560 Notices respecting New Books. 



by the authors themselves. The remainder of the work deals with 

 ultra-microscopic objects. A chapter is devoted to the study of the 

 minute particles embedded in transparent solids — such as particles 

 of gold in ruby -glass — and those forming a thin deposit on their 

 surfaces — such as metallic deposits on glass. Particles contained 

 in liquids are next dealt with. Three chapters are devoted to the 

 study of colloidal solutions. It is shown how the properties of 

 such solutions may be accounted for by supposing them to possess 

 a granular structure. The electrical behaviour of such solutions is 

 studied in detail, and an account is given of the indirect methods 

 which enable us to form some sort of notion regarding the form 

 and properties of colloidal granules. The concluding chapter deals 

 with the ultra- microscopic particles of interest to biologists. 



The authors are to be congratulated on having produced so 

 readable and simply written a book on a very difficult subject, 

 and no person interested in microscopic work can afford to be 

 without it. 



Lectures on the Theory of Functions of Real Variables. Vol. I. 

 By James Pierpont. Ginn & Co. : Boston, New York, <fcc. 1905. 



The distinguished Professor of Mathematics in Yale University 

 has in these published lectures given an altogether unique book to 

 the English-speaking student of pure mathematics. The idea at 

 the basis of the treatment of what the ordinarily trained student 

 will soon recognize as his familiar friends in the infinitesimal 

 calculus is rigorous demonstration. More particularly stated, the 

 problem is to examine the condition under which the recognized 

 theorems and processes can be applied correctly. The foundations 

 of the whole doctrine of mathematical continuity must be laid firm 

 and sure ; and to effect this the author is compelled to begin with 

 the modern theory of rational and irrational numbers. Following 

 mainly along the iines initiated by Cantor, he devotes several 

 chapters to the theory of the elementary functions ; and it is not 

 till Chapter VIII. that the subject of differentiation of functions 

 of one variable is taken up. On the basis of the doctrine of limits 

 already established, the differential coefficients of the fundamental 

 formulae of differentiation are deduced with rigour, an important 

 educative feature being the indication in certain cases of the 

 incorrectness of the more usual methods of establishing these 

 formulae. From Chapter XII. to the last chapter (XVI.) inte- 

 gration is discussed with the same uniform method — clear, rigorous, 

 and brief. At times, indeed, the discussion seems almost too brief, 

 partaking more of the character of lecture notes than of a 

 systematic treatise. This, however, has its advantages in making 

 the reader complete the demonstration in his own way. There 

 can be but one opinion as to the value of the work ; and the 

 publication of the second volume will be awaited with interest and 

 expectation. 



