262 Onan Instrument for the Mechanical Trisection of an Angle. 
of the paper. Thelinks DE, DF, DH, D G lie below the rods 
AB and AC at their respective points of connexion, so that the 
plane of ABC always hes parallel to the plane of the paper. 
The instrument is used in the following manner. 
Let K D L be the angle to be trisected. Produce K D to H, 
and LD to G. Laying hold of the extremities B and C of the 
rods AB and AC, move the links DH and DG until their inner 
edges coincide with the lines forming the angle GDH; draw 
lines D E, D F along the inner edges of the links DE and DF; 
then these lines will trisect the given angle K D L, as required. 
Demonstration.—In all positions of the instrument, AEDF is 
an equilateral parallelogram, and DGE, DHF are equal isosceles 
triangles. 
Because DG=DE, ZGED= ZDGE; and because DF is 
parallel to EG, 2LDF= 2 DGE; therefore 2 LDF= 2GED; 
but because DF is parallel to EG, 2 EDF= 2 GED; therefore 
ZLDF= ZEDF. 
In like manner it may be shown that 4KDE= ZEDF; 
therefore the lines DF and DE trisect the angle KDL. 
I have made this instrument of lance-wood, with brass pivots. 
The links are each 8 inches long, 4 of an inch in width, and 
yzths of an inch in thickness. By means of this instrument, 
any angle not much exceeding two right angles and not less than 
nine degrees, may be at once trisected with great precision; but 
by an obvious mathematical artifice it may be used for the tri- 
section of angles, however small or large they may be. 
It is scarcely neces- 
sary to observe, that 
all general methods 
for the multisection 
of a given arecan only 
be approximate. In 
such cases there can 
be no objection to the 
use of approximate 
methods of construc- 
tion, provided that 
they are given assuch. 
By the following me- 
thod of construction, 
a given arc may be 
divided intoany num- 
ber of equal parts, 
with an approach to 
truth which is only 
limited by the inac- 
