AucustT 27, 1897. ] 
the sign ,, as (w+ yy + ez) ,, (cosI+ 
e sin I) and says that it has only imper- 
fectly the signification of a sign of multi- 
plication, for the operation leaves unchanged 
that one of the segments occurring in the 
multiplicand which is outside of the plane 
corresponding to the rotation indicated by 
the multiplier. He calls attention to the 
fact that the factors must be used in order 
from left to right. Similarly when the ex- 
tremity of the radius moves through an are 
of II degrees parallel to the vertical plane 
we have 
(@ + yy + <2) ,, (cos II + 7 sin IL) = 
ez + « cos II — y sin II + 7x sin II + 
ny cos II. 
Tt follows at once that 
(a + 7y + <2) ,, (cos I +<sinI),, 
(cos IIT + «sin III ) = (w+ yy + <2) ,, 
(cos (I+ III) + «sin (1+ IIL) ) 
and 
(@ + yy + ez) ,, (cos II + 7sin IT) ,, 
(cos IV + esin IV ) = (@+ ny + «2),, 
(cos (II + IV ) +¢sin (II+I1V) ) 
also that 
w+ yy + c2= (a+ ay + €z ),, (cos I+ 
esinI),, (cos l— esinI) = (w+ wy + «2),, 
(cos II + y7sin IL) ,, (cos I1—7sin IL ). 
Wessel then studies the effect of alternate 
horizontal and vertical rotations. Repre- 
senting the radius in its first position by s 
and in its final position by S, and denoting 
the arcs in order by 1, II, III, * * * VJ, he 
obtains the formula 
S65 1, dOU INO I NW WE 
In this connection he observes that such 
factors as V’,, VI’ can be transferred to 
the first member by using their reciprocals 
in inverse order, as 
Sop WAY oy WY on UW 9 5 WOE? 5 INI 
These results are applied to the solution 
of spherical polygons and the determination 
of the properties of spherical triangles. As 
in the case of plane polygons, I, IT, III, etc., 
SCIENCE. 
303 
represent the exterior angles and sides in 
order, the odd numbers the angles, and the 
even numbers the sides. Supposing the 
angles and the sides of a polygon known 
except one angle and two sides, or two an- 
gles and a side, or three angles, or three 
sides, the unknown parts can be deter- 
mined by the equation 
8 ”? BU ” Il’ ” Ti’ 7 Iv’ ”? Ww ) 
NAW os 6 ogg VSS 
where s is indeterminate, and may be sup- 
posed equal to v, <r, or yr. The effect of 
the rotations indicated by this equation is 
to submit the sphere alternately to rota- 
tions about the axis of the horizon and the 
axis of the vertical circle so that each point 
of the sphere describes first a horizontal are 
which measures the first exterior angle of 
the polygon, then a vertical arc containing 
as many degrees as the first side of the 
polygon, then a new horizontal arc which 
measures the second angle, etc. The sphere 
finally returns to its original position, while 
each of its points has described as many 
horizontal arcs as the polygon has angles 
and as many vertical arcs as it has sides. 
While Wessel’s results, as obtained by 
these alternate rotations, are correct so far 
as they go, he fails to observe that a gen- 
eral rotation must be compounded of three 
rotations about the axes «, 7, © or 4%, ¢, 7. 
Stranger still he makes no study of rota- 
tions about the real axis. Thiele, in his in- 
troduction to Wessel’s memoir, shows how 
easy it would have been to go a few steps 
further and arrive at the notion of quater- 
nions. But be that as it may, Wessel de- 
serves great credit for having devised the 
only successful method of dealing with line- 
segments in space previous to the work of 
Hamilton beginning in 1843. 
Unmindful of EKuler’s demonstration of 
the real value of (~—1)’— Argand en- 
deavors to show that such an expression 
may be used to represent a directed line in 
