THE LIBRARY 
 OF 
 
 THE UNIVERSITY 
 OF CALIFORNIA 
 
 GIFT OF 
 
 Prof. G. C. Evans 
 
mm 
 
Linear Associative Algebra. 
 
 By BENJAMIN PEIRCE, LL. D. 
 
 I \ 
 
 LATE PERKINS PROFESSOR OF ASTRONOMY AND MATHEMATICS IN HARVARD UNIVERSITY 
 AND SUPERINTENDENT OF THE UNITED STATES COAST SURVEY. 
 
 New Edition, with Addenda and Notes, by C. S. PEIRCE, Son of the Author. 
 
 [Extracted from The American Journal of Mathematics^ 
 
 NEW YORK : D. VAN NOSTRAND, PUBLISHER. 
 
 1882. 
 
PRESS OF ISAAC FRIBDBNWALD, 
 Baltimore, Md. 
 
ERRATA. 
 
 Page 10, 31. The first formula should read 
 
 (AB) C=ACBC. 
 Page 30. The third formula should read 
 
 k(i k)=j. 
 Page 36. Foot-note, second line of second paragraph, read 
 
 J=~(h ^), ^-^(l+jy. 
 
 Page 40. Last line of foot-note. For e , read I . 
 
 Page 52. Multiplication table of ft) . For ji = i, read ji=j. 
 
 Page 75. Last line of foot-note, insert I, at beginning of line. 
 
 Page 86. Foot-note. Add that on substituting k + vj for k , the algebra 
 ) reduces to (oar B ) ; and the same substitution reduces (ay s ) to (ag B ) . 
 
 Page 91. Last line of foot-note. For i, read I. 
 
PREFACE. 
 
 Lithographed copies of this book were distributed by Professor Peirce among his 
 friends in 1870. The present issue consists of separate copies extracted from The Ameri 
 can Journal of Mathematics, where the work has at length been published.* 
 
 The body of the text has been printed directly from the lithograph with only slight 
 verbal changes. Appended to it will be found a reprint of a paper by Professor Peirce, 
 dated 1875, and two brief contributions by the editor. The foot-notes contain transforma- 
 
 * 
 
 tions of several of the algebras, as well as what appeared necessary in order to complete 
 the analysis in the text at a few points. A relative form is also given for each algebra ; for 
 the rule in Addendum II. by which such forms may be immediately written down, was 
 unknown until the printing was approaching completion. 
 The original edition was prefaced by this dedication : 
 
 To MY FEIENDS. 
 
 This work has been the pleasantest mathematical effort of my life. In no other have 
 I seemed to myself to have received so full a reward for my mental labor in the novelty 
 and breadth of the results. I presume that to the uninitiated the formulae will appear cold 
 and cheerless ; but let it be remembered that, like other mathematical formulae, they find 
 their origin in the divine source of all geometry. Whether 1 shall have the satisfaction of 
 taking part in their exposition, or whether that will remain for some more profound 
 expositor, will be seen in the future. 
 
 B. P. 
 
 * To page n of this issue corresponds page +96 of Vol. IV. of The JoumaT~ 
 
 342 
 
LINEAR ASSOCIATIVE ALGEBRA. 
 
 1. Mathematics is the science which draws necessary conclusions. 
 
 This definition of mathematics is wider than that which is ordinarily given, 
 and by which its range is limited to quantitative research. The ordinary 
 definition, like those of other sciences, is objective ; whereas this is subjective. 
 Recent investigations, of which quaternions is the most noteworthy instance, 
 make it manifest that the old definition is too restricted. The sphere of mathe 
 matics is here extended, in accordance with the derivation of its name, to all 
 demonstrative research, so as to include all knowledge strictly capable of dog 
 matic teaching. Mathematics is not the discoverer of laws, for it is not 
 induction ; neither is it the framer of theories, for it is not hypothesis ; but it is 
 the judge over both, and it is the arbiter to which each must refer its claims ; 
 and neither law can rule nor theory explain without the sanction of mathematics. 
 It deduces from a law all its consequences, and develops them into the suitable 
 form for comparison with observation, and thereby measures the strength of the 
 argument from observation in favor of a proposed law or of a proposed form of 
 application of a law. 
 
 Mathematics, under this definition, belongs to every enquiry, moral as well 
 as physical. Even the rules of logic, by which it is rigidly bound, could not be 
 deduced without its aid. The laws of argument admit of simple statement, but 
 they must be curiously transposed before they can be applied to the living speech 
 and verified by,observation. In its pure and simple form the syllogism cannot 
 be directly compared with all experience, or it would not have required an 
 
2 PEIRCE : Linear Associative Algebra. 
 
 Aristotle to discover it. It must be transmuted into all the possible shapes in 
 which reasoning loves to clothe itself. The transmutation is the mathematical 
 process in the establishment of the law. Of some sciences, it is so large a 
 portion that they have been quite abandoned to the mathematician, which 
 may not have been altogether to the advantage of philosophy. Such is the 
 case with geometry and analytic mechanics. But in many other sciences, as in 
 all those of mental philosophy and most of the branches of natural history, the 
 deductions are so immediate and of such simple construction, that it is of no 
 practical use to separate the mathematical portion and subject it to isolated 
 discussion. 
 
 2. The branches of mathematics are as various as the sciences to which they 
 belong, and each subject of physical enquiry has its appropriate mathematics. 
 In every form of material manifestation, there is a corresponding form of human 
 thought, so that the human mind is as wide in its range of thought as the 
 physical universe in which it thinks. The two are wonderfully matched. But 
 where there is a great diversity of physical appearance, there is often a close 
 resemblance in the processes of deduction. It is important, therefore, to separate 
 the intellectual. work from the external form. Symbols must be adopted which 
 may serve for the embodiment of forms of argument, without being trammeled 
 by the conditions of external representation or special interpretation. The 
 words of common language are usually unfit for this purpose, so that other 
 symbols must be adopted, and mathematics treated by such symbols is called 
 algebra. Algebra, then, is formal mathematics. 
 
 3. All relations are either qualitative or quantitative. Qualitative relations 
 can be considered by themselves without regard to quantity. The algebra of 
 such enquiries may be called logical algebra, of which a fine example is given 
 by Boole. 
 
 Quantitative relations may also be considered by themselves without regard 
 to quality. They belong to arithmetic, and the corresponding algebra is the 
 common or arithmetical algebra. 
 
 In all other algebras both relations must be combined, and the algebra must 
 conform to the character of the relations. 
 
 4. The symbols of an algebra, with the laws of combination, constitute its 
 language ; the methods of using the symbols in the drawing of inferences is its 
 art; and their interpretation is its scientific application. This three-fold analysis 
 of algebra is adopted from President Hill, of Harvard University, and is made 
 the basis of a division into books. 
 
PEIRCE : Linear Associative Algebra. 3 
 
 BOOK I.* 
 THE LANGUAGE OF ALGEBRA. 
 
 5. The language of algebra has its alphabet, vocabulary, and grammar. 
 
 6. The symbols of algebra are of two kinds : one class represent its 
 fundamental conceptions and may be called its letters, and the other represent 
 the relations or modes of combination of the letters and are called the signs. 
 
 7. The alphabet of an algebra consists of its letters ; the vocabulary defines 
 its signs and the elementary combinations of its letters ; and the grammar gives 
 the rules of composition by which the letters and signs are united into a 
 complete and consistent system. 
 
 The Alphabet. 
 
 8. Algebras may be distinguished from each other by the number of their 
 independent fundamental conceptions, or of the letters of their alphabet. Thus 
 an algebra which has only one letter in its alphabet is a single algebra ; one 
 which has two letters is a double algebra ; one of three letters a triple algebra ; 
 one of four letters a quadruple algebra, and so on. 
 
 This artificial division of the algebras is cold and uninstructive like the 
 artificial Linnean system of botany. But it is useful in a preliminary investiga 
 tion of algebras, until a sufficient variety is obtained to afford the material for a 
 natural classification. 
 
 Each fundamental conception may be called a unit; and thus each unit has 
 its corresponding letter, and the two words, unit and letter, may often be used 
 indiscriminately in place of each other, when it cannot cause confusion. 
 
 9. The present investigation, not usually extending beyond the sextuple 
 algebra, limits the demand of the algebra for the most part to six letters ; and 
 the six letters, i, j, k, I, m and n, will be restricted to this use except in 
 special cases. 
 
 10. For any given letter another may be substituted, provided a new letter 
 represents a combination of the original letters of which the replaced letter is a 
 necessary component. 
 
 For example, any combination of two letters, which is entirely dependent 
 for its value upon both of its components, such as their sum, difference, or 
 product, may be substituted for either of them. 
 
 Only this book was ever written. [C. S. P.] 
 
of letter* is radically important, and is a 
 
 of originality in the present investigation: and without it. such 
 investigation would have been impossible. It enables the geometer to 
 analyse am algebra, reduce h to its simplest and characteristic forms, and 
 compare it witk offer algebras. It involves in its principle a corresponding 
 substitution of wok of which it is in reality the formal representative. 
 
 There is. however, no danger in working with Hie symbols, irrespective of 
 the t4r?t* allai**^ to Ihr , and the consideration of die change of the original 
 be safety reserved for the boat of tmkrpntatitm. 
 
 11. In making die substitution of letters, the original letter wffl be preserved 
 with the distinction of a subscript number. 
 
 That, for the letter t there may successively be substituted h, s,, s, 7 etc. In 
 the fe^l forms, the sobscript numbers can be omitted, and the y may be omitted 
 at any period of the investigation, when it will not produce confusion. 
 
 It win be practically found that these subscript numbers need scarcely ever 
 be written. They pass through the mind, as a sure ideal protection from erro 
 neous substitution, but disappear from the writing with the same fecility with 
 which those evanescent chemical compounds, which are essential to the theory 
 of tnmsformation, escape the eye of the observer. 
 
 12. A JHOK algebra is one in which every letter is connected by some 
 indissoluble relation with every other letter. 
 
 13. When the letters of an algebra can be separated into two groups, which 
 are mutually independent, it is a sttzx*? oJ.yJjrfj. It is mixed even when there 
 are letters common to the two groups, provided those which are not common to 
 the two groups are mutually independent. Were an algebra employed for the 
 simultaneous discussion of distinct classes of phenomena,, such as those of sound 
 and light, and were the peculiar units of each class to have their appropriate 
 letters, but were there no recognized dependence of the phenomena upon each 
 other, so that the phenomena of each class might have been submitted to 
 independent research, the one algebra would be actually a mixture of two 
 algebras, one appropriate to sound, the other to light 
 
 It may be farther observed that when, in such a case as this, the component 
 algebras are identical in form, they are reduced to the case of one algebra with 
 two diverse interpretations. 
 
PEIRCE : Linear A*-*jci4tice Algebrn. 5 
 
 The Vocabulary. 
 
 14. Letters which are not appropriated to the alphabet of the algebra * 
 may be used in any convenient sense. But it is well to employ the *mafl letters 
 for expressions of common algebra, and the capital letters for those of the algebra 
 under discussion. 
 
 There must, however, be exceptions to this notation : thus the letter D will 
 denote the derivative of an expression to which it is applied, and ^ the summa 
 tion of cognate expressions, and other exceptions will be mentioned as they 
 occur. Greek letters will generally be reserved for angular and functional 
 notation. 
 
 15. The three symbols J . d . and (3 will be adopted with the signification 
 
 J= V 1 
 
 9 = the ratio of circumference to diameter of circle = 3.l4l592t ~ 
 6 = the base of Xaperian logarithms = 2.7182818285, 
 
 which gives the mysterious formula 
 
 J = v/ 6 9 =4.810477381. 
 
