Review of the Cambridge Course of Mathematics. 31 1 



effectually prepares a young man to pursue a course of dis- 

 covery of [lis own, after becoming so thoroughly acquainted 

 with the discoveries of others. 



2. He avoids repetitions, (les doubles emplois* Essais. 

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

 the recent progress of the mathematical and physical sci- 

 ences has greatly increased the mass of objects of instruc- 

 tion. He seldom employs different demonstrations to come 

 io the same result ; and never, unless it be in a succeeding 

 part to give a more rigorous demonstration to a principle 

 of extensive use which could not at firstbe demonstrated in a 

 rigorous manner ; or unless the second demonstration gives 

 him occasion to make some new remark, or deduce some 

 interesting principle. 



3. He always chooses the most general methods. This 

 rule is in some degree a consequence 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 

 simple manner, and you will find at the same time, that 

 they are always the most easy."* (Essais p. 181.) It is 

 time to distrust this predilection for particular methods, 

 under 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 

 explained, and do not meet, in the minds of those who stu- 

 dy them, or judge of them, with old ideas to be effaced, 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, and that the writings of the great mathe- 

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

 methods which it is absolutely necessary to study, as soon 

 as we 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 instrument of 

 invention at all times in mathematical science, and has 



