SOLT7TIONS TO THE EXERCISES.] MATHEMATICS. ALGEBRA. 



631 



lo.r'y + G! 



6<Xry' 



5-ry 



4 1 5x 3 y + 25 j-j/ 



10.ry 3* 4 



-2y 



3x J 15j:y 2y' 



Hence the cube root of the proposed polynomial is 

 x j 5xy 2y 2 , with GSz 3 */ 3 for remainder ; so that the 

 expression proposed differs by Cox 3 ?/ 3 from the complete 

 cube of x 2 5xy 2j/ 2 . If the remainder be intro- 

 duced, with changed sign, into the polynomial, it will 

 then become a complete cube : that is, 

 ar" 15yfy + 69-c'y 2 65*V 138xV SOxi/ 4 8x 6 

 = (x* 5zy V) 3 - 



And in this way we may always ascertain what expres- 

 must be introduced into a proposed polynomial, in 

 order to render it a complete cube. 



NOTE. The process adopted above, in imitation of 

 that at page 485, of the Algebra, occupies a good deal 

 more space +han the work by the old method ; but it 

 must be observed that here there are no bye-operations : 

 the whole is placed fairly before the eye, and every step 

 may be performed with great ease and rapidity, and may 

 be readily revised in case of mistake advantages not to 

 be overlooked. 



The solutions now completed will, we think, afford to 

 tl^e learner all the aid ho can reasonably expect in his 

 efforts to make himself acquainted with the PRINCIPLES 

 OF ALGEBRA. Wherever his own attempts to accomplish 

 the satisfactory solution of a question fail, he will find it 

 of much service to him to have at his command the 

 assistance here offered ; and even when his endeavours 

 are successful, he will often gain instruction by com- 

 paring his own work with the details thus given. In 

 the expectation that the present collection of solutions 

 will be consulted with a view to such instruction, we 

 have frequently given different methods of working out 

 the same example, and of thus exhibiting certain alge- 



braical artifices and expedients familiar enough to pro- 

 ficients, but which can be acquired by a learner only by 

 practice, and the careful examination of illustrative 

 models. No one can become an algebraist by Rule : all 

 that rules can do is to dictate the mode of applying 

 general principles ; while, in many special cases, inde- 

 pendent judgment and ingenuity must be exercised, in 

 reference to the particular circumstances of the inquiry, 

 in order to avoid an unnecessary accumulation of sym- 

 bols, and a needless outlay of time and trouble in fact, 

 in order to give to the algebraic process that compact- 

 ness of form, and neatness of finish, which constitute 

 what is called an elegant solution. This mastery over 

 his subject the student can acquire only by consulting 

 the best models, and by cultivating the power, himself, 

 of proceeding under the guidance of his own free judg- 

 ment and penetration, unfettered by the shackles of 

 rules. A learner should always be on the look-out for 

 those facilitating expedients, which are often to be 

 brought into operation in cases, in reference to which 

 rules can afford but imperfect guidance. He would do 

 well, however, to keep in remembrance that a short 

 solution is not necessarily the best solution : what looks 

 short upon paper may be the result of more mental 

 exertion than what looks long ; the art is so to manage 

 that the expenditure of time and thought may be re- 

 duced to the smallest possible amount : a solution thus 

 characterised is entitled to be called an elegant solution. 

 To those merely mechanical operations the performance 

 of which it is the business of the early parts of algebra 

 to teach these remarks do not, of course, apply ; they 

 refer exclusively to the subject of problem-solving, and 

 to the more advanced portions of algebraical research. 



CHAPTER VI. 

 SERIES AND LOGARITHMS. 



ON SERIES. 



1. The Principle of the Permanence of Equivalent Forms. 



IT was stated at page 451, that "the processes of 

 Algebra are, for the most part, only processes of Arith- 

 metic, extended and rendered more comprehensive by the 

 aid of a new set of symbols, taken in combination with 

 the well-known symbols of Arithmetic ;" and in the ex- 

 planations following, it is made to appear that Algebra is 

 a generalisation of Arithmetic that whereas 2, 5 ... 

 represent certain special numbers, a, 6 ... represent any 

 numbers. This is professedly an elementary view of the 

 case ; and, as an elementary view, is quite sufficient. 

 But when the nature of the generalisation is more closely 

 considered, it appears that, in what is commonly called 

 Algebra, there are really two distinct, though closely 



connected sciences, which may be called respectively 

 Arithmetical Algebra, and Symbolical Algebra. In 

 Arithmetical Algebra, "the symbols represent numbers, 

 whether abstract or concrete, whole or fractional, and 

 the operations to which they are subject are assumed to 

 bo identical, in meaning and extent, with the operations 

 of the same name in common arithmetic. The only dis- 

 tinction between the two sciences consists in the substi- 

 tution of general symbols for digital numbers." 



Thus, in arithmetic, it is impossible to subtract 7 from 

 5 : so that 5 7 is impossible ; and hence in arithmetical 

 Algebra, when we write a b, we do so with the tacit 

 assumption that a > b. If we generalise a step farther 

 than this, and allow ourselves to write o b for all 

 values of a and b, then it is clear that the negative 

 sign has a more extended meaning than that of mero 