 16. All the signs of common algebra will be adopted: but any signification 
 will be permitted them which is not inconsistent with their use in common 
 algebra : so that, if by any process an expression to which they refer is reduced 
 to one of common algebra, they must resume their ordinary signification. 
 
 17. The sign =. which is called that of equality, is used in its ordinary sense 
 to denote that the two expressions which it separates are the same whole. 
 although they represent different combinations of parts. 
 
 18. The signs > and < which are those of inequality, and denote "more 
 than " or " less than " in quantity, will be used to denote the relations of a whole 
 to its part, so that the symbol which denotes the pan shall be at the vertex of 
 the angle, and that which denotes the whole at its opening. This involves the 
 proposition that the smaller of the quantities is included in the class expre> 
 
 by the larger. Thus 
 
 B<A or A>B 
 
 denotes that .4 is a whole of which B is a part, so that all B is -4> 
 
 formula in the text implies, also, that some A is not B. [C. S. P.] 
 
t> PEIRCE : Linear Associative Algebra. 
 
 If the usual algebra had originated in qualitative, instead of quantitative, 
 investigations, the use of the symbols might easily have been reversed ; for it 
 seems that all conceptions involved in A must also be involved in B , so that B 
 is more than A in the sense that it involves more ideas. 
 
 The combined expression 
 
 5> G<A 
 
 denotes that there are quantities expressed by G which belong to the class A 
 and also to the class B. It implies, therefore, that some B is A and that some A is 
 B* The intermediate G might be omitted if this were the only proposition 
 intended to be expressed, and we might write 
 
 In like manner the combined expression 
 
 B < G > A 
 
 denotes that there is a class which includes both A and B,-\- which proposition 
 might be written 
 
 B<>A. 
 
 19. A vertical mark drawn through either of the preceding signs reverses its 
 signification. Thus 
 
 A^- B 
 
 denotes that B and A are essentially different wholes ; 
 
 A^>B or B^A 
 
 denotes that all B is not A , J so that if they have only quantitative relations, 
 they must bear to each other the relation of 
 
 A = B or A < B . 
 
 20. The sign -|- is called plus in common algebra and denotes addition. It 
 may be retained with the same name, and the process which it indicates may be 
 called addition. In the simplest cases it expresses a mere mixture, in which 
 
 * This, of course, supposes that Cdoes not vanish. [C. S. P.] 
 t The universe will be such a class unless A or B is the universe. [C. S. P.] 
 
 I The general interpretation is rather that either A and B are identical or that some B is not A. 
 [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 7 
 
 the elements preserve their mutual independence. If the elements cannot be 
 mixed without mutual action and a consequent change of constitution, the mere 
 union is still expressed by the sign of addition, although some other symbol is 
 required to express the character of the mixture as a peculiar compound having 
 properties different from its elements. It is obvious from the simplicity of the 
 union recognized in this sign, that the order of the admixture of the elements 
 cannot affect it ; so that it may be assumed that 
 
 A-}- B = B + A 
 and 
 
 21. The sign is called minus in common algebra, and denotes subtraction. 
 Retaining the same name, the process is to be regarded as the reverse of 
 addition ; so that if an expression is first added and then subtracted, or the 
 reverse, it disappears from the result ; or, in algebraic phrase, it is canceled. This 
 gives the equations 
 
 A+BB=AB+B=A 
 
 and 
 
 B B = Q. 
 
 The sign minus is called the negative sign in ordinary algebra, and any term 
 preceded by it may be united with it, and the combination may be called a 
 negative term. This use will be adopted into all the algebras, with the provision 
 that the derivation of the word negative must not transmit its interpretation. 
 
 22. The sign x may be adopted from ordinary algebra with the name of 
 the sign of multiplication, but without reference to the meaning of the process. 
 The result of multiplication is to be called the product. The terms which are 
 combined by the sign of multiplication may be called factors ; the factor which 
 precedes the sign being distinguished as the multiplier, and that which follows it 
 being the multiplicand. The words multiplier, multiplicand, and product, may 
 also be conveniently replaced by the terms adopted by Hamilton, of facient, 
 faciend, andfactum. Thus the equation of the product is 
 
 multiplier X multiplicand == product ; or facient X faciend = factum. 
 
 When letters are used, the sign of multiplication can be omitted as in ordinary 
 algebra. 
 
8 PEIRCE : Linear Associative Algebra. 
 
 23. When an expression used as a factor in certain combinations gives a 
 product which vanishes, it may be called in those combinations a nil/actor. 
 Where as the multiplier it produces vanishing products it is nilfacient, but where 
 it is the multiplicand of such a product it is nilfaciend. 
 
 24. When an expression used as a factor in certain combinations over 
 powers the other factors and is itself the product, it may be called an idem/actor. 
 When in the production of such a result it is the multiplier, it is idem/orient, 
 but when it is the multiplicand it is idemfaciend. 
 
 25. When an expression raised to the square or any higher power vanishes, 
 it may be called nilpotent; but when, raised to a square or higher power, it gives 
 itself as the result, it may be called idempotent. 
 
 The defining equation of nilpotent and idempotent expressions are respec 
 tively A n =. , and A n = A but with reference to idempotent expressions, it 
 will always be assumed that they are of the form 
 
 A*=A, 
 
 unless it be otherwise distinctly stated. 
 
 26. Division is the reverse of multiplication, by which its results are verified. 
 It is the process for obtaining one of the factors of a given product when the 
 other factor is given. It is important to distinguish the position of the given 
 factor, whether it is facient or faciend. This can be readily indicated by com 
 bining the sign of multiplication, and placing it before or after the given 
 factor just as it stands in the product. Thus when the multiplier is the given 
 factor, the correct equation of division is 
 
 dividend 
 quotient = , 
 
 divisor x 
 
 and the equation of verification is 
 
 divisor X quotient =: dividend. 
 But when the multiplicand is the given factor, the equation of division is 
 
 dividend 
 
 quotient = =7 ; 
 X divisor 
 
 and the equation of verification is 
 
 quotient X divisor = dividend. 
 
 27. Exponents may be introduced just as in ordinary algebra, and they 
 may even be permitted to assume the forms of the algebra under discussion. 
 
PEIRCE : Linear Associative Algebra. 9 
 
 There seems to be no necessary restriction to giving them even a wider range 
 and introducing into one algebra the exponents from another. Other signs will 
 be defined when they are needed. 
 
 The definition of the fundamental operations is an essential part of the 
 vocabulary, but as it is subject to the rules of grammar which may be adopted, 
 it must be reserved for special investigation in the different algebras. 
 
 The Grammar. 
 
 28. Quantity enters as a form of thought into every inference. It is 
 always implied in the syllogism. It may not, however, be the direct object of 
 inquiry ; so that there may be logical and chemical algebras into which it only 
 enters accidentally, agreeably to 1. But where it is recognized, it should be 
 received in its most general form and in all its variety. The algebra is 
 otherwise unnecessarily restricted, and cannot enjoy the benefit of the most 
 fruitful forms of philosophical discussion. But while it is thus introduced as a 
 part of the formal algebra, it is subject to every degree and kind of limitation in 
 its interpretation. 
 
 The free introduction of quantity into an algebra does not even involve the 
 reception of its unit as one of the independent units of the algebra. But it is 
 probable that without such a unit, no algebra is adapted to useful investigation. 
 It is so admitted into quaternions, and its admission seems to have misled some 
 philosophers into the opinion that quaternions is a triple and not a quadruple 
 algebra. This will be the more evident from the form in which quaternions 
 first present themselves in the present investigation, and in which the unit of 
 quantity is not distinctly recognizable without a transmutation of the form.* 
 
 29. The introduction of quantity into an algebra naturally carries with it, 
 not only the notation of ordinary algebra, but likewise many of the rules to 
 which it is subject. Thus, when a quantity is a factor of a product, it has the 
 
 * Hamilton s total exclusion of the imaginary of ordinary algebra from the calculus as -well as from 
 the interpretation of quaternions will not probably be accepted in the future development of this 
 algebra. It evinces the resources of his genius that he was able to accomplish his investigations under 
 these trammels. But like the restrictions of the ancient geometry, they are inconsistent with the 
 generalizations and broad philosophy of modern science. With the restoration of the ordinary imaginary, 
 quaternions becomes Hamilton s biquaternions. From this point of view, all the algebras of this research 
 would be called bi-algebras. But with the ordinary imaginary is involved a vast power of research, and 
 the distinction of names should correspond : and the algebra which loses it should have its restricted 
 nature indicated by such a name as that of a semi-algebra. 
 
10 PEIECE : Linear Associative Algebra. 
 
 same influence whether it be facient or faciend, so that with the notation of 
 S 14, there is the equation 
 
 O - 1 4 A 
 
 Aa = a A , 
 
 and in such a product, the quantity a may be called the coefficient. 
 
 In like manner, terms which only differ in their coefficients, may be added 
 by adding their coefficients ; thus, 
 
 (a 6) A = a A it bA Aa =b Ab = A (a =b ft) . 
 
 30. The exceeding simplicity of the conception of an equation involves the 
 
 identity of the equations 
 
 A = B and B = A 
 
 and the substitution of B for A in every expression, so that 
 
 MA C=.MB C, 
 
 or that, the members of an equation may be mutually transposed or simultaneously 
 increased or decreased or multiplied or divided by equal expressions. 
 
 31. How far the principle of 16 limits the extent within which the 
 ordinary symbols may be used, cannot easily be decided. But it suggests limi 
 tations which may be adopted during the present discussion, and leave an ample 
 field for curious investigation. 
 
 The distributive principle of multiplication may be adopted ; namely, the 
 principle that the product of an algebraic sum of factors into or by a common 
 factor, is equal to the corresponding algebraic sum of the individual products of 
 the various factors into or by the common factor ; and it is expressed by the 
 
 equations i(\ 
 
 (AB)C=A$ EG. 
 
 C(AB)=CA CB. 
 
 32. TJie associative principle of multiplication may be adopted ; namely, that 
 the product of successive multiplications is not affected by the order in which the 
 multiplications are performed, provided there is no change in the relative position 
 of the factors ; and it is expressed by the equations 
 
 ABC=(AB)G=A(BC). 
 
 This is quite an important limitation, and the algebras which are subject to it 
 will be called associative. 
 
PEIRCE : Linear Associative Algebra. 11 
 
 33. The principle that the value of a product is not affected by the relative 
 position of the factors is called the commutative principle, and is expressed by the 
 equation 
 
 AB BA. 
 
 This principle is not adopted in the present investigation. 
 
 34. An algebra in which every expression is reducible to the form of an 
 algebraic sum of terms, each of which consists of a single letter with a quanti 
 tative coefficient, is called a linear algebra* Such are all the algebras of the 
 present investigation. 
 
 35. Wherever there is a limited number of independent conceptions, a 
 linear algebra may be adopted. For a combination which was not reducible to 
 such an algebraic sum as those of linear algebra, would be to that extent 
 independent of the original conceptions, and would be an independent conception 
 additional to those which were assumed to constitute the elements of the 
 algebra. 
 
 36. An algebra in which there can be complete interchange of its indepen 
 dent units, without changing the formulae of combination, is a completely 
 symmetrical algebra; and one in which there may be a partial interchange of its 
 units is partially symmetrical But the term symmetrical should not be applied, 
 unless the interchange is more extensive than that involved in the distributive 
 and commutative principles. An algebra in which the interchange is effected in 
 a certain order which returns into itself is a cyclic algebra. 
 
