May 2, 1 901] 



.VA TURE 



Die wisseiischaftlichen Grundlagen der analyiischen 



Chemie elementar darcrestellL Von W. Ostwald. 



Dritte Auflage. Pp. xi + 221. (Leipzig : Engelmann, 



1901.) Price M. 7. 

 The services' that Prof. Ostwald has rendered to physical 

 science during the last quarter of a century are so 

 numerous and so valuable that his writings cannot fail to 

 e.Kert considerable influence. In working out and ex- 

 tending the theories of van 't Hoff and Arrhenius he 

 played a leading part in laying the foundations of 

 physical chemistry ; and in applying these principles to 

 the consideration of the problems of analytical che- 

 mistry, he has effected a complete revolution in the 

 methods of approaching that subject. In 1894 he pub- 

 lished the first edition of the "Wissenschaftliche 

 GiTindlagen," and thus furnished us with scientific ex- 

 planations of much that up till that time had been little 

 more than mere empiricism ; analytical processes were 

 interpreted by him in the light of the theory of 

 solutions and the ionic hypothesis, and thus new life 

 was infused into a branch of science that had become 

 almost moribund. 



It is gratifying to think that Prof. Ostwald's efforts 

 have been appreciated ; and the fact that a third edition 

 of this striking work has been called for is sufficient 

 evidence of its success. The new ideas are beginning 

 to take a firm root, and are already finding their way 

 into the latest te.xt-books on the subject. 



It is to be hoped that teachers of practical chemistry 

 will study the pages of this last edition of the " Grund- 

 lagen der analytischen Chemie," and arrange their 

 methods of instruction on the new lines it suggests. 

 With this end in view Prof Ostwald has added a 

 chapter containing descriptions of a number of experi- 

 ments illustrating some of the more important principles 

 on which analytical chemistry is based. 



In conclusion, we would draw attention to the closing 

 words in which the author advocates the use of as simple 

 apparatus as possible, that the attention of the student 

 may be concentrated on the chief features of the experi- 

 ment. Coming from so brilliant an e.vperimenter and 

 so popular a teacher, the advice is worthy of special 

 emphasis. 



An Introduction to Modern Scientific Chemistry. By Dr. 



Lassar-Cohn. Translated by M. M. Pattison Muir.M.A. 



Pp. viii -t- 348. (London : H. Grevel and Co.) 

 The German original of this book has already been 

 noticed in these columns (vol. Ixi. p. 51, 1899). It has 

 been translated into smooth English by Mr. Pattison 

 Muir, and it may be cordially recommended as a clear 

 exposition of the leadmg facts and principles of 

 chemistry, well adapted to the class of readers for whom 

 it was written, namely. University extension students and 

 general readers. It must be borne in mind that the book 

 is not intended for those who are able to study chemistry 

 with their own liands. The fifty-eight illustrations in the 

 book are its worst feature, but they are by the author, 

 and no doubt the translator had no choice but to repro- 

 duce them. A. S. 



First Aid to the Injured. By H. Drinkwater. Pp.104. 



(London : J. M. Dent and Co. ; no date.) Price, 



\s. net. 

 The number and excellency of the illustrations are 

 special features of this little book, and increase its 

 interest and clearness, doing away also with the need of 

 lengthy explanations. The proportion between the 

 theoretical and practical parts is well maintained. The 

 anatomical details are not by any means unduly pro- 

 minent, but are only introduced in so far as they are 

 necessary to enable the practical directions to be intelli- 

 gently followed. The book can be strongly recommended 

 as a clear and trustworthy instruction in "first aid." 



XO. 1644. VOL. 64] 



LETTERS TO THE EDITOR. 



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

 pressed by his correspondents. Neither can he underta/te 

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

 manuscripts intended for this or any other part of NATURE. 

 No iwtice is fatten of anonymous communications.^ 



Solution of Cubic and Biquadratic Equations. 



