1833.] Defence of Li. Burt’s Trisection Instrument. 487 
gradually approaches, and ultimately (when the angle trisected equals 180°) passes 
through the centre. 
Mr. B. says that the fourth leg is absolutely necessary to the first construction 
of the instrument, but it has occurred to me that by forming the scale upon AD 
in a different mannerit might be dispensed with altogether. For since AG is always 
equal to AF, and consequently the angles AFG and AGF also equal, it follows 
thatif the arm AB be turned round till AG coincides with AF, that the point G will 
also coincide with the point F, and the line FL would form an angle with AG; as 
for instance the angle AGp equal to the angle AFL or AGF. The instrument 
would therefore I think be equally correct if the divisions upon AB were first 
drawn in the way that Mr. B proposes, as above described, and, if this leg when 
complete were afterwards converted into the chord AD by reversing the inclinations 
of all the lines Gg, making them form equal angles on the opposite side of a per- 
pendicular to AB, for then “GA would be equal / BGg=Z FGA=/Z AFG, 
As I before observed, the locus of the point F (fig. 2) is a curve passing through 
the centre C. A representation of this is given in fig. 3, which also shows it conti- 
. nued, and passing through the extremity of a diameter at right angles to GC, which 
it again meets at M, GM being equal toGL, the diameter of the circle GDL. From 
the circumstances of the distance DK being always equal to 2 vers-sin. Z DCG 
(which may be easily deduced from Mr. Burr’s theorem) may be derived an equa- 
tion to the curve when the co-ordinates originate at the centre (r being =GC) 
pa a a 
y= era—ax? + ——r rf tra+ 
2 4 
*, 
As it is also easily described geometrically, it affords a very simple form for the 
construction of an instrument for trisecting any angle from 0 to 180°, and consist- 
ing of a single piece only. A representation of one which I have lately made up, 
and found to answer my expectations fully, is given in fig. 4. It consists simply 
of an ivory scale, whose edge is sloped off, and accurately formed to the figure of 
the curve GKC (fig. 3), and a small part of the diameter GL produced on each 
side to ensure its accurate adjustment to one of the sides containing the given 
angle, for which purpose also small portions of the edge at C and G are cut away, 
in order that the coincidence of these two points with the centre and point G of 
the chord of the given angle may be accurately determined. As no graduation 
whatever is necessary the instrument is very easily made, and the application of it, 
which is also extremely simple, will be understood from the following example : 
IT must first mention, however, that for more convenient measurement the exact 
length of the radius GC is laid off on the centre of the scale between the points 
M and N. 
Let GCD (fig. 4) be any angle to be trisected. 
From the point C with the distance CG or MN as a radius, describe an are GLD. 
Draw the chord GD, then apply the scale so as to make its edge coincide with the 
side CG of the given angle, and the point C with the centre of the circle 
2 
: 
. 
* From this equation may be derived the other properties of the curve just men- 
tioned. For instance if x be taken equal to 0, theny becomes = oorr = CH. Ifx= 
r, then y also becomes = 0 or 7 4/3 = GF;; and lastly, if x be taken equal to 3 7, then y 
becomes = 0 or an imaginary quantity, The curve will therefore pass through the points 
G,C,H and M. 