 Thus, quaternions is a cyclic algebra, because in any of its fundamental 
 equations, such as 
 
 # = 1 
 ij = ji = k 
 ij~k = 1 
 
 there can be an interchange of the letters in the order i , j , k, i, each letter 
 being changed into that which follows it. The double algebra in which 
 
 * In the various algebras of De Morgan s " Triple Algebra," the distributive, associative and com 
 mutative principles were all adopted, and they were all linear. [De Morgan s algebras are " semi- 
 algebras. 1 See Cambridge Phil. Trans., viii. 241.] [C. S. P.] 
 
! 
 
 is oyolio hooauso the letters :nv intorohangoahlo in the ordor . ./. . Hut neither 
 of those algobras is commutative. 
 
 ST. \Vhou an algebra ran bo roduood to a form in whioh all the KM tors :\ro 
 expressed :is powON of MUM ono of thorn. it may bo oallod a potential nbjelmi. 
 If the po\\ors ;uv all squares. it may bo railed ,/mK/r,r/V : if thoy aro oubos. it 
 max bo oallod eiibie ; and siiuilarlv in oihor OMM 
 
 l.ine<ir 1- . .r/iiv 
 
 s. j4// ^ MMNM 
 
 ij eb /o (i// // 
 
 For it is ob\ious that iu tho equation 
 
 eaoh letter eau be multiplied by an inteirer. whieh jrives the form 
 
 -f /;) = fie^ + frtf* -h 
 
 in whieh i. /> . <^ anvl <1 aro r The integers oan have the ratios of any 
 
 four ival number, so that by simple division they oan be roihiooil to suoh real 
 uumbei-s. Other similar equations oan also be formed by writing for ii and b. >i 
 and />,. or for , and ,/. ^ and </ t . or by making; both these substitutions simulta- 
 neously. If then the two first of those new equations are multiplied by J and 
 the last b\ 1 : the sum of the four equations will be the same as that whieh 
 ild be obtained by substituting for <i . />. < and </. i + J^,. b -h J^. o + J<\ 
 ami ^4- Jt/ v Hemv <i , b. c and </ may be any numbers, real or imaginary, and in 
 gvneral whatever mixtures .1. />. ( x and /) may IN of the original 
 
 units under the form of an algehraio sum of the letters \ /. If. ^o.. we shall 
 
 have 
 
 ( A + B)(C + /)) = AC + KC -h AD -f Bl 
 
 whioh is the oomp.- \ rossion of the distributive prinoiple. 
 i ? 
 
 For if .-t = i (<ii) = <ii -f rtj -h <t^ ^ A 
 
 fi= v^> = W^ftJ + \fr+ A, 
 ^ = n" -h r^y -h Jt -f A 
 
ri-:nu K: Linear Axxoeiatirt A/</>!>ni. 
 il is obvious that AB = ^.al^ij] 
 
 (AB)C= (a(>,e,ijk) = A(BC) = ABC 
 
 which is tin 4 general ex pression of tin associative principle. 
 
 40. lu wry linear atwocnitirt alyehra, there i* at lea*t one nl> inj>ot> nt or one 
 nil potent c.ry>/v.vs /<>//. 
 
 Take anv combination of letters at will and denote it by A. Its square 
 is generally independent of A, and its cube may also be independent of A 
 and J 8 . Hut the number of powers of A that are independent of A and of 
 each other, cannot exceed the number of letters of the alphabet ; so that there 
 must be some least power of A which is dependent upon the interior powers. 
 The mutual dependence of the powers of A may be expressed in the form of an 
 equation of which the tirst member is an algebraic sum. such as 
 
 V ( n Am\ () 
 -m\ (t m a ) U> 
 
 All the terms of this equation that, involve the square and higher powers of A 
 may be combined and expressed as />.!, so that II is itself an algebraic sum of 
 powers of A, and the equation may be written 
 
 BA + l A = (B + a l )A = 0. 
 
 It is easy to deduce from this equal ion successively 
 
 B + <iA n = 
 
 "l/ ! 
 
 so that is an idempotent expression. Hut if a l vanishes, this expression 
 
 becomes infinite, and instead of it we have the equation 
 
 so that B is a m lpotent expression. 
 
 41. When there is an icU M/wfent expression in a linear associative algebra, it 
 
 can be assumed as one of the independent units, and be represented by one of 
 f/it letter* of tin alfthalxt ; and it may be called the httsi*. 
 
 The nniaininy unit* ean i>< *<> ..< l<ctnl a* to />t M />arat>h into four dixftnet (jroni>*. 
 
 With /< ft /> ne-i to tJi< l><i*i*, f/it unfa of the first group an ! l< mfat fnr* ; fh<>*< of 
 
 the MCOIH/ <//<>///> <in idemfaeicnd and ni/faeienf ; those of Me third <jroii)> are id< in- 
 
 farirnf <(/K/ ni/fan t iul : ami fho*e of /At fourth </roi<fi an /n /farf<>r. 
 
1 4 PEIRCE : Linear Associative Algebra. 
 
 First. The possibility of the selection of all the remaining units as idem- 
 faciend or nilfaciend is easily established. For if i is the idempotent base, its 
 definition gives 
 
 $ = i . , 
 
 The product by the basis of another expression such as A may be represented 
 
 
 
 by B, so that 
 
 iA = B, 
 
 which gives 
 
 iJB = i*A = iA = B 
 
 whence it appears that B is idemfaciend and A B is nilfaciend. In other 
 words, A is divided into two parts, of which one is idemfaciend and the other is 
 nilfaciend ; but either of these parts may be wanting, so as to leave A wholly 
 idemfaciend or wholly nilfaciend. 
 
 Secondly. The still farther subdivision of these portions into idemfacient and 
 nilfacient is easily shown to be possible by this same method, with the mere 
 reversal of the relative position of the factors. Hence are obtained the required 
 four groups. 
 
 The basis itself may be regarded as belonging to the first group. 
 
 42. Any algebraic sum of the letters of a group is an expression which 
 belongs to the same group, and may be called factorially homogeneous. 
 
 43. The product of two factorially homogeneous expressions, which does not 
 vanish, is itself factorially homogeneous, and its faciend name is the same as that 
 of its facient, while its facient name is tJie same as that of its faciend. 
 
 Thus, if A and B are, each of them, factorially homogeneous, they satisfy 
 the equations 
 
 i(AB) = (iA)B , 
 (AB)i = 
 
 which shows that the nature of the product as a faciend is the same as that of 
 the facient A, and its nature as a facient is the same as that of the faciend B. 
 44. Hence, no product which does not vanish can be commutative unless both its 
 factors btlong to tlw same group. 
 
PEIRCE : Linear Associative Algebra. 
 
 15 
 
 45. Every product vanishes, of which the facient is idemfacient while thefaciend 
 is nilfaciend ; or of which the facient is nilfacient while the faciend is idemfaciend. 
 For in either case this product involves the equation 
 
 46. The combination of the propositions of 43 and 45 is expressed in the 
 following form of a multiplication table. In this table, each factor is expressed 
 by two letters, of which the first denotes its name as a faciend and the second as 
 a facient. The two letters are d and n, of which d stands for idem and n for nil. 
 The facient is written in the left hand column of the table and the faciend in the 
 upper line. The character of the product, when it does not vanish, is denoted 
 by the combination of letters, or when it must vanish, by the zero, which is 
 written upon the same line with the facient and in a column under the faciend. 
 
 dd dn nd 
 
 nn 
 
 dd 
 
 dn 
 
 nd 
 
 nn 
 
 dd 
 
 dn 
 
 
 
 
 
 
 
 
 
 dd 
 
 dn 
 
 nd 
 
 nn 
 
 
 
 
 
 
 
 
 
 nd 
 
 nn 
 
 
 
 i 
 
 47. It is apparent from the inspection of this table, that every expression 
 which belongs to the second or third group is nilpotent. 
 
 48. It is apparent that all commutative products which do not vanish are 
 restricted to the first and fourth groups. 
 
 49. It is apparent that every continuous product which does not vanish, has 
 the same faciend name as its first facient, and the same facient name as its last 
 faciend. 
 
 50. Since the products of the units of a group remain in the group, they 
 cannot serve as the bond for uniting different groups, which are the necessary 
 conditions of a pure algebra. Neither can the first and fourth groups be con 
 nected by direct multiplication, because the products vanish. The first and fourth 
 groups, therefore, require for their indissoluble union into a pure algebra that there 
 should be units in each of the other two groups. 
 
16 PEIRCE : Linear Associative Algebra. 
 
 51. In an algebra which has more than two independent units, it cannot 
 happen that all the units except the base belong to the second or to the third group. 
 For in this case, each unit taken with the base would constitute a double algebra, 
 and there could be no bond of connection to prevent their separation into 
 distinct algebras. 
 
 52. The units of the fourth group are subject to independent discussion, as if they 
 constituted an algebra of themselves. There must be in this group an idempotent 
 or a nilpotent unit. If there is an idempotent unit, it can be adopted as the 
 basis of this group, through which the group can be subdivided into subsidiary 
 groups. 
 
 The idempotent unit of the fourth group can even be made the basis of the 
 whole algebra, and the first, second and third groups will respectively become 
 the fourth, third and second groups for the new basis. 
 
 53. When the first group comprises any units except the basis, there is beside* 
 the basis another idempotent expression, or there is a nilpotent expression. By a 
 process similar to that of 40 and a similar argument, it may be shown that for 
 any expression A, which belongs to the first group, there is some least power 
 which can be expressed by means of the basis and the inferior powers in the 
 form of an algebraic sum. This condition may be expressed by the equation 
 
 If then h is determined by the ordinary algebraic equation 
 
 s. (**-)+&=<>, 
 
 and if 
 
 A l = A hi 
 
 is substituted for A , an equation is obtained between the powers of A , from 
 which an idempotent expression, B, or else a nilpotent expression, can be 
 deduced precisely as in 40.* 
 
 54. Wlien there is a second idempotent unit in the first group, the basis can be 
 changed so as to free the first group from this second idempotent unit. 
 
 Thus if i is the basis, and if j is the second idempotent unit of the first 
 group, the basis can be changed to 
 
 * The equation in h may have no algebraic solution, in which case the new idempotent or nilpotent 
 would not be a direct algebraic function of i and A. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 17 
 
 and with this new basis, j passes from the first to the fourth group. For 
 First, the new basis is idempotent, since 
 
 *! = (* y) 2 = ** 2i? +y s = i y = i i ; 
 and secondly, the idempotent unit y passes into the fourth group, since 
 
 hj=(ij)j= *y y 8 =y y = o , 
 y% =y (* y) =y* y 2 =y y = o . 
 
 55. TF^A Ae preceding change of basis, expressions may pass from idemfacient 
 to nilfacient, or from idemfaciend to nilfaciend, but not the reverse. 
 
 For first, if A is nilfacient with reference to the original basis, it is also, by 
 45, nilfacient with reference to the new basis ; or if it is nilfaciend with 
 reference to the original basis, it is nilfaciend with reference to the new basis. 
 
 Secondly, all expressions which are idemfacient with reference to the 
 original basis, can, by the process of 41, be separated into two portions with 
 reference to the new basis, of which portions one is idemfacient and the 
 other is nilfacient ; so that the idemfacient portion remains idemfacient, and the 
 remainder passes from being idemfacient to being nilfacient. The same process 
 may be applied to the faciends with similar conclusions. 
 
 56. It is evident, then, that each group* can be reduced so as not to contain 
 more than one idempotent unit, which will be its basis. In the groups which 
 bear to the basis the relations of second and third groups, there are only 
 nilpotent expressions. 
 
