346 



NA TURE 



[February 7, 1901 



Of course it is not .to be expected that such a work 

 should be altogether perfect, and if we indicate some of 

 the points which strike us as susceptible of improvement, 

 we do so in the hope that the work may gain still further 

 in value when future editions are called for. 



Whilst many of the illustrations are distinctly good, 

 some are very much the reverse, and as an example of the 

 latter class Fig. 40 may be cited, which is excessively 

 bad, and can hardly be said to illustrate the text (which 

 it certainly does not adorn) in any sense whatever. The 

 story of the digestive functions attributed to the leaves 

 of Lathraea is now generally discredited, and might 

 as well have been omitted from the text, whilst the 

 somewhat teleological explanation of the red colour in 

 leaves perhaps might at least have been accompanied by 

 suggestions as to the proximate causes of its appearance 

 such as are indicated by Overton's recent experiments. It 

 is, however, against the short chapter on the influence of the 

 environment on plants that we incline to take the greatest 

 exception. The subject is a large one, and can only be 

 adequately, or even usefully, treated by the aid of copious 

 illustrative examples, without which as in the present 

 instance it is apt to degenerate into rather senseless 

 cramming. 



Apart, however, from what after all are but minor and 

 easily remedied faults, the book is, as we have already 

 said, a decidedly good one, and its author has displayed 

 such excellent judgment in the selection of his materials 

 in order to meet the special needs of the class of readers 

 for whom it is primarily designed that there will in 

 future be no excuse for that neglect of vegetable physiology 

 which is at present but too common with junior students 

 of botany. T- B. F. 



A Text-book of Important Minerals and Rocks, with 

 Tables for the Determination of Minerals. By S. E. 

 Tillman, Professor of Chemistry, Mineralogy and 

 Geology, U.S. Military Academy, West Point, N.Y. 

 Pp. 176. (New York : John Wiley and Sons. London : 

 Chapman and Hall, Ltd., 1900.) 



In this little manual, Prof Tillman has brought together 

 such fundamental instructions as are necessary to enable 

 a beginner to determine the most commonly occurring 

 minerals and rocks. Three short chapters on crystallo- 

 graphy, the chemical characters, and the physical 

 properties of minerals are followed by a series of tables 

 for the determination of 135 common species. In the 

 choice of these species a considerable amount of judg- 

 ment is shown, though it is obvious that the opinions of 

 an American mineralogist as to what should be regarded 

 as the most important species differ from those of 

 workers in Europe. The tables are on the familiar 

 plan of those of Weisbach, Persifor Eraser, Brush and 

 Penfield, and other well-known authors, and the arrange- 

 ment adopted is a very simple one. The twenty pages 

 devoted to rocks at the end of the volume are only 

 sufficient to enable the author to give a very slight 

 sketch of petrographic science. The work is worthy of 

 the attention of teachers organising a system of very 

 elementary instruction in determinative mineralogy. 



Laboratory Companion for Use 7vith Shenstone'sbiorganic 

 Chemistry. By W. A. Shenstone, F.R.S. Pp. vi -f 117. 

 (London: Edward Arnold, 1901.) 



Mr. Shenstone's course of work in inorganic chemistry 

 was noticed in these columns a few weeks ago (January 10, 

 p. 249). Most of the experiments in that book are 

 reprinted in the present volume, together with a number 

 of exercises, and other experiments have been added. 

 A volume suitable for use as a laboratory manual, that is, 

 containmg directions and suggestions, without theoretical 

 considerations, has thus been produced. On p. 117 

 reference is made to a frontispiece showing Fraunhofer 

 lines, but the picture has been omitted. 



NO. 1632, VOL. 63] 



LETTERS TO THE EDITOR. 



\The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of Nature. 

 No notice is taken of anonymous communications. "^ 



A Compact Method of Tabulation. 



In arranging tables of successive values of a variable quantity, 

 it is often difficult to find a middle course between making the 

 entries too numerous and making the intervals too large. I 

 wish to call attention to a mode of tabulation which, although 

 compact, provides facilities for the accurate deduction of inter- 

 mediate values. 



For convenience of description we may regard the tabulated 

 values as equidistant ordinates of a curve. If the common 

 distance is small enough (which implies that the number of 

 ordinates is large), intermediate values can be deduced by the 

 ordinary method of "proportional parts" — in other words by 

 employing first differences only. If the number of ordinates is 

 diminished by largely increasing the common interval, it becomes 

 necessary to take account of differences higher than the first. 

 We shall supppse the interval to be so chosen that the first three 

 orders of differences — and no more — require to be considered. 



A table showing the given values accompanied by three 

 columns of differences presents a formidable aspect ; and on the 

 other hand, if the user of the table is left to compute these 

 differences for himself, his labour is materially increased. What 

 I wish to point out is that, without any sacrifice of accuracy, 

 the first and third orders of differences can be omitted, the 

 second only being retained ; as in the following table of sines, 

 which is suitable for computing the sine of any angle to four 

 places of decimals. The differences entered opposite the sines 

 are the " central " second differences ; for example, - 104, 

 which stands opposite to sin 20°, is (sin 30° - sin 20°) ^ (sin 20° 

 - sin 10°). 



Let «/o «i be any two consecutive tabulated ordinates (sines) 

 between which it is desired to interpolate a new ordinate u ; 

 x^ x-y X being the corresponding abscissas (angles). Putting h 



for the common interval x^ - x^, let / stand for -, and q 



h 



for _L — , so that /-I- 17= I. Also let n^' z^/' denote the cen- 

 h 



tral second differences of ti^ ti^ respectively. Then it can be 



shown that the value of u true to third differences is 



P"i + 



■hqUfs + 



p(p+l)(p-l)j 



1.2.3 

 q{V+l){q-l),^ 



The sum pu^-Vqu^ of the two terms in Mj and «/q, though it 

 does not put first differences in evidence, really includes them, 

 and is the exact value of u when the connecting curve is a 

 straight line. In like manner, though third differences are not 

 in evidence, they are implicitly contained in the sum of the 

 two terms in u{' Uq. 



The coefficients of m^" m^" are identical in form, and are easily 

 computed. The following list of their values for each tenth of 

 an interval will serve to check mistakes. Their values (neglect- 

 ing sign) are always less than '065. 



p{p+l){p-i) 



