324 Review of the Cambridge Course of Mathematics, 



the present state of mathematical science. They are taken 

 principally from the writings of Newton, Clairaut, Euler 

 and Lagrange. On the " resolution of Numerical Equations" 

 by approximation, Sir I. Newton's " method by successive 

 substitutions" is given with the improvements of Lagrange, 

 by which it is simplified, and the degree of approximation at 

 each step made known. Several other principles are in- 

 troduced from Lagrange's " resolutions des Eqtiations Nu- 

 meriques," which in some measure prepare the student to 

 engage in the study of that very complete and profound 

 treatise. 



After equations, the leading principles in the theory of 

 proportion and equidifFerence, are demonstrated with an 

 ease and conciseness, which must surprise those who are 

 acquainted only with the tedious method of Euclid and oth- 

 er ancient mathematicians. As proportion and equidifFer- 

 ence may be expressed by equations with perfect conven- 

 ience, we think it would be well to supercede our present 

 parade of proportions by substituting the corresponding 

 equations. Such a form of expression would be more sim- 

 ple as well as precise, and would at the same time, give 

 greater uniformity to our methods. Progression both 

 by differences and by quotients, is made to depend immedi- 

 ately upon the preceding principles, and general formulas 

 are investigated to determine any particular term, the sum 

 of any number of terms, &lc. in each of them. Some ele- 

 mentary principles of the general theory of series, are like- 

 wise derived from the doctrine of progressions 



In his exposition of logarithms, M. Lacroix has adopted 

 the view of them presented first by Euler in his " Introduc- 

 tion a ['analyse de Pinfini," and of which the elementary 

 part is developed with great care in the first volume of his 

 algebra. In this view, all numbers are considered as pro- 

 duced by a constant number raised to a variable power ; 

 and logarithms are the exponents of the powers to which a 

 constant number must be raised, in order that all possible 

 numbers may be successively deduced from it. The cele- 

 brated Lagrange contemplates logarithms as derived from 

 the same origin. To the objection, that these views are 

 not sufficiently elementary, Euler long since answered, by 

 treating the subject without any other preparatory knowl- 

 edge than the arithmetic of powers. The treatise is con 