 57. In a group or an algebra which has no idempotent expression, all the 
 expressions are nilpotent. 
 
 Take any expression of this group or algebra and denote it by A. If no 
 power of A vanished, there must be, as shown in 40, some equation between 
 
 the powers of A of the form 
 
 2 m a m A m =0, 
 
 in which a l must vanish, or else there would be an idempotent expression as is 
 shown in 40, which is contrary to the present hypothesis. If then m denote 
 
 * That is, the first group as well as each of the subsidiary groups of \ 52. [C. S. P.] 
 
18 PEIRCE : Linear Associative Algebra. 
 
 the exponent of the least power of A that entered into this equation, and m + h 
 the exponent of the highest power that occurred in it, the whole number of 
 terms of the equation would be, at most, h + 1 . If, now, the equation were 
 multiplied successively by A and by each of its powers as high as that of which 
 the exponent is (w 1)A, this highest exponent would denote the number of 
 new equations which would be thus obtained. If, moreover, 
 
 B = A m , 
 then the highest power of A introduced into these equations would be 
 
 The whole number of powers of A contained in the equations would be mji -f 1 , 
 and 7i + 1 of these would always be integral powers of B ; and there would 
 remain (m l)h in number which were not integral powers of B. There 
 would be, therefore, equations enough to eliminate all the powers of A that 
 were not integral powers of B and still leave an equation between the integral 
 powers of B and this would generally include the first power of B. From 
 this equation, an idempotent expression could be obtained by the process of 40, 
 which is contrary to the hypothesis of the proposition. 
 
 Therefore it cannot be the case that there is any equation such as that here 
 assumed ; and therefore there can be no expression which is not nilpotent. The 
 few cases of peculiar doubt can readily be solved as they occur; but they 
 always must involve the possibility of an equation between fewer powers of B 
 than those in the equation in A* 
 
 58. When an expression is nilpotent, all its powers which do not vanish are 
 mutually independent. 
 
 Let A be the nilpotent expression, of which the ?i th power is the highest 
 which does not vanish. There cannot be any equation between these powers 
 of the form 
 
 * In saying that the equation in B will generally include the first power of B , he intends to waive 
 the question of whether this always happens. For, he reasons, if this is not the case then the equation 
 in B is to be treated just as the equation in A has been treated, and such repetitions of the process must 
 ultimately produce an equation from which either an idempotent expression could be found, or else A 
 would be proved nilpotent. [C. S. P.] 
 
PEIRCB : Linear Associative Algebra. 19 
 
 For if w were the exponent of the lowest power of A in this equation, the 
 multiplication of the equation by the (n w ) th power of A reduces it to 
 
 a m A n = , a m = , 
 
 n 
 
 that is, the w th power of A disappears from the equation, or there is no least 
 power of A in the equation, or, more definitely, there is no such equation. 
 
 59. In a group or an algebra which contains no idempotent expression, any 
 expression may be selected as the basis; lid one is preferable which has the 
 greatest number of powers which do not vanish. All the powers of the basis which 
 do not vanish may be adopted as independent units and represented by the 
 letters of the alphabet. 
 
 A nilpotent group or algebra may be said to be of the same order with the number 
 of powers of its basis that do not vanish, provided the basis is selected by the 
 preceding principle. Thus, if the squares of all its expressions vanish, it is of 
 the first order ; if the cubes all vanish and not all the squares, it is of the second 
 order, and so on. 
 
 60. It is obvious that in a nilpotent group whose order equals the number 
 of letters which it contains, all the letters except the basis may be taken as the 
 successive powers of the basis. 
 
 61. In a nilpotent group, every expression, such as A, has some least 
 power that is nilfacient with reference to any other expression, such as B , and 
 which corresponds to what may be called the facient order of B relatively to 
 A ; and in the same way, there is some least power of A which is nilfaciend with 
 reference to B , and which corresponds to the faciend order of B relatively to A. 
 When the facient and faciend orders are treated of irrespective of any especial 
 reference, they must be referred to the base. 
 
 The facient order of a product which does not vanish, is not higher than that of 
 its facient ; and the faciend order is not higher than that of its faciend. 
 
 62. After the selection of the basis of a nilpotent group, some one from 
 among the expressions which are independent of the basis may be selected by the 
 same method by which the basis was itself selected, which, together with all its 
 powers that are independent of tlie basis, may be adopted as new letters ; and again, 
 from the independent expressions which remain, new letters may be selected by the 
 same process, and so on until the alphabet is completed. In making these selections, 
 regard should be had to the factorial orders of the products. 
 
20 PEIRCE : Linear Associative Algebra. 
 
 63. In every nilpotent group, the facient order of any letter which is indepen 
 dent of the basis can be assumed to be as low as the number of letters which are 
 independent of the basis. 
 
 Thus, if the number of letters which are independent of the basis is denoted 
 by n , and if n is the order of the group (and for the present purpose it is suffi 
 cient to regard n as being less than n), it is evident that any expression, A , with 
 its successive products by the powers of the basis i , as high as the w th , and the 
 powers of the basis which do not vanish, cannot all be independent of one 
 another ; so that there must be an equation of the form 
 
 n n 
 
 I o m 
 
 Accordingly, it is easy to see that there is always a value of A l of the form 
 
 which will give 
 
 i m A t = , 
 
 which corresponds to the condition of this section. 
 
 There is a similar condition which holds in every selection of a new letter by the 
 method of the preceding section. 
 
 64. In a nilpotent group, the order of which is less by unity than the number of 
 letters, the letter which is independent of the basis and its powers may be so selected 
 that its product into the basis shall be equal to the highest power of tlw basis which 
 does not vanish, and that its square shall either vanish or shall also be equal to the 
 highest power of the basis that does not vanish. Thus, if the basis is i , and if the 
 order of the algebra is n , and if j is the remaining letter, it is obvious, from 63, 
 that/ might have been assumed such that 
 
 #=r&, 
 
 which gives 
 
 iji = ij* ; 
 
 and therefore, ji = ai n + bj , 
 
 j* = a i n + bj, 
 = ji n + x bji n b n ji b , 
 ji ai n , 
 
 j*i = ajP = = b j* = //, 
 / 2 a i n 
 
PEIRCE : Linear Associative Algebra. 21 
 
 so that if 7 _ 
 
 we have 
 
 j\i l = i?=.j\, 
 
 and it and/! can be substituted for i and/, which conforms to the proposition 
 enunciated. 
 
 It must be observed, however, that the analysis needs correction when the 
 group is of the second order. 
 
 65. In a nilpotent group of the first order, the sign of a product is merely 
 reversed by changing the order of its factors. Thus, if 
 
 A* = B* = (A + B) 2 = 
 it follows by development, that 
 
 (A -\- B} 2 = A z -j- AB -\- BA -f- B? = AB -f- BA = 
 BA = AB, 
 
 which is the proposition enunciated. 
 
 66. In general, in any nilpotent group of the ?i th order, if (A 8 , B*) denotes the 
 sum of all possible products of the form 
 
 A p B q A p B q A p "B q " . . . 
 in which 
 
 2p = * , 2q = t, 
 
 and if 
 
 i , ___ i -i 
 
 it will be found that 
 
 For since (A + xB) n + 1 = 
 
 whatever be the value of x , the multiplier of each power of x must vanish, which 
 
 gives the proposed equation 
 
 (A 8 , B t )=Q. 
 
 67. In the first group of an algebra, having an idempotent basis, all the expres 
 sions except the basis may be assumed to be nilpotent. For, by the same argument 
 as that of 53, any equation between an expression and its successive powers 
 and the basis must involve an equation between another expression which is 
 
22 PEIRCE : Linear Associative Algebra. 
 
 easily defined and its successive powers without including the basis. But it 
 follows from the argument of 57, that such an equation indicates a corres 
 ponding idempotent expression ; whereas it is here assumed that, in accordance 
 with 56, each group has been brought to a form which does not contain any 
 other idempotent expression than the basis. It must be, therefore, that all the 
 other expressions are nilpotent. 
 
 68. No product of expressions in the first group of an algebra having an idem- 
 potent basis, contains a term which is a multiple of the basis. 
 
 For, assume the equation 
 
 AB- xi+ C, 
 
 in which A, B and C are nilpotents of the orders m , n and p, respectively. 
 
 Then, 
 
 = xA m + A m C 
 
 that is, the term xi vanishes from the product AB. 
 
 69. It follows, from the preceding section, that if the idempotent basis were 
 taken away from the first group of which it is the basis, the remaining letters of the 
 first group would constitute by tliemselves a nilpotent algebra. 
 
 Conversely, any nilpotent algebra may be converted into an algebra with an 
 idempotent basis, by the simple annexation of a letter idemfaciend and idemfacient 
 with reference to every other* 
 
 70. However incapable of interpretation the nilfactorial and nilpotent 
 expressions may appear, they are obviously an essential element of the calculus 
 of linear algebras. Unwillingness to accept them has retarded the progress of 
 discovery and the investigation of quantitative algebras. But the idempotent 
 basis seems to be equally essential to actual interpretation. The purely nilpotent 
 algebra may therefore be regarded as an ideal abstraction, which requires the 
 introduction of an idempotent basis, to give it any position in the real universe. 
 In the subsequent investigations, therefore, the purely nilpotent algebras must 
 be regarded as the first steps towards the discovery of algebras of a higher 
 degree resting upon an idempotent basis. 
 
 * That every such algebra must be a pure one is plain, because the algebra (a a ) is so. [C. S. P.] 
 
PEIRCB : Linear Associative Algebra. 23 
 
 71. Sufficient preparation is now made for the 
 
 INVESTIGATION OF SPECIAL ALGEBRAS. 
 
 The following notation will be adopted in these researches. Conformably with 
 9, the letters of the alphabet will be denoted by i, j , k, I, m and n. To 
 these letters will also be respectively assigned the numbers 1 , 2 , 3 , 4 , 5 and 
 6. Moreover, their coefficients in an algebraic sum will be denoted by the 
 letters a, 6, c, d, e and/. Thus, the product of any two letters will be 
 expressed by an algebraic sum, and below each coefficient will be written in 
 order the numbers which are appropriate to the factors. Thus, 
 
 jl = a u i + 6 24 j + C 24 & + ^24 Z + e 24 m + / 24 n , 
 while 
 
 lj = a&i + & 42 j H- c 42 k + c? 42 I + e 42 
 
 In the case of a square, only one number need be written below the coefficient, 
 
 thus 
 
 tf =a 3 i + b s j + c 3 k + d 3 l + e 3 m+f 3 n. 
 
 The investigation simply consists in the determination of the values of the 
 coefficients, corresponding to every variety of linear algebra ; and the resulting 
 products can be arranged in a tabular form which may be called the multipli 
 cation-table of the algebra. Upon this table rests all the peculiarity of the 
 calculus. In each of the algebras, it admits of many transformations, and much 
 corresponding speculation. The basis will be denoted by i . 
 
 72. The distinguishing of the successive cases by the introduction of 
 numbers will explain itself, and is an indispensable protection from omission 
 of important steps in the discussion. 
 
 SINGLE ALGEBRA. 
 
 Since in a single algebra there is only one independent unit, it requires no 
 distinguishing letter. It is also obvious that there can be no -single algebra 
 which is not associative and commutative. Single algebra has, however, two 
 
 cases : 
 
 [1], when its unit is idempotent ; 
 
 [2], when it is nilpotent. 
 [1]. The defining equation of this case is 
 
24 PEIRCE : Linear Associative Algebra. 
 
 This algebra may be called (a-^ and its multiplication table is * 
 
 [2]. The defining equation of this case is 
 
 * 2 = . 
 This algebra may be called (^) and its multiplication table is f 
 
 DOUBLE ALGEBRA. 
 There are two cases of double algebra : 
 
 [1], when it has an idernpotent expression ; 
 [2], when it is nilpotent. 
 
