28 



NA TURE 



[November 14, 1907 



therefore recommend Mr. Palmer's book with confi- 

 dence to those teachers who take a special interest in 

 and make a special study of the teaching of arith- 

 metic. They will probably find rules and methods 

 which they do not approve of, but these can be 

 neglected without any loss. The method of dealing 

 with the multiplication of decimals is open to the 

 objection that without any gain a much more difficult 

 method than the direct one is given. The author 

 makes use of rough approximations before and rough 

 checks after working out an example. These are very 

 good, and should be used in all working, but they 

 should not be made the means of finding the decimal 

 point in approximations. The placing of the point 

 should give no difficulty if a logical method has been 

 adopted throughout the study of decimals. 



(3) Mr. Jones's book is a laudable attempt to remove 

 the study of arithmetic from its commercial trammels 

 and widen its scope. We are afraid that, in the 

 attempt, he has overburdened his book. Practical 

 work is introduced at all stages of the work, and the 

 numerous explanatory diagrams will be a useful addi- 

 tion to the teaching of the subject. There are one or 

 two things which strike us as being out of place in a 

 book which is intended for a general course in arith- 

 metic. Thus the tables of weights and measures in- 

 clude some units which arc not in general use. The 

 introduction of these tends to specialise the work, 

 a thing which Mr. Jones claims, in his preface, that 

 he desires to avoid. We are sorry, to see in an arith- 

 metic of this type the instruction to " move the point." 

 It is always difficult for a teacher to keep before young 

 pupils the reason for the step, and he is not aided 

 when the text-book adopts the mechanical method. 

 Mr. Jones has added an index, an example that ought 

 to be followed by all writers of school text-books. 



F. L. G. 



OVR BOOK SHELF. 



Die typischcn Gcomctricn iind das Unendliche. By 

 B. Petronievics. Pp. viii + 88. (Heidelberg: C. 



Winter, 1907.) Price 3 marks. 

 The author of this curious work asserts (p. 86) that 

 it is impossible to make a one-one correspondence 

 between the points of a linear segment and the 

 elements of the arithmetical continuum (o, i) ; in other 

 words, he not only declines to accept the Dedekind- 

 Cantor axiom, but asserts that it is illogical. His 

 attempted proof (p. 85) involves the assumption of 

 actual infinitesimal segments; thus he says "so 

 entspricht dem ersten Punkte, der sich niit dem 

 o-Punkte beriihrt, gar keine Zahl in der Zahlmenge 

 o ... I, da das entsprechende .Segment unendlich 

 klein ist, und dasselbe wird auch fiir den zweiten, 

 dritten usw. Punkt gelten." 



This idea of immediately adjacent yet different points 

 pervades the whole tract, and leads to wonderful para- 

 doxes ; an attempt is made to remove the most obvious 

 difficulties by a distinction between real and unreal 

 points (pp. 9, lo), but this is not satisfactory. There 

 is a continual confusion between the idea of space 

 consisting of points and that of points forming 

 " parts " of space. You cannot eat your cake and 

 then look at it; if in one context "point" means 

 something with extension, it should not be treated 

 NO. 1985, VOL. 'jy] 



elsewhere as having position only. Moreover, no in- 

 tuition, logic, or metaphysic can get a geometrical 

 thing having extension from two points devoid of it. 



Unless something better than this can be said for 

 it, the assumption of actual infinitesimals of different 

 orders in geometry is not likely to be accepted, and 

 the Dedekind-Cantor axiom will probably be retained 

 as the simplest way of connecting geometry with 

 analysis. From the metaphysical side we want 

 something better than a puerile criticism of Cantor's 

 transfinite number-system, vitiated by misunderstand- 

 ings. Extensional quantities (lengths, volumes, &c.) 

 can be arithmetically defined for figures in an arith- 

 metical space; but no one with an active geometrical 

 imagination can enjoy this way of treating the sub- 

 ject, although he may admire it as a logical feat. 

 Again, take the connectivity of Riemann surfaces, or 

 the classification of knots ; here are things with char- 

 acteristics easily recognised by inspection, but diffi- 

 cult to specify by the arithmetical method ; cannot we 

 find some means for testing our intuitions without 

 putting them into this newly invented arithmetical 

 machine? To give a satisfactory answer to the ques- 

 tions arising from the modern aspects of mathematics, 

 is a task sufficient to strain the highest philosophical 

 powers ; and although Dr. Petronievics has the 

 temerity to declare that Hilbert's " Grundlagen der 

 Geometrie" is logically defective (p. 24, end), he has 

 added little, if anything, which is of value or interest 

 to the discussion. 



G. B. M. 



Engineering ]]'orksliop Practice. By Charles C. 



Allen. Pp. vii + 254. (London: Methuen and Co., 

 ' n.d.) Price 3s. 6d. 



A BOOK for students on engineering workshop 

 practice is, in many ways, more difficult to write than 

 one addressed to those who, from years of actual 

 practice, have gained an intimate knowledge of the 

 elaborate processes by which engines and other 

 machines are produced. The beginner requires ample 

 explanations of processes, which he has probably never 

 seen carried out, but which to the workman are as 

 familiar as his daily paper. 



This book, good as it is, would have been much 

 more useful if no attempt had been made to write 

 for the information of both the beginner and the 

 skilled workman ; their needs are so different that the 

 result cannot be satisfactory to either class. A typical 

 instance of the consequences of such an attempt occurs 

 on p. 159, with reference to the cutting of vee 

 threads in a lathe. In a short paragraph the author 

 points out, quite properly, that, in taking a cut over 

 the whole form, there is a great tendency to rip the 

 thread, and then goes on to state that the diagrams 

 indicate the proper method, but offers no further ex- 

 planation of them. To a skilled workman these 

 diagrams are quite unnecessary; to a student they 

 arc merely perplexing. He is left to discover, if he 

 can, that one diagram is intended to indicate that the 

 roughing cut is to be taken on one side of the vee, 

 while in a second diagram a tool, apparently floating 

 in mid-air, lies between two objects, which he may or 

 may not recognise as rake gauges. In other cases 

 where explanations of the diagrams are given they are 

 far from being clear; thus on p. 191', in the instruc- 

 tions for cutting helical gears, we are told that " The 

 cutter used must be selected for the number of 

 teeth there would be in a gear with outside diameter 

 equal to the diameter of a circle determined by the 

 curvature of the gauge in this way." But the 

 author gives no intelligible explanation of what " this 

 way " is. 



While it is proper to direct attention to blemishes 



