March 28, 



NATURE 



II 



If N is prime, it is expressible as the difference of two squares 

 in only one way, viz. {^(N + i)}^ - {4(N - i)}-. To prove 

 that N is prime by this method, the number of additions 

 required is i(N + i) — «, which is \{{n — i)- - r„}. 



It may be noticed that when n + m and r,„ have a common 

 -measure, it must be a factor of N, and the additions need be 

 continued no further. 



For example, 



N 



91 



= 92 



= 8281, 

 = 150 

 183 



= 133 

 185 



+ 2 = 93 



5'^ 

 187 



^^^ « + 3 = 94 '-3 = 70s 



H|K and 705 have a common measure, 47 ; therefore 8131 is 

 fj^^divisible by 47, and the other factor is then found to be 173. 

 Mr. Busk's method of shortening, exemplified on p. 414 by 

 his proof that 73 = 37- - 36', depends upon the following : 



Let r„ + 2w« + w2 = (/. + mf^ then ;// = |(F - r„)/(« - k) : 

 ■.ince k' - Tg is even, /& is even or odd according as ;-„ is even or 

 odd ; it is necessary only to try values of /c descending by 

 differences of 2 ; the greatest possible number of operations is 

 i(n — I - ^,1), when /•„ is the value ofk, with which we begin. 

 The process may conveniently be arranged as in the following 

 example : 



Let N = 6667, «- = 6724 = 82-, r„ = 57. 



-0 82 - A i(/y- - 57) Quotient 



15 ... 67 ... 84 ... a fraction 



32 



23 ... 59 ... 236 ... 4 



therefore 6667 = (82 + 4)'- - (23 + 4)- = 113 X 59. 



If N is composite, this method is not always shorter than the 

 former. It will be shorter whenever 2m>k - k^, but it is not 

 easy to see how to determine a priori whether this is the case. 



The method by decreasing squares is not one of general 

 application. For instance, the factors of 323,171 cannot so be 

 found. It is the difference of two squares, each more than ten 

 times as large as the first square used. 



W. H. H. Hudson. 



King's College, London, March 15, 



Dolomedes fimbriattis, Clerck, at Killarney. 



It may interest some of your readers to know that this rare 

 and fine aquatic spider occurs on Cromaglaun Mountain, near 

 Killarney Lakes. I first found it when collecting the little 

 shell, Limnccus involutus, and though I had it two or three times 

 in my hand, it was so active that it escaped, and I, not knowing 

 its powers of diving, never thought of looking for it under water. 

 The following year I again visited the little lake, which is called 

 Crincaum, with some friends, and this time we fairly captured 

 the spider, which I then easily identified as Dolomedes fimbriattis. 

 There is a good account of it in Blackwall's "British Spiders," 

 and also in Andrew Murray's "Economic Entomology — Aptera," 

 but I am not aware that it had been observed in Ireland before 

 I found it. A. G. More. 



March 18. 



BEECH- WOOD. 



T T is so characteristic of the science of to-day to find 

 -*■ specialists narrowing their field of research, and con- 

 iining their investigations to a deep narrow channel, that 

 aio surprise can be felt that two able men should devote 



their energies for two years to the examination of the 

 biology and chemistry of the wood of a single tree. It is 

 not so easy to avoid astonishment at the results of the 

 two years' work, however, appearing as they do in the 

 form of a large book ^ of 238 pages of close description 

 and argument, interspersed with long tables of figures, 

 abounding in interesting information when properly read. 

 The authors have divided their work very fairly, the 

 botanist having set himself the task of elucidating in 

 detail the histology of the wood, the distribution of water, 

 starch, and other contents, the formation of annual rings, 

 and the growth in thickness of the trunk, and a number 

 of other problems throwing light on the growth of the 

 beech in the forest ; while the chemist has confined him- 

 self to the task of analyzing the timber, so as to discover 

 (i) the quantities of total ash, water, nitrogen, &c., in 

 different parts of the tree ; (2) the percentage composi- 

 tion of the ash, and the manner of distribution of the 

 individual constituents ; (3) the absolute quantities of 

 each ash-constituent in 1000 parts, and other chosen 

 quantities of dry substance of the wood ; (4) the annual 

 in-take and out-put of these constituents on a hectare of 

 beech forest ; and (5) similar particulars for the nitro- 

 genous constituents. 



The authors have by no means spared their trees. It 

 is enough to make one envious to read of the trees cut 

 down at all ages from 15 to 1 50 years, and of the speci- 

 mens selected at all heights from each ; how the research 

 was extended to good, bad, and indifferent soils, and how 

 trees in shade and in the open, trees entire and trees 

 pruned, &c., were all laid under contribution as required. 

 More than 100 stems of all ages were thus employed. 



The manner of utilizing this enormous mass of material 

 is worth noticing, for every kind of determination was 

 made that would yield practical information. 



The height of the trees was found, as the best indica- 

 tion of the value of the situation ; the number of stems 

 on a given area, their surface, contents, &c., were also 

 determined ; the age of the trees, their physiological con- 

 dition, &c., were all considered in due course. The 

 selected stems were then cut up as follows : transverse 

 disks were cut at the successive heights of i'3, 5*5, 107, 

 i5"9, 2ri,and 26*3 metres, and separate determinations 

 made of the specific gravity, histological peculiarities, 

 analysis, &c., and these not only for wood and cortex 

 separately, but also for each 30 annual rings of the stem. 

 The thickness, density, «&;c., of the annual rings were 

 also tabulated, and attention paid to north, south, east, 

 and west sides of the stem. 



Not only are all these data given in detail in the tables, 

 but other tables are provided showing the mean densities, 

 cubic contents, &c., &c., of whole trees, or of the trees 

 on given areas ; and the patient compilation and ingenious 

 methods here displayed reflect the greatest credit on the 

 authors. It is, in fact, especially in the application of 

 their measurements, &c., to the forest as a whole that 

 the tables will find their greatest practical value. There is 

 also much of more abstract scientific interest to be learnt 

 from the results. 



On examining the histology of the wood, several new 

 facts were discovered. The curious dipping in of the 

 annual rings where they cross the broader medullary 

 rays, and the deposits of grains of calcium carbonate on 

 the septa of the vessels, may be mentioned by the way ; 

 but the most important results are those relating to the 

 length of the elements, the lumina of the vessels, and the 

 relative numbers and distribution of the latter on a square 

 millimetre of transverse section. 



The wood of the beech consists of the usual elements — 

 vessels, tracheides, libriform fibres, and wood parenchyma, 

 with transitional elements difficult to classify under any 

 one of these heads. As was long ago pointed out by 



' "Das Holz der Rothbuche," by Profs. R. Hartigand R. Weber. (Berlin : 

 Springer, 1888.) 



