486 Defence of Lt. Burt’s Trisection Instrument. [Szpr. 
ing be a correct explanation of the meaning, there can, I imagine, be no hesitation 
in admitting the conclusion he has drawn. In forming the scale of equal parts 
upon this fourth leg AB (fig. 2) each point in the scale is to be successively brought 
to the circumference by turning the scale round the point A, so as that each divi- 
sion shall in turn terminate a chord of the variable arc AG, and the line marking 
the division is then to be cut on it, in the direction of the radius passing through 
it. At the same time the leg AD being placed in its corresponding position (viz. 
at an equal distance on the other side of a perpendicular to CG), its divisions 
will be marked by the same radius, and this is to be done for every point of the 
circumference AGgB’. 
The divisions upon AD, therefore, form a scale of chords equal in length to 
those of the corresponding arcs AG, Ag, and each of the lines forming them, will 
by the construction tend to the centre when AD is so situated as to cut off an are 
three times the extent of that of which AG is the chord ; and the application of the 
instrument merely consists in adjusting the line AD to the chord of the given arc, 
and then turning round the movable radius CG till it coincide with the division, 
which in that position would if produced pass through the centre, and which, if 
the coincidence be exact, will of course direct the radius to an arc one-third of 
AD. It must however be shown that in any position of AD there can be only one 
of the divisions which tends to the centre (or can be made to coincide with the 
radius), this may be easily proved ; for if FL (fig. 2) be the correct division on the 
scale, cutting off (by radius passing through it) the are AG = one-third of the are 
AD, and if f7be any other division belonging to anare A g the whole of the divisions 
having been marked off in the manner above described, then it may be demonstrated 
that the radius Cf will, if drawn through f, form with f/an angle / f n equal to the angle 
FCF plus half the anglegCF*. The instrument therefore seems to be complete enough 
in theory without the fourth leg, but in use, it appears to me that the want of it 
would considerably diminish its accuracy, as it must be very difficult to hit upon the 
exact coincidence when the divisions are very numerous, and as any error at the point 
F would be multiplied at G in the proportion of the two distances CF: CG, this 
would be a serious evil in large angles, as the focus of the point F is a curve which 
* The demonstration of this is as follows, vide fig. 2. 
First 7 CGA= 7 GCM + 7 CMG 
= / GCM+ / CgA — Z gAM 
= Z GCM + Z CgA — § Z GCM (Zi. III. 20.) 
=/7CgA + Z$GCM 
eae ae c ae (by hypothesis) 
Therefore substituting these values in above equation 
Z AFL= Z Afl + 4 Z GCM (No. L) 
Again 7 CfD orAfn= Z CFF + Z fCF 
Or Z AFl+ Z nfl = Z AFL + 7 fCF 
Or, by substituting the value of / AFL found at (No I.) 
=ZAfl+ 7 fCF + 47 6CM 
And subtracting 7 Af/ from each side of the equation 
Lnfli=ZfCF+% 7 GCM 
Q.E.D. 
