48o SCIENCE PROGRESS 



a transformation theory of the first ; the pieces are now uncoloured, but are 

 no longer of the same shape, the external boundary being modified in accordance 

 with the colour of the corresponding piece and the contact-law assumed, 

 but so that the new pieces will still fit together. A bewildering variety of 

 curious shapes and jig-saw puzzles is thus obtained, and the author has to 

 call in the help of Latin and Italian quotations to express his feelings. " Quel 

 che 6 nuovo e sempre bello." 



Part III examines the problem of fitting together pieces of the same size 

 and shape so as to cover the whole plane, a subject on which (and on the 

 similar problem for space) the author has recently communicated two 

 learned papers to the Royal Society and the London Mathematical Society. 

 He calls attention to the sad lack of initiative on the part of designers of 

 parquet floors, carpets, and wallpapers which involve repeating patterns, 

 and certainly anyone who possesses this book will be overwhelmed and 

 possibly scared by the patterns which he might have to live with. 



This is an extremely well written and entertaining book ; we share the 

 author's regret that it was impracticable to print Part I in colours, but 

 the reproduction of the many complicated diagrams leaves nothing to be 

 desired. 



F. P. W. 



Frequency Arrays. By H. E. Soper, M.A. [Pp. 48.] (Cambridge: at the 

 University Press, Price 35. 6d. net.) 



The author of this little book of less than fifty pages illustrates the uses 

 to which his symbolic method of treatment of various statistical concepts 

 may be put, by demonstrating many well-known propositions in the theory 

 of mathematical statistics. His aim has been merely to show that, by the 

 use of logical symbols and " ascribing to such symbols the laws of common 

 algebra in their combination, the description, analysis, and derivation of fre- 

 quency distributions are often much simplified." He has certainly succeeded 

 in arousing interest, if only on account of the fact that he has concentrated 

 in his few pages matter which, when treated by the usual methods, requires 

 ten times as much algebraic analysis. This pamphlet should be read by all 

 mathematical statisticians. 



E. C. Rhodes. 



Practical Least Squares. By O. M. Leland, B.S., C.E. [Pp. xiv + 237.] 

 (New York : McGraw-Hill Book Co., 1921. Price 15s. net.) 



The aim of this book is essentially practical, as its title states ; it is intended 

 to show the engineer and surveyor how to adjust his observations. The 

 knowledge of mathematics assumed is very small indeed ; it is even con- 

 sidered necessary to explain in a footnote what is meant by the geometric 

 mean of two quantities. Any integral which occurs is examined in detail, 

 but, surely, even an engineer nowadays ought not to accept without question 

 the inversion of the order of integration in an infinite repeated integral 



which is assumed in the evaluation oi J e dx on p. 214. But these are 



details ; the main idea is to show how the calculations can be done with the 

 minimum of labour and the minimum chance of mistake, and this is 

 successfully carried out. 



A general introduction classifies errors of observation into systematic, 

 theoretical, instrumental, personal, and accidental, and explains that it 

 is only the last which are considered in adjustments. The law of error which, 

 says the author somewhat confusedly, " has been completely derived and 

 tested, later, in a multitude of cases, with entire satisfaction," is then 

 assumed (Gauss's derivation is given in an appendix), and the principla of 



