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XLYIII. Notices respecting 



The Theory of Determinants in the Historical Order of Development. 

 By Thomas Muir, M.A., LL.D., F.R.S. Second Edition. 

 London : Macmillan & Co., Ltd. 1906. 



TN August 1890, the first edition of this work was reviewed in 

 these pages. The present work is nearly twice the size of the 

 earlier volume ; but that is almost entirely because of the new 

 matter added under Part II., which treats of the development of 

 special determinants down to the year 1841. Part I. gives an 

 account of general determinants down to the same date, and is, 

 with a few additions, essentially the same as the first edition. A 

 suggestion made at the end of the former review that the names 

 and dates of authors being discussed should be mentioned in the 

 head-lines of the pages has been followed, greatly increasing ease 

 of reference. The different chapters of Part II. deal with 

 Axisymmetric Determinants, Alternants, Jacobians, Skew Deter- 

 minants, Orthogonants, etc., all of which received attention at 

 the hands of mathematicians before the specific name had been 

 invented. This indicates one great merit of Dr. Muir's labours, 

 which — to quote his own words — show " the extent to which the 

 study of special forms of determinants had been carried prior to 

 184i, an extent probably hitherto unsuspected." In these earlier 

 days authors had a tendency to use their own particular notations, 

 which in certain cases differed markedly from the notations now 

 familiar ; and here again mathematicians cannot be too grateful to 

 Dr. Muir for the masterl} r way in which he has brought the whole 

 literature into line. He quotes important parts of the original 

 papers, explains the notations, and elucidates the results obtained 

 in many cases by translating them into modern notation. The 

 broad features are clearly brought out in three short chapters 

 termed ' Retrospects.' In recent papers communicated to the 

 Eoval Society of Edinburgh, Dr. Muir has continued his historical 

 studies down to a later date than 1841 ; but there is no hint in 

 the present volume that he contemplates a discussion of the more 

 modern developments in the same systematic and thorough manner. 

 The closing sentence of the book states that " The fruitful era of 

 nomenclature, but not of that alone, was ushered in by Sylvester 

 shortly after the date with which " the present volume closes. It 

 may be doubted whether an historic sketch, concise yet complete, 

 of these later developments can be given ; but if it can, then no 

 one could do it better than Dr. Muir. 



Lectures on the Method of Science. Edited hy T. B. Strong. Oxford : 

 at the Clarendon Press. 1906. Pp. viii+249. 



The present volume is the outcome of a new plan adopted by the 

 Delegates for the Extension of University Teaching during the 

 Summer Meeting at Oxford, in August 1905. Instead of pro- 

 viding a course on some special branch of science, it was suggested 



