1815.] Scientific Intelligence. 155 
4. The extremities, Vand V’, of the axes are the interior and ex- 
terior vertices. 5, The angle contained by a chord and the axis is 
the vertical inclination of that chord. © 
' Corollary 1.—The interior axis is equal to the radius of the 
ircle, and the exterior axis is equal to thrice that radius, 
Corol, 2,—If radius equal unity, any interior chord, F P, is 
equivalent to the difference between the interior axis and the 
rectangle under twice that axis, and the cosine of the vertical 
inclination of that chord. 
Corol. 3.—Any exterior chord is equivalent to the interior axis 
plus or minus the same rectangle, according as the chord is greater 
or less than the interior axis. 
Corol. 4.—lf a tangent, G H, to the circle at the focus be 
limited by the curve at G and H, it is equal to twice the interior 
axis. 
Problem.—Any obtuse angle, A BF, being given, to trisect it 
by means of the radiatrix. 
Produce AB one of the sides of the angle, and upon the other 
side let a segment of a circle be described, which shall contain an 
angle equal to the supplemental angle C B F: through F as a focus, 
and the centre, V, as an interior vertex, let a radiatrix be described: 
through the point P, where the curve is cut by the diameter, draw 
ey join DB and DV. The angle DBP is one-third part of 
For since D V is equal te B V, the angle VDB is equal to 
V BD, and the exterior angle D VP to double of the interior 
angle D BP: and because D P is in the direction of the focus F, 
DP is equal to the radius D V: therefore the angle D P V is equal 
to D V P, or is double of V BD. Now the angle CBF being 
equal to B DP in the alternate segment, the angles D.BP»and 
DPB are together equal to A.3F: and consequently the angle 
DBP is one-third, and DPB two-thirds, of ABE, 
Corollary.—The éxcess of 60° above one-third of A BF is one- 
third of the supplemental acute angle C BP, 
