March 13, 1902] 



NA TURE 



439 



(i) By adopting Euclid's construction of cutting off equal 

 segments AD, AE from the sides and proving as he does that 

 the triangles ABE, ACD are equal in all respects, and then 

 making D, E coincide with A, B respectively. The method 

 of passing to limiting cases is highly instructive, the only 

 question being as to the advisability of trying to introduce it to 

 beginners, except as an experiment. 



(2) By drawing a duplicate DEF of the triangle ABC (this 

 operation being only a slight extension of the method of super- 

 position used by Euclid in I. 4), and proving first that 



/ A = z E and _ B = - p', and second that _ A = .£ F 

 and J B = - E. 



(3) ^y folding the triangle so as to bring .\B into coincidence 

 with AC. This is practically equivalent to bisecting the vertical 

 angle, but it replaces a "hypothetical construction" by an 

 operation which the beginner can easily perform. 



The method of folding has many obvious advantages, and 

 much would be gained if beginners could be taught at once to 

 lecognise cases in which one half of a figure could be brought 

 into coincidence with the other half by folding. For example, 

 the property that the common chord of two circles is bisected at 

 right angles by the line joining the centres is obvious when it is 

 recognised that one half of the figure is the fold of the other 

 half. The method is, moreover, hardly more artificial than the 

 method of superposition which Euclid himself employs. 



It should be noticed that Euclid's proof of I. 4 involves an 

 assumption which I have never seen pointed out, namely, that 

 two straight lines cannot totuh one another. If this be not 

 assumed, then when the sides DE and AB are brought into 

 coincidence, the sides DF and AC do not necessarily coincide 

 even though they make the same angle with the same straight 

 line and on the same side of it. G. H. Brv.'^n. 



The Zodiacal Light and Sun Pillars. 



The appearance on clear evenings of the zodiacal light after 

 sunset at this season of the year in this latitude is usual, and it 

 has been frequent and beautiful to observe in this district for 

 many nights. It would be interesting if the readers of Nature 

 could detect any definite movement of the arm of light, for much 

 yet remains to be discovered about this phenomenon, and any 

 observer can make this point a siudy. From a short half-hour 

 after sunset to from 8 to 9 p.m., a straight line drawn from the 

 sun's po.'iition at sunset to the Pleiades will not remain the centre 

 line of the zodiacal light. It appears to emanate from the sun 

 and move as the luminous spoke of a wheel which has the sun 

 for centre frequently, but not invariably. What makes the light 

 apparently fade away? Is it that the motion of the earth has 

 drawn with it the arc of volcanic or meteoric particles, which 

 may be the medium of the light, away fnm the sun 'i or is it not 

 po.ssible that such a band of dust is lit with a degree of earth- 

 shine? If so, may not this account for that other phenomenon 

 of the Gergershein, which is usually brilliant in proportion to 

 the brilliancy of the zodiacal light ? Doubtless these phenomena 

 are always present, but their visibility depend's on the magnetic 

 or electric condition of our atmosphere. Irritated by either of 

 these conditions, the belts of dust would alter positions of all 

 the bodies forming them, and so lie at a different angle and be 

 clearer or dimmer accordingly. 



The very remarkable sunset of March 6 has probably been 

 observed by many readers of Nature. The " fire-finger " left in a 

 perfectly perpendicular position for upwards of fifty minutes after 

 sunset was visibly withdrawn, without losing colour or size or 

 changing from the perpendicular, and was a vivid and beautiful 

 adjunct to a sunset afterglow strangely reminding one of the 

 " Krakatoa sunsets" of years ago. This finger of fire the 

 writer has only ob.servcd once before, after a similar-coloureil 

 sunset over the estuaries of the Medway and Thames last 

 summer, but London smoke dimmed the efiect. This rare 

 appearance seemed on March 6 to resolve or be replaced by five 

 vivid white rays with slight wheel motions from north to south, 

 the largest of which eventually seemed the zodiacal light itself. 



Guv J. Bridges. 



.Sutton Mandeville Rectory, Salisbury. 



A I'HENOMENON was vi.sible here this evening which I think 

 deserves to be recorded. We have an uninterrupted view of the 

 western sky, and about a hundred pairs of sharp eyes are avail- 

 able, so that any unusual sunset is pretty certain to be noticed. 



