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Rtviexo of the Cambridge Course of Mathematics. 31 1 



cifectually prepares a youns; man to pursue a coarse of dis- 

 covery of liis own, after becoming so thoroughly acquauitod 

 with the discoveries of others. 



2. He avoids repetitioiis. (les doubles emplois* E«sais. 

 p. 180,) This becomes so Riuch the more necessary, as 

 the recent progress of the mathematical and physical' sci- 

 ences has greatly increased the mass of objects ofinstriic- 

 tiou. He seldom emp'oys different d'emonstrations to come 

 to the same result; and never, unless it be in a sncceeduie: 

 part to give a more rigorous demonstration to a principle 

 of extensive use which could r»ot at first be demonstrated i..a 

 rigorous manner 5 or unless the second demoiic^tration *;iv€s 

 him occasion to make some new remark^ or deduce some 

 interesting principle. 



, 3* He always chooses tlie most general methods. Tin's 

 rule is in some degree a const^quence of the former, since 



by means of such methods, repetitions are most effectual- 

 ly avoided. *' In instruction,'' says Laplace, *' prefer 

 general methods, take care to present them in the most 

 smiple maMier, and you will (ind at the same time, that 

 they are always the most ca^y.'^* (Essais p. 181.) It is 

 time to distrust this predilection for particular methods, 

 Qnder the idea that they are more elementary than gene- 

 ral methods ; whereas the truth is, that they are preferred 



because more ancient, and more agreeable to habits pre- 

 viously acquired, and which are not easily reformed. It is 

 erroneous and contrary to established experience, to sup- 

 pose that general methods must be preceded by an expo- 

 sition of particular methods. General methods have no 

 need of any assitance of this kind, when they are suitably 

 <^xplained, and do not meet, in the minds of those who stu- 

 % theyn^ or judge ofthem^ with oid ideas to be eflliced, or 

 old prejudices to be destroyed. If we prefer the synthet- 

 ic methods, because we think them attended with more 

 complete evidence, and that they speak more to the sen- 

 ses ; we must recollect that the analytical methods are vast- 

 ly more fertile, aiid that the writings of the great mathe- 

 maticians of our age, are composed in the style of these 

 methods which it fs absolutely necessary to study, as soon 

 as wc rise above the elements.* (Essais p. 183.) 



4. He makes use, as far as possible, of the analytic 

 method. This method has been the great instrumeut of 

 invention at all times in mathematical science, and has 



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