IN 



ALGEBRA. 



/, it sunwU iuelf to let + 1 and 1 mou, not merely the addi- 

 tion tad subtraction of 1, but any two opposite kind* of uniu whal- 

 erer, addition and aubtnction being only one of the infinite number of 

 opposite meaning* which may be ugnitied. 



5. Extensive induction ahow, not only that the five rule* remain 

 true when the but augmentation of meaning is adopted, but that the 

 want of lymbolic distinction between the different meaning* of -I- and 

 M of no consequence whatever. Complete opposition of cliaracter 

 in + and U all that U necessary to the permanence of the rules ; 

 and + 1 may stand in the same problem for a unit of gain, a unit of 

 ascent, a unit of future time, a unit of addition, Ac. ; while 1 

 stand* for a unit of loss, a unit of descent, a unit of past time, a unit 

 of subtraction, Ac. 



. The algebra thus established, having all the processes of pure 

 arithmetic, ad no mart, and completely containing pure arithmetic 

 within iU range of subject* (for + 1 and 1 mean addition and sub- 

 traction of 1, as well as other things), may be called tingle algebra, 

 Space of one dimension (length) is, so to speak, wide enough for it : 

 if -I- 1 signify a unit measured in one direction upon a straight line 

 from a given point, then 1 signifies a unit measured in the opposite 

 direction. Every positive and negative quantity may find ito inter- 

 pretation in this line. 



7. On proceeding to ulterior result*, it is found that the extension of 

 meaning which is sufficient to admit such symbols as 3 7 or 4 

 within the range of intelligible quantities will not do the same for 

 their roots of the second, or fourth, or any even order. Thus, even in 

 the complete single algebra, </ 1, V 1, &., are as void of meaning 

 as 4 7 in pure arithmetic : they are not positive quantities ; they are 

 not negative quantities ; and there are no other sorts. Under the 

 name of impouible quaxtilia they were long used without being under- 



Ktood, because they were found to lead to true results. Some defended 

 them on different theories, but the sentiment of Professor Woodhouse 

 must have been the one which guided the class of mathematicians who 

 think on first principles ; " since they lead to true results, they mutt 

 hart a logic." The question was to find that logic : and till it was 

 found, a great part of algebra was art, not telenet. 



8. Since it has appeared that extension of meaning has succeeded in 

 clearing away one class of difficulties, the means of removing the rest 

 must probably be sought in further extension ; but it will give a better 

 idea of the subject to begin by throwing away all meanings, treating 

 the symbols as symbols only, subject to certain five rules of operation. 

 Let the only symbol which retains its meaning be = , denoting the 

 identity of what precedes with what follows. We then stand thus : 

 Let it be granted that there are certain symbols, 0, 1, a, 6, e, Ac., and 

 symbols of connection, + x -t-a* (we cannot here dispense with the 

 letters, since the symbol consists in the manner of placing the letters) : 

 let + a and a be synonymes, and also a and a, and 1 xo and a, 

 and a -7-1 and a: let 1+1 be abbreviated into 2, 2 + 1 into 3, Ac.: 

 let +o a be 0, and let xa-~a be equivalent to xl: let the 

 five rules in 3 be perfectly and universally true, and let a, b, Ac., be 

 competent to represent (among other things, perhaps) 0, 1, 2, Ac., 

 or any formations from them or from one another. Let no idea of 

 meaning be attached, for the present, to any symbol or to any of the 

 words by which operations are described. Required all manner of 

 methods of forming symbols which must be considered as identical. 

 The collection of such methods is tymbolical algebra, nothing but 

 symbols and prescribed laws of use. For instance, required a necessary 

 transformation for (a + 6) (e + d). By the fourth rule this is (a + b) 

 c + (a + b)d, which, by the third rule, is r(a + 4) + rf(o + A), which again, 

 by the fourth rule, is ac + bc + (ad + db), or ac + bc + ( + ad) + ( + db), 

 which, by the first rule, is ae + be + ad + db. This symbolical algebra 

 is not a science, but an art : it may be illustrated as follows : Suppose 

 a person to join the pieces of a boy's dissected map by the bocks, 

 without looking at the countries engraved on the fronts ; he i* then 

 going through the dead and (by itself) uninstructive process of sym- 

 bolical algebra ; the pieces are his symbols, the forms of the edges are 

 his rule* for guidance. But another, who turns the map the right 

 way, and puts the countries which he knows into their right places at 

 once, and helps himself to the position of those which lie does not 

 know by trying to fit edges together, is going through the improving 

 process of geographical acquirement, which corresponds to an algebra 

 with meanings attached to the symbols. Symbolical algebras may be 

 invented without end : how many of them would be worth anything 

 for the value of their possible systems of meaning, in another question. 

 We avowedly make a retrograde step when we introduce symbolical 

 algebra ; we throw away the arithmetic and single algebra which sug- 

 gested the symbols and their rules, and retain for the moment only the 

 unmeaning symbols and their laws of combination. 



9. The next step is, given the symbols, rules, and consequent trains 

 of legitimate transformation, of a system of symbolic algebra, how 

 many and what systems of meanings may be attached to the symbols, 

 so that all the fundamental rules may be true of those meanings, and all 

 the symbolic identities which follow from those rule* may be necessary 

 consequences, also with intelligible meanings. Every such system of 

 meanings, tuperadded to the symbolical algebra, U a loffical o/i/r/.m. r 

 one in which every process of transformation is a reasoning or a 

 collection of reasonings. In all probability there a an infinite number 

 of logical algebras to every self-consistent system of symbolical algebra : 



ALGEBRA. loo 



we merely note down the list of those which have hitherto been traced 

 from the ordinary system, originally suggested by arithmetic. Since 

 theee sentences were first written, Professor Boole, in his ' Laws of 

 Thought,' ha* shown that all the processes of logic, actual or possible, 

 are capable of representation under the symbols of common algebra, 

 and the rules of common algebra. 



