1016 



RELATION. 



RELATION. 



1018 



admit of the word sameness being used instead of similarity. Thus 

 when we say that antwUnce is formed from lulatarr in a nmnnor 

 similar to that in which distance U fonned i. the analogy 



snirted is one of absolute identity (of mode nf derivation). 



Reasoning by analogy is either the same thing as common reasoning; 

 or else analogy i but another word for induction. If A give B, and c 

 hare something in common with A, it may be a necessary consequence 

 that c give* o ; n being connected with < in the same manner as 11 with 

 A. But this happens only when tin- following of B from A is a neces 

 sary consequence of that which A and have in common, and of that 

 only : in which case the deduction of n from c by analogy with the 

 deduction of B from A, is only an assertion of the possibility of applying 

 the same mode of proof to that part or property of c which v. 

 viously applied to the same part or property of A. But when we 

 conclude by analogy of a horned animal that it is not carnivore ' 

 is said; that is, when we conclude that the horned animal of which 

 we speak will resemble all other horned animals which we know, in 

 every point in which they resemble each other, we apply no other 

 process than the establishment of a highly probable result by in- 

 duction. 



Reasoning by pure analogy, U then, not absolutely demonstrative 

 reasoning, except in the cane above described, in which we want no 

 new name for the process. But attention to analogy in the structure 

 of definitions, and in the route of investigation, is necessary to the 

 surcees of many inquiries, and gives clearness and saves time in all. 

 Indeed it may be taken as a maxim that whenever there is any species 

 of resemblance pervading the results of two branches of inquiry, there 

 ought to be a reason for that resemblance in the nature of the two 

 subjects, expressed by a resemblance of the notations used ; and this 

 reason ought to be made prominent and insisted on. 



For instance, we have two distinct algebra* [ALGEBRA], which, for 

 temporary distinction, we may call arithmetical and geometrical, using 

 the same symbols in the same manner, but proceeding upon meanings 

 given to those symbols which appear altogether different. 7'1 

 reason given to the student in the article cited, to justify the definitions 

 of the latter, or geometrical algebra, was that they would be found to 

 answer a certain purpose, namely, to make all theorems in the earlier 

 algebra true, when no other alteration was made than that of the 

 meanings of the symbols. It is now to be asked, why have the new 

 definitions that property ? what relation have they to the old ones which 

 gives the results of the two a perfect community of form ? The answer 

 to this question is not very difficult ; but it will require us first to 

 consider what are the operations of common arithmetic, and how they 

 are to be described in terms of the simplest notions of the science. 



This fundamental operations of arithmetic are addition, subtraction, 

 multiplication, and division. Of these we may make the definitions of 

 subtraction. and division follow from those of addition and multipli- 

 cation : thus subtraction is the process which destroys the effect of 

 addition, and division that which destroys the effect of multiplication. 



The fundamental ideas of arithmetic are, first, that absence of all 

 magnitude which must precede the consideration of any particular 

 number; secondly, the particular magnitude which we choose for 

 repetition, and to which we refer other magnitudes. Nothing and unity 

 are the names of these ideas ; and and 1 are their well-known symbols. 

 The first, 0, reminds him who uses it, of the state in which he is 

 antecedently to thinking of any number ; the second, 1, of the succes- 

 sive accessions by which he passes from one object of consideration 

 to another. If do not present itself before we can think of any number, 

 it is that we avoid it by an act of memory ; but if, for instance, a 

 person had forgotten what seven was, as a young child might do in 

 learning arithmetic, he would be obliged, beginning from 0, to con- 

 struct 7 by repeated accessions of a unit each time. 



Now addition of one number to another is a process which merely 

 puts a number in the place of nolhin;/, and proceeds to count from 

 that number in the same manner as when we form the number to be 

 added from 0. Thus to add 6 to o we do with a what we should have 

 done with to form 4 : to add 4 to 3, we do with 8 what we should 

 have done with to form 4. If this last operation were performed on 

 the fingers, we should first complete three, and then count the fingers 

 which make four from and after the completion of the three; thus 



4=0+1+1+1+1 

 3+4=3+1+1+1+1 



This definition of addition, namely, that " a + l> is a direction to 

 do that with a, which would give 6 if o were nothing," will now be 

 put by for a moment, until we are ready to apply it in the construction 

 of the new algebra. 



