Mi 3 



RELATION. 



RELATION. 



1014 



performed, and on the other aide the result of performing that opera- 

 tion by general rules, as in 



-o) = a; ! -a' ' -7- a' = a". 



Whenever the resulting form is intelligible both in form and magni- 

 tude, the resulting relation is equality of magnitude under difference 

 of form, independently of the particular values of the symbols ; but 

 when the result is unintelligible, as is the second of the preceding 

 results when first obtained, this relation no longer exists : the process 

 :'>d in INTERPRETATION makes it exist. In all such cases the 

 relation is that of sameness of value and properties, sameness in fact of 

 everything but form ; and the relation is independent of the magnitude 

 of the algebraical symbols. 



But did no relation exist in a'-7-o' = a, until we had interpreted 

 the then unknown symbol o to mean unity ? We answer, that a 

 relation did exist namely, sameness of properties. The value of the 

 first side is unity ; the unknown symbol of the second side would be 

 found on trial to have all the properties of the unit, when common 

 algebraical rules are applied. If we were to refuse the interpretation, 

 and consider o as a self-contradictory symbol, we could not deprive it 

 of the properties of a unit, or rather, we could not deprive ourselves of 

 the knowledge that the algebraical use of it would produce the same 

 results as the algebraical use of a unit. 



To the same heat! may be referred the meaning of =, as connecting 

 an infinite series with its finite source of development (or its invelop- 

 rnent). Arithmetical equality may not exist, for the series may be 

 divergent ; but between the development and its invelopment exists 

 thfi relation of sameness of properties, and the relation of sameness of 

 source. The infinite series 1 1+1 1+, &c., is equated to half a 

 unit ; that is, the sign = is put between { and 1 1 + 1 1 + , Ac., ad 

 inf. The relation of sameness of magnitude has no existence, for 

 1 - 1 + 1, Ac., ad iafnitum. furnishes no definite idea of magnitude ; 

 but in properties, the two are the same. 



2. The sign = means the relation of sameness of magnitude, without 

 reference to form ; and in this sense its use generally imposes conditions 

 on one or more of the symbols employed, and always does so unless 

 when the sign might also truly have the meaning described under the 

 first head. Thus 2x+3=x+i+x\ imposes no condition on the 

 value of x, because the first side is only a more simple performance of 

 the operations indicated on the second side; but 2x + 3 = 21 xis the 

 assertion cf a relation existing which is not true of the forms, and is 

 not generally true as to the magnitudes. The condition x=6 is neces- 

 siry to the truth of the relation asserted to exist. Relations of this 

 ort, under the name of equations, are the first which meet the 

 student at his entrance into algebra ; and he frequently has a subse- 

 quent difficulty in extending the use of the symbol = . Being accus- 

 tomed to see it impose conditions of magnitude, he cannot easily cease 

 to imagine that it always does so; and he looks upon the two 

 equations 



=]+: T + + Ac., and .T + 1 = 2, 



as things of the same kind, differing only in complexity. To prevent 

 thU, the distinction between idinlical equations (so called), namely, 

 assertion* of the relation described under the first head, and equations 

 . should be strongly marked at the outset of his course. It 

 i even bo wise to use somewhat different symbols for the two 

 relation* : thus* === might denote the first described relation, and 

 = the second. The learner might drop the slight distinction which 

 exisU between the two symbols when he finds himself able to do with- 

 out it ; but we are satisfied that those who had once learned to use it 

 would never think the time was come when they might safely drop it. 



8. The sign = means the relation of algebraical identity between 

 the results of different operations, when the symbols are not symbols 

 of magnitude, but of OPERATION : that is, it asserts the relation of 

 nmnnqnn of effect between the two operations which are written on 

 one side and the other of it. And here it is in truth used in the first 

 sense described, the difference being in the meaning of the symbols, 

 not in that of the relation. And here again there is the distinction 

 between the case in which the relation is explicable from definitions, 

 and that in which it requires interpretation. Thus, in the relation 

 (I + A) 2 = 1 + 2A + A*, we can prove and verify that the operation 1 + A 

 la of that sort which, if performed twice following, will yield the same 

 result a* the turn of the results of the operations 1, 2 A, and A 2 . But 

 when, having established, as in the article cited, a right to the use of 

 all the ordinary transformations of algebra, we come to 1 + A = E" 

 and r> = log (1 + A), we have results of which the first side only is 

 explicable, and the second requires interpretation. It might be satis- 

 factory to consider such symbols as log (1 + A), Ac., in no other light 

 than as abbreviations of the series into which they might be developed 

 in common algebra ; hut as such a use of interpretation seems to a 

 beginner to be more arbitrary than it really is, we may point out how 

 to make the passage in a somewhat more guarded manner, presuming 

 the re.i.l<T to be perfectly well acquainted with the results of the 

 article UrniATlox. 