I. 1'nrr arithmetic, in which the subject matter U number. 

 II. f>inyle alutbra, in which the symbols represent number* derived 

 from concrete magnitude, considered as being, in every case, of one or 

 another of two diametrically opposite kinds. III. DoMt algebra, the 

 main subject of the present remarks, in which each symbol represent* 

 a line of definite length, and in some definite direction out of tin- miinii.- 

 number which may be taken in one jJanr. IV. Triple and qiiadriijtlr 

 algebra, in which the directions are not confined to one plane (this 

 subject is in its earliest infancy). V. The geometry of the second book 

 of Euclid, and the corresponding solid geometry. Here AB actually 

 means the area of the rectangle under the lines A and B, and ABC the 

 content of the rectangular parallelopi|>ed (or right solid) un<l> r tin- 

 lines A, B, and C : the fundamental rules can be demonstrated, and 

 propositions can rigorously be proved. The ordinary and incomplete 

 mode of demonstrating the second book of Euclid might thus be 

 rendered unobjectionable. VI. The calrttliu o/ ojxrativm, in which 

 the symbols, or as many of them as we choose, are not magnitude* at 

 all, but directions to perform certain operations on a variable quantity. 

 [OPERATION.] 



10. It does not follow that every species of logical algebra adi. 

 explanation for every symbol or combination of symbols. Thus I. just 

 mentioned rejects 1; II. rejects / 1 ; III. is perfect; ami IV. 

 may perhaps be mode so : in V. AB + C means no more than AB, if it have 

 meaning, and A B U wholly inexplicable ; and VI. is incumbered with 

 difficulties of new and serious kinds as soon as its elements are passed. 

 It may happen that the proper meaning of a symbol or formula cannot 

 be assigned at the commencement of a logical algebra, but can after- 

 wards be deduced from its symbolic consequences. When this is the 

 case, the deduced meaning must not disturb any one of the five rules. 

 This process is the interpretation to which we have alluded, and symbols 

 of which the meaning is laid down from the commencement may be 

 said to be explained. 



II. It is impossible that a perfect algebra can be founded on ideas of 

 time, loss and gain, or any in which only two directions can be 

 imagined. Space, from the infinity of directions which it admit*, i, 

 as yet, the only perfect medium of explanation. Time before ami time 

 after a certain epoch may be represented by tin- |>-ithv and nejsitive 

 quantity, but what is there in the iden of time to which / 1 can 

 possibly apply? Again, show us a connn :;ition which, per- 

 formed upon a gain, produces a sort of result which can neither IK- 

 called gain nor loss, but which repeated tin or more times upon a gain, 

 turns it into a loss and we can immediately construct a system of 

 commercial algebra, in which J 1 shall be intelligible. But, as yet, 

 the necessary ideas are found in geometry only, which causes some 

 persons to object to the extension of algebra. But these surely forget 

 that even common single algebra must derive its theory of oppo- 

 from concrete quantity ; 1, standing alum-, i* unintelligible in the 

 science of pure number. 



12. Remark the manner in which [Hi:i. ATIOX] the definitions of 

 a + b and ab can be given, even in arithmetic, in terms of process, 

 without mention of niibjcet -matter : a +b requires us to proceed from 

 a as we proceed from t> form b : ul> it-quires us to proceed with a as 

 we should with 1 to form b. 



13. Let the common symbol of algebra signify a length in a direction 

 in a certain plane, change either of length or direction demanding 

 change of symbol. Take a point for tin- '-., /.Hint or origin, and let 

 signify that we do not leave that jioint. From O draw any line at 

 pleasure for the ajrit / Irnyth, ami take a length Oil upon it for 1 : 

 continue the unit axis both ways. Draw a porpendirul.u though O to 

 the axis of length, and call it the a.ri / tlinciinii : the reasons for 

 these terms will appear in the sequel. 



14. Let A + B denote the distance and direction from the origin 

 which is gained by going over first A and then B ; and let A B 

 denote that gained by going over first A and then :i line equal and 

 contrary in direction to B. Let AB denote a line uli.pue length ha* 

 units equal to the product of the units in the lengths of A and B, and 

 the sum of the angles which they make with t In- unit line !t the angle 

 it makes with the unit Inn-. Similarly, M A -^B have the quotient of 

 the lengths of A and B for its length and the difference of the angles 

 for its angle. These definitions are fully explained in NEGATIVE, Ac., 

 and RELATION. 



15. From the last it appears a* an actual consequence of definition. 

 that A means simply a line equal and opposite to A : and al 



+ 1 + 1 <>r + 2 can moan nothing but 2 units extended on the unit- 

 side of the axis of length, and so on ; while 2 means 2 units 

 extended on the opposite side of the axis of length. It also appears 

 that A + B and A B are the operation* of single algebra wh 



* Iy the quadruple algebra we mrsn Sir \V. Hamilton'* quattrniotu, which 

 we call quadruple becaune it author calli it *o ; but Inanmuch a* it findf It* 

 complete interpretation in ipace of three dimension*, we consider it an triple. 

 Iti symbolic rules are not altogether thow of ordinary algebra. 