NO. 1689, VOL. 65] 



Solar halos are comparatively common occurrences, and I have 

 come to the conclusion that lunar rainbows are not so rare as 

 is believed, but the " pillar of fire " which has been visible here 

 for at least forty minutes is the most brilliant sight I or any of 

 my oldest friends have ever witnessed. It was first observed at 

 6 p.m. just after the sun had disappeared, and was exactly vertical 

 over the sun. The colour was at first silvery (resembling a 

 searchlight) and later a golden yellow, the width equal to the 

 sun's diameter, and the length 18' to 20°. A few light clouds 

 seemed to pass behind it. Some observers noted a flickering 

 and also a swaying motion, but this may have been an optical 

 efl^ect. At about 6. 30 the colour had changed by gradations into 

 a deep crimson-red, and for the next ten minutes it gradually 

 became deeper in colour and shorter, disappearing at 6.40. 



I may add that on February 20, 1901, I observed a very faint 

 trace of a similar phenomenon. 



Can any of your readers direct me to any literature bearing on 

 the matter ? W.\r. A. K.night. 



Sexey's Trade School, Bruton, Somerset, March 6. 



The Quadrantid Meteors. 



Mr. John R. Henry, according to his letters in Nature 

 of January 2 and 23, unfortunately looked out too late for the 

 (Juadranlids, owing to having miscalculated the time of maxi- 

 mum, the approximate probable time of which might also have 

 been obtained from the British Astronomical Association. He 

 is right in saying that the date of the shower is advancing 

 into the year, but the advance is slower than he thinks. Taking 

 the data be gives, which, however, are only very rough, and 

 also a consideration of the sun's longitude as given in the 

 ' ' Nautical Almanac, " there is an advance of only five hours in the 

 thirty-seven years 1825 to 1862. It would appear that in the 

 forty years from then a further advance of probably about eight 

 hours has taken place. It is unfortimate that Prof. A. S. 

 Herschel has not published the exact number of meteors he 

 saw from hour to hour on January 2, 1900, when he watched 

 from iih. to i6h. 30m. He, however, states that the frequency 

 continued about the same during the whole period, and seeing 

 the radiant point was rising all that period, this would mean that 

 the maximum was near the beginning of his watch. 



Mr. Henry may be right in saying that the period of maximum 

 fluctuates somewhat from year to year, though the data he gives 

 are not sufficiently accurate to prove this ; but I fail to see what 

 ground he had for expecting the maximum so late as he did 

 this year. Taking all the data into consideration, we might have 

 expected the maximum to be about 23h. on the 2nd. I see no 

 reason to doubt that this expectation was fulfilled ; but as the 

 maximum would occur in the daytime, observations in other 

 countries would be necessary to prove this. The cjuadrantids 

 as seen here were most numerous on that morning. 



It seems probable, therefore, that the time of the next maximum 

 will be about 5h. on January 3, 1903. T. W. Backhouse. 



West Hendon House, Sunderland, March 5. 



Elementary Mathematics. 



I WAS very glad to read in N.ature of January 30 (p. 297) 

 the letter of Mr. J. W. Marshall on elementary mathematics, 

 because all his suggestions referring to elementary algebra have 

 already been realised in my book, " Applied Algebra," pub- 

 lished in 1900 at St. Petersburg, in Russian. 



I send, therefore, two copies of my book, one for the editor, 

 the other for Mr. Marshall. The algebraical characters will 

 permit every mathematician to judge somewhat of the character 

 of a mathematical book without a knowledge of the Russian 

 language. My compatriots have already condemned my 

 heresy ; but I hope that my ex-compatriots (our family is of 

 Scottish origin, derived from Lehrmouth) will judge more 

 liberally my attempt to improve the old method of teaching 

 the prolific science of algebra. 



The peculiarities of my exposition are explained by the fact 

 that I wished to say all that was needed, and only what was 

 needed. For that purpose it was necessary to put in the first 

 place the systematical exposition of all the most fundamental 

 methods of algebra which are sufficient for ordinary applications, 

 and to postpone to a supplement all complicated questions 

 that are usually combined with fundamental notions in the 

 existing manuals, only producing a confusion in the minds of 

 the pupils. 