 [1]. The defining equation of this case is 
 
 By 41 and 50, there are two cases : 
 
 [I 2 ], when the other unit belongs to the first group ; 
 [12], when it is of the second group. 
 
 The hypothesis that the other unit belongs to the third group is a virtual 
 repetition of [12]. 
 
 [I 2 ]. The defining equations of this case are 
 
 It follows from 67 and 69, that there is a double algebra derived from (JJ 
 which may be called ( 2 ) , of which the multiplication table is J 
 
 * This algebra may be represented by i A : A in the logic of relatives. See Addenda. [C. S. P.] 
 
 t This algebra takes the form i A : B , in the logic of relatives. [C. S. P.] 
 
 I This algebra may be put in the form i = A : A + B : B , j A : B . [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 25 
 
 [12]. The defining equations of this case are, by 41, 
 
 ij =j , ji = ; 
 whence, by 46, 
 
 /*=0. 
 
 A double algebra is thus formed, which may be called (b z ), of which the multi 
 plication table is * 
 
 * j 
 
 [2]. The defining equation of this case is 
 
 i n = 0, 
 in wliich n is the least power of i which vanishes. There are two cases : 
 
 [21], when n 3 ; 
 [2 s ], when n =2. 
 
 [21]. The defining equation of this case is 
 
 ?=0, 
 
 and by 60, 
 
 *=/- 
 
 Tliis gives a double algebra which may be called (V 2 ), its multiplication 
 table being f 
 
 * This algebra may be put in the form i = A : A , j. A : B . [C. S. P. ] 
 t In relative form, i A:B+B:C. j A\C, [C. S. P.] 
 
26 
 
 PEIRCE : Linear Associative Algebra. 
 
 [2 2 ]. The defining equations of this case are 
 
 and it follows from 64 and 65 that 
 
 so that there is no pure algebra in this case. 
 
 TRIPLE ALGEBRA. 
 
 There are two cases : 
 
 [1] , when there is an idempotent basis ; 
 [2], when the basis is nilpotent. 
 
 [1]. The defining equation of this case is 
 
 * 
 
 There are, by 41, 50 and 51, three cases : 
 
 [I 2 ], when./ and /,: are both in the first group ; 
 
 [12], whenj" is in the first, and 7^ in the second group ; 
 
 [13], when./ is in the second, and Je in the third group. 
 
 The case of j being in the first, and & in the third group, is a virtual 
 repetition of [12]. 
 
 [I 2 ]. The defining equations of this case are 
 
 *This case takes the form i A : B, 3 C : D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 27 
 
 It follows from 67 and 69, that the only algebra of this case may be derived 
 from (c 8 ) ; it may be called (a 3 ), and its multiplication table is * 
 
 
 
 [12]. The defining equations of this case are 
 
 ji ij j , ik = 7f. , Id = ; 
 whence, by 46 and 67, 
 
 y* = /^ = Zy = 0, J7s = cuk, 
 j*7c = c 23t /& = c| 3 fc = c 23 =// 
 
 and there is no pure algebra in this case.f 
 
 [13]. The defining equations of this case are 
 
 ij = j, Id = 7c , ji i7c = ; 
 whence, by 46, 
 
 j 9 = /.: 2 = 7ej = , jk = a. 23 i , 
 
 jig = o = a a j = 23 =y/i-, 
 
 and there is no pure algebra in this case.J 
 
 [2]. The defining equation of this case is 
 
 in which n is the lowest power of i that vanishes. 
 
 There are three cases : 
 
 [21], when n = 4 ; 
 
 [2 1 ], when n 3; 
 [23], when w = 2. 
 
 *In relative form, i A :A+B:B-\- C : (7, j= A :B + B:C, Tc A:C. [C. S. P.] 
 t That is to say, i and j by themselves form the algebra a 2 , and i and k by themselves constitute the 
 algebra & 2 , while the products of j and k vanish. Thus, the three letters are not indissolubly bound 
 together into one algebra. In relative form, this case is, i A:A-\-B:B,jA:B, k A:C. 
 
 [C. S. P.] 
 
 I In relative form, i=A:A+D:D,j=.A:B,*=C:D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 [21]. The defining equation of this case is 
 
 and by 60 
 
 i 4 = , 
 
 This gives a triple algebra which may be called (b 3 ), the multiplication table 
 being * 
 
 Jc 
 
 
 
 
 
 
 
 
 
 [2 2 ]. The defining equation of this case is 
 
 i 3 = , 
 and by 59 and 64, observing the exception, 
 
 ? ^*- ^ o /" * (I 
 
 There is no pure algebra when b 31 vanishes,*)- and there are two 
 
 [2 2 1], when b 3 does not vanish ; 
 [2 3 ], when b 3 vanishes. 
 
 [2 2 1]. The defining equation of this case can, without loss of generality, 
 be reduced to 
 
 This gives a triple algebra which may be called (c 3 ), the multiplication table 
 being J 
 
 cases : 
 
 Inrel a tiveform,i = A:B + B:C+C:D,j = A:C+B:D, Je = A:D. [C. S P] 
 This case takes the relative form, i=A:B + B:C, j = A:C, Jc = b 3 A : D + D :C. 1C S P 1 
 lIvtetiv e foi-m,i = A:B+B:C,j = A:C,lc = a.A:B + A:D + D-C. [C S P ] 
 
PEIRCE : Linear Associative Algebra. 
 (c 3 ) i j Jc 
 
 29 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 aj 
 
 
 
 j 
 
 An interesting special example of this case is afforded by a . = 2 , when 
 
 (k + i) i = j 
 (* ?= 0, 
 
 so that 7j + i might be substituted for It, and in this form, the multiplication 
 table of this algebra, which may be called (c 3 ), is * 
 
 j 
 
 
 
 j 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 * In relative form, i = A:B + B:C, jA:C, k = A :B+B:C+A :D+D:C. 
 When a = + 2 , the algebra equally takes the form (c 3 ) , on substituting k i for k. On the other 
 hand, provided a is neither 2 nor 2 , the algebra may be put in the form 
 
 To effect the transformation, we write a 6 and substitute i + bk and i + ^ k for i and &, and 
 V ~ 6/ J for<? " Thus the al S ebra ( c a) has two distinct and intransmutable species, (c s ) and (c 3 ). [C. S. P.] 
 
30 
 
 PEIRCE : Linear Associative Algebra. 
 
 [2 3 ]. The defining equation of this case is 
 
 P = , 
 
 and l sl may be reduced to unity without loss of generality, giving a triple 
 algebra which may be called (cZ 3 ), the multiplication table being 
 
 i j & 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 Iii this case 
 
 = 0- 
 
 (i-k? =0, 
 
 so that i 7c may be substituted for i, and in this form the multiplication 
 table is * 
 
 MO * J * 
 
 
 
 
 
 
 
 
 
 
 
 
 
 .; 
 
 
 
 
 
 [23]. The defining equations of this case are 
 
 .-2 . 2 13, n 
 
 I J Iv U , 
 
 and by the principles of 63 and 65, it may be assumed that 
 
 ij = ji = ilf, = Id = , 
 
 *In relative form, i B\C, j A:C, k = A:B. This is the algebra of alio-relations in its 
 typical form. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 31 
 
 We thus get a triple algebra which may be called (e 3 ), its multiplication table 
 being* 
 
 M i J & 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 i 
 
 
 
 QUADRUPLE ALGEBRA. 
 There are two cases : 
 
 [1], when there is an idempotent basis ; 
 [2], when the base is nilpotent. 
 
 [1]. The defining equation of this case is 
 
 2 
 
 There are six cases : 
 
 [I 2 ], when/, &, and I, are all in the first group ; 
 
 [12], when/ and It, are in the first, and I in the second group ; 
 
 [13], when/ is in the first, and It and I in the second group ; 
 
 [14], when/ is in the first, It, in the second, and I in the third group ; 
 
 [15], when/ and le are in the second, and I in the third group ; 
 
 [16], when /is in the second, It, in the third, and I in the fourth group. 
 
 The other cases are excluded by 50 and 51, or are obviously virtual repeti 
 tions of those which are given. 
 
 [I 2 ]. The defining equations of this case are 
 
 and from 60 and 69, the algebras (6 3 ), (c 3 ), (c/ 3 ), and (e s ), give quadruple 
 algebras which may be named respectively (# 4 ), (6 4 ), (c 4 ), and (W 4 ), their 
 multiplication tables being 
 
 * In relative form, i = A : D , j= A : B C:D. 
 numbers. [C. S. P.] 
 
 A:C+B:D. This is the algebra of alternate 
 
32 
 
 PEIRCE : Linear Associative Algebra. 
 
 Tf 
 
 i 
 
 j 
 
 It 
 
 i 
 
 j 
 
 k 
 
 1 
 
 
 
 If 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 (c 4 ) i j Tf I 
 
 i 
 
 j 
 
 Tf 
 
 1 
 
 j 
 
 Tf 
 
 
 
 
 
 Tf 
 
 
 
 
 
 
 
 1 
 
 Tf 
 
 
 
 
 
 i 
 
 j 
 
 If 
 
 1 
 
 j 
 
 Tf 
 
 
 
 
 
 Tf 
 
 
 
 
 
 
 
 1 
 
 ale 
 
 
 
 Tf 
 
 i 
 
 j 
 
 Tf 
 
 j 
 
 j 
 
 
 
 
 
 
 
 Tf 
 
 
 
 
 
 > 
 
 1 
 
 
 
 J 
 
 * 
 
 
 
 The special case (c 3 ) gives a corresponding special case of (Z> 4 ), which may 
 be called (& 4 ), of which the multiplication table is 
 
 i 
 
 j 
 
 & 
 
 1 
 
 j 
 
 Tf 
 
 
 
 Tf 
 
 Tf 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
PEIRCE : Linear Associative Algebra. 
 
 33 
 
 The second form of (d 3 ] gives a corresponding second form of (c 4 ). of which 
 the multiplication table is 
 
 (c 4 ) i j Jc I 
 
 I 
 
 i 
 
 j 
 
 7c 
 
 I 
 
 j 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 I 
 
 * 
 
 
 
 
 
 [12]. The defining equations of this case are 
 
 if = ji =. j, ik = Id = It , il ==. I, li =. , 
 and it follows from 67 and 69, that (c 2 ) gives 
 
 and from 46, 
 whence 
 
 = Ik = Z 2 = , jl = 
 
 kl = 
 
 and there is no pure algebra in this case.* 
 
 [13]. The defining equations of this case are 
 
 ij =ji =j , Hi = If, il = I, Id = li = , 
 which give by 46 and 67 
 
 and it may be assumed that 
 
 jk = I, whence jl = . 
 
 This gives a quadruple algebra which may be called (e 4 ), its multiplication 
 table being f 
 
 *In relative form, i A : A + B :B + C : C+ D : JD, j = A:B + B:C, k = A:C, l=D:C. 
 [C. S. P.] 
 
 t In relative form, i= A : A + B : B , j A : B , k B : C, I = A : C. [C. S. P.] 
 VOL. IV. 
 
34 
 
 PEIRCE : Linear Associative Algebra. 
 (e 4 ) i j k I 
 
 k 
 
 I 
 
 i 
 
 .; 
 
 k 
 
 I 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [14]. The defining equations of this case are 
 
 if = ji = j, ik = 7c, li = I, Id = il = ; 
 which give, by 46 and 67, 
 
 =j* =jl = kj 7t* = lie = P, 
 jk = c^k, 7j = d^l, Id a u i + 
 =j z 7c = c z3 j7t = cffl = c 2S =j7c, 
 = If = d^j = dU = d 4Z = Ij, 
 
 =kl=0 
 
 and b 3i cannot be permitted to vanish,* so that it does not lessen the generality 
 to assume 
 
 7d = j. 
 
