August 12. 1Q09] 



NA TURE 



the general problem above mentioned which render it 

 impossible of complete treatment by any known 

 method of analysis. The condition that the first n- 

 natural numbers must appear one, and only one, in 

 each cell, is, to begin with, of a most difficult char- 

 acter, and is made more so by the importation of the 

 diagonal conditions. Many kinds of magic squares, 

 moreover, involve additional properties connected with 

 broken diagonals, nuclear squares, symmetrically 

 placed cells, &c., which mathematically are of a most 

 arduous nature. It thence arises that though many 

 mathematicians of repute from the earliest times have 

 studied the subject, they have devoted their labours 

 mainlv to observational and tentative methods of 

 actuallv constructing different classes of these squares, 

 and have seldom seriously attempted the enumeration. 



Dr. Willis has put together in this work all the 

 best-known methods of constructing the various types, 

 and he gives a good idea of the magnitude of the 

 numbers tha? may be reached when attempt at 

 enumeration is made. There is one magic square of 

 order 3, and 880 of order 4 ; beyond this huge numbers 

 may be expected. As an illustration he takes the 

 case of the " square of eight in magic quarters " 

 (chapter viii.), and in forming an estimate he takes as 

 unit the number of different squares which could be 

 printed in a year; if this could be done at the rate of 

 a million per second, by keeping the press at work 

 day and night without ceasing during 365^ days ; 

 his conclusion, "at a very moderate estimate," is 

 that 150,000 years would be required. A particular 

 kind of pernuclear square of order 20 is similarly shown 

 to have a number of forms approximately given by 

 2953 followed by 135 zeros. 



Chapter x., on magic cubes and geometric designs, 

 takes the reader somewhat out of the beaten track, 

 and will be found particularly interesting. In intro- 

 ducing the subject of magic designs, he says : — 



" By making use of magic cubes in which the hori- 

 zontal squares are magic . . . we may form a mosaic 

 design which is magic in the sense that the n hori- 

 zontal squares, from which the design is formed, may 

 be found by inspection of the number of tessera of n 

 different colours contained in each of the n- squares of 

 which the complete design consists." 



The author gives five examples of these beautiful 

 patterns printed in colours. Anyone who will take the 

 pains to master the method explained will find himself 

 - able to add to these to an unlimited extent, and prob- 

 ably possessed of the power of evolving new principles 

 of mosaic design. 



The late Prof. Sylvester was enthusiastic about 

 mosaic designs based on mathematical principles, and 

 one of his " anallagmatic squares," carried out in 

 marble, was for many years in the hall of the United 

 Service Club. 



This book will be useful to the numerous persons 

 who find the subject an amusement and recreation. 

 The mathematical equipment required for its com- 

 prehension is limited to the multiplication table. The 

 book is well and appropriately printed, and the different 

 types used for both letters and numbers are of much 

 assistance to the reader. P. A. M. 



NO. 2076, VOL. Si] 



PHYSICAL CHEMISTRY. 

 Grundriss der aUgemeinen Chemie. By Wilhelm 

 Ostwald. Vierte, voUig umgearbeitete Auflage. 

 Pp. ix + 66i. (Leipzig: W. Engelmann, 1909.) 

 Price 20 marks. 



TWENTY years have passed since the appearance of 

 the original edition of " Ostwald's " Grundriss," 

 in which the principles of modern physical chemistry 

 were first competently placed before the general 

 scientific public. From time to time new editions 

 have appeared, but comparatively little change was 

 made in the original form, the lines of development of 

 physical chemistry having been determined by the 

 work of van 't 'Hoff, Arrhenius, Nernst, and the 

 author at a period immediately preceding the composi- 

 tion of the book. 



The fourth edition, which is now before us, has in 

 many respects been profoundly modified. The dis- 

 covery of radio-activity and the investigation of gaseous 

 ions give a new point of departure for chemical 

 research and theorisation, and this the author has 

 fully recognised in the present volume. It says much 

 for his open-mindedness that he now relinquishes 

 definitelv and explicitly his former position that the 

 existence of atoms and molecules received no cogent 

 experimental proof from chemical or physical science. 

 In his preface he writes : — 



" Even the most cautious experimental man of 

 science is now justified in speaking of the atom- 

 istic nature of extended matter— on the one hand 

 by the isolation and enumeration of the gaseous 

 ions, the crowning-point of the excellent and persever- 

 ing work of J. J. Thomson ; and on the other by the 

 correspondence of the Brownian movements with the 

 demands of the kinetic theory which has been shown 

 to exist by a number of investigators, finally and most 

 completely by J. Perrin." 



With the further remark of the author that this, 

 for him, new point of view should be first 

 properly introduced in the chapters devoted to the 

 results of these researches, one may be disposed 

 to agree, in view of the real pedagogic danger 

 of allowing the student to believe that the stoichio- 

 metric laws can only be valid owing to the existence 

 of atoms, instead of being, as they are, generalisations 

 from experiment, and independent of any hypothesis 

 as to the constitution of matter. The author has, 

 therefore, made only a subsidiary use of the atomic 

 hypothesis in the earlier part of the work. 



The volume is now subdivided into the following 

 books :— i., substances; ii., stoichiometry ; iii., chemical 

 thermodynamics; iv., electrochemistry; v., micro- 

 chemistry; vi., photochemistry; vii., chemical affinity. 



Chemical kinetics and equilibrium receive treatment 

 in book iii., colloidal solutions in book v., and radio- 

 activity in book vi. 



The author has made a deliberate attempt to meet 

 the criticism that former editions of the book were too 

 difficult for the ordinary scientific reader. The extreme 

 compression and logical treatment aimed at certainly 

 required an amount of concentration beyond that at 

 the disposal of the student when he began the study 

 of physical chemistry. It may be said at once that 