The historical note in your last number by Si(;. Vacca regard- 

 ing the graphical solution of a cubic, given by Mr. T. Hayashi, 

 reminds me that I had intended, when Mr. G. B. Matthews 

 published his suggestion for the graphical solution of a biquad- 

 ratic by means of two parabolas (Nature, Nov. 16, 1899), 

 to point out that he too had been anticipated, as will be seen by 

 referring to a paper by Mr. R. E. AUardice in the Proceedings 

 of the Edinburgh Mathematical Society (April 7, 1890), where 

 it is shown that, with the exception of the case where the roots 

 of the biquadratic are equal in pairs, the real roots of the 

 general biquadratic can be found graphically by means of two 

 equal parabolas having their axes at right angles, the one fixed 

 and the other movable ; and also that every cubic can be 

 reduced to the form y"±^-l-;' = o ; and then solved graphically 

 by means of the fixed curve y-.x'^ and the movable straight 

 line .x±,y — r. 



I may take this opportunity of calling the attentioti of 

 elementary teachers to the fact, also dwelt upon in Mr. Allardice's 

 paper, that the most convenient method of discussing the alge- 

 braic solution of the general biquadratic, and of testing whether 

 any particular biquadratic is soluble by means of quadratics or 

 not, depends on the familiar theorem that ax- .^ ih.xy + by- + 

 2g.x + 2fy + c is decomposable into linear factors if «Ai' + 275-/2 - 

 af--bg--ch- = o, and not unless. Along with the biquadratic 

 x-^+px^ + q.\~ + rx + s = o (i) consider the equation y^--y = o (2). 

 By interequational transformation it is obvious that the system 

 (i), (2) is equivalent to the system composed of (2) and 

 qx-+p.xy+y- + rx + s = o (3). Again, the system (2), (3) is 

 equivalent to the system composed of (2) and (</ - K)x-+pxy-^ 

 y- + rx + \y + s — o (4), where \ is a constant at our disposal. 

 If \ be so chosen that the left hand side of (4) breaks up into 

 linear factors ; that is, if A. be a root of the cubic 



K^-,jX- + [pr-C[s)K + ^i/s-r--p-s = (5), 



then the system (2), (4) will be equivalent to two systems 

 y + lji.x + v = o,y = .V-, and J' -I- p.i- -1- ir = o, ;• = .v-. In other words, 

 the four roots of (i) are the roots of the two quadratics 

 .i-4-jn.v + v = o, x- + px + i! = o. 



The cubic (5) is not in general soluble by means of quadra- 

 tics without the adjunction of a cube root : hence the solu- 

 tion of a biquadratic in general depends on the solution of a 

 cubic and two quadratics. 



The necessary and sufficient condition that the cubic be 

 soluble by means of quadratics is that it have a commensurable 

 root, which, if it exist, can be readily found by finding an inte- 

 gral root of another cubic of the form x'^ + ax"- + bx -^ c , where 

 a, h, ,- are all integral. The determination of ft, v, p, <r then 

 requires, in addition to rational operations with p, q, r, s, K, 

 merely the extraction of a square root. 



To the tyro who is familiar with the elements of the coordm- 

 ate geometry of the conic sections the rationale of the above 

 process can be made evident by the consideration of the two 

 line-pairs which contain the four points of intersection of^ two 

 conies. It may be noted that, instead of the parabola / = .r-, we 

 may use the rectangular hyperbola xy=l, the only difference 

 being that we are led to a different cubic resolvent. 



Considering the space usually given in English text-books of 

 algebra to the discussion of equations which are soluble by means 

 of' quadratics, it is strange that few, if any, of their authors 

 emphasise the fundamental fact that the reduction of a biquad- 

 ratic which is soluble by means of quadratics can be effected by 

 finding the rational root of a cubic equation. I fear that I 

 too must plead guilty to this omission, which among other 

 things I propose to make good in the next edition of vol. i. 

 of my "Algebra." G. Chrystai.. 



Edinburgh, April 26. 



Electro-Chemistry. 

 Allow me to point out an omission unnoticed by your 

 reviewer of Mr. Bertram Blount's book on pr.actical electro- 

 chemistry (p. 5S2). Mr. Blount refers to the electrolysis of gold 

 ore as a failure (Haycraft's method). 



