides in order beginning 
with the first are designated by the even 
numbers II, IV, VI, VIII, while I, III, V, 
VII, represent their deviations (in degrees) 
each with respect to the preceding side pro- 
longed, regarding these deviations as posi- 
tive or negative according as they have the 
same sense as the diurnal motion of the 
sun or the opposite; I’, III’, V’, VII’ de- 
note the expressions cos I+. sin I, etc., 
while I~, III~, V-, VII- denote the ex- 
pressions cos (—I)-++e sin (—I) or cos I—« 
sin I, ete. 
The author then deduces the two for- 
mule, 
THESE INP INOW WARSIANe Go WEG 4 TO” - 
Vv’. VIl/=0, 
DEO IOUU Oo WANe 5 IO 6 Wo Wade ae 
VII’ + VIII=0, 
and proves that two equations of this form 
will suffice for the solution of any polygon 
in which the only unknown parts are three 
angles, or two angles and a side, or an 
angle and two sides. 
Wessel next attacks the problem of repre- 
senting the direction of any line segment 
in space by taking it as the radius, 7, of a 
sphere. Assuming three perpendicular 
radii as axes and denoting positive unit 
lengths upon these, to the left by 1, for- 
ward bye and upward by 7 respectively, 
where « = —1, and 7” = —1, he concludes 
that a radius whose extremity has for co- 
ordinates x,y, ¢z will be properly desig- 
nated byx+y7y-+e«2. Defining the plane 
of r and er as the horizontal plane and that 
of r and 7ras the vertical plane, he ex- 
amines the effect of moving the extremity 
through an are of I degrees parallel to the 
horizontal plane and obtains for x + ny + «z 
the new value, 
ny + (« + ez) (cos I + esin 1) = yy + 
x cos I—z sin I + ex sin I 4+ «z cos I, 
in which the term 7y remains unchanged. 
This operation he indicates by the use of 
