CORRESPONDENCE. 



I)l\iniN(^, ANGLKS INTO THREE EOLAL I'ARTS. 

 To the Editors of " Knowi.eijgk." 



Sirs, — Vour article last month on the above subject I at first 

 thought to be a claim to have solved the outstanding classical 

 problem in Pure Geometry '' To trisect a given rectilineal 

 angle," a problem which has interested many of us and 

 wasted many an hour. The approximation of Mr. Bingle\'s 

 reminds me of one of my attempts which may be sufficiently 

 interesting to record. .-Ml your geometrical readers (not their 

 shape), will know that by joining the bisection of a side of a 

 parallelogram to one of the opposite angles, the diagonal is 

 trisected. I fondly hoped that it might be possible to trisect 

 an arc in somewhat the same way with curved if not with 

 straight lines. If we include a right angle in a circle with its 

 centre at the angular point and join through the bisection of 

 one side with the extremity of the diameter of which the other 

 side is a part, the 90' arc is not trisected though there is a 

 suggestion that the line may cut the circle at the apex of the 

 equilateral triangle standing on the other side (see Figure 153>, 

 but the angle tan ''■% is formed. 



This is not an encouraging start, but on applying this to 

 smaller angles good approximations well within the draughts- 

 man's error are reached I see Figure 154). 



Let A B C be the given angle. Describe the circle A C 

 with centre B. Produce C B to the circumference at D. 

 Bisect A B at E. Join D E and produce to the circum- 

 ference at F. The arc A C is approximately trisected at F. 

 The word in italics is unfortunate but necessary. 



If we try an angle of 120° it is actually bisected, whereas 

 one of 90", as we have found, comes within 7' of the proper 

 place. Calculation shows that at 60' the error has fallen 

 to 1° 47' ; for 45° it is 43' only, then it rapidly decreases and 

 at 30° it is less than 1 1', which is already negligible ; the two- 

 thirds angle always being slightly too small. 



Mr. Bingley's method. I find, gives precisely the same results, 

 his E J, E K, lines corresponding to my D F though obtained 

 by a more elaborate construction. He, however, bisects 

 his angle first, so that when I speak of an angle 30° his 

 would be 60°. 



To pro\e the relationship of the two methods let us take a 



Figure 153. 



FiGURi; 154. 



part of his figure, but with a larger angle and a complete circle 

 (see Figure 155 where A D. E C, and G D are bisected). 



Take the centre of rectangular co-ordinates at A. the axis of 

 X, A C, and the co-ordinates of D (a, b) where a" + b' =1. 



The co-ordinates of the points will be : — 



A, (0,0); C, (1,0); D, (a,b); 



5a + 2 5bN 



" 8 rs) 



Equation to E I 



b^5b" 

 2 8 



5a + 2 



a 5a + 2 



8y — 5b _ 8x — 5a — 2 

 -b ' - a - 2 



b 



••■y=^^^»x + i). 



This and the equation to the circle x' + y" = 1 , are 

 obviously both satisfied by y = 0, x = — 1. 



Therefore, K E when produced cuts the circle at Z, and the 

 two constructions produce the same results. 



In the above investigation the co-ordinates a, b, do not 

 necessarily refer to the circle ; the a^ + b" = 1, is 

 not used ; showing that the point D may be any- 

 where and K E always cuts the circle at Z ! (see 

 Figure 156). Of course the approximation is not 



FlGL'RE 155. 



FlGURK 156. 



obtained unless D is taken on the cuxumference. There is 

 no particular mystery about the process finding the circle at 

 last, for C is of necessity on the circle. 



The construction of Figure 156 doubles a straight line. C -A. 

 by a system of bisections — a rather curious result. 



H. F. CHESHIRE, B.Sc. F.I.C. 



Hastings. 



THE TRISECTIOX OF ANGLES. 

 To the Editors of '' Knowledge." 

 Sirs. — I am much obliged for the interest that your 

 correspondents, Messrs. Thomson and '' Computer," have 

 taken in this matter, originated by my little article in your 

 November. 1911, issue, and for their corrections and remarks 

 thereon. I only intended the method to apply to small angles 

 of 45° or less, as then stated, as it is not applicable to larger 

 angles. With the means at my disposal, which are. unfor- 

 tunately, very primitive, I could not detect any error, but I 

 quite accept the corrections, of course, and a"m only sorry 

 that the method turns out to be inaccurate, or, in other 

 words, not a method at all. As to angles of exactly 45°, 90°, 

 135° and ISO', surely these are mathematically divisible by 

 a much simpler method than Mr. Thomson's No. 2, viz.: the 

 angles 45°. 90° and 180° by that of 60°, and the angle of 135° 

 by that of 90° ? I have always so treated thein. and should 

 be glad to be shewn any correction that they require. 



,„. , „ .. CHAS. S. BINGLEY. 



18d, Albion Road. N. 



THE FLIGHT OF BIRDS. 

 To the Editors of " Knowledge." 

 Sirs, — The methods of birdsin maintaining or rather recover- 

 ing energy of position during flapless or soaring flight is very 

 easy of observation in the plains of India during the beginning 

 of the hot weather, and the following notes may be of interest 

 to some of your readers less favourably placed as regards 

 opportunities for watching them. The two conditions required, 

 namely, ascending currents of air and a plentiful supply of 

 large soaring birds, are both present. At the beginning of the 

 hot weather, before the more regular winds set in, the air near 



135 



