REVIEWS 479 



Charlier.'and uses the method of Semi-Invariants and illustrates his method by 

 many examples, generally contrasting his results and the length of time of 

 working, with the results obtained by submitting the same material to the 

 Pearsonian methods. We may note in this connection that he does not 

 submit the results he obtains to the test of goodness of fit, but probably he 

 will discuss this in his second volume. It appears that for the most part 

 he uses curve-fitting methods as a means of graduation rather than with the 

 idea of obtaining theoretical laws which his data might obey, and which could 

 probably be more easily represented by expressing his theoretical distribution 

 in the Pearsonian manner rather than by means of the Laplacean Probability 

 curve and its derivatives. It may be pointed out that the Pearsonian curves 

 are unimodal, and will only fit observational data well when these are unimodal 

 in form, whereas his theoretical curve is multimodal in form and will naturally 

 fit a frequencydistribution of the type he shows on p. 258, where the Pearsonian 

 method would fail. (" Age Distribution of Male Employees in the Bell 

 System.") 



We would rather the author had given more discussion on the question 

 of the method of least squares ; we disagree here with Mr. Fisher and hold 

 that the principle of least squares is invariably interwoven with the Normal 

 Law of Errors. On the whole, however, we have to thank the author for his 

 very able and successful attempt at a good book on statistics, and only wish 

 that he will follow it up with the second volume at an early date. 



E. C. Rhodes. 



New Mathematical Pastimes. By Major P. A. Macmahon, R.A., D.Sc, F.R.S. 



[Pp. vii + 116.] (Cambridge: at the University Press, 1921. 

 Price 125. net.) 



Major Macmahon has generalised the game of dominoes and has produced 

 a variety of extremely diverting new mathematical pastimes. In the simp- 

 lest form there are twenty-four pieces, each an equilateral triangle, divided 

 into three by lines joining the angular points to the centre, the compart- 

 ments being coloured with all possible combinations of four colours. The game 

 is to fit the pieces together to form a large regular hexagon, with certain 

 contact conditions and certain external boundary conditions. For example, 

 we may lay down that a compartment shall be adjacent to a compartment 

 similarly coloured, and that all the external compartments shall be coloured 

 red. The boundary conditions with the above contact conditions must, 

 of course, be restricted so that each colour occurs in the boundary an even 

 number of times. In the table at the top of p. 5 it is stated that there 

 are eight such schemes for the boundary, but the author then proceeds to 

 give as an illustration a case not included in the table, with six of one colour 

 and six of another in the boundary. Further varieties may be obtained 

 by selecting certain ten of the pieces containing only three colours, or by 

 selecting the thirteen which contain a particular colour, the shapes into 

 which the pieces are to be fitted being then a flat hexagon and a blunted 

 triangle. It is not always as simple as it sounds to fit the pieces together 

 correctly, and we are encouraged by a quotation from Holinshed : " And 

 when he had taryed there a long time for a convenable wind, at length it came 

 about even as he desired." The present writer has made himself a set of 

 pieces and has wasted a good deal of his own time and that of his friends, 

 both mathematical and non-mathematical, very pleasantly on the above 

 pastimes alone. But this is only the fringe of Major Macmahon's subject ; 

 he goes on to squares, right-angled triangles, cubes, and hexagons with ever- 

 increasing complexity and with a wealth of quotation from the most diverse 

 sources, rare and familiar. 



The second part, heralded by a passage from The Consolation, deals with 



