AvuGuUST 27, 1897. ] 
For these reasons I have proposed to myself : 
1° to give the rules of operations of this nature ; 
2° to show by examples the application to cases 
where the segments are found in the same plane ; 
3° to determine by a new method not algebraic the 
direction of segments situated in different planes ; 
4° to deduce the general solution of plane and 
spherical polygons ; 
5° to deduce in the same way the known formule 
of spherical trignometry. 
This, in brief, is an outline of the present memoir. 
I was led to write it by my desire to find a method 
which would enable me to avoid impossible operations; 
having discovered it I have made use of it to con- 
vince myself of the generality of certain known 
formule. 
How well the author succeeds in carrying 
out his plan is shown by the memoir itself. 
Wessel says: 
The addition of two segments is effected in the 
following manner: we combine them by drawing the 
one from the point where the other terminates ; then 
we join by a new segment the two ends of the broken 
line thus determined. 
He extends the definition to more than 
two segments and affirms: 
In the addition of segments, the order of terms is 
arbitrary and the sum always remains the same. 
His definition of the product of two seg- 
ments is especially noteworthy : 
The product of the two line-segments ought in 
every respect to be formed with one of the factors in 
the same way as the other factor is formed, with the 
positive or absolute segment taken equal to unity ; 
that is to say : 
1° The factors ought to have such a direction that 
they can be placed in the same plane as the positive 
unit ; 
2° As to length the product should be to one of 
the factors as the other is to the unit ; 
3° As to the direction of the product, if we draw 
from the same origin the positive unit, the factors 
and the product, the latter ought to be in the plane 
of the unit and the factors, and ought to deviate from 
one of the factors by as many degrees and in the same 
sense as the other deviates from the unit so that the 
angle of direction of the product or its deviation with 
respect to the positive unit is equal to the sum of the 
angles of direction of the factors. 
Let us designate by -+-1 the positive rectilinear 
unit, by -+-« another unit perpendicular to the first 
and haying the same origin ; then the angle of direc- 
SCIENCE. 
301 
tion of +1 will be equal to 0°, that of —1 to 180°, 
that of +-e to 90° and that of —e to —90° or to 270° ; 
and according to the rule that the angle of direction 
of the product is equal to the sum of the angles of the 
factors, we shall have: (+1). (+1)==+1, (+1). 
(—1) ==], (+1) >(-D=H1, (1) > G2) =—s, 
(=1)- (42) >==e, (—1)- (—=)= £4, ($2) = (Fe) = 
—1, (+) -(—«)=+1, (—e) : (—=) 1. Hence it 
follows that « is equal to )/—1 and that the devia- 
tion of the product is determined so that we violate 
none of the ordinary rules of operation. 
It is interesting to note that while Wes- 
sel makes the addition and multiplication 
of directed lines a matter of definition, Ar- 
gand, in his famous memoir of 1806, Hssat 
sur une maniere de représenter les quantités 
imaginatres dans les constructions géométriques, 
says: ‘‘Inasmuch as these principles de- 
pend upon inductions which are not se- 
eurely established, they cannot as yet be 
considered as other than hypotheses whose 
acceptance or rejection should depend upon 
either the consequences which they entail 
or a more rigorous logic,” although in his 
last contribution to the Annales de Gergonne 
he grants that this difficulty will vanish if 
with M. Franeais we define what is meant 
by a ratio of magnitude and position be- 
tween two lines. 
After explaining that if v represents 
any angle, and sin v a segment equal in 
length to the sine, positive when the meas- 
uring are terminates in the first semicir- 
cumference and negative when it termi- 
nates in the second, «sin v will express the 
sine of the angle v in direction and magni- 
tude, Wessel shows that any radius making 
the angle v with the positive unit will equal 
cosv+esinv. In the multiplication of two 
radii cos v+e sin v, cos u+e sin u, he es- 
tablishes the distributive law by reference 
to the formule, 
sin (v-+w) =sin vcos u+ cosvsin u, 
cos (v+u) = cos 7 CoS U— Sin v sin u, 
in contrast to Argand, who assumes the dis- 
tributive law and then derives the trigono- 
metric formule. 
