346 On the Curves of Trisection. 
dius be supposed to move from F at an equal distance from 
the centre C and B in the circumference of the outer semi- 
circle, and as it moves towards D in the moving radius, al- 
ways to keep at an equal distance from the centre C and 
from the arc BHAD, this last distance being measured on 
the moving radius ;—the point thus carried around from 
to D, continually receding from the inner semi-circle and ap- 
proaching D until it touches the outer semi-circle at D, will 
describe the trisecting curve of secants. 
r the generation of this curve may be expressed as fol- 
lows: Let the radius DB revolve on D, and the radius CB 
on €, in such a manner as that the distance from F to Bon 
DB shall be always equal to the distance from F to € on UB. 
The point of intersection of these radii describes the curve. 
When DB is in the position DG, and CB in the position 
CA, FB will be enlarged to oG and FC to oC. When 
is in the position Dg, and CB in the position CD, FB be- 
comes Dg, and FC becomes DC. 
2. This curve may be described by points, thus. Take 
two thirds of the exterior semi-circle, which i ig found by ex- 
nae the radius twice along the arc fromB. In figure 1, 
wo of the exterior semi-circle will be the are BGe. 
Divide this are into any number of equal parts, and to each 
point of division draw a straight line from D. Divide the 
whole interior semi-circle into the same number of equal 
parts, and from the centre through each point of division 
draw a straight line to the exterior semi-circle : or, whichis 
the same thing, divide the whole exterior semi-circle into 
the same number of equal parts, and draw the lines ~~ 
the centre. The intersection of these lines from D and fro 
the centre, will give points of the Curve of Secants, dashes 
which + spa with a steady hand the curve may be drawn. 
his curve may be described mechanically, by a con- 
tinued motion, as follows. In figure 2, let CG bea aia 
rule, moveable about the centre C, where it is fastened by 4 
Let this rule have a fixed part, or perpendicular rule, 
EiK, attached to it at H, and let there be a slit through this 
perpendicular rule, which oltenea Gat Hatta equal 
ce from C and G. 
Let CA be another rule, moveable about C, fastened by 
the pin, which fastens CG, and having a slit through a little 
more than half of it from A towards C. 
