March 21, 1895] 



NATURE 



48J 



I 



of HNO3 has to accumulate, that is, but little decom- 

 position has to take place, before the solubility product is 

 reduced to the critical value, and no more precipitate is 

 produced. 



If Pb(C.,H30.,)_, be used, since acetic acid is but feebly 

 dissociated, much more precipitate can be produced 

 before the H ions introduced into the solution by the 

 acetic acid are numerous enough to reduce the solubility 

 product to the critical value, and thus stop further pre- 

 cipitation. 



Reactions may also be due to the formation of com- 

 plex ions. Alumina in presence of water is in equili- 

 brium with the ions Al and 3OH. If KOH be added, 

 K3AIO3 is produced, of which the ions are 3K and AIO3. 

 The latter is not available for equilibrium with alumina, 

 hence the solubility-product is reduced, and more 

 alumina must dissolve. 



With a chapter on quantitative estimations, such is the 

 subject-matter of the theoretical part of the book. The 

 second, or practical part, is concerned with the important 

 reactions of qualitative analysis, treated as m'lch as 

 possible from the above standpoints. 



The book is not written for the beginner, but to supply 

 adequate theoretical support to the routine work of 

 general analytical chemistry. Enough has been said 

 to show that it is in great part unique, and through- 

 out it is refreshing and profitable reading ; indeed, 

 it affords the easiest means yet offered to the general 

 reader of getting the gist of current views, not only of 

 the theory of analytical processes, but of the con- 

 stitution of solutions, and of chemical mechanics — 

 the law of mass action and the velocity of chemical 

 change. 



The ideas put forward relating to the nature of 

 analytical processes are, of course, open to differences of 

 opinion. The time has no doubt passed when the con- 

 ceptions of ions and electrolytic dissociation were re- 

 garded as but the vain imaginings of mathematicians 

 who had dabbled in chemistry. Nevertheless the view is 

 still widely spread, at any rate in this country, that 

 although they serve as a means of correlating otherwise 

 isolated properties, and of predicting phenomena in 

 solution, they are altogether too artificial to be anything 

 but the crudest picture of what actually exists. Be this 

 as it may, the book before us serves a good purpose, 

 inasmuch as it puts in a clear light the fact that the 

 direction of analytical reactions is determined by the 

 solubility of the possible products of the interaction, 

 and that when the reaction has run its course we have 

 a system in equilibrium again governed by solubility. 



In the common method of presenting qualitative 

 analysis to the student, theoretical considerations are 

 either entirely ignored, or are supposed to be supplied by 

 statements asserting that a precipitate AB is formed in a 

 given solution because the affinity of A for B is greater 

 than its affinity for the other radicles present. It is this 

 system, coupled with the abuse of the ordinary chemical 

 equation, whereby all reactions are regarded as perfect, 

 that fills the ranks of budding chemists with disciples of 

 Bergmann,the gospel according to Berthollet, Wenzel, 

 and Guldberg and Waage, notwithstanding. 



J. W. Rodger. 



NO. 1325, VOL. 51] 



ANALYTICAL GEOMETRY. 



An Introductory Account of certain Modern Ideas ana 

 Methods in Plane Analytical Geometry. By Charlotte 

 Angas Scott, D.Sc, Professor of Mathematics in Bryn 

 Mawr College, Pennsylvania. Pp. xii. -|- 288. (London: 

 Macmillan and Co., 1894.) 



DR. SCOTT'S treatise is a welcome addition to the 

 many excellent text-books on analytical geometry 

 which have been published during the last few years. 

 But while most of the text-books in use at the present 

 time adequately explain the initial difficulties of the sub- 

 ject, scarcely one can be regarded as a satisfactory book 

 for those students who wish to go beyond the elements 

 as treated with the use of Cartesian coordinates. For such 

 students Salmon's "Conies" is still the standard work. 

 But although every fresh edition of this is carefully 

 revised to meet modern requirements, there are many 

 beautiful geometrical methods and theories which are 

 only briefly noticed, but which should be fully discussed 

 in a standard work. The book under revievv does not, 

 indeed, aim at replacing Salmon's, but it is admirably 

 adapted to be used as a companion to it. It aims at 

 giving a concise account of the principal mo iern develop- 

 ments due to Cayley, Clebsch, Reye, Klein, and a few 

 other continental geometers. As an introduction to the 

 study of the higher branches, it may be confidently 

 recommended to students as a clear, full, and well- 

 arranged exposition of the leading principles of the sub- 

 ject. At the same time the book is sometliing more 

 than a text-book for students. Those who wish to keep 

 up their mathematics, and have no time to spare to read 

 the various papers and memoirs that are published 

 every year, will find much that will interest them — many 

 beautiful geometrical ideas that are here published for 

 the first time in an English text-book. 



The author starts with the general idea of coordinates, 

 and develops simultaneously the theory of point-coordi- - 

 nates and line-coordinates, with a full explanation of 

 the special peculiarities of the two systems. The prin- 

 ciple of duality is then explained, and the descriptive 

 properties of curves discussed. After a short but ade- 

 quate chapter on curve tracing, the theory of projection 

 and homography is introduced, leading up to the theory 

 of correspondence. The chapter devoted to this theory 

 is very well written, and is in fact one of the principal 

 features of the book. In it will be found a full discus- 

 sion of quadric inversion, based on a memoir by Dr. 

 Hirst. The remaining chapters are devoted to a dis- 

 cussion of the generalisation of metrical properties of 

 curves obtained by replacing the circular points by a 

 conic, and a brief explanation of the connection which 

 subsists between geometry and the algebraical theory of 

 invariants. 



The treatise is confined to plane geometry, in which 

 figures are considered as combinations of points and 

 straight lines. In constructing a system of coordinate 

 geometry for such figures, two theories naturally present 

 themselves ; the two in which the point and the straight 

 line are, respectively, the primary elements, and the 

 straight line and point the secondary elements. But 

 although Dr. Scott states that any other geometrical 

 entity might be taken as the primary element, she makes 



