FEBRI'AUV, l<Jli 



;\(n\Li:nr,K. 



63 



MeTHonl 



filament becomes red or white-hot. If the wires conducting 

 the current were continuous, there would not be friction 

 sufficient to cause heat or light, so that in passing thnuigli tln' 

 filament the velocity of current must be 

 retarded. 



An electric lamp can be seen for 

 miles in a clear atmosphere, but it 

 cannot pierce a London fog for any 

 great distance. Is not the velocity of 

 the electric current reduced in this and 

 similar cases, owing to the obstruction 

 of the filament, or of matter in the 

 earth's atmosphere ? 



And so, in the case of a flash of 

 lightning from cloud to earth : the air 

 being a bad conductor of electricity may 

 retard its velocitx' more or less according 

 to the dryness and (|uantity of floating 

 matter in the air. If the atmosphere 

 oft'ered no obstruction to the friction or 

 impulses which cause light, there would 

 be no light : otherwise, would not the 

 ether in the depths of space always be 

 illuminated by the light of the millions 

 of suns in the heavens to such an 

 extent, by its accumulation, as to turn 

 our night into day ? 



Of course, if I am mistaken in sup- 

 posing that there is an appreciable 

 lapse of time between the commence- 

 ment of a flash in a thunder-cloud and 

 the end of the same flash at or near 

 the surface of the earth, no retardation A 

 in the velocity of light may be caused 

 in the waj' I ha\'e suggested. 



As an enquirer I must beg your kind 

 indulgence tor thus troubling you witli 

 the expression of my thought. 



G. K. (ilHBS, Colonel (Retiredi. 

 Canmori:, 



Westbourne Pauk ROAIl. 

 BorRNEMorni. 



I)(" in X 

 loiu M \ 

 unci K res 



THE 



"KISECTION 

 ANGLE. 



OF AX 



To the Editors of '" Knowledge." 



Sirs, — Mr. Hingley's method of 

 dividing angles into three equal parts. 

 published in the November number of 

 " Knowledge," is, so far as I am 

 aware, original, and gives accurate 

 results for small angles. In attempting 

 to find an approximately correct solu- 

 tion of the problem for larger angles. I 

 have hit upon two simple constructions. 

 to which I now venture to call attention. 

 The first is an alternative method of 

 trisecting small angles, which is not 

 only simpler, but also more nearly 

 e.\act, than that of Mr. Hingley ; the 

 second is a method applicable to I'lCL' 



larger angles. 



Method I. — Let B .\ C be the angle 

 to be divided. I'rom .\ describe arc B C. Bisect arc B C 

 in D, and join DA. Bisect .A B in E, and A C in F. Join 

 E D and F D. Bisect E D in G, and F D in H. Join A G 

 and AH, and produce them to meet arc B C in J and K 

 respectively. Then the angle B .\ C is divided into the three 

 eqnal angles, BAJ, J A K, K.A.C. 



Method II. — .•^s before, let BAG be the angle to be 

 divided. From .\ describe arc B C. Bisect arc B C in D. 

 and join D .\. Bisect .\ B in E, and A C in F. Join E D and 

 FD. Bisect ED in G. and ED in H. Join BD and DC. 

 Join .\ G and .A H. and produce them to meet chords B D and 



uul r respectively. Bisect A D in L, and .\ L in M. 

 and \! P, and produce tlicm to meet arc 1! C in J 

 |)ectively. Join J .A and K A. Then the angle B A C 

 is divided into the three equal angles, 

 BAJ. JAK, KAC. 



It is obvious that the smaller the 



angle B A C, the more nearly do A J 



1 and A K coincide with A N and A 1' 



^ Kspcclixely, and thus approximate to 



^^ the solution by .Method 1. 



In order to ascertain the degree of 

 u accuracy of the three methods — that 

 of Mr. Binglej' and the two described 

 above — I have examined each of them 

 an.dvticallv. If 2a be the angle B AC, 

 and 2ff be the middle (J A K) of the 

 thiie component angles, then 

 By the Bingley method — 



cot a -f- -2 cosec a = cot f^— J cosec /i. 

 I5y Method I— 



cot a -f- 2 cosec o = cot 1^. 

 By Method II— 



cot a + j cosec o = cot (3 — I cosec /^. 

 If each of these equations be solved 

 for different values of o, the error 

 inherent in each method can be deter- 

 mined. Some of these are given in the 

 table. It is seen that the error due to 

 using Method I is exactly half that 

 present in Mr. Bingley's method, and 

 _ that neither of them can be used with 

 U reasonable accuracy for angles much 

 above 45°. On the other hand, Method 

 II is sufficiently exact for angles up to 

 135\ and even ;it 180 the error is only 

 thirty-nine minutes. In Mr. Bingley's 

 method the middle angle is loo large, 

 and in the other methods it is too small. 



D. HALTON THOMSON. 

 Kknleith, Broadlands Road, 



HiGHGATE. N. 



THE TRISI-XTION ()!•" AN 

 ANGLi:. 

 To the Editors of " Knowledge." 

 Sirs, — In the issue of ''Know- 

 ledge " for November, 1911, I noticed 

 a problem on the trisection of an 

 angle by a geometrical method. 



As the geometrical trisection of an 

 angle has long been supposed to be 

 impossible, I thought at first (for I am 

 naturally credulous) that a great dis- 

 covery had been made ; but when I 

 tested the matter by a numerical 

 example, the construction, as given by 

 Mr. Bingley, proved to be incorrect 

 from a mathematical point of view, 

 although it gives a rough approximation 

 to the truth. Suppose the angle to be 

 ,. f:,^ divided is 22° 30'. 



According to the Bingley construction, 

 we obtain the following values for the 

 three divisions of the angle: — 



7° 31' 18" 

 7 27 24 



7 31 18 



The middle angle shows a deviation from the others of 

 nearly four minutes, and the deviation would be still greater 

 for a larger angle. I leave to the geometrical expert the task 

 of pointing out the fallacy in the construction ; for some fallacy 

 there must be if the construction is supposed to be exact, and 

 not merely an approximation. COM lM"ri~K 



Boston, Mass. 