 This gives a quadruple algebra which may be called (/ 4 ), its multiplication 
 table being f 
 
 i 
 
 j 
 
 k 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ; 
 
 I 
 
 
 
 
 
 
 
 * For then the algebra would split up into three double algebras. [C. S. P.] 
 tin relative form, i = A : A + B :B , j= A :B , k = A:C, l~C:B. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 35 
 
 [15]. The defining equations of this case are 
 
 # y> ik k, H = I, ji = Id = il = 
 
 which give, by 46, 
 
 =j*jk = lj = V = lj = lk = I*, 
 
 jl = a u i , Id = a^i , 
 
 =r /// = a 3 Jc = # 34 = kl , 
 
 and there is no pure algebra in this case.* 
 
 [16]. The defining equations of this case are 
 
 *y = y &* = * > y* = ^ = ^ = n = o , 
 
 which give, by 46, 
 
 o=y = x*==^ 
 
 J7c = a 23 i, jl=b H j, 7y = dnl, lk cdc, I* = JJ , 
 
 llej = dftdj, = c 43 4 3 ?, pie = c&k = c 43 djc, a Z3 = l^d^ , 
 = a 23 (c 43 5 24 ) = Z> 24 (5 24 ^ 24 ) = d 32 (6 24 c^ 4 ) = d& (c 43 c? 4 ) =c 43 (c 43 
 
 There are two cases: 
 
 [161], when 4,2 d e s not vanish; 
 [162], when d 3Z vanishes. 
 
 [161]. The defining equation of this case can be reduced to 
 
 <4 = 1 , 
 which gives 
 
 43 
 
 There are two cases : 
 
 [161 8 ], when d^ does not vanish ; 
 [1612], when d vanishes. 
 
 [161 2 ]. The defining equation of this case can be reduced to 
 
 which gives 
 
 * In relative form, i=A:A,j = A:B, k = A:C, l = D:A. There are three double algebras of 
 theform(6 2 ). [C. S. P.] 
 
36 
 
 PEIRCE : Linear Associative Algebra. 
 
 and there is a quadruple algebra which may be called (<7 4 ), its multiplication 
 table being 
 
 i j If I 
 
 It 
 
 I 
 
 i 
 
 j 
 
 
 
 
 
 
 
 
 
 i 
 
 j 
 
 7c 
 
 I 
 
 
 
 
 
 
 
 
 
 Jc 
 
 I 
 
 This is a form of quaternions* 
 
 [1612]. The defining equation of this case is 
 
 which gives 
 
 * In relative form, i A: A, j A:B , k B:A, l = B:B. This algebra exhibits the general 
 system of relationship of individual relatives, as is shown in my paper in the ninth volume of the 
 Memoirs of the American Academy of Arts and Sciences. In a space of four dimensions, a vector may 
 be determined by means of its rectangular projections on two planes such that every line in the one is 
 perpendicular to every line in the other. Call these planes the .A-plane and the B-plane, and let v be 
 any vector. Then, iv is the projection of v upon the 4-plane, and Iv is its projection upon the B-plane. 
 Let each direction in the A -plane be considered as to correspond to a direction in the .B-plane in such a 
 way that the angle between two directions in the A -plane is equal to the angle between the correspond 
 ing directions in the J5-plane. Then, jv is that vector in the 4-plane which corresponds to the projection 
 of v upon the B-plane, and lev is that vector in the 5-plane which corresponds to the projection of v upon 
 the 4-plane. 
 
 Professor Peirce showed that we may take i 1 , j\ , fc : , as three such mutually perpendicular vectors 
 
 in ordinary space, that i ~ (1 ji,} , ; = A (^ jfcj), *=i{ j\ jfc,), 1 = ^(1^. jj). [See, also, 
 
 Spottiswoode, Proceedings of the London Mathematical Society, iv, 156. Cayley", in his Memoir on the 
 Theory of Matrices (1858), had shown how a quaternion may be represented by a dual matrix.] Thus 
 i, ji k, I, have all zero tensors, and j and k are vectors. In the general expression of the algebra, 
 q xi -\-yj-\-zk-\-wl, if x + w = l and yz x a? 2 , we have g 2 =g; if x = w = /yz, then 
 q 2 = . The expression i + I represents scalar unity, since it is the universal idemfactor. We have, also, 
 
 Tq = 
 
 yz (i 
 
 The resemblance of the multiplication table of this algebra to the symbolical table of 46 merits 
 attention. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 37 
 
 and there is a quadruple algebra which may be called (7i 4 ), its multiplication 
 table being* 
 
 i j Jc I 
 
 If- 
 
 i 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 If 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [162]. The defining equation of this case is 
 
 which gives 
 
 and there can be no pure algebra for it.f 
 
 [2]. The defining equation of this case is 
 
 i n = . 
 
 There are four cases : 
 
 [21], when n = 5; 
 [2 2 ] , when n = 4 ; 
 [23], when n = 3 ; 
 [24], when w = 2. 
 
 [21]. The defining equation of this case is 
 
 and by 60, #=j, i^ 1^ i*=l. 
 
 This gives a quadruple algebra which may be called (i 4 ), its multiplication table 
 being J 
 
 * In relative form, i = A:A, j A :B, k C: A, l C:B. [C. S. P.] 
 
 tin this case, i A:A, l = d 4 (B:B+C:C),j=A:Boi: = A:D,k=:C:Aor = E:A. [C. S. P.] 
 Jin relative form, i=.A:B + B:C+C:D-}-D:E, j A :C+ B :D + C : E , k= A :D-\- B:E, 
 l A:E. [C. S. P.] 
 
38 
 
 PEIRCE : Linear Associative Algebra. 
 (i 4 ) i j Jc I 
 
 J 
 
 j 
 
 Jc 
 
 I 
 
 
 
 Jc 
 
 I 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 [2 8 ]. The defining equation of this case is 
 
 ; 4 = o, 
 
 and by 59, **=/, i 3 = k. 
 
 There are then, by 64, two quadruple algebras, which may be called (j 4 ) and 
 , their multiplication tables being * 
 
 4 ) i j Je I and (Jc 4 ) i j Jc I 
 
 I 
 
 j 
 
 Jc 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 Jc 
 
 Jc 
 
 j 
 
 Jc 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 [23]. The defining equation of this case is 
 
 i 3 = , 
 and by 59 
 
 */ 
 
 and it may be assumed from the principle of 63 that 
 which gives 
 
 * In either of these algebras, i A : B + B : C-f C:D, j=A:C+B:D, k= A : D; and 
 in (j\) l = A:E + E:D + A,:C, while in (fcj l = A:C. [C. S. P.] 
 
PEIRCE: Linear Associative Algebra. 39 
 
 There are two cases : [231], when il = k 
 
 [232], when il=Q. 
 [231]. The defining equation of this case is 
 
 U = k , 
 which gives 
 
 y?=w=a?=o, 
 
 Id a 3l i -f 6 3i y 4- c 3l k 4- d sl l , 
 
 Q = iki=:a 31 j+d 3l k, a 31 =0, d 3l = Q, Id = b Bl j + c 31 k . 
 So, because it* = , F = b 3 J 4- c 3 & , 
 
 and because iU = , fc? = 6 34 y + c 34 & , kj = Idi = cjd = b sl c 3l j + c|^ . 
 = kj i = clJd , c 3l = Q = kj, 
 
 Hi = Id = b 31 j , li = b 3l i + buj + %& , y = & = (&8i + 
 = F = c 3 7<^ = c 3 , t tt = ^ = b 3 j , 7A - J 3 i + 5 43 y + c 
 
 But M contains no term in I , so that d 4 = . 
 M = il* = a j, b 3 ,= a,, c 34 = 0, 
 
 = Z 3 = M + CAJ , ^34 = 4 = = U , I* = bj + C 4 
 
 M =tf = bnjl =0, = M = 6 81 fe = IU = b 3l = Id = 
 U = 5 41 y + c 41 / 6 . ; fa in _ o 
 
 There are two cases : 
 
 [2 3 1 2 ], when c 41 does not vanish ; 
 [2312], when c 41 vanishes. 
 
 [2 3 1 2 ]. The defining formula of this case is 
 
 c , 
 and if ^ is determined by the equation 
 
 we have 
 
 so that I 4- ^?i and A; + pj may be substituted respectively for /and & , which is 
 the same as to make 
 
40 
 
 PEIRCE : Linear Associative Algebra. 
 
 and there are two cases : 
 
 [2 3 1 3 ], when c 4 * does not vanish ; 
 [231 2 2], when c 4 vanishes. 
 
 [22 1 3 ]. The defining equation of this case can be reduced to 
 
 P = 7c. 
 This gives a quadruple algebra which may be called (7 4 ), its multiplication 
 
 table being f 
 
 (Z 4 ) i j Jc I 
 
 i 
 
 j 
 
 
 
 
 
 k 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 I 
 
 ck 
 
 
 
 
 
 1 
 
 [231 2 2], The defining equation of this case is 
 
 * 7. e. the new c 4 , or what has been written c t +pc 41 . In all cases, when new letters of the alpha 
 bet of the algebra are substituted, the coefficients change with them. [C. S. P.] 
 
 fWhen 6 = 0, c rr 1 , we have Z (i I) = (i I) I = ; so that by the substitution of * I for i , the 
 algebra is broken up into two of the form (c 2 ) . Whenb=.0, c : 1 , on substituting i l =i I, 
 j 1 =jck,k 1 = (c l} 2 k, *! = (<! 1)Z, we have i\ =j\ , i 1 Z 1 = 0, I l i 1 ~l 2 1 =k l ; so that the 
 algebra reduces to (r 4 ) . When & =. 1 , c =. , on putting i 1 i I , j\ j k , we have if = ij = , 
 Zij jj , I 2 k ; so that the algebra reduces to (g 4 ) . When &r= 1 , c 4= , on putting ^ = s/c- 1 (i I) , 
 jj J+ (c 1) fc, we have if = Z 2 =A;, ij = 0, Zij ^ ; so that the algebra reduces to (p t ). 
 When 6(6 1) (&c + & 1) : 0, on putting ij = (1 b) 6t (1 b) Z, J>i = (1 &) 2 (1 & &c) ^, 
 fci =b 2 (1 b) (1 b be) jf -b(l b)(l & c + c 2 b)A;, l l b (1 b) i bcZ , we get the multipli 
 cation table of (o 4 ). When6(b 1) ^i , 6c + b = l ; on putting t 1 =b(t Z), j\=b 2 (l b}jb 2 cJc , 
 k l =6(1 b c)fc, Z x bi Z, we get the following multiplication table, which may replace that in 
 
 the text : 
 
 (Z 4 ) i j Tc I 
 
 i 
 
 3 
 k 
 I 
 
 J 
 
 
 
 
 
 / 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fc 
 
 
 
 
 
 In relative form, i -A :B+B: C+ A:D,j=A:C, k = A:E, e=A:B + D:E. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 There are two cases: 
 
 [231*21], when 5 41 does not vanish; 
 
 [231 2 2 2 ], when 6 41 vanishes. 
 [231 2 21]. The denning formula of this case is 
 
 41 
 
 There are two cases : 
 
 [23P21 2 ], when c 41 -f 1 does not vanish; 
 [23P212], when c 41 + 1 vanishes. 
 
