104 



NATURE 



[November 29, 1900 



upon the help of those who have paid special attention to 

 particular lines of research ; and it is with the intention 

 of doing a service of this kind that the remarks which now 

 follow have been made. 



On p. 137 "the form ;r = a (mod. b\ identical with 



^=j + a" should be corrected, at the end, by printing 



x=by-\-a. On p. 142 Reuschle's tables of 1856 are men- 

 tioned, butnot his " Tafeln complexer Primzahlen" (Berlin, 

 1875). By an extraordinary oversight, it is said, on p. 207, 

 that " we can construct a regular polygon of n sides only 

 when n-i=2^^ {p an arbitrary integer)," although a 

 correct statement (so far as it goes ^) is given, pp. 161-2. 

 On page 162, again, it is apparently said that Baltzer was 

 the first to notice that 2"+ i is not always prime when n 

 is a power of 2 ; as a matter of fact, Euler proved that 

 2^2+1 is divisible by 641 (^.Smith's "Report on the 

 Theory of Numbers," Art. 61). 



On page 259, after explaining von Staudt's interpretation 

 of " imaginary points " as double elements of involution- 

 relations (which is not strictly correct : the involution 

 itself, plus a distinguishing " sense," is the imaginary 

 point), the author says, "This suggestion of von Staudt's, 

 however, did not become generally fruitful, and it was 

 reserved for later works to make it more widely known 

 by the extension of the originally narrow conception." 

 Besides being rather disparaging in tone, this is likely to 

 convey a wrong impression. It is true that Kotter and 

 others, in trying to extend von Staudt's theory to curves 

 of higher orders, have been led to introduce involutions 

 of a more general kind than his ; but this does not affect 

 his definition of an imaginary point, which is perfectly 

 general and complete. The imaginary points in which 

 a curve of any order is met by any line must admit 

 (theoretically) of representation by involutions in von 

 Staudt's sense : just as an equation with ordinary com- 

 plex coefficients has a set of ordinary complex roots. The 

 equation may be, from some points of view, insoluble or 

 irreducible, and we may find it convenient to keep all its 

 roots together ; it is this which corresponds to the case 

 of these " higher " involutions. 



There are some obscurities which may be due to the 

 author or translators or both. Thus, p. 250, " Mobius 

 started with the assumption that every point in the plane 

 of a triangle ABC may be regarded as the centre of 

 gravity of the triangle : " (this is partially cleared up by 

 the context). On p. 205, line 4, the sentence beginning 

 " The semiparameter " is unintelligible, and is probably 

 a mistranslation. Page 147, "the theory of binary forms 

 has been transferred by Clebsch to that of ternary forms 

 (in particular for equations in line co-ordinates)" is a very 

 inadequate account of Clebsch's " Uebertragungsprincip," 

 and will hardly convey any definite idea to the average 

 reader. 



Two obvious slips in translation may perhaps be men- 

 tioned. On p. 270, through not noticing an idiomatic 

 inversion, the subject of a sentence has been treated as 

 the predicate, and vice versa : read " this point is offered 

 by the eleventh axiom." On p. 203, for " and also with 



1 The necessary and sufficient condition that a regular polygon of 

 n sides may admit of Euclidean construction with rule arid compass is that 

 the " totient " of « is a power of 2 ; in other words, n = 2"t/i7r. . . , where 

 m is zero or any natural number, and/, g, r . . '. are different odd primes, 

 ffl^A of the form 2'*+i. The values of « below 100 are excluding 2) 3, 4, 

 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 30, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96. 



NO. 1622, VOL. 63] 



the normals " read " that is to say, with the normals : " 

 also has been confused with auch, or rather with our 

 " also." Finally, by the omission of an " s," Pliicker has 

 been made to say that "he (Monge) introduced the 

 equation of the straight line into analytical geometry." 



At the end of the book there are short biographical 

 notices of a number of mathematicians : the list has been 

 recast by the translators. Whether it is worth the space 

 it occupies (26 pp.) is rather doubtful. Many entries are 

 either trivial, or anticipated in the previous part of the 

 book. Some of the notes are misleading, to say the 

 least. Cauchy is said to have "contributed" to the 

 theory of residues, the fact being that he invented it. 

 All that is said of Eisenstein is that "he was one of the 

 earliest workers in the field of invariants and covariants " ; 

 this is true in a sense, but his fame rests principally on 

 his arithmetical memoirs, and his researches on doubly 

 infinite products and elliptic unctions. Sophie Germain 

 " wrote on elastic surfaces." Legendre "discovered the 

 law of quadratic reciprocity," an erroneous statement 

 which may be corrected by p. 138 of the book itself. And 

 what is the use of such entries as " Donatello, 1386— 1468. 

 Italian sculptor"? It would be an improvement to cut 

 down this list to the really important names, and to give 

 indications of such trustworthy biographies, or other 

 sources of information, as may be available. G. B. M. 



THE SCIENCE OF COLONISATION. 

 New Lands: their Resources and Prospective Advantages^ 

 By H. R. Mill, D.Sc, LL.D. Pp. xi -I- 280. (London : 

 Charles Griffin and Co., Ltd., 1900.) 



THE present is a very appropriate time for the publi 

 cation of this book. Public attention is occupied 

 with Imperialism and colonial development, so that a 

 trustworthy statement of the resources and conditions of 

 life in the countries of the temperate zone, where there is 

 still an opening for the energies of English-speaking 

 people, should be of real service. |The colonies and 

 countries described from this point of view are Canada, 

 Newfoundland, United States, Mexico, Temperate 

 Brazil, and Chile, Argentina, the Falkland Islands^ 

 Australia and Tasmania, New Zealand and South Africa. 

 To intending settlers and capitalists desiring to know the 

 prospects of success in these countries the book will be 

 invaluable ; for it brings together in a convenient and con- 

 cise form all the essential particulars available in official 

 reports and other authoritative works. 



This is what the practical man wants, and he will 

 probably not concern himself seriously with the chapter 

 in which the development of new lands is considered in 

 its scientific aspects, yet to our minds this chapter is the 

 most valuable in the book, and every statesman and 

 colonial official anxious that the progress of his country 

 shall be steady and permanent should be familiar with 

 the principles it contains. It is an instructive statement 

 of the factors which ought to be considered in connec- 

 tion with the development of every land, but are often 

 neglected. 



Take, for instance, the subject of geographical boun- 

 daries. It is the British habit not to give any serious- 

 attention to this subject until forced to do so by a dispute 

 with a neighbouring nation. As Dr. Mill remarks : 



