April 2b, 1888] 



NATURE 



605 



which can still be seen going on around the coast and harbour. 

 At Mokullo, at a depth of 20 feet, I observed masses of coral 

 (Aperosa) almost perfect in shape, covered up with alluvium. It 

 is probable that the whole coast from the mountains has been 

 reclaimed by the action of coral builders, and that eventually 

 the group of islands outside will be joined to the mainland." 



I noticed a similar formation of the coral reefs in Suakim 

 Harbour ; while at Key West, Florida, there was no lessening 

 of the depth of the water on the edge of the reefs. 



David Wilson-Barker. 



The following table, showing some of the results of work 

 done in connection with the solubility of carbonate of lime in 

 sea-water will be of interest. The difference in solubility be- 

 tween heavy dense corals and the lighter porous varieties is very 

 marked. 



Table I. — Shmuing Solubility oj Carbonate of Lime, under 

 different forms, in Sea-water, in grammes per litre. 



of it to acyclic quadrilaterals given in Todhunter's "Euclid,' 

 p. 318, and at the same time generalize the problem thus — 



Tofnda point Ewilhin a triangle such that I . AE + m . BE 

 + n . CE may be a minimum ; I, m, n being such that any two 

 are together greater than the third. 



Table II. 



Weathered oyster-shells 



Mussels allowed to rot in sea-wafer seven days 



Crystallized carbonate of lime 



a Amorphous carbonate of lime (freshly prepared) 

 b Ditto ditto ditto 



Melobesia, Kilbrennan Sound, Scotland 



a and b. The carbonate of lime was added as long as it dissolved. 



The figures in Table II. will give Mr. T. Mellard Reade 

 facts (so far as laboratory experiments may) upon which to found 

 reasonable views. Mr. George Young, who has made all the 

 determinations under my direction, is one of the chemical staff 

 attached to the Marine Station here. 



Robert Irvine. 



Royston, Granton, near Edinburgh, April 16. 



Note on a Problem in Maxima and Minima. 

 I suppose most lovers of elementary geometry who read the 

 communication on the above subject from Mr. Chartres in 

 Nature of February 2 (p. 320) admired the simple investigation 

 he gave of the problem. 



I should like, however, to point out — 



(i) That it might be made still more elementary by proving 

 EB -f EC = ED without the aid of Book VI. 



Let E be any point on the arc of the circumcircle of an equi- 

 lateral triangle BDC on which the angle D stands, and on ED 

 as diameter describe a circle cutting EB, EC in B', C. 

 Then / B'C'D = ^ BED -^r / BCD. 

 Similarly z C'B'D = / CBD ; 

 .-. z B'DC = z BDC ; 

 .•. B'C'D is equilateral. 



Hence B'E, EC are sides of a regular hexagon inscribed in the 

 circle B'C'D. 



.-. B'E + EC = ED. 

 Again, BD, DB' = CD, DC, 

 and / BDB' = ^ CDC ; 



.-. BB' = CC ; 

 .-. BE -}- EC = B'E -t- EC 

 = ED. 



(2) If we assume Ptolemy's theorem (conventionally quoted 

 as Euclid, VI. D) we may as well assume the known extension 



On BC describe a triangle BCD such that BC : CD : DB :: 

 I \m:n ; the point required will be the intersection E of AD 

 with the circumcircle of BCD if E is within the triangle ABC. 



For BE . CD -1- CE . BD = ED . BC, 

 .-. /w , BE-t- «. CE = /. ED; 

 .-. / . AE -^ ;« . BE -f « . CE = / . AD. 



But if G is any other point on the arc BEC, 



»/ . BG + « . CG = / . GD ; 



.-. / . AG -^ w . BG -f « . CG = / . AG -f / . GD ; 



.-. /. AG-H w . BG-f « . CG>/. AD. 

 And if P be any point within the triangle ABC, but not on the 

 circumference — 



BP . CD-f CP . BD>PD . BC (Todhunter's "Euclid," 

 .-. w . BP4-« . CP >/. PD; [p. 318); 



.-. /. AP-f;« . BP-t-w . CP >/. AP -1- /. PD; 

 .-. /. AP-l-;« . BP + « . CP >/. AD. 



If /, w, n are proportional to a, b, c, E is the orlhocentre of 

 ABC. 



If /, ;;/, n are proportional to c, a, b, or b, c, a, E is one of the 

 Brocard points of ABC, and the construction for E is equivalent 

 to that of Mr. R. F. Davis for the Brocard points (" Reprint of 

 Mathematics from the Educational Tivies," vol. xlvii. App. II.). 



It will, of course, be seen that the triangle formed by drawing 

 perpendiculars to AE, BE, CE through A, B, C, is the maxi- 

 mum triangle with its sides proportional to /, m, n and passing 

 through A, B, C. Prof. Genese has kindly supplied me with 

 an elementary investigation of the problem, depending on the 

 construction of that triangle. 



It may also be seen that the question has an intimate con- 

 nection with one proposed by Mr. Morgan Jenkins in the 

 Educational Times for August i, 1884 : — 



If on the three sides of a triangle, ABC, there be described 

 any three triangles, BDC, CEA, AFB, either all externally or 

 all internally having their angles in the same order of rota- 

 tion, and the angles which are contiguous to the same corner of 

 ABC equal to each other, prove that AD, BE, and CF nieet in 

 a point O, which is also the common point of intersection of 

 the circumcircles of BDC, CEA, AFB (" Reprint," vol. xliii. 

 pp. 88-91). Edward M. Langley. 



Bedford, April 14. 



Self- Induction. 



I FIND I am being quoted as having said that an iron con- 

 ductor has less self-induction than a copper one. You will 

 perhaps spare me a line to disclaim any such statement. It is 

 one which seems to me on the face of it absurd. 



Oliver J. Lodge. 



