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VI. Notices respecting New Books. 



Arithmetic in Theory and Practice for advanced Pupils. Part the 

 First. By J. Brook Smith, M.A. London: Macmillan and 

 Co. 1860. 



ALTHOUGH Mr. Smith's arithmetic can scarcely be said to meet 

 any urgent demand, inasmuch as we already possess several ex- 

 cellent works of the same kind, it deserves to take its place amongst the 

 small number of sound educational works. The first part, the only one 

 yet published, contains a lucid and correct exposition of the prin- 

 ciples of numeration, addition, subtraction, multiplication, and divi- 

 sion, of the theories of the greatest common measure and least 

 common multiple, and of some of the more elementary properties 

 of numbers with respect to their divisibility. 



In such a treatise we do not, of course, expect to find new matter ; 

 our demands are limited to a few necessary qualities, such as clear- 

 ness, precision, conciseness, good arrangement of the several parts, 

 and at least so much originality in the general treatment of the sub- 

 ject as is necessary to support the book's claim to a separate existence. 

 With respect to the two first of these qualities, this treatise has far more 

 than average merits; in the third quality it does not excel so much, 

 for the numerous theorems scattered throughout the book might, we 

 think, have been better " welded " together. In point of arrange- 

 ment there is much merit and some originality, a greater space than 

 usual having been devoted to the very useful and interesting sub- 

 jects of the criteria of the divisibility of numbers and of their decom- 

 position into prime factors. 



With respect to general treatment, one of the author's chief objects 

 has been to render his demonstrations purely arithmetical. To do 

 so he has been induced to discard all symbols and to conduct his proofs 

 by means of particular numbers, striving always to make the general 

 law apparent in the particular example. In so'me cases, and for 

 young pupils especially, this method has its advantages, but in 

 other cases it is clumsy and open to suspicion. Mr. Smith appears 

 to have mistaken a little the true character of an arithmetical, as 

 distinguished from an algebraical proof ; he must admit that symbols 

 may be introduced without interfering in the least with the purity or 

 the strictly arithmetical character of the reasoning. Thus used, 

 symbols are merely representatives of the subjects (numbers) dis- 

 cussed ; no material diminution in the length of the proof is produced 

 by their introduction, but the latter gains in elegance, in generality, 

 and sometimes even in simplicity. Some of the author's proofs might, 

 we think, have been improved in this manner. 



Mr. Smith professes to write for advanced pupils, but in all pro- 

 bability his work will find the greatest number of readers amongst 

 junior masters, the advanced pupils of our schools being in general 

 too much occupied in breaking new ground to be able to return and 

 inquire so closely into the principles upon which the very rudiments 

 of their science is based. To the junior master, however, this re- 

 examination is absolutely indispensable : he has to launch the young 

 intellect entrusted to his care into the wide ocean of mathematics, 



