314 Review of the Cambridge Course of Mathematics. 



the French method, the 10th figure is the place of bilhons, 

 the 13th of triUions. and each succeeding addition of three 

 places, gives a new denomination.* 



Fractions are introduced immediately after division, and 

 are very naturally considered as deriving their origin from 

 imperfect divisions. He explains the changes w^hich a 

 fraction receives from operations performed upon its nu- 

 merator and denominator; and in this way collects a few 

 principles upon which the whole theory of fractions is 

 made to depend. Indeed, these princi{)les might be redu- 

 ced to one, were it not that the subject would thus be ren- 

 dered unnecessarily and unprofitably abstract. 



A circumstance over which the greatest part of authors 

 have passed too superficially, is, the application of the defi- 

 nitions of multiplication and division relative to whole 

 numbers, to fractions. These definitions applied to whole 

 numbers, comprise only the most simple cases of the ope- 

 rations which they express, while as applied to fractions, 

 the terms multiplication and division have a general accep- 

 tation, in which new cases are comprised, connected with 

 the first only by simple analogies. Our author has, there- 

 fore, given new definitions of multiplication and division, 

 which appear a little abstract before reflection, but which 

 are applicable to all possible cases of these operations. By 

 this instance, also, the student is taught in a striking man- 

 ner, the signification of the term generalization in mathe- 

 matical and philosophical writings. 



The complication which the diversity of denominators 

 introduces into operations by common fractions, leads nat- 

 urally to the invention of decimal fractions, which removes 

 this complication. Decimal fractions are therefore, here 

 introduced in the order of invention. The student is pre- 

 pared by his own experience of the inconveniences attach- 

 ed in practice to the use of vulgar fractions, to seize com- 

 pletely the advantages of the decimal system, although this 

 system generally gives only approximate instead of rigorous 

 values. This disadvantage, however, of the decimal sys- 



* The French method ol estimating numbers is adopted in the Art. Arith- 

 metic of the Edinb. Encyc. as being less complicated than the English meth- 

 «d. As the difference between the two methods is only in the higher de- 

 nominations, which seldom occur, the difference for practical purpoees will 

 Dot be great. 