 [231 2 21 2 ]. The defining formula of this case is 
 
 C41 + 1 t 
 
 so that 
 
 c 41 -f 1 
 
 c 41 
 
 7 _ 
 
 ~ 
 
 41 
 
 so that the substitution of 4 ^ , " , 4lJ , " C41 " , and - . respectivelv, for 
 
 41 + 1 C 41 -f- 1 C 41 -f 1 
 
 i, y, and k, is the same as to assume 
 
 c 41 = o, z> 41 =y, 
 
 which reduces this case to [2312]. 
 
 [231 2 212], The denning equation of this is easily reduced to 
 
 This gives a quadruple algebra which may be called (??? 4 ), its multiplication 
 table being 
 
 
 
 ./ 
 
 
 
 
 
 & 
 
 .; 
 
 
 
 
 
 
 
 
 
 A- 
 
 
 
 
 
 
 
 
 
 z 
 
 ./-* 
 
 
 
 
 
 
 
42 
 
 PEIRCE : Linear Associative Algebra. 
 
 The substitution of i I and/ It, respectively, for i and/ transforms this 
 algebra into one of which the multiplication table is * 
 
 (w 4 ) i j Jc I 
 
 
 
 
 
 
 
 It 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 [231 2 2 2 ]. The defining equation of this case is 
 
 This gives a quadruple algebra which may be called (%), its multiplication 
 table being f 
 
 W i j -k. .1 
 
 j 
 
 
 
 
 
 & 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ck 
 
 
 
 
 
 
 
 [2312]. The defining equation of this case is 
 which gives 
 so that the substitution of I b^i for I passes this case virtually into [232], 
 
 *i = A:B+C:D, j = B:D, k = A:C, l B:C. [C. S. P.] 
 
 t In relative form, i = A:B + B :C+D :E, j=A :C, k= A : E , l B:E+cA:D. When c 
 the algebra reduces to (# 4 ). [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 43 
 
 [232]. The defining equation of this case is 
 
 il , 
 and it may be assumed that 
 
 Td = , 
 Q=jl = kj = iW tfi = ild = Hi = ilk = M i7 
 
 U = 6 <?& <Z 
 
 There are two cases : 
 
 [2321], when c 41 does not vanish; 
 [232*], when c 41 vanishes. 
 [2321], The defining equation of this case is easily reduced to 
 
 U = / , 
 which gives = lik = 1? = HI = Id 
 
 = Z% = dfk = d\lk = d, , Ik = a J = Pi , 
 
 Z 2 = a 4 i + 6 4 y + c/- , 
 
 = / 3 = ajc -f c 4 a 4 y = a 4 = Ik . 
 
 There are two cases : 
 
 [2321 2 ], when c 4 does not vanish ; 
 [23212], when c 4 vanishes. 
 [2321*]. The defining equation of this case can be reduced to 
 
 which gives a quadruple algebra which may be called (o 4 ), its multiplication 
 table being* 
 
 (oj i j * Z 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 ty+ft 
 
 *In relative form, i A :E +E : D + B : C, j A : D , k=A:C, l A:B + B:C+bB:D. 
 Wlien &rz , this algebra reduces to (r 4 ). When 6 = 1 , the substitution of i I for Z reduces it to [L). 
 [C. S. P.] 
 
44 PEIRCE : Linear Associative Algebra. 
 
 [23212]. The defining equation of this case is 
 
 There are two cases : 
 
 [232121], when Z> 4 does not vanish ; 
 
 [23212 2 ], when ?> 4 vanishes. 
 [232121]. The defining equation of this case can be reduced to 
 
 This gives a quadruple algebra which may be called (^ 4 ), its multiplication 
 
 table being* 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 * 
 
 [23212 2 ]. The defining equation of this case is 
 
 / 2 = 0. 
 
 This gives a quadruple algebra which may be called (q), its multiplication 
 table being f 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 & 
 
 
 
 
 
 
 
 * In relative form, t = .4 :B + B :D+ C:E, j = A:D, k = A:E, l = A:C+C:D. [C. S. P.] 
 t In relative form, i=.A:C+C:D,j A:D, k = B:D, I ~ B : C. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 45 
 
 [232 2 ]. The defining equation of this case is 
 
 ft = b 4 ij 
 and we have 
 
 ^ = 1> 3 j + c 3 7f + d 3 l 
 U b + cc + dl 
 
 P = bj + cjc + dj 
 
 so that there can be no pure algebra in this case if 5 41 vanishes, * and it may be 
 assumed without loss of generality that 
 
 U=j. 
 There are two cases : 
 
 [232 2 1], when d 3 does not vanish ; 
 
 [232 3 ], when d 3 vanishes. 
 [232 2 1]. The defining equation of this case can be reduced to 
 
 which gives 
 
 = 7j 3 = M=m=J*l= 
 
 and there is no pure algebra in this case.f 
 
 [232 3 ]. The defining equation of this case is 
 
 4 = 0, 
 which gives = 1 = c 3 tf =. c 3 , k z = b 3 j , 
 
 = m C 34 /^ H- d*Jd = f?34 = 
 
 = 1L* G^ H- djd d^ = 
 
 There are two cases : 
 
 [232 3 1], when b 3 does not vanish; 
 
 [232 4 ], when b 3 vanishes. 
 [232 3 1]. The defining equation of this case can be reduced to 
 
 which gives 
 
 this case, y, k and Z, might form any one of the algebras (& 3 ), (c 3 ), (d 3 ) or (e 3 ). [C. S. P.] 
 t The case is impossible because fci = and k 2 i j. [C. S. P.] 
 
46 
 
 PEIRCE : Linear Associative Algebra. 
 
 so that / b 3i j can be substituted for I without loss of generality, which is the 
 same as to assume 
 
 and this gives 
 
 = Z 8 = dp = d A = cjk = c,b 43 = Pk = c 4 , 
 
 so that there is no pure algebra in this case.* 
 
 [232 4 ]. The denning equation of this case is 
 
 which gives 
 
 #==0, 
 
 o x = # = **_=%/=*, 
 
 = Id* =: c 3 Jd =. c u , Id =. bftj , 
 
 and there can be no pure algebra if c 4 vanishes, so that it may be assumed, 
 without loss of generality, that 
 
 ? = *, 
 
 which gives 
 
 = l 3 = Uc = Jd. 
 
 This gives a quadruple algebra which may be called (/- 4 ), its multiplication 
 table being f 
 
 (n) * j h i 
 
 j 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 A- 
 
 [24]. The defining equations of this case are 
 
 Substituting i I for i , this case is, i = B : D , j = A : D , k = A : C + C : D , I A : B [C. S. P. 1 
 
 :E. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 47 
 
 and it may be assumed, from 63 and 65, that 
 
 ij = It = ji , il = U = , 
 which give 
 
 = ilt = Id = jit = // = Id = 770 , 
 
 so that there is no pure algebra in this case.* 
 
 QUINTUPLE ALGEBRA. 
 There are two cases : 
 
 [1], when there is an idempotent basis ; 
 [2], when the algebra is riilpotent. 
 
 [1]. The defining equation of this case is 
 
 There are eleven cases : 
 
 [I 2 ], when/, It, I and m are all in the first group ; 
 
 [12], when/, It and I are in the first, and m in the second group ; 
 
 [13], when/ and It are in the first, and I and m in the second group ; 
 
 [14], when/ and It are in the first, I in the second, and m in the third group ; 
 
 [15], when/ is in the first, and It, I and m in the second group ; 
 
 [16], when/ is in the first, It and I in the second, and m in the third group ; 
 
 [17], when/ is in the first, It in the second, I in the third, and m in the fourth 
 
 group ; 
 
 [18], when /, It and I are in the second, and m in the third group ; 
 [19], when/ and /.; are in the second, and I and m in the third group ; 
 [10 1 ], when/ and k are in the second, I in the third, and m in the fourth group ; 
 [II 1 ], when/ is in the second, It in the third, and I and m in the fourth group. 
 
 [I 2 ]. The defining equations of this case are 
 
 */ /^ y, ik Id = It , il li l, im mi m. 
 
 The algebras deduced by 69 from algebras (i 4 ) to (r 4 ) may be named (a 5 ) to (/ 5 ), 
 and their multiplication tables are respectively 
 
 i = A:C+B:E,j=A :+ C:E+ cD \E, k = A :E, I A :D + cB : E. [C. S. P.] 
 
48 
 
 i j 
 
 3 
 
 PEiRCft : Linear Associative Algebra. 
 k I m (/> 5 ) * j k I m 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 j 
 
 k 
 
 I 
 
 m 
 
 
 
 k 
 
 I 
 
 m 
 
 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 (c 5 ) i j ~k I m 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 j 
 
 k 
 
 I 
 
 
 
 
 
 ~k 
 
 I 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 / 
 
 
 
 
 
 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 Jc 1 
 
 
 
 
 
 
 
 m 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 j 
 
 k 
 
 I 
 
 
 
 
 
 k 
 
 I 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 Z 
 
 
 
 
 
 I 
 
 or (<? 5 ) i j 
 
 k 
 I 
 
 m 
 
 i j k I m 
 
 i 
 
 j 
 
 ~k 
 
 / 
 
 m 
 
 j 
 
 Je 
 
 
 
 
 
 I 
 
 Tc 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 aJc 
 +U 
 
 
 
 
 
 I 
 
 I m 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 j 
 
 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 k 
 
 
 
 
 
 
 
 
 1 
 
 
PEIRCE : Linear Associative Algebra. 
 
 49 
 
 m 
 
 i 
 
 j 
 
 Jc 
 
 1 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 1 
 
 Jc 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 m 
 
 al 
 
 
 
 
 
 
 
 Jc I m 
 
 i 
 
 i 
 
 j 
 
 Jc 
 
 1 
 
 m 
 
 j 
 
 j 
 
 Jc 
 
 
 
 
 
 
 
 Jc 
 
 Jc 
 
 
 
 
 
 
 
 
 
 1 
 
 1 
 
 
 
 
 
 
 
 
 
 m 
 
 m 
 
 1 
 
 
 
 
 
 Jc 
 
 J 
 
 Jc 
 
 I 
 
 m 
 
 O s) i j Jc I m 
 
 i 
 
 j 
 
 i 
 Jc 
 
 1 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 
 
 
 
 
 
 m 
 
 Jc 
 
 
 
 
 
 1 
 
 Jc 
 
 m 
 
 i 
 
 j 
 
 Jc 
 
 1 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 
 
 * 
 
 
 
 
 
 
 
 
 
 L 
 
 6 
 
 
 
 
 
 
 
 n. 
 
 1 
 
 
 
 
 
 1+aJc 
 
 Jc I 
 
 m 
 
 i 
 
 j 
 
 Jc 
 
 I 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 
 
 K 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 m 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
50 PEIRCE : Linear Associative Algebra. 
 
 [12]. The defining equations of this case are 
 
 ij = ji =j, ik = ki = k , il = li = I , tm = ?ri , m^ , 
 
 which give, by 46 , 
 
 = mj = mJc = ml = m?, 
 
 and if A is any expression belonging to the first group, but not involving i , we 
 
 have the form 
 
 Am am , 
 
 and by 67, A is nilpotent, so that there is some power n which gives 
 
 = A n = A n m = aA n ~ l m = a n m = a Am , 
 =jm = km =1 Im ; 
 
 and there is no pure algebra in this case.* 
 
 [13]. The defining equations of this case are 
 
 ij ji /, Ht ki = k, il =-1, im = m , li = mi = , 
 which give, by 46, 
 
 = Ij = Ik = P = Im = mj = mk = ml = m* ; 
 and it may be assumed from (c/ 3 ), by 69, that 
 
 / = , y 3 = o. 
 