If A, B, Ac. , stand for symbols of operation, then A + , A B, A -t- B, 

 are compound resulU of operation, which are capable 6T and actually 



e ceive a distinct definition. Similarly, A" is also deducible in meaning 

 ' r om the definition when n is any number, whole or fractional, positive 

 or negative ; but A ', where B is also a symbol of operation, cannot 

 be immediately explained from definition. But it is to be remembered 

 that au algebraic quantity may be susceptible of different definitions, 

 though really amounting to the same definition. Sometimes nothing 

 more than a mere change of the form of words will render a notion 

 capable of being rationally extended further than it could have been 

 before the change was made. For instance, in FRACTIONS, we under- 

 stand the division of 7 into 3 equal parts, and into 4 equal parts ; but 

 a division into 3-J equal parts is a set of words without meaning. But 

 if we only speak of taking parts of which three make 7, and other parts 

 of which four make 7, it is perfectly easy to imagine parts such that 

 three parts and half a part make 7. Can we not, then, take such a 

 method of denning A B as, without in any way altering its common 

 meaning, shall present that common meaning in a form which will be 

 intelligible when A and B are symbols of operation ? 



In BINOMIAL THEOREM it is proved that the equation <t>xx<pz= 

 <t> (x + z) can only be satisfied for all values of x and z, by <px= d* where 

 c is independent of x. 



If, then, we propose the equation 



(1) 



the only solution must be c*. It is easy enough to show that 



the proof referred to shows that c* is the only solution of this equation. 

 If : and x be symbols of operation, and if by <fx we mean a combination 

 of operations performed with x, and by <px . <pz the result of succes- 

 sively performing the operations <pz and tj>x, we may denote by y* an 

 operation which is such, that calling it <fix, the successive performance 

 of <p .< and fyz is equivalent to that of q> (x + z) ; and that, calling it tyy, 

 the successive performance of tyy and tyz is equivalent to that of $(yz). 

 If we want to define the particular operation A *, we must add to the 

 equation (1) the following : 



Thus, let it be the definition of t", D being a symbol of operation, that 

 we have here an operation such that if it aud e"' were successively per- 

 formed, the result would be the same as if +' were performed at 

 once ; this last symbol implying that the operation D + D' is used in the 

 same way as D in the first. Moreover, let it be understood that if D 

 were 1 that is, if the operation D produced no alteration in the func- 

 tion operated on the result of e u would be simple multiplication by e. 

 There is nothing in this definition which is unintelligible, though there 

 is something unknown. An operation is defined by means of itself ; 

 the definition must then be developed before its object can be under- 

 stood, but it is not the less a definition that is, a description of some 

 one operation, and a distinction between it and every other. Thus, in 

 common algebra, the magnitude of x may be defined by au equation, 

 say J"=12 r. Here x is only given in terms of its unknown self, but 

 it is not the less defined to be 6, and nothing but 6. When the step 

 above described has been made, it is (owing to the demonstrated con- 

 nection of the rules of common algebra with those of the calculus of 

 operations) the same process to prove that 



when D signifies an operation, as when it signifies a quantity. 



The definition of log D is that this operation is the inverse of c" with 

 respect to D; so that log e means D. Those functions which in 

 common algebra are trigonometrical [SINE] cannot be defined in the 

 subject of which we are speaking otherwise than by reference to the 

 well-known exponential forms. Thus, D denoting an operation 



Cos D 



Sin D means 



^Vl-D + f'V(- j- 



2 .// 



/ tV(-i) e-W(i 



i-) j- 



It might perhaps be said that though we have constantly used the 

 word relation, yet we have considered nothing but identity, that is, 

 either identity of magnitude, form, process, or properties ; but that 

 the term in common life refers to something short of complete identity, 

 frequently meaning mere connection, and sometimes only analogy, or 

 even nothing more than resemblance. We answer, that relation always 

 refers to identity of some sort. For example, there is a relation be- 

 tween the position of the Rim and moon and the state of the ocean. 

 Here the word means merely a connection ; but this connection in- 

 volves an absolute identity : having given the position of those heavenly 

 bodies with respect to any place, together with the direction aud 

 quantity of their motions, the height of the water at that place is 

 connected with the quantities which express those positions and mo- 

 tions by an equation or a mathematical identity. Resemblance again 

 means identity in some respect, or near approach to identity : analogy, 

 a term generally applied to relations of similarity, will be found to 



