August 28, 1890] 



NATURE 



415 



compared with the ascertained direction of the earth-waves, con- 

 firm or upset my general supposition. But, if this were found 

 to be correct, such observations would furthermore constitute, 

 even in the absence of special seismic instruments, a certain 

 amount of evidence as to the actual direction of the earth-waves 

 on any particular occasion. 



Similarly, a record of the exact direction of recumbent ob- 

 servers in regard to the points of the compass, might, when 

 compared with their respective descriptions of the movements of 

 objects about them, serve a similar purpose. 



Man himself would thus to a certain extent — that is, as regards 

 the local direction of the earth-waves — be his own seismo- 

 meter. Possibly, some evidence on this subject might even 

 now be obtained by comparing the descriptions of the appear- 

 ances with the ascertained directions of the outlook of different 

 observers. John Marshall. 



92 Cheyne Walk, Chelsea. 



On a Problem in Practical Geometry. 



In treatises on practical geometry rules are given by which 

 an arc and its chord or an arc and its tangent may be divided 

 proportionally, but they leave an error which is often too great. 



By the following method the points of division move step by 

 step towards their required positions until errors are of less 

 than any assigned amount. 



Let GMH be the chord (Fig. l), M its middle point, 

 AOB the perpendicular diameter. In AOB produced take 

 a series of points B'B"B"'..., determined thus : BB' = BG, 

 B'B" = B'G... Then evidently circles with centres at BB'B'..., 

 passing through G, form a series of which each has on its cir- 

 cumference the centre of the succeeding one. These circles 

 cut the line AM in a series of points A' A"..., and the arcs 



Fig I. 



GAH, GA'H, GA"H... get rapidly nearer the straight line 

 GH. Any point C on the given arc GAH may now move to 

 its destination 7 on the line GH by stepping up to each circle in 

 the direction of its centre; along a path CC'C"..., made up of 

 CC' tending to B, C'C" tending to B', and so on. 



That all the arcs which this path crosses and the chord to which 

 it tends are divided proportionally at the points CC'C"... 7 

 follows from the almost obvious theorem that if the centre of 

 one circle is on the circumference of another, lines drawn from that 

 centre intercept arcs of the circles having a constant ratio. 



If the circles become inconveniently large before the required 

 approximation is reached, we may use the following : C'C"... 



NO. 1087, VOL. 42] 



are the centres of the circles inscribed in GCH, GC'H..., 

 which are easy to construct. Also it may be noticed that the 

 angle made with .\0 by each of the parts of CC'C... is 

 half that made by the preceding part, and the process may be 

 brought to an end at any stage with progressive accuracy by 

 making the last angle one- third instead of half of the preceding 

 one. 



In Fig. I the process is closed after the second stage by draw- 

 ing the last line (C"7) not towards B", but towards the middle 

 point P of B"B"'. 'This may be done as soon as the last arc 

 GC"H comes to be less than a quadrant. 



When the chord becomes the tangent AT at A, the points 

 AA'A"... coincide, all the circles have AT touching them at 

 A, the radius of each is half that of the succeeding one, the 

 arcs intercepted AC, AC, AC"... are equal, and so we get 

 in the limit A7, the length of the arc AC laid out on the 

 tangent. 



But in Fig. 2 an alternative construction is shown. Bisect 

 TAC by AC, TAC by AC", and so on. Draw CC, 

 CC"... at right angles to AC, AC... The process is shown 

 closed after the third stage by drawing C"'7 at right angles, 

 not to AC", but to a line AF such that C"AF is one-third of 

 C"'AT. In the result A7 is equal to arc AC. 



John Bridge. 



Caught by a Cockle. 



I HAVE often intended writing to you describing a curious 

 occurrence which I witnessed on the coast of Queensland in 

 September 1889, but I have as often forgotten to do so when 

 the opportunity came. While out shooting, along a sandy 

 beach, I noticed a small muddy patch just covered by the rising 

 tide. In this I observed a bird, a sand-piper, which seemed to 

 be striving in vain to rise. I could not think how the bird had 

 become caught, but on coming up to it I found that one claw of 

 one foot was firmly held by a large cockle (about i^in. by 2 in.). 

 Of course the bird would have been drowned eventually (though 

 the benefit to be derived by the cockle seems rather problem- 

 atical) ; and though it seemed to be aware of its danger, yet it 

 had made no attempt to free itself by trying to bite through the 

 claw, as one sometimes reads of animals doing when caught in 

 traps. As I believe this is rather an uncommon incident, I must 

 make that my excuse for troubling you. D. McNabb. 



H.M.S. Dart, New Hebrides, July 3. 



ON STELLAR VARIABILITY. 



ON the hypothesis of the meteoritic origin of the 

 various orders of cosmical bodies there is a grand 

 and orderly variation, both in light and colour, in the 

 case of every undisturbed swarm during its condensation 

 from its most nebulous condition to that of a cool dark 

 globe. 



As by virtue of the ordinary evolutionary process an ««- 

 disturbed swarm successively passes through the changes 

 the results of which define the various groups, the light 

 will wax through Groups I. to IV., and then wane till it is 

 finally extinguished ; at the same time the colour se- 

 quences will be successively passed through. But with 

 such a variability as this, compared with the period of 

 which our annus viagnus is but a point of time, we have 

 now nothing to do. We have to deal really with diS' 

 turbed s\va.rms or with double or multiple swarms through 

 their various stages of condensation. 



Let us take the purely disturbed swarms first. Imagine 

 a nebula, sparse, and therefore so dim as hardly to be 

 visible at all. Then, further, imagine the appulse of 

 another, or the approach of a meteoritic stream. We 

 shall have the condition which must bring about increased 

 luminosity ; the outburst may be short or sudden ; the 

 greater luminosity may last a short or a long time ; the 

 dying down of the light may be fast or slow. In that we 

 shall have the possibilities of new and dying stars. 



If the spectrum of the light produced by this clashing 

 be observed, we may not have precisely the same pheno- 

 menon as that observed in the various groups defining 



