August ii, 1898J 



NATURE 



341 



In order to secure for a work of this kind the fair 

 trial which it so thoroughly deserves, we venture to 

 make an appeal to the great body of examiners, in 

 whose hands lies so much power for influencing, either 

 for good or ill, the character of mathematical teaching 

 in schools. A paper on elementary algebra is too often 

 a medley of questions, generally of a stock type, which 

 do, indeed, test the candidate's familiarity with certain 

 set rules, and to some extent his ingenuity in applying 

 them, but are very far from gauging his powers of 

 mathematical reasoning. So long as this is the case, a 

 premium is offered to radically bad methods of teaching. 

 A boy can be taught the rule for algebraic long division 

 in a very short time, without any attempt to make him 

 understand its object or principle ; and what is the use 

 of wasting time upon such superfluities, when we can 

 take him on to the practice of G.C.M., and thus enable 

 him to make sure of answering two questions in his 

 examination ? Now it is quite possible to combine 

 questions on set rules (and it would be absurd to 

 propose the entire omission of them) with fair and 

 simple questions on matters of principle : if this were 

 done, it would be a great encouragement to a good 

 teacher, and tend to raise the average standard of 

 instruction. 



The book being so good, it is worth while to call 

 attention to the points in which it appears capable of 

 improvement. First of all, sufficient emphasis is not 

 laid on the fact that in applications of algebra the signs 

 + and - are used both as symbols of operation and 

 also as indications of quality, or "sense" : that this is 

 possible, without causing confusion, is not obvious a 

 priori. Thus, in the case of steps, let ira mean a step of 

 a units to the right, vb a step of b units to the left, and 

 let + and - refer, in the usual way, to the composition 

 of steps : then we have formulae such as tra ■^- -nb = 

 17 {a 4- b\ ira - vb = IT {a + b), n a + vb = n (a - b) 

 OT V {b - a) according as ^ > or < <^, and so on. If we 

 write + and - for n and v throughout, apply the formal 

 rules + (+«) = «, + (-«)= -a, Sec, and then in- 

 terpret the sign of any result qualitatively, i.e. as 7r or «/ 

 according as it is + or - , the conclusion is correct, and 

 the same as if the complete notation had been used 

 throughout. This remark is due to De Morgan, and has 

 been strangely ignored by subsequent writers. 



The expression " latent sign " occurs without explana- 

 tion, and apparently for the first time on p. 64. This is 

 a point which often puzzles beginners, and might well 

 receive a little attention. 



Chapters xvi. and xvii., on irrational functions and 

 surds, are a miserable compromise, as Prof Chrystal is 

 evidently aware. Arts. 169-74, 181-84, should have been 

 omitted altogether ; this would leave room for other 

 illustrations, especially of Art. 172. Most of the ex- 

 amples, too, are of a thoroughly unpractical type ; they 

 might, perhaps, be put in an appendix as samples of the 

 curious trifling of examiners. 



Arithmetical Progression is without value in itself, but 

 affords capital exercise in what may be called algebraical 

 counting [99*9 per cent, of ordinary students say that the 

 nth term is a -I- mi], in the derivation and use of a 

 general formula, and many things besides. For these 

 reasons it might be discussed at an earlier stace ; the 

 NO. 1502, VOL. 58] 



formula s = ^n (a + f), which, by the by, is not given, 

 may be illustrated by two pieces of paper cut in the 

 shape of the side elevation of a staircase. 



In treating Geometrical Progression, it might be well 

 to prove, without using the binomial theorem, that as n 

 increases indefinitely r« becomes infinite or infinitesimal 

 according as [ri exceeds or falls short of unity. This 

 would enable the teacher to take it earlier, if he wished. 



Two additions might very well be made in the interest 

 of technical or scientific students. The principle used in 

 calculating the slope of a graph from its equation might 

 be explained and illustrated ; and it might be stated, 

 without proof, that the binomial theorem is true for all 

 rational values of n if jr is a proper fraction, and hence 

 deduced, or proved separately, that (i + xy = i + nx 

 approximately, whenever x and nx are both small. 



Another thing that might easily be done would be to 

 introduce examples involving complex quantities in the 

 later chapters, for instance those on partial fractions, on. 

 proportion, and on series. Purely algebraic work with 

 complex quantities is too much neglected, and the sooner 

 a student becomes familiar with it the better. 



As might be expected, there are very few definite in- 

 accuracies ; there is, however, a rather striking one at 

 the top of p. 68. It is, of course, untrue that " the larger 

 n the more slowly does Xn increase between x = o and 

 r = -I- I " ; and this slip is the more remarkable because 

 it is contradicted by the figure on p. 67. The tyro may 

 amuse himself by finding the value of;»r for which x"^ and 

 x'^ are increasing at the same rate. G. B. M. 



T//E CUNEIFORM INSCRIPTIONS OF 

 WESTERN ASIA. 

 First Steps in Assyrian. By L. W. King. Pp. cxxxix 

 + 399- 8vo. (London : Kegan Paul and Co , Ltd., 

 1898.) 



THE appearance of Mr. King's volume, with its 

 modestly worded title, is opportune, and we think 

 it likely that it will be welcomed by every student of the 

 literatures of the East. The author's avowed object is 

 to help the student of the cuneiform inscriptions who has, 

 as yet, made but little progress in his difficult work, but 

 there is little doubt that Mr. King's stout volume will be 

 of considerable use to others besides him. 



The readers of Nature will remember that attention 

 has been called in these pages to the series of important 

 texts which the Trustees of the British Museum have 

 recently issued, and those who have taken the trouble to 

 examine the various parts as they appeared will have 

 found that, with the exception of short prefaces which 

 roughly classify the texts, no detailed information of their 

 contents has been given. Any translations, or even good 

 summaries of the contents of most of the texts, are, in the 

 present state of Assyriological knowledge, impossible ; and 

 if we consider for a moment that not only is the language 

 in which a large section of the documents is written 

 imperfectly known, but also that the readings of several 

 of the signs are doubtful, this fact will not appear won- 

 derful. It must not, however, be imagined that Assyri- 

 ologists are beaten, far from it ; but they ask for time, and 

 time must be given to them. Their chief necessity is, of 

 course, the texts, and the, sooner these are put into 



