Jati. 27, 1887] 



NA TURE 



393 



termination of labours which must have occupied most 

 if not all of his leisure time from other more regular 

 work. 



But Mr. Carr has not confined his work to the limits of 

 his previous title-pages : he has " supplemented " it " by an 

 index to the papers on pure mathematics which are to be 

 found in the principal Journals and Transactions of 

 learned .Societies, both English and foreign, of the 

 present century." Such indices have, in these busy 

 times, great value for students in all branches of know- 

 ledge, and this one is, we think, very accurate for a first 

 edition. There are, however, some defects. On p. 720 

 occurs an historical note on the cycloid ; no reference is 

 made to the exhaustive treatise, on the cycloid and all 

 forms of cycloidal curves, by Mr. Proctor (Longmans, 

 1878). In the like case of Cartesian ovals there is no 

 account taken of Prof Williamson's paper on these curves 

 in Hermathena (No. iv. p. 509 ; subsequently given in the 

 author's " Differential Calculus "). For the length of an 

 arc of an oval expressed by elliptic functions, Mr. Carr 

 (p. 731) cites a paper by Mr. S. Roberts, F.R.S., in the 

 London Mathematical Society's Proceedings (vol. v. p. 6), 

 but does not mention that Mr. Roberts (vol. vi. p. 200) 

 subsequently found that he had been anticipated 

 by Prof Genocchi (1S55, see " II Cimento," Turin). 

 The name of Desargues, rightly given p. 917, is twice 

 printed Desarques on p. S58. On pp. 913 and 935 we 

 have " polyzonal," " zonal," and " tetrazonal " : the author 

 of the paper in question has "m" in place of "n" 

 throughout. " Nicomaque " (p. S53), " implexe " (p. S59), 

 " pseudosfera " (p. 915), are easily traceable; " coplana- 

 tion " and " complanation " are also to be traced to the 

 titles of the original papers. Cases of wrong spelling 

 occur in the names of Hesse (p. 890), Kronecker (p. 890), 

 Pliicker (p. 916), Rodrigues (p. 921), Lissajous (p. 900), 

 and there are various forms of MacCuUagh ; but " Tetra- 

 tops " (p. 931) eludes us. These are trifling matters, and 

 in the text in such parts of XIII. and XIV. as we 

 have read we have not observed any errata of con- 

 sequence. 



It would be a boon to students if Mr. Carr would issue 

 this supplement in a separate form, and add to his exten- 

 sive list of thirty-two periodicals references to the papers 

 on pure mathematics which occur in the Philosophical 

 Magazine, Lady's and Gentleman s Diary, the Mathema- 

 tician, " Reprint from the Educational Times" (a limited 

 selectionhere of general results and the occasional papers), 

 and Mathesis. For instance, on the " 1 5-girr" problem 

 there is a good article in the Diary, ovi. the "chess- 

 men " in the Philosophical Magazine. 



Mr. Carr will understand that our remarks are made in 

 no captious spirit : we are very grateful for the trouble 

 he has taken, and desire only that a second edition 

 may be made even more valuable from its increased 

 accuracy and stores of information than the present 

 one is. 



He is to be congratulated on the arrangement 

 of his text, the several different kinds of types which 

 have been put at his disposal by the printers, and the 

 excellent diagrams. It only remains to express the wish 

 that what the author has done for one side of mathe- 

 matics he may be encouraged to do for the other, i.e. for 

 applied mathematics. 



COMMERCIAL ORGANIC ANAL YSIS 

 Commercial Organic Analysis. By Alfred H. Allen, 

 F.I.C., F.C.S., &c. Second Edition, Revised and 

 Enlarged. Vol. II. Fixed Oils and Fats, Hydrocarbons, 

 Phenols, &c. (London : J. and A. Churchill, 18S6.) 



THIS work has been so much enlarged and so tho- 

 roughly revised that it has become almost a new 

 book, and certainly its value has been greatly en- 

 hanced to all analysts and others interested in the 

 special points discussed. These copious additions 

 have rendered it necessary to divide the present edi- 

 tion into three volumes, the first of which appeared 

 some little while ago, whilst the third (on Aromatic Acids, 

 Tannins, Colouring Matters, Cyanogen Compounds, Or- 

 ganic Bases, Albumenoids, &c.), is now in course of pre- 

 paration, and is promised shortly. Of the sections treated 

 in the present volume it may be said generally that the 

 author has collected together and systematically digested 

 almost all the available information extant scattered 

 about in text-books and numerous papers read before 

 scientific Societies, and that he has largely contributed 

 personally to the mass of information by means of experi- 

 mental and observational work carried out in his own 

 laboratory at the cost of much time and labour. More- 

 over, he has sought and obtained the aid of several well- 

 known chemists possessing special knowledge and skill 

 in certain kinds of work, by whose assistance many of the 

 more important articles have acquired an almost ex- 

 haustive character. As a result, the treatise has become 

 a most valuable hand-book and book of reference with 

 respect to the class of matter coming within its scope ; 

 and it may now be fairly said to be an essential item in 

 the list of works requisite in the library of an analytical 

 chemist. 



Amongst the numerous special points of research to 

 which the author has devoted much personal attention, 

 a notable one is the examination of various of the 

 physical properties of fatty and oily matters with a view 

 to their discrimination and identification, and more espe- 

 cially the methods in use in the determination of their 

 specific gravity. In order to avoid complications arising 

 from differences in physical state — some being solid, others 

 pasty, and-others fluid at the ordinary temperature — he 

 recommends, as previously proposed by Estcourt, that 

 comparisons should always be made at the temperature 

 of boiling water, the melted fat or fluid oil being placed 

 in a test-tube heated in a water-bath (of such construction 

 that no steam escapes in the vicinity of the operator), 

 and the indications of a Westphal's hydrostatic balance 

 noted when the plummet is immersed in the hot oil. The 

 figvires thus obtained on repetition of experiments show 

 very little divergence ; and characteristic values are thus 

 obtainable for certain kinds of fats, e.g. butter-fat when 

 genuine. Owing to the fact that the Westphal balance 

 as sold is constructed to give specific gravity indications 

 at the ordinary temperature {i.e. that the plummet is 

 adjusted so as to lose a particular weight, conveniently 

 5 or 10 grammes, when immersed in water at, say, 15° C), 

 it is obvious that the numerical values obtained on im- 

 mersion in hot material at close upon 100' C. represent 

 neither the true densities at 100° of the substances 

 (weights per cubic centimetre) nor the ratios between 



