Review of the Cambridge Course of Mathematics. 319 



We may pursue the same process with any other frac- 

 tion, and obtain a series of approximate values, alternately 

 greater and less than its true value, if it is a fraction prop- 

 erly so called, or alternately less and greater, if as in the 

 preceding example, the numerator exceeds the denominator. 



The developments which I have now found for the ex- 

 pression ^Yi ^^^ called continued fractions, which may be 

 defined in general thus :— -Fractions whose denominator is 

 composed of a whole number and a fraction, which fraction 

 has for a denominator also a whole number and afraction,i^c.^^ 



Of the introduction to the elements of Algebra taken 

 from Euler, much need not be said. The original treatise 

 of Euler especially with Lagrange's additions, is very ex- 

 tensive, and is certainly one of the most luminous and com- 

 plete treatises that have ever been written, and of course, 

 any selection from it must partake of the merits of the orig- 

 inal work. But as is always the case in such selections, 

 the parts selected are not perfectly adjusted to each other, 

 which gives rise to some abrupt transitions. It forms no 

 necessary part of the course, and maybe read previous to 

 Lacroix's Algebra, or may be passed over without any oth- 

 er inconvenience than the necessity of dwelling somewhat 

 longer upon Lacroix. It is, it must be confessed, read 

 with more facility than Lacroix's, and is exceedingly well 

 adapted to primary instruction. But we do notkiiow (hat 

 any point is explained by Euler which is neglected by La- 

 croix, much less that any one is better explained. The 

 selections here published, comprise the greatest pari of 

 Euler's first volume, and are made from a translation pub- 

 lished in England. 



The Algebra of Lacroix next claims our attention. He 

 introduces his subject by some preliminary remarks upon 

 the transition from Arithmetic to Algebra, which are fol- 

 lowed by observations on its nature and object. Several 

 problems involving an equation are tirst solved entirely by 

 common language, for which algebraic signs are immedi- 

 ately substituted, and the solution performed by means of 

 them. In this way, the reason of the employment of Al- 

 gebraic signs is from the first clearly made known, their 

 utility and necessity become manifest, and Algebra i« shewn 

 to be a branch of universal language, differing from com- 

 mon language principally in the circumstances, that every 

 sign has an invariable meaning, that it is of a far more ab- 



