February 15, 19 12] 



NATURE 



513 



The section on live stock runs on lines similar to 

 those adopted for crops ; tables are given showing 

 the composition of feeding stuffs, typical rations, and 

 methods of computing variants, together with much 

 information about the animals themselves. Alto- 

 gether the book will be found very useful for reference 

 purposes, and, as it is well indexed, it is very easily 

 consulted. 



Mineralogy. By Dr. F. H. Hatch. Fourth edition, 

 entirely rewritten and enlarged. Pp. ix+253 

 (London : Whittaker and Co., 1912.) Price 45. net. 



In this "fourth edition" a revision has for the first 

 time been undertaken. The consequent doubling of 

 its size and price is fully justified by the enhanced 

 value of the work, which ^for twenty j^ears has been 

 handicapped by its modest size. The addition of 

 eighty pages to the section dealing with descriptive 

 mineralogy has allowed a much fuller treatment of the 

 ores, this portion being trebled in length, while ore- 

 dressing processes (electromagnetic, oil-concentration, 

 &c.), find brief reference under properties of minerals. 

 The portion on optical properties, formerly relegated 

 to a couple of pages, is enlarged sevenfold, thus per- 

 mitting of an explanation of double-refraction 

 phenomena. Coupled with the fuller description of 

 rock minerals, this renders the book of some use in 

 microscope work. The use of the letters a, b, C to 

 indicate elasticity axes is regrettable, owing to the 

 likelihood of confusion (both in writing and speak- 

 ing), with the a, b, and c crystallographic axes; the 

 substitution of X, Y, Z, as adopted in Iddings's "Rock 

 Minerals," avoids this difficulty. 



The arrangement of . the descriptive portion under 

 the four heads Rock-forming Minerals, Ores, Other 

 Salts, and Gems is convenient, if inconsistent, and it 

 is supplemented by a list of mineral species, chemically 

 classified. We are surprised to find so small a book 

 including among " the more important minerals " 

 metacinnabarite, hauerite, &c. The treatment of 

 mineral names is not always satisfactory ; thus 

 nowhere is mention made of the name kupfernickel, so 

 commonly used as a synonym of niccolite ; dialogite 

 appears in the text as such, but in the index as dial- 

 logite. Wolfram and wolframite (though used as 

 synonyms) are used apparently indiscriminately in the 

 text, but are separately indexed. 



The typography is good and misprints very rare 

 (on p. 57 statical changes evidently means charges), 

 but some illustrations of crystals, like Figs. Cit, and 

 75, might be improved. R. F. G. 



Revolving Vectors, with Special Application to Alter- 

 nating-current Phenomena. By Prof. Geo. W. Pat- 

 terson. Pp. vi + 89. (New York: The Macmillan 

 Company; London: Macmillan and Co., Ltd., 

 Tqii.) Price 4s. 6d. net. 



This brief but excellent little treatise can be recom- 

 mended as a good introduction to the modern topic of 

 revolving vectors, and particularly to the use of the 

 symbolic notation in the development of the subject. 

 It opens with a brief historical note on the discovery 

 in 1797, by Wessel, of the use of the imaginary \''-i 

 as an operator having a geometric function of rotation 

 through a right angle. From this the author loads 

 on to the treatment of complex quantities, and their 

 use in representing harmonic motion. The latter half 

 of the book deals with the application to alternating 

 electric currents and other electrical matters. It is 

 satisfactory that the author conforms to the conven- 

 tion adopted by the International I-'lrcirotrrhnical 

 Commission in its recent session in Turin, in u>.int; 

 }!ir ( nunter-clockwise sense of rotation as positive. 



NO. 2207, VOL. 88] 



LETTERS TO THE EDITOR. 



[The Editor does not hold himself responsible for opinions 

 expressed 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.] 



Contour Diagrams of Human Crania. 

 In the last number of Biometrika (viii., i, 2, 1911) Prof. 

 Karl Pearson has edited and published a very valuable 

 paper by the late Dr. Crewdson Benington on cranial 

 type-contours. The work is based on long series of skulls 

 of various races, e.g. English of the seventeenth centurv 

 (from the Whitechapel Plague Pit), modern English (Royal 

 Engineers), various Negro races from the Congo, Guanche, 

 Egyptian, Eskimo, and the prehistoric Cro-magnon skull. 

 For each series three typical contours are selected, viz. : — 

 (i) a " transverse vertical," passing through the auricular 

 points and the apex of the skull ; (2) a sagittal or median 

 section ; and (3) a horizontal section through the glabella ; 

 and in each case all the individual skulls of a series are 

 combined into a single " type " by a process of arithmetic 

 averages. Lastly, the diagrams thus obtained are repro- 

 duced on tissue-paper, so that one may be superposed upon 

 another, and the characteristic differences easily compared. 



I venture to think that we may go a little' further, and 

 may, by a simple device, get a new series of diagrams 

 which shall throw into still greater relief the presence and 

 the amount of essential difference of form : for, after all, 

 comparison of the two superposed contours is a matter 

 of individual judgment, and there is a lack of fixity and 



RA. 

 Fig. I. — English crania, t7th century. Tr«nsver»e contour. 



precision in our interprel.iticui nf ilio ir^uli. Moi.-i 

 is obvious that we !i:i\''. in ih. lii^i in>.i;iiico, i 

 and easy distinction between dilfereDifs of >ize and <iiit' 1 

 ences of form. 



Taking the case of the transverse, or vertical inter- 

 auricular, section of the skull, I^r. Benington's diagrams 

 represent it for us as in lit;. 1. where a median verticil 

 axis is divided into ten equal parts (the uppermost .'f 

 these also by a point one-quarter of a division from t! 

 apex), and at each of these points of division tli 

 zontal distance to the contour-line is measured .1 

 corded. Thus we are in possession of such tabulm 

 ments as the following : — 



A B 



Vertical Height ... "^■^ «^« 



II 

 10 

 9 



8 



7 

 6 



5 

 4 

 3 



2 



521 



60 '4 

 65 o 



673 

 68 6 

 685 

 66-6 



63*4 

 612 



20 

 37 

 55 

 (>7 



7i 

 74 



(>Q 



"I 



Dimensions (in mm.) of the Iransvcr-* cranial ■if. tion (riijht Milel. 

 English skull* of 17th century ; M, Cro-magnon skull. 