Multiplication of one number by another is a process which puts a 

 number in the place of unity, and proceeds to use that number in the 

 same manner as we use unity when wo make another number. Thus, 

 to multiply a by 4, we do with a what we should have done with unity 

 to make 4 ; to multiply 8 by 4, we do with 3 what we should have 

 done with unity to make 4. Thus, 



4 = 1 + 1 + 1 + 1, 

 3x4 = 3 + 3 + 3 + 8. 



The definition* o/ subtraction ami division are then obtained, as before 



"escribed, by the supposition nf inverse operation*, or operations 

 i -'nil-five of the effects of addition and multiplication. 



In consequence of the preceding conai.ln.it i< n-. we shall pass from 

 the limited to the more extended algebra without anything of any 

 arbitrary character, except only the choice of a meaning for tlm 

 fundamental symbols. In arithmetic, the symbols , '/. r. &c., mean 

 simply numbers ; let their meaning in the geometrical algebra be not 

 numbers but lengths, or if numbers, let them be numbers of lengths, 

 a given length being taken as the unit. And let each symbol be ex- 

 pressive not only of a length, but of a length in some p.utirn'.-ir 

 direction ; by which we mean that two lines are not to be d,-n. ; 

 the same symbol, unless they have not only the same lengths but the 

 same direct: 



This one fundamental change in the meaning of the things signified 



by letters is now all that need be made ; for all the rest is absolutely 

 the same as in arithmetic. For esampje, what is o + 4 ? Let <> \ and 

 o B be the lines represented in length and direction by a and '/ ; i 

 completed o A, that is, having passed from o to A through the ] 

 length and in the proper direction, do that which would have nivi-n 

 o B if o A had been nothing, or if A had been at o ; that is. dn 

 equal and parallel to OB, and in the same direction. The point thus 

 attained terminates a lineoc, the length and direction of whi.h is 

 therefore to be denoted by a + It, since every process is to res. 

 th.it of arithmetic in everything but the meaning of the ol>i- 

 calculation. 



-\ -: iiu, required the meaning of b a ! We must now choose a length 

 and direction which is to be represented by 1 ; let this be o u. We 

 are now to do with o A or a what we should have done with o u to 

 make o B. Suppose for simplicity that o B is double of o f. To turn 

 o u into o B, we must double its length, and let it revolve through a 

 certain angle u o B. Do this with o A ; that is, double its length, and 

 make it, thus doubled, revolve to the position o c, so that the angles 

 u o B and A o c are equal. Then o c must be that which is repre- 

 sented by b a ; and the angle u o c is the sum of the angles u o B and 

 no A. 



If we examine the fundamental definitions of the geometrically 

 defined system of ALGEBRA, we shall find that we have here des 

 enough to deduce them all, and that we have done it l>y pure an 



<> remark that analogy here means nothing but an identity ol 

 process; we have described the proces-i-s .if addition and multiplication 

 in terms which connect them so closely with the objects to whirl) 

 are applied, and at the same time make the process so distinct from 

 the subject-matter of the process, that when we change the subject- 

 matter, we can still preserve the process. If then any other subject- 

 matter could be found, such that, with reference to meanings of a and 

 h derived from it, a + 4 and a 4 could be consistently defined, or rather 

 deduced, from and 1, a new application of the rules of algebra would 

 follow. 



In the calculus of operations, the same steps might be made ; and 

 when this branch of algebra consisted of nothing but the separation of 

 the symbols of operation and quantity in an arbitrary manner 

 OPERATION], analogy was the species of relation by which the deduc- 

 tion of results from this separation was connected with the results of 

 common algebra. But this analogy was only the guide to the results, 

 and not the proof of them : it became a proof when it was shown tliat 

 the validity of the common .-il^el.ra itself depended, not upon the 

 whole meaning of the symbols, but upon that part of it which was 

 preserved in the meanings of the symbols of operation. 



RELATION (Logic). In the article LOGIC (cols. 345, 346) we have 

 contended that any composition of relations is syllogism, an.: 

 stated our objection to tho mode used by logicians of reducing such 

 compositions to tlteir syllogism, in which the only relation is identity. 

 Without further controversy we shall proceed to stateim -in, 

 lothing more, of a few heads of the general doctrine of syllogism. 

 Those who desire more must consult a paper on the logic of n-' . 

 which will appear in vol. x., p. 2, of the Cambridge Philosophical 

 Transactions. 



Let L and M denote relations: X, v, z individual objects of thought, 

 classes or attributes, it may be, considered as unite of thought. 

 LI denote anything which is an L of x, in the relation L to x. 



