356 On the Curves of Trisection. 
means of the one curve, (as the curve of secants) the con- 
tiguous angles ACD, HCD muy be trisected by means of 
the other ; and thus by means of these curves, as AH re- 
volves, all the angles, which it makes with DB at C, may 
be trisected. 
[The following additions to the above Memoir have been lately received 
in a letter from the author. ] 
In re-examining the instruments for describing the curves 
of trisection, it has occurred to me, that they might be im- 
proved. 
In the instrument for describing the curve of secants, (fig- 
ure 2,) the part AC is wholly unnecessary for the purpose 
of describing the curve; and it is obvious, that it renders 
the instrament more complex. Removing this part, the 
pencil will pass through the slits of only the two rules HK 
and DG. 
Another change may be made in this rule. Let the pin 
at G, instead of moving in the slit of the rule DG, be made 
fast to the rule DG, only allowing the two rules to move upon 
G as the centre. , Let the rule, with its slit, be lengthened 
beyond D. At D let there be a fixed pin, on which the 
rule GD shall move by its slit. By this change the motion 
of the rule is on a fixed pin at D, instead of hetnesee a moy- 
ing pin at G, 
The instrument for describing the curve of sines algo ad- 
mits of improvement, and may be rendered more:simple. 
Let the rule EG, (figure 4;) be entire from H to G, without 
the rule Hm. Let it be extended beyond E, with slit from 
H, so as to allow its motion on a fixed pinat E. The rule 
CH is retained with pins atC and H.. The distance Hm is 
made equal toCH. The point m describes the curve. By 
this change the rule EG slides upon a fixed pin at Eyinstead 
of having the rule Hm slide within it. 
Even without the central rule CH, if a perpendicular rule 
be attached to the rule EG at H, extending through the 
point F, with a slit init so asto move upon a fixed pin at F, 
while the part EG moves or slides in the same manner on a 
pin at E, it is obvious, that the result will be the same, as if 
there was a central rule. As this square rule is moved 
by its legs on the fixed pins at E and F, the point of the 
