Mahcii 13, 1885.] 



KNOWLEDG R 



209 



ones) being equal, and divided b_v the pivots in the same 

 ratio, and all jointed togotlicr as shown. Tims we have 

 the first liuk on the left side eijual to the tirst part of the 

 second link, so that, in all positions, angle (1) = angle (2) ; 

 the remainder of the second link equal to the first part of 

 the third, so that angle (3) = angle (4), and so on. But it 

 is evident that (2) - (3). Therefore, we have (1) = (2) 

 = (3) = (4), which = (.')), and so on, all the numbered 



Fig. 1. 



angles being equal. It will thus be clearly seen that the 

 angles AO p, HO q, C O r are equal, and the angles ji O B, 

 5 0, rOD, also. Therefore the whole angles A B, 

 B C, 00 D are necessarily always equal, and hence, if 

 A D be made equal to any angle, either of the above is 

 equal to one-third of that angle. 



The apparatus devised by Mr. Kempe presents a some- 

 what different appearance. The dark lines in Fig. 2 form 



t ^ 



Fig. 2. 



what is called a conira-paralleloj ram of rods pivoted at 

 0, B, A, and E ; E A being equal to O B, and E B to O A. 

 The triangles O A E, E B O having three corresponding 

 aides equal, we obviously have for all positions angle 

 A E = annle E B O, and, by a simple deduction, angle 

 B E A = angle A B. If, now we take a point F in B E 

 such that OB:BF = OA:AE, and in the next long rod a 

 point C such that O = F B, and connect F and by a 

 link F equal to O B, we evidently obtain a jointed system 

 of a similar character to the preceding, so that, if we sup- 

 pose the angles OOF, B F to be equal to A E or 

 B E, the two systems of rods form precisely similar 

 contra-parallelograms, having their other corresponding 

 angles equal. .-..^ ;^.- 



xvow the angles DBF, QBE arc one and tlio same; 

 therefore the corres|iondiiig angle.s 1!, B0,\, uiiist 

 necessarily always be equal, tlio two contra parallelograms 

 being similar in all ))Ositions. In the same way, by taking 

 G iu F, bearing the same ratio to O that B F does to 

 B, or A E to O A, and proceeding as before, wo obtain a 

 still smaller system of jointed rods, D G 0, similar in 

 character to O F B, and having the angle D O neces- 

 sarily always equal to the angle COB, owing to the equality 

 of the angles O G and O B F = F 0. Tims wo see that 

 the angles D O 0, O 1>, BOA must be equal for all jjosi- 

 tions, so that this apparatus, like .Sylvester's, eniiblea us to 

 trisect any angle, by niitkiiig A O D equal to it. The appa- 

 ratus has tlii.s advantage over Sylvester'.s, that the extent 

 to which it may be opened is not limited ; but this advan- 

 tage is not great, since the trisection of an angle nearly 

 equal to two rij;ht angles is evidently effected when the 

 angle representitig its defect thf^efrom has been trisected. 



The two j)ieces of apparatus just described consist 

 entirely of rigid rods pivoted together. The following, 

 for which the writer is responsible, depends for its efficiency 

 on the addition of two sprhx/s. It consists of two equal 

 jointed rhombuses (Fig. 3), 6 A E 0, O B F D, pivoted 



Fig. 3. 



together at O. B is connected with E, and with F, by 

 some kiiid of springs (shown by dotted lines), the effect 

 ~ in a line with O and E, and 

 F. Now, a rhombus consisting 

 diagonal O E bisects the angle 

 O F bisects the angle BOD. 

 on E and on O F, it is 

 the dark rhombus, the angle 



of which is to keep B 

 in a line with O and 

 of four equal side.", the 

 A C, and the diagonal 

 If, then, B always lies 

 evident that, considering 



B O C is equal to B O A for all positiotis, and to O D, con- 

 sidering the other rhombus. Thus O B, O always divide 

 the angle A D inlo three equal part--. No difficulty will 

 be found in making the above apjiaratus, the rods (which 

 may be of sheet brass) being all equal in length ; the 

 springs BE, OF, may be stretched between rings slipped 

 over the pivots at B, E, 0, and F, their tension being of 

 little importance. Two more links, 0, DO, forming a 

 third rhombus O C G D, enable us to divide an angle into 

 jive equal parts ; for, the angle COG being half of O i), 

 is one quai ter of A O 0, and is therefore one- fifth of A ( ! 

 in all positions. 



Another trisection apparatus, also due to the writer, 

 may be mentioned, which involves the use of a single 

 rhombus, such as A E in the lust figure, a lamp and a 

 small plane mirror. 



We will suppiise that the operation is conducted in a 

 darkened room, the trisection depending on the reflection 

 of a beam of liyht from the mirror, which is set perpen- 

 dicular to the rhombus, and is capable of turning about 

 so that the apex of the angle AOC always lies on its 



