NATURE 



[February 13, 1902 



and were the discovery (now fifty years old) of the 

 test-bearing protrochal stage of Dentalium, lately 

 observed by Drew to be passed through by Yoldia and 

 by Pruvot by a Dondersia, but recognised, we should not 

 find the Chitons referred to a subclass of the Gastropoda 

 and the " Solenogastres " accorded a class distinction. 

 To this developmental stage, the discovery of which has 

 dealt the death-blow to the idea of a Rhipidoglossan 

 affinity of the Pelecypoda, and which, we trow, will ere 

 long be extended to other groups, our authors should 

 have directed attention. Had they done so, but three 

 lines would not have sufficed for the Scaphopoda, and 

 Spirula would not have been dismissed as a mere name. 



Turning to paheontology, the non-recognition of the 

 recent discovery in the Trilobites of nauplius characters 

 deprives the authors' treatment of this larva of all force. 

 And, similarly, had the Eurypterid forms recently de- 

 scribed by Holm from the Russian Silurian, by Beecher 

 from the Cambrian, and the Scorpionid genus Pahco- 

 phonus, met with recognition, Limulus could not in 

 justice have been once more relegated to the Arach- 

 nida. The absence in the present book of all 

 mention of the Odontorinthes and Archseopteryx, 

 of the Anomodontia, the Plesiosauria, and other 

 leading fossil forms which might be named, is a serious 

 omission, but even this does not excuse the non-reference 

 to so important a group as the living Sphargidae. 

 Embryology and palaeontology are branches of mor- 

 phology coequal with the rest, and, so far as they reveal 

 facts of primary significance, they should be dealt with as 

 elementary subjects. Lack of appreciation of this 

 principle is the weakest feature of the present work, 

 which is, curiously enough, written with a special view 

 to the requirements of the American student, who, of all 

 beginners, is brought up in a pala^ontological air, and for 

 whose benefit e.xamples, wherever possible, are drawn 

 from American as well as British animals. 



Allowing for this serious defect, the book can be 

 confidently recommended as well written and trust- 

 worthy, so far as it goes. It has been compiled at great 

 pains, and its style leaves little to be desired. We wish it 

 success and a speedy passage into a second edition ; and, 

 in anticipation of this, we would recommend to the authors' 

 consideration the need of revision of such definitions as 

 that of the endoderm cell (p. 48) as " tall " ; of the 

 blood-vessels (p. 89) as "chinks"; the replacement of 

 the term " rudiment " on p. 259 by blastema ; and certain 

 other loosenesses which are self-evident. It is pertinent 

 to this to remark that in some of their recent attempts 

 at revised terminology, the zoologists of the Cambridge 

 school have been none too successful. Thus, we note in 

 the account of the life-history of the New Zealand 

 reptile Sphenodon, given in the recently published 

 natural history volume on "Amphibia and Reptiles," 

 that the writer has substituted the word "aestivation" 

 for what its discoverer rightly termed a hibernation. Is 

 it possible that he has temporarily confused the southern 

 summer with our own ? 



Of the illustrations, it may be said that figs. 266, 

 289 and 299 are examples which are poor, and might 

 well be replaced ; the statement that of the 32,000 

 " known species of Vertebrata " some 10,000 are Teleostei 

 is surely excessive. 



NO. 1685, VOL. 65] 



MATHEMATICAL TEXT-HOOKS IN THE 

 UNITED STATES. 

 College Algebra. By J. H. Boyd, Ph.IX Pp. xxii-t-788. 

 (Chicago : Scott, Foresman and Co., 1901.) 



WE cannot obtain a complete view of the state of 

 mathematical studies in a country merely by 

 examining the text-books and treatises which are in 

 vogue there ; but we do, in this way. gain a good deal of 

 information about the aims and standards of its mathe- 

 matical teachers. Dr. Boyd's treatise illustrates very 

 well the qualities and defects of American methods, and 

 suggests a few general remarks, as well as particular 

 criticisms, which may not be out of place. 



First of all, it must be acknowledged that the excel- 

 lences of the better class of mathematical authors in the 

 United States greatly outweigh their deficiencies. The 

 American student is alert and inquisitive ; he is neither im- 

 pervious to new ideas, nor unwilling to make experiments. 

 Moreover, teachers and students alike regard mathe- 

 matics in the proper spirit — as a science which has, 

 indeed, a venerable history, but is at the same time 

 living and progressive, with ever new developments and 

 ever fresh applications to the needs of man. Many, if 

 not most, of the leading mathematicians in the States 

 have studied in (Germany, and have thus become ac- 

 quainted with the work of Kronecker and Weierstrass 

 and tlie far-reaching influence of this upon function- 

 theory and the foundations of analysis. In elementary 

 geometry, too, they are not the slaves of tradition, as we 

 are ; and it is not impossible that they may ultimately 

 give us the ideal class-book in geometry for which we 

 are waiting. 



Dr. Boyd, in his preface, accepts the modern standard 

 of rigour, and in his choice of topics combines the indis- 

 pensable rudiments with those developments and appli- 

 cations which are really important. The general scope 

 of his book may be indicated by saying that Book I. 

 deals with the fundamental laws of operation ; II. with 

 equations of the first degree ; III. with indices, surds 

 and complex quantities ; IV. with quadratic equations ; 

 V. with proportion, progressions and logarithms ; \ I. 

 with induction, permutations and combinations, and the 

 binomial expansion for a positive integral exponent ; \'1I. 

 with limits and series ; VIII. with the properties of deter- 

 minants and the elementary theory of etjuations. 



After proving the fundamental laws of operation for the 

 cases where they are arithmetically intelligible, the 

 author extends them by purely formal definitions ; thus 

 (a-i) is defined by the formal equivalence {a-d)+d = (i. 

 This is unobjectionable, but seems to us to require more 

 justification than Dr. Boyd explicitly gives. He appeals 

 to the " principle of permanence of form," but this 

 "principle" remains practically an assumption. No 

 doubt it would be extremely tedious to give (what we 

 think has never been done) a complete logical proof that 

 the application of the generalised laws of operation 

 never involves an inconsistency ; still, something more 

 might have been done to help the reader to apprehend 

 the reasonableness of the assumption. 



Again, Dr. Boyd is not always consistent with himself. 

 Thus, in the chapter on fractions, he begins with the 



formal definition jx6 = a; he subsequently says that 



