February 13, 1902] 



NA rURE 



339 



4 7 means that a group of 7 things is regarded as a unit 

 group out of which 4 things are taken ; and finally gives 

 a proof of the equivalence of 47 and 12/21 by means of a 

 graduated scale. This is mixing up three different ways 

 of looking at the matter in a fashion which is very likely 

 to cause confusion. And, so far as his " group " defini- 

 tion goes, he gives it in an imperfect form which is not 

 immediately applicable to improper fractions and which 

 fails to account for the equivalence of a pair such as 4/7 

 and 12/21. 



Another chapter to which we naturally turn is that on 

 irrational numbers and limits. Irrational numbers are 

 treated, after Cantor, as the limits of sequences ; and 

 the discussion is satisfactory so far as it goes, though it 

 might well be made rather more complete and is occa- 

 sionally rather illogical. Thus, for instance, in the early 

 part of the chapter it is said that the ordinary rule for 

 finding a square root, when applied to 2, leads to the 

 inequalities 



i<V2<2. i'4<v'2<i-5, I-4K ^/2-;i-42, 

 and so on. As thus stated, the proposition is a pure 

 petitio principii. The sequence (i, 14, r4l, . . . ) is 

 convergent, and may be rationally combined with other 

 such sequences according to Cantor's rules ; therefore it 

 may be regarded as a number. By definition 



(I, 1-4, 1-41, . . . )-=(r-, I■4^.I■4I^ . . . ), 

 and this sequence can be proved to be equivalent to 

 2; therefore ^/2 is an appropriate symbol for (i, i'4, 

 r4i, . . . ). We must not begin by assuming the exist- 

 ence of si- as an arithmetical quantity. The proof that 

 sequences obey the laws of operation is put very briefly, 

 and when we turn to the chapter on surds, we find that 

 such an equivalence as ^'2 . V3= \'6 is justified, not by 

 the use of sequences, but by a reference to the purely 

 formal law of indices. Here, again, we have a rather 

 unfortunate association of two entirely different notions. 

 If, for any purpose, we like to introduce a symbol 8 such 

 that 6-=2, every rational function of 5 can be reduced, by 

 formal processes, to the shape P-I-Q5, where P and Q 

 are independent of 6 ; this is quite independent of the 

 question whether 6 can be properly regarded as a 

 number or net ; still less does it assign to 6 its place in 

 the arithmetical continuum. 



Dr. Boyd's chapter on the binomial theorem for any 

 exponent deserves attention, because, although it requires 

 supplementing, it is novel, at least in a text-book, and 

 may prove to be a good way of explaining the theorem 

 to the college student. Let plq be a positive rational 

 fraction ; then 



(l+xY'i=lj{.\+px-ir\p'^p-l)x'--V . . . +XP). 



Now it can be shown, as Dr. Boyd indicates without 

 going into detail, that we can, by a process which is, in 

 fact, Horner's method, determine a polynomial 



y=l+-x + f J.V- + c^' + 

 1 



+ <rmX' 

 1 

 such that 



(i-l-jr)''-j« = R = A.v"'+' + Bjr"+=-F . . . -1-L.i^, 



where m is any positive integer assigned beforehand. 



The coefficients c.^, C3, &c., are numerical, and it can be 



proved by the method of undetermined coefificients that 



<i=hP(P-q)l'i'', 



NO. 1685. VOL. 65] 



.~-(^-r..y.,r. 



for i<r<OT4-i. By making tn an indefinitely large 

 integer, J* becomes an infinite series, which is convergent 

 for I r I <i. It remains to be proved that the sum of 

 the infinite series y, when convergent, represents that 

 branch of the function (i-t-x)"'" which reduces to i when 

 X is zero. This last part of the proof Dr. Boyd has 

 failed to supply or even to indicate ; the need of it will 

 be seen when it is observed that when y becomes an 

 infinite series, the remainder R is also an infinite series 

 and it is essential to prove that, as w increases indefin- 

 itely, the limit of R is zero. 



It will not be amiss to observe that these criticisms, 

 offered with all friendliness and sympathy, are provoked 

 just because Dr. Boyd aims at a high standard of logical 

 exactitude. Many a worse book than his may be said 

 to have fewer faults — faults, that is, which lie on the sur- 

 face and can be pointed out in a few words. To write a 

 really sound book on algebra, not incomprehensible to the 

 ordinary college student, and not hopelessly unscientific 

 when judged from the standpoint of contemporary 

 analysis, is a very difficult task. But it is a worthy one ; 

 and the attempt justifies itself, even if it is not crowned 

 with unqualified success. The reader of Dr. Boyd's 

 book cannot fail to gain many fruitful ideas ; if he has 

 mathematical capacity he will very likely apprehend 

 them in a substantially correct form, even when the 

 author's exposition is not entirely rigorous. 



To sum up, we find in this treatise, as in others of its 

 class, much that is fresh, vital and stimulating ; an interest 

 in the progress of research, and in the development of 

 new conceptions ; together with a style that is neither 

 frivolous nor pedantic. What we miss is, on the one 

 hand, the German thoroughness which spares no pains 

 to make the logical cham of an argument complete, and, 

 on the other, our English dexterity of manipulation. This 

 last faculty is not of much importance, truly, but is worth 

 reasonable cultivation. It is strange to us, for instance, 

 to find a whole page spent on the decomposition of 

 x^-\-px- + g into a product {x"- + a)(x'^ + ^) without any 

 reference to the fact that x^+px- + g is a quadratic in 

 x'^. It is only fair to say that, in this instance, the con- 

 text partly accounts for the phenomenon ; but other 

 examples of needlessly complicated work could easily be 

 given. G. B. M. 



A CANADIAN PIONEER IN SCIENCE AND 

 EDUCATION. 



Fifty Years of Work in Canada, Scientific and Educa- 

 tional. By Sir William Dawson, C.M.G., LL.D., 

 F.R.S. Pp. viii H- 308. (London and Edinburgh : 

 Ballantyne, 1901.) 



LITTLE more than a year has passed since the friends 

 of science and of education in Canada had to mourn 

 the death of Sir William Dawson. Though for the last 

 six years of his life he had retiredfrom his active official 

 duties, his pen was not allowed to remain idle, but con- 

 tinued to throw off papers for scientific journals, addresses 

 to societies and books of a more or less popular kind. 

 One of the occupations of these closing years appears 

 to have been the preparation of a sketch of his own 

 career, which he left complete even to the dated preface 



