464 



NATURE 



{March 18, 1880 



Again, the tables virtually furnish lists of primes to their 

 full extent. We need not remind mathematical readers 

 how often it is important to know, concerning certain 

 results of calculation, whether a number is prime or not, 

 this being the necessary preliminary to further inferences 

 from the processes which give rise to it. As an easy 

 of the consequence of knowing how a number 

 splits up into prime factors, we may mention the ele- 

 mentary theorem, that any recurring decimal whose period 

 consists of five figures must have one or more of the 

 numbers 3. 4.1, or 271, as factors of its divisor. This is 

 simply a consequence of the numerical identity — 

 99999=3.3.41.271. 



Now, as to relative utility: We are inclined to think 

 that the utility of such tables is measured by the index of 

 the power of 10, to which they extend — that this rule 

 represents the advantage of Vega's table, up to io8,ooo ; 

 over Barlow's, up to 10,000; of Chernac's, up to 1,012,000, 

 over Vega's ; and of this set of tables, when completed up 

 to ten millions, over Chernac's. We think this estimate 

 holds for theoretical questions relating to the enumera- 

 tion and distribution of primes, and cognate questions 

 relating to the theory of numbers, as well as for the 

 practical command they give us over the numbers them- 

 selves. Nevertheless, it would not be right to under- 

 estimate the important point that this work does give us a 

 command over numerical magnitude such as we did not 

 possess before. In that view he would be a bold man 

 who should say that the money cost of the production and 

 printing of these tables was a bad investment for science, 

 i when the directing labour was gratuitously 

 given. What that directing labour involves can be under- 

 stood by those alone who have worked upon millions. 

 None others know what an awful factor a million is, when 

 applied to the multiplication of the simplest process. 

 We shall heartily congratulate Mr. Glaisher on the 

 termination of his labours, and we no less heartily 

 congratulate our mathematical friends upon their good 

 fortune in having found such a man to undertake such 

 a task. 



We conclude by reminding our mathematical readers 

 that all the processes by which these tables have been 

 formed are but skilful adaptations of the well-known 

 CRIBRUAI Eratosthenis, of which the analytical ex- 

 1 was first given by Euler [introductio in 

 ■ infinitorum) in his remark that the harmonic 

 series — 



l+J + i + l+i + >+ • • • to infinity 

 reciprocal of the continued product — 



(I -*){!-*) (I -£)(!-*)• • -, 



h the primes only enter. 



C. W. M. 



WHO ARE THE IRISH? 

 e the Irish f By James Bonwick, F.R.G.S. 

 (London : David Bogue, 1S80.) 

 r T" , IlIS little work is issued as the first of a series on 

 J- li pur Nationalities," to be followed by three others 

 on the Scotch, Welsh, and English. It does not appear 

 from the prospectus whether the rest of the series is to 

 be entrusted to Mr. Bonwick ; but if they are it is to 



be hoped that he will qualify himself for the task by 

 a preliminary study of at least the first principles of eth- 

 nology. The present volume, with all its good inten- 

 tions and praiseworthy industry, must be regarded as 

 a hopeless failure, owing entirely to the neglect of 

 this necessary precaution. For many years ethnology, 

 anthropology, and philology were subjects which any one 

 seemed competent to deal with, who had got hold of a 

 few lists of words in some obscure African or Polynesian 

 dialects (the obscurer the better), or who had desecrated 

 a sufficient number of ancient barrows, or posed to 

 admiring circles under the shadow of some Druid's 

 in Cornwall or Brittany. But those halcyon days of the 

 amateur ethnologist are no more, though the writer, un- 

 fortunately, seems scarcely alive to the fact. Almost 

 every page of his little tractate betrays solecisms and 

 crudities, such as one naturally looks for in the writings 

 of the Pinkertons, Vallanceys, Vans Kennedys, Bethams, 

 and other obsolete writers of the old Keltic school, but 

 which have become anachronisms since Keltic studies 

 have been placed on a solid basis by the labours of 

 Pritchard, Pictet, Zeuss, Ebel, Lottner, Diefenbach, 

 Whitley Stokes, and Dr. W. K. Sullivan. 



A great many authorities are quoted, some, it may be, 

 at first hand, but most of them vicariously, some good, 

 some of no account, some utterly worthless. But all 

 are treated with equal deference, and nowhere is there 

 betrayed the least sense of discrimination as to their 

 respective merits. Thus at p. 27 we have " Betham makes 

 them Teutons, and Wilde, Celts," as if the opinion of a 

 keltomaniac like Sir William Betham could matter a 

 straw one way or the other, and as if in the writer's view 

 it commanded as much weight as that of the distinguished 

 member of the Royal Irish Academy, with whom he is 

 here strangely associated. This vice pervades the entire 

 work, and of itself alone reveals the utter incapacity of 

 the author to deal with such a theme as that of the 

 affinities of the Irish race. Hence it is not perhaps 

 surprising to find ethnical terms treated quite as wildly 

 as ethnological authorities. At p. 19 occurs the following 

 passage, which is quite a curiosity in its way : — " The 

 Basques are believed to be of Turanian origin, while the 

 Celts are Aryans, like most of the Europeans, as well as 

 Persians, Hindoos, &c. Some Turkish and Finnish 

 tribes, with ancient races in Greece, Italy, and Assyria, 

 have been deemed Turanian with Tartar (sic) sym- 

 pathies. The Etruscans of Tuscany were leaning to the 

 Iberian." For wild incoherence and confusion this will 

 surely hold its own with anything to be found in the lucu- 

 brations of the most popular exponents of Keltic ethno- 

 logy in the present or past generation. Frequent use is 

 naturally made of the convenient but dangerous term 

 "Turanian," but its meaning is nowhere defined. Careful 

 writers, if they use it at all, at least restrict it to the 

 Finno-Tataric or Ural-Altaic family. But it is here 

 apparently separated from that connection, so far at least 

 as regards the Tatars, while the Tatars themselves are 

 spoken of as something distinct from the "Turkish" 

 (read Turki) tribes, with whom they are nevertheless 

 identical. Why or when " the Etruscans of Tuscany were 

 leaning to the Iberian" we are not informed, nor are we 

 told by whom "the Basques are believed to be of Turanian 

 origin." Meantime it may be well to remind the author 



