INTRODUCTION. 



MATHEMATICS. ALGEBRA. 



451 



CHAPTER II. 



ALGEBRA. 



WE need scarcely remark, that the study of Algebra 

 requires a previous acquaintance with the principles of 

 common Arithmetic ; it is upon these that its opera- 

 tions are based, and the processes of Algebra are, for 

 the most part, only the processes of Arithmetic, ex- 

 tended and rendered more comprehensive by the aid of 

 a new set of symbols, taken in combination with the 

 well-known symbols of Arithmetic. The symbols of 

 Arithmetic are the figures of Arithmetic 1, 2, 3, ic. ; 

 and the new symbols introduced by Algebra are the 

 letters of the alphabet, a, 6, c, <tc. As noticed at page 

 4133, the peculiar marks or symbols employed in any 

 science constitute the notation of that science ; the 

 notation of Algebra is therefore of a mixed character, 

 consisting of the figures of Arithmetic and of the letters 

 of the alphabet. The letters may be regarded as stand- 

 fur figures ; and the reason why figures alone will 

 unable us to do all that Algebra will, is that & figure 

 ivs has a fixed and invariable signification : thus 

 r a 5, or a 7, conveys the same meaning to every- 

 no one uses the mark or symbol 3 to denote auy- 

 tjiit three. But a Ittt-'i; as a, or 6, may be used 

 note any figure whatever : a 3, or a 4, or a 7, <tc. ; 

 and it may therefore be made to stand even for a figure 

 whose value may be unknown to us, and which value it 

 may be the object of the inquiry to discover. This is 

 one of the principal advantages of algebraical notation ; 

 that part of this notation, which is borrowed from 

 Arithmetic, enables us to express the knoicn figures in 

 any mathematical investigation by marks intelligible to 

 all ; while the other part, the letters, enables us to re- 

 present the figures or quantities, at the outset n- 

 maown, but which it is the object of the problem to 

 determine. 



But besides tho marks or symbols of quantity the 

 figures and letters other marks are introduced into the 

 notation, as signs of operation. Some of these are used 

 alike, both in Arithmetic and Algebra as the sign (+) 

 for addition, the sign ( ) for subtraction, and one or 

 two others with which the reader of tho Arithmetic 

 must be ahva'ly familiar. The signs of operation in 

 most frequent request in Algebra we shall now explain, 

 and shall give a few easy examples of their application 

 before entering on the formal rules of the science. We 

 wish the reader to understand, at the outset, that we 

 ibout to exhibit the first principles of Algebra to 

 mure beginners to persons who, as yet, know nothing at 

 all of the subject. Some experience in elementary 

 teaching has convinced us, that to succeed in an under- 

 taking of tins kind, the instructor must forego all self- 

 importance, descend to the level of his pupil, and, as it 

 were, sit familiarly down by his side, aud address him 

 in that simple and unadorned style that no intelligent 

 individual can fail to understand. We shall endeavour, 

 therefore, to be very plain and simple in the language 

 vc<l. 



i 'tions and Explanations of First Principles. You 

 already know, from arithmetic, that numbers are of two 

 kinds : abstract numbers, and concrete numbers. An 

 abstract number is simply a figure, or number formed 

 by two or more figures: thus, 3, 23, 147, <fec., are 

 abstract numbers ; but 3 oz , 23 ft., 147, are concrete 

 number*. Abstract numbers merely denote how many 

 times or repetitions ; and, accordingly, whenever you 

 use a multiplier, you use an abstract number. Concrete 

 numbers denote how many things ; and some mark or 

 symbol, to tell us what the things are, must always be 

 joined to the figure or figures which tell us the number 

 of them. ,The mark or symbol 02. , as you know, means 

 ounces; the symbol ft., feet ; and stands for po 

 in inoii-y. In algebra, letters are used to stand for 

 numbers, whether they be abstract or concrete. Instead i 



of writing down the abstract number, 23, a letter, or, for 

 instance, may be put to represent 23 ; in like manner, 

 instead of writing down the concrete number 23 ft., a 

 single letter may be made to stand for it. You see, 

 therefore, that a letter serves for a number of either kind, 

 while a figure must have a particular symbol joined to 

 it when a concrete quantity is to be represented. 



It is the business of a teacher of algebra to show how 

 the operations carried on in arithmetic, by the help of 

 figures alone, may be conducted with figures and letters 

 both. The letters, as just noticed, representing numbers, 

 whether abstract or concrete, are called symbols of quan- 

 tity ; and the marks or signs which indicate operations 

 performed with the letters, are called signs of operation. 

 We are now to explain to you some of these signs of 

 operation : most of them, however, have been used in 

 the Arithmetic. 



Tho mark + is the sign of addition : whenever you 

 see it put before a number or letter, you are to under- 

 stand that the addition of whatever that number or 

 letter signifies is meant : thus, 7+3, which is read 

 7 j'lus 3, denotes that the 3 is to be added to the 7 ; 

 in like manner, a -f- 6, that is, a plus b, signifies that the 

 6 ia to be added to the o ; but you will ask how can a 

 6 be added to an o ? Our reply to this is, that a and 6 

 both stand for numbers, abstract or concrete ; and 

 although we say, according to custom, that " 6 is to be 

 added to a," what we really mean is, that the number 

 represented by 6 is to bo added to tho number repre- 

 sented by a. In the case of 7 plus 3, that is, of 7 + 3, 

 we can obey the direction of the sign of operation, and 

 we know that 10 is the result ; but in the case of a + 6, 

 we cannot obey the direction of the sign till we know the 

 interpretation of a and 6: we can only indicate the addi- 

 tion but cannot perform it. 



The sign is the sign of subtraction : whenever it is 

 placed before a quantity, it indicates that the quantity 

 is to be subtracted : thus, 7 3, which is read 7 minus 3, 

 means that 3 is to bo subtracted from 7 ; and, in like 

 manner, a b means that 6 is to be subtracted 

 from a. In tho case of 7 3, the result of the sub- 

 traction is 4 ; the result of a 6 cannot be given 

 so long as the numbers represented by a, and b remain 

 unknown. 



Tho sign X placed between two quantities denotes the 

 in a /H plication of those quantities together: it is called 

 the sign of multiplication ; thus, 7X3 means that 7 and 

 3 are to be multiplied together ; and, in like manner, 

 a X b means tho product of a and 6. In the case of 

 7x3, the result or product is 21 ; in the case of a x b, 

 the result remains unknown till tho number represented 

 by the multiplier is stated. Every multiplier, as you 

 are aware, must be an abstract number ; if a stand for 

 the abstract number 5, then a X 6 is 5 X b ; that is, 

 5 times b, whatever 6 may stand for. The sign just 

 explained is not always used to indicate multiplication ; 

 instead of it a dot placed between the two quantities, is 

 often employed to mean the same operation: thus, 7.3 

 and a.b mean the same as 7 X 3 and a X b ; and, in the 

 case of letters, even this sign is usually omitted, and the 

 letter* simply written side by side, without any inter- 

 vening mark for multiplication at all ; thus a&, a.b, and 

 a X b, all equally mean the product of a and b. When 

 figures are to be multiplied, some sign for multiplication 

 must be put between them, to prevent misunderstand- 

 ing ; if, when we meant 7X3, or 7.3, we put merely 

 73, we should be thought to mean seventy-three, instead 

 of seven times three. 



The sign -T-, placed before a quantity, denotes division 

 In/ Unit quantity : it is called the sign of division; thus, 

 12-5-4 means 12 divided by four, and a -~- b means a 

 divided by 6, Division is othenvi^p indicated by writing 



