1833.] Defenee of Lt. Burt’s Trisection Instrument. 485 
V.—Defence of Lt. Burt's Trisection Instrument. 
To the Editor of the Journal of the Asiatic Society. 
Sir, 
The accompanying observations regarding the correctness of Mr. Burr’s instru- 
ment for trisecting angles, described in No. 11, suggested themselves to me, in 
consequence of my attention having been drawn to it by some remarks contained 
in a note at page 159 of No. 15, and I take the liberty of sending them to you, in 
the hope, that should you think them likely to be of interest to any of your 
readers, you will give them a place in the Journal of the Asiatic Society. As it is 
not improbable, however, that ere this letter reach you the subject may have been 
taken up bya more able correspondent, or that its object may have been anticipat- 
ed by Mr. Burr himself having forwarded a reply in defence of his invention, I 
hope that in either of these cases you will have no hesitation whatever in laying 
this communication aside. 
In the note above alluded to, it is objected to Mr. Burr’s demonstration 
of the correctness of his instrument, that the rad. bo is not proved equal 
to rad. ao, and that it is in consequence imperfect. In the way however in 
which I understood the description, it appeared to me that the length of o 2 was 
constant, the leg a d being confined to a fixed point of it by a groove; and although not 
so expressed, I imagine it must have been intended that, that point should be at an 
equal distance from the centre with the point a. Should this supposition be cor- 
rect, the demonstration would, I imagine, be complete, without the necessity of 
proving that the locus of the point 4 is the circumference of the circle ; but that 
such is the case whenever the angle is trisected, would be easily demonstrated as 
follows :—Let BAF (fig. 1) beany angle whereof BF is the chord, and let AC be 
the line trisecting the angle BAF and crossing the chord BF in D. It is required 
to prove that if from the point B with the radius BD an arc be described cutting 
AC in C (whence BC=BD) then, that the point C shall be situated in the cir- 
cumference of the circle whose centre is at A and radius—=AB, or that AC will equal 
AB. 
First 2 BDA= 7 DAF + / AFD (EI.I. 32) 
= Z DAF + Z ABD (Hyps.) (No. 1.) 
Again 7 BDA= / DBC+ / BCD 
= Z DBC + 7 BDC 
= Z DBC+ / DAB+ Z ABD 
Equating these two values of / BDA, we have 
Z DAF + Z ABD=/ DBC + / DAB + 7 ABD 
Taking Z ABD from each side, 7 DAF = / DBC + / DAB 
But /£ DAB=#% Z DAF (by hyp.) therefore Z DBC also =} Z DAF 
Whence also DAB= / DBC 
But the angle ADF or its equal 7 BDC, or / BCA 
= Z DAB + ¥ ABD or 
= ZDBC+ 7 ABD or 
== # ABC 
Whence AB= AC, 
With regard to the latter part of Mr. B.’s paper, concerning the remoyal of the 
fourth leg of the instrument, I am not quite sure that I fully comprehend the 
mode in which the construction of the scale is detailed, If, however, the follow- 
