n 



314 Review of the Canibrulgc Course of Mathemaiics. 



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

 the 13th of trillions, and each succeediiig aduition of three 

 places, gives a new denominatioiu* 



Fractions are introduced immediately after division, and 

 are very naturally considered as deriving their origin from 

 imperfect divisions, lie explains the changes which 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 13 

 made to depend. Indeed, these principles mi^ht be redu- 

 ced to one, wcvQ 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 deh- 

 nitions of muUipllcation and division relative to whole 

 numbers, to fractions. These definitions applied to 1^'hole 

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

 rations which they express, while as applied to fractions, 

 the terms mnltiplication 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 tauglit in a striking man- 

 ner, the signification of the term generalization in mathe- 

 matical and pliilosophical writings. 



The complication which tlie diversity of denominators 

 introduces into operations by common fractions, leads nat- 

 urally to tlie invention of decimal fractions, which removes 

 tins 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 coni- 

 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- 



+ 



* Thf? French method oJ estimating numbers ig adopted in the Art. Arith- 

 metic oi'tbe Edinb. Encyc. as beins^less complicated than th*=? English mi^th- 

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

 nominations, which seldora occur, the difFerence for practical purposes wiU 

 opt he great 



