458 



NATURE 



[August 3, 19 16 



lodes, but, in view of the fact that two of them 

 are Germans, it may be interesting to note briefly 

 what they say about the mineral resources of 

 " German " colonies. Gold-bearing lodes occur in 

 the contact-belts around different eruptives, mostly 

 of a dioritic nature, near the village of Sekenke, 

 in East Africa. They are lenticular in form, and 

 five of them are payable, three of these con- 

 stituting the Dernberg lode. The average assay 

 of sixty samples, after rejecting those which 

 yielded abnormally high results, gave 47 gm. per 

 ton. These samples were taken from the cementa- 

 tion zone, which is of no great depth. The gold 

 content of the primary zone does not appear to 

 be sufficient to pay for working. In West Africa 

 gold-copper ore is won on Swakop River, where 

 a garnetiferous layer in gneiss is sparsely im- 

 pregnated with copper. Auriferous copper de- 

 posits of a more important character occur on 

 the Groot and Klein Spitzkop, some 20 km. to 

 the north-west of Rehoboth. The copper-ore 

 occurs sometimes as malachite, sometimes as 

 chalcocite, bornite, or chrysocolla. The primary 

 ore probably consists of pyrites and chalcopyrite. 

 The gold occurs either as free gold or associated 

 with pyrites. Wedges of country rock between 

 converging veins have assayed 3 gm. to 4 gm. of 

 gold and 20 gm. of silver per ton. Auriferous con- 

 glomerates have been observed in the Ussungo 

 district, but they have not as yet proved to be of 

 any economic importance. 



In dealing with the world's production of gold 

 and silver the authors estimate that the total yield 

 from 1493 to 191 1 was 20,737 tons, representing 

 2838 millions sterling, a small sum compared with 

 the cost of the present war. 



The volume concludes with an account of ore- 

 bearing rocks interstratified with sedimentary de- 

 posits. This part commences with a description 

 of the conditions under which stratified rocks are 

 formed, and especially of those chemical and 

 physical processes which throw light on the 

 origin of ore-deposits. Then follow descriptions 

 of iron-ore beds, of manganese beds, of copper- 

 shale beds, of auriferous conglomerates, and 

 finally of placer deposits yielding tin, gold, and 

 platinum. 



The treatise is a valuable addition to the litera- 

 ture of ore-deposits, and the translator deserves 

 high praise for the way in which he has done 

 his work. 



NAPIER AND HIS LOGARITHMS. 

 Napier Tercentenary Memorial Volume. Edited 

 by Dr. C. G. Knott. Pp. xi + 441. (Published 

 for the Royal Society of Edinburgh by Long- 

 mans, Green and Co., London, 1915.) Price 

 215. net. 

 'X'HE first place in this miscellany is naturally 

 -*- assigned to Lord Moulton's inaugural 

 address. For once in a way, this is not an 

 empty compliment; for the address is a model 

 of what such an oration should be. There is only 

 one mathematical formula in it, and this so 

 simple and familiar to the audience that it did 



NO. 2440, VOL. 97] 



not need lo be written down, while several im- 

 portant points are brought out with convincing 

 lucidity. Of these are (i) that Napier, before 

 publishing his "Canon," had arrived at the notion 

 of a logarithm as a continuous function — we may 

 even say, as one defined by a differential equa- 

 tion ; (ii) that the essential property of the log- 

 arithm, in Napier's eyes, is that, if a : b = c : d, then 

 log a ~Iog b = log c ~ log ci, so that a table with 

 numbers as entries, and logarithms as extracts, 

 will economise labour in doing rule of three sums.^ 



The papers contributed are, on the whole, more 

 interesting and appropriate than is usual in pro- 

 ductions of this kind. Of course, some of the 

 contributors, however eminent, have little know- 

 ledge, and less interest, about the history of 

 logarithms ; so they either write an original note 

 on an irrelevant subject (such as spherical har- 

 monics) or a perfunctory page or so on relevant 

 but well-known topics. As there are twenty-six 

 technical papers, we cannot notice them all, but 

 have to select those which seem to us most worthy 

 of attention. 



Among these are the two brief contributions 

 by Prof. G. Vacca. One of these recalls the work 

 of Pietro Mengoli ; the other is, we think, vital 

 to the whole question of what was the induction 

 that led Napier to his goal. In Fra Luca Paciolo's 

 "Summa de Arithmetica " (Venice, 1494) there is 

 the following statement : — 



" If you wish to know in how many years a 

 sum of money will double itself at compound 

 interest (paid per annum), divide 72 by the rate 

 per cent. For example, if the rate of interest 

 is 6 per cent., the number of years is 12." 



No doubt this rule was obtained empirically ; 

 but the interesting thing is that we have a formula 

 implying that the number of years required is 

 inversely as the rate per cent. Now, Napier was 

 a business man, and his constructio is essentially 

 the formation of a table of compound discount 

 at a very small rate per cent. We are convinced 

 that this mercantile method contains the germ 

 of Napier's invention, and not any trigonometrical 

 formula. If we assume that, for a small fixed 

 rate r, 



A = (i+r)«=i + ar, 

 then with 



B = {i+r)P, C = {i+r)y, D = (i+r), 



we have approximately 



and now, if A: B = C: D, we have, to the same 

 degree of approximation, o-j8 = y-5, which is 

 Napier's fundamental theorem. We now kno\ 

 that if 



</>(.r/y) = «^(x)-<^(y) + <^(i), 



then cf>(x) = p\ogeX + q, where p, q are constants. 

 In Napier's original system, as Prof. Gibson' 

 points out (p. 128), 



p= - 10^, g' = 7.io^log« 10. 



1 For rea.<ons given later, we entirely disagree with Lord Moulton'» 

 suggestion that the first germ of Napier's discovery is to be found in the 

 expression for the difference of two cosines as the product of two sines. 



