April 30, 1908J 



NA TURE 



603 



scmi-Kymric dialect." Mr Nicholson, on the other 

 hand, claims to have shown that Pean is ' a Goidelic 

 borrowing from the Latin pcniia or piiuia.' " 



The astonishing- allusion here to the luestcrn ter- 

 mination of the wall of ^events might seem at first 

 >ight to be a mere misprint for eastern ; but, on 

 looking at the original, it turns out to be something 

 more, something calculated to create serious uneasiness 

 ns to other statements which one has not had time to 

 verify in this volume. Bede's words, as given in 

 riummer's edition, I. xii. (p. 26), run thus: — 



" Incipit autem duorum ferme milium spatio a 

 monasterio Aebbcrcurnig* ad occidentem in loco, qui 

 sermone Pictorum IVnnfahel, lingua autcm Anglorum 

 Pcnneltun appellatur; et tendens contra occidentem 

 tcrminatur ju.\ta urbcm .Mcluith. " 



Dr. Holmes rightly acquiesces in the view thai 

 faJicl is an old form of the Irish genitive fiiil match- 

 ing a nominative fiil, "a hedge, a wall"; but this 

 docs not, to say the least of it, help the theory that 

 Pictish phonetics were like those of \\"elsh rather than 

 of Gaelic. As to Bede's pean from Latin pinna, the 

 author proceeds to show how absurd it is to think 

 that this word "could beget a geographical name." 

 In any circumstances whatsoever, that sort of state- 

 ment must be hard to prove, so the argument comes 

 dangerously near mere quibbling, and the appeal to 

 Cx-sar should have been an appeal to the German 

 Diez, who derives from Latin pinna (" Zinne der 

 mauer ") the Italian pcnna, " the top, height, or peak 

 of a hill or mountain," and the Spanish pciia, "a 

 rock, a cliff," instances of which Diez finds in the 

 oldest Spanish records as Latin penna. This is not all; 

 a passage in the second volume of Stokes and 

 Strachan's "Thesaurus Palaohibernicus," from a 

 famous Irish MS. written in the earlv vears 

 of the ninth century, has the words a pinna 

 mantis Berbicis usque ad niontem Mis. The 

 latter height was probably Slemish Mountain, 

 in CO. .\ntrim; the Top of the Mountain of the 

 Wether (verve.x) remains to be identified. But its 

 name in the Book of Armagh shows that pinna was 

 current in Irish L.-itiiiity, and was capable of forming 

 part of a place-name. From Latin it passed into the 

 Goidelic language, whence Bede's Pean-fahel. which 

 is accordingly neither Kymric nor even semi-Kymric. 

 One of the case forms of a feminine pinna in modern 

 Irish would be pinn, and it was known to O'Reilly, 

 who gives it in his dictionary as a feminine meaning 

 " the summit of a hill or headland." 



The foregoing instances will serve to show that the 

 author has not been quite happy in his treatment of 

 the philologists ; whether he has been happier w-ith 

 the geologists and astronomers, the ethnologists 

 and arch<eologists, they could best tell. We regret 

 to be unable for want of space to pass under review 

 the rest of the second part of the work : we have 

 drawn on the excursus treating of the ethnolog)" of 

 ancient Britain. There arc others, however, on such 

 attractive subjects as the Cassiteridcs, the configura- 

 tion of the coast of Kent in the time of Caesar, Portus 

 Itius, the place of Caesar's landing in Britain, and 

 many minor themes. The Clarendon Press has done 

 xo. 2oog, vor,. 77] 



its part with its \^ onted success, and the reader has 

 th'^ aid of useful maps, together with good illustra- 

 tions. As to the work as a whole, one may say that, 

 in spite of certain grave defects and a uniform lack 

 of originality, it is a great monument to the author's 

 industry. 



LINEAR ALGEBRA. 

 Synopsis of Linear Associati~ee Algebra. By J. B. 

 Shaw. Pp. 146. (Washington : Carnegie Institu- 

 tion, 1907.) 

 THIS work serves three purposes : it gives a biblio- 

 graphy of the subject ; a synopsis of the various 

 algebras considered, in a fairly uniform notation, with 

 a classification into families and types ; and, in the 

 introduction and § xiii. especially, some general re- 

 marks on algebra and its development. Part iii. (pp. 

 1 13-134) deals with applications. 



Prof. Shaw points out that there arc two views of 

 complex algebra : — 



" the one regards a number in such an algebra 

 as in' reality a duplex, triplex, or multiplex 

 of arithmetical numbers or expressions ; . . . the other 

 regards the number in a linear algebra as a single 

 entity, and multiplex only in that an equality between 

 two such numbers implies n equalities between certain 

 coordinates or functions of the numbers." 



On this it may be remarked that both views are 

 equally legitimate, and equally useful, but in different 

 ways. The formulae of a special algebra which are 

 most characteristic and most powerful are those which 

 most naturally associate themselves with the second 

 point of view; an example is afforded by the qua- 

 ternion formula \\aVffy) = ySali-liSay. On the other 

 hand, the place of quaternion algebra among its 

 fellows is most clearly shown w'hen we consider a 

 quaternion as a complex (a, b, c, d) of four ordinary 

 numbers, with rules for the addition and multiplica- 

 tion of two such tetrads. 



The general impression produced by reading the 

 synopsis is that, after Grassmann and Hamilton, the 

 most remarkable work has been done by Benjamin 

 Pierce. By developing his methods it has been possible 

 to make a classification of linear associative algebras 

 which, so far as it goes, is really exhaustive, and may 

 be said, also, to be a natural classification. Of recent 

 papers, those of Cartan, Frobenius, and Poincar^ de- 

 serve particular mention ; they tend to show that the 

 characters of special algebras can be included in the 

 all-embracing theory of groups. 



.\ few lines (p. 18) are given to a definition of com- 

 plex numbers by Mr. Bertrand Russell, in terms of 

 logical constants. This is certainly interesting from a 

 philosophical point of view, but it illustrates a tend- 

 ency on the part of what may be called the Peano 

 school to over-refine their definitions, and become 

 verbose if not tautological. When the theory of real 

 numbers has been logically established, it is sufficient 

 to define a complex algebra in arithmetical terms, 

 without bringing in logical notions already used in 

 defining number and arithmetical operations. Why 

 not make use of a symbolism which has been fully 



