May 19, 1887] 



NA TURE 



51 



a way that not many English words will brook, and 

 certainly not a word of such wide meaning and acceptance 

 as this. In the same way the book-names of the Stone- 

 Curlew— Thickknee, Norfolk Plover, and the rest, to the 

 number of half a dozen— are gravely printed as if they 

 were "provincial" ; and here we may remark that Mr. 

 Swainson applies (p. 200) the Arabic name of this species 

 " Karrawan" (as he prints it) to its namesake the Long- 

 billed Curlew or Whaup, which must be wholly unknown 

 to most of the descendants of Ishmael, 



We very much regret that we have to express ourselves 

 in such terms of this book. We doubt not that the author 

 has done the best that in him lies, and we are especially 

 sorry to find from his preface that the delay in its appear- 

 ance (for it had been long looked for) is due to his ill 

 health. It is on this last account that we are indisposed 

 to drive home many charges of carelessness which might 

 easily be established, and .we part from him trusting 

 that in another edition he may have the opportunity of 

 justifying his selection by these two learned Societies for 

 the duties that we strongly suspect he must already regret 

 having undertaken ; but to do this he should acquire 

 some knowledge of the ways and looks of birds, and learn 

 the rudiments of etymology. 



RECENT WORKS ON THE THEOR Y OF 



DETERMINANTS. 



Primeiros Principios da Theorta dos Determinantes. Por 



J. -A. Albuquerque. (Porto, 1884.) 

 Die Determinanten in genetischer Behattdlung. Von 



Adolf Sickenberger. (Miinchen, 1885.) 

 Vorlesungen iiber Invariantetitheorie. Bd. I. Deter- 



mitianten. Von Paul Gordan. (Leipzig, 1885.) 

 Elemente der Theorie der Determinanten. Von Paul 



Mansion. 2te vermehrte Auflage. (Leipzig, 1886.) 

 An Elementary Treatise on the Theory of Determinants. 



By Paul H. Hanus. (Boston, 1886.) 

 Beitrdge zur Theorie der Determinanten. Von Wilhelm 



Schrader. (Halle, 1887.) 



THREE of these works are introductory text-books of 

 from fifty to eighty pages, and may consequently 

 be dismissed in a few lines. The first is a skilfully 

 arranged and well-written manual, furnished with suitable 

 exercises, and ought to be found exceedingly serviceable 

 in the secondary schools of Portugal. That by Prof. 

 Mansion, of Ghent, has already been referred to in 

 Nature ; the fact that it is now in the fourth French 

 edition and second German edition is sufficient proof of its 

 value. The third, by Gymnasial-Professor Sickenberger, 

 is the largest and yet the most elementary, having been 

 intended (not very wisely, we are disposed to think) for 

 pupils very imperfectly prepared in algebra. Twelve 

 pages, including a collection of thirty exercises, are 

 devoted to determinants of the second order, thirty-eight 

 pages to those of the third order, and the remaining thirty 

 pages to determinants in general, the whole being pre- 

 pared with endless pains and much preceptorial skill. 

 Introductory works of this kind have for a number of 

 years been appearing in Germany at the rate of 1.3.... 

 per annum : in England we have not had one since 1875. 



Our insular way of doing things, however, is so different 

 from that of the Germans that it may be fairly questioned 

 whether we are any the worse for the deficiency. It 

 would certainly be very erroneous to conclude that the 

 advance of the theory of determinants in the two 

 countries during the period referred to has shown the 

 same marked contrast. 



Gordan's "Vorlesungen" is a book on the lines ot 

 Salmon's " Modern Higher Algebra." The theory of 

 determinants is consequently not taken up in its entirety, 

 the design having been to give the main propositions 

 regarding general determinants and to include the dis- 

 cussion of only those special forms which are connected 

 with the chief subject of the work — to give, in fact, such 

 a knowledge of determinants as would enable the student 

 to prosecute investigation in the theory of invariants. It 

 is nevertheless a very full exposition— much fuller than 

 Salmon's, and much more methodical. The section on 

 Permutation and Substitution should be carefully noted : 

 although in essence it dates from the time of Cauchy, it 

 will be none the less fresh to many readers. 



" Beitrage " is rather a misnomer for the remaining 

 German work on our list. The book is nothing more nor 

 less than an ordinary, or very ordinary, text-book^ contain- 

 ing three chapters, the first on determinants in general, the 

 second on the adjugate determinant, and the third on 

 determinants of special form. The author attaches con- 

 siderable importance to a new notation which he intro- 

 duces and uses throughout, and to various new theorems 

 which he enunciates and proves ; indeed, the title-page 

 bears the intimation, " Neue Satze und eine neue Bezeich- 

 nung." The said new notation is obtained by placing 

 the lengthy but excellent umbral notation — 



2,3, 

 2,3, 



atop of another notation, which is itself none too compact, 

 viz. D{a. ^), the outcome being 



I, 2, . . . . , » 



2?(«.^^x, ........ 



It is a little hard to see that this piling of Pelion on 

 Ossa results in " grossere Einfachheit." As for the new 

 theorems, we are satisfied that the author will change his 

 mind in regard to them as his range of reading widens. 

 All the results on pp. 98-1 11, for example, are perfectly 

 well known in England and America : indeed, the whole 

 of them which concern alternants of the third order are 

 included in a single theorem of Prof. Woolsey Johnson's. 

 None the less credit, however, is due to the author for 

 the work he has done ; and we trust that, having exa- 

 mined the literature of his subject, he will continue his 

 investigations and attain a more enduring success. 



The new American text-book stands out in marked 

 contrast with the preceding. First of all, it is a book ot 

 good outward appearance ; paper, printing, and binding 

 being unexceptionable. In the second place, the author 

 makes no pretensions to originality : in his preface he 

 enumerates a few manuals, English and Continental, to 

 which he is indebted, and frankly states that he has used 

 them all freely. Some of them, we should say, he has 

 used more freely than others, but, on the whole, with 

 good judgment, and in such a way as to show that he 