 It may also be assumed that 
 
 jl = m , whence f kl= jm = . 
 
 We thus obtain a quintuple algebra which may be called (7r 5 ), its multiplication 
 table being this : J 
 
 *In fact i and m, by themselves, form the algebra (& 2 ), while i, j, fc, Z, by themselves form 
 one of the algebras (aj, (& 4 ), (c 4 ), (d 4 ), the products of m withy, k and I vanishing. [C. S. P.] 
 
 fThisis proved as follows: = J 3 lj 2 m = d 2 -.jl + e, -,jm d 25 e 25 l-{- (d 25 + e 2 2 5 )m. Thus d 2S e 25 
 = 0; ord 25 =:0, e 25 = 0, jm U Q. [C. S. P.] 
 
 :C, k A:C,l B\D, m A:D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 i j Jc I m 
 
 51 
 
 i 
 
 i 
 
 Jc 
 
 I 
 
 m 
 
 j 
 
 Jc 
 
 
 
 m 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [14]. The defining equations of this case are 
 
 ij = ji / , ik = ki = k, il = I , mi = m , li = im = , 
 
 which give, by 46, 
 
 = jm = km = Jj=z lit = P = ml == m?. 
 
 It may be assumed from 69 and ( 3 ) that 
 
 /=*, y = o, 
 
 whence 
 
 j7 = /*;/ = wy = w/^ =jlm = a 45 y H- 6 45 7^ =: a 45 = & 45 . ??rz = c^Jc , 
 
 and there is no pure algebra in this case.* 
 
 [15]. The defining equations of this case are 
 
 ij = ji = j , iJc = k , il = I , im = m , ld = li-=. mi = , 
 which give, by 46 and 67, 
 
 =y 2 = lg =J<?=Jd = Jem = Ij Ik = P = Im = mj = mk = ml = m?. 
 It may be assumed that jlc = I, jm = ,f 
 
 whence, jl=Q, 
 
 and there is no pure algebra in this case.J 
 
 *iA: A-\-B:B + C:C, j A :B + B:C, k A:C, l A:D. m = cD:C. [C. S. P.] 
 
 t We cannot suppose j k = fc , because j-k = Q. We may, therefore, put I for jk. Then.;7 = 0. Then, 
 
 Q-=.j 2 m =^c 25 e 25 k-{- (d 25 e 2 5 + c 25 ) Z + e 2 ^m . It follows that jwi = d 2 s^ an( i substituting m d 25 fc for 
 
 m , we have jm = . The algebra thus separates into (& 2 ) and (e 4 ). [C. S. P.] 
 :,y--^l:5, k = B:C, l=A:C, m = A:D. [C. S. P.] 
 
52 PEIRCE : Linear Associative Algebra. 
 
 [16]. The defining equations of this case are 
 
 *)* =y* y ^ & , il = l, mi = 
 which give, by 46 and 67, 
 
 = li = im = 
 
 = =m = g = = 
 7cm = a 35 i 
 
 and it may be assumed that 
 
 = a^i -f 
 
 and cZ 23 cannot vanish in the case of a pure algebra,* so that it is no loss of 
 generality to assume 
 
 ... . /*= . 
 
 which gives 
 
 There are two cases : 
 
 [161], when 35 does not vanish ; 
 [162], when 35 vanishes. 
 [161]. The defining equation of this case can be reduced to 
 
 %> = 1 , 
 which gives lm=.j, 7cm = i + & 35 y, 
 
 and i + 6 35 y can be substituted for i, and this gives a quintuple algebra which 
 may be called (/ 5 ), of which the multiplication table is 
 
 ft) i j Ik I m 
 
 m 
 
 < 
 
 j 
 
 7e 
 
 I 
 
 
 
 i 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 j 
 
 m 
 
 
 
 
 
 
 
 b 35 l. Hence a, 5 = and either c* 2 8 or 6 3 5 = , 
 and m either case there is no pure algebra. The two algebras (l s ) and (m 5 ) are incorrect, as may be seen 
 by comparing k . mk with km . k . [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 [162], The defining equation of this case is 
 
 35 = , 
 
 b 35 j, m 0; 
 
 53 
 
 which gives 
 
 and 6 35 cannot vanish in the case of a pure algebra, so that it is no loss of 
 generality to assume 
 
 km = j . 
 
 This gives a quintuple algebra which may be called (m 5 ), of which the multipli 
 cation table is 
 
 * j & 
 
 m 
 
 i 
 
 j 
 
 * 
 
 I 
 
 
 
 j 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 [17]. The denning equations of this case are 
 
 *y =y* y & = & , u = ? , 
 
 which give, by 46 and 67, 
 
 = 17 = ^ = M* = o , 
 
 = 
 
 Jd = 
 
 = 
 
 lid = 
 
 11cm = 
 
 7m? = 
 
 =y* =jl=jm = kj =tf = Ij = l* = l m = mf 
 
 = = c^ml = c 43 4 4 , We = b^jk =0 = 
 = c 43 7?i 2 = c 35 c 43 7?2 = = c 43 e 5 , 
 = c^ld , ((7 54 c 35 ) & 34 = , to 8 = e 5 to 
 = cl^ml , (e B rf 54 ) 6? 54 = . 
 
 = c 35 to 
 
 C 3 5 )% = 0, 
 
54 
 
 PEIRCE : Linear Associative Algebra. 
 
 There are two cases : 
 
 [171], whene 5 = l; 
 [172], 
 
 [171]. The defining equation of this case is 
 
 771 ~~~ 77? 
 
 = c 43 = Ik . 
 
 which gives 
 
 There can be no pure algebra if either of the quantities 6 34 , c 35 or d^ vanish, 
 and there is no loss of generality in assuming 
 
 Jcl=j, km = 7c, ml=l. 
 
 This gives a quintuple algebra which may be called (n$), its multiplication table 
 being 
 
 (%;) i j k I m 
 
 J 
 
 i 
 
 j 
 
 7c 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 7c 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 m 
 
 [172]. The defining equation of this case is 
 
 m 2 0, 
 = c 35 d$ = 7;7n = ml ; 
 
 which gives 
 
 * But on examination of the assumptions already made, it will be seen that if e 5 is not zero, and 
 consequently c 43 =0 , the algebra breaks up into two. Accordingly, the algebra (i 5 ) is impure, for i, 
 j , k and I , alone, form the algebra (/ 4 ), "while m , I , k , j , alone, form the algebra (7t 4 ), and im = mi 
 . [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 55 
 
 and there can be no pure algebra if either 5 34 or c 43 vanishes, and it may be 
 
 assumed that 
 
 Id =.j, Uc = m . 
 
 This gives a quintuple algebra which may be called (0 5 ), its multiplication table 
 
 being as follows : * 
 
 (o 5 ) i j Jc I m 
 
 i 
 
 i 
 
 j 
 
 7c 
 
 
 
 
 
 j 
 
 j 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 j 
 
 
 
 I 
 
 I 
 
 
 
 m 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 [18]. The defining equations of this case are 
 
 ij = j , ik = k , il = I , mi = m , ji =. Id = li = im = , 
 which give, by 46 , 
 
 But if A is any expression of the second group, 
 
 Am = ai ; 
 which gives 
 
 = Amj = aj = a = Am =jm = 7cm = Im , 
 
 and there is no pure algebra in this case. 
 
 [19]. The defining equations of this case are 
 
 ij = j, i7c = k , li = l, mi = m , il ==. im =ji = Id , 
 which give, by 46, 
 
 =y 2 =jk = l?j = 1<? = lj = Ik = 1* = lm = mj = mk = ml = m 9 . 
 
 B:B+D:D+F:F, j D:F, k = B:C+D:E, l A:B + E:F, m = A:C. [C. S. P.] 
 
56 PEIRCE : Linear Associative Algebra. 
 
 But if A is an expression of the second group and B one of the third, 
 
 AB = ai , 
 which gives 
 
 = ABj = aj = a = AB =jl =jm = Id =.lm , 
 
 and there is no pure algebra in this case. 
 
 [10 ]. The defining equations of this case are 
 
 ij = j, ik =. k , li = I , ji =. lei = il im = mi =. , 
 which give, by 46, 
 
 and it is obvious that we may assume 
 We have, then, 
 
 = &&JI =jml = c 25 Id = 
 
 There are two cases : 
 
 [lO l], when a 34 does not vanish ; 
 [10 2], when a 34 vanishes. 
 
 [lO l]. The defining equation of this case can be reduced to 
 
 Id = i , 
 which gives 
 
 There are two cases : 
 
 [10 P], whene B = l; 
 [10 12], when e 5 vanishes. 
 
 [lO l 2 ]. The defining equation of this case is 
 
 m* ==. m ; 
 and we assume 
 
 jm =j, ml = I, km = k, 
 
 because otherwise this case would coincide with a subsequent one. We get, then, 
 
 =JU = e&jm = e 42 = 1j , =jlk = e^jm = e 43 = Ik , 
 which virtually brings this case under [10 2].* 
 
 * This does not seem clear. But i i 2 = klkl = , which is absurd. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 57 
 
 [10 12J. The defining equation of this case is 
 
 m$ f\ 
 
 m u , 
 which gives 
 
 =jm z = b^jm = 6 25 =jm, = m*1 = d^ml = a 54 = ml, 
 = km* = c^km = c 35 , km = b^j , M li = l c ^ml = , 
 
 which is impossible, and this case disappears. 
 
 [10 2]. The defining equation of this case is 
 
 There are two cases : 
 
 [10 21], when e 5 =. 1 ; 
 [10 2 2 ], when e 5 vanishes. 
 
 [10 21]. The defining equation of this case is 
 
 m z =: m, 
 and if we would not virtually proceed to a subsequent case, we must assume 
 
 jm =j, km = k, ml = I, 
 and there is no loss of generality in assuming 
 
 so that there is no pure algebra in this case.t 
 
 [10 2 2 ]. The denning equation of this case is 
 
 m*=0, 
 which gives 
 
 = mH = d M ml.= d^ = ml 
 and we may assume 
 
 % = , 
 which gives 
 
 =jm? = b m jm = b, 5 =jm, = km 2 = c 35 km = c ss , km 
 
 = e 43 m* = Ikm = b m e, z m = 5 35 e 42 ; J 
 
 * In this case, the algebra at once separates into an algebra between j\ k , Z and m . and three double 
 
 algebras between i and j , i and ft, and i and Z , respectively. [C. S. P.] 
 
 t In fact, = Iklk = e&m = e 43 =lk. So that the algebra falls into six parts of the form (& 2 ) . [C. S. P.] 
 t The author omits to notice that = klk = e 4 3 km = e 4 ,6 3 5 . Thus, either km = or lj Ik = . The 
 
 algebra (jo f) ) involves an inconsistency in regard to klk. [C. S. P.] 
 
58 
 
 PEIRCE : Linear Associative Algebra. 
 
 and we have without loss of generality 
 
 lj = , km = j, Ik = m . 
 This gives a quintuple algebra which may be called (p s ), of which the multipli 
 
 i j k I m 
 
 cation table is 
 
 k 
 
 i 
 
 j 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 I 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 mk = 
 
 [11 ]. The denning equations of this case are 
 
 ij =y, Id = k, ji = ik = il = im = li = mi Q; 
 
 which give, by 46, 
 
 =j* 12 Id = km = Ij = mj , 
 
 jk = a 23 i , jl = l u j, jm = & 85( /, kj = d^l + e&m , Ik = c^k, 
 There are two cases : 
 
 [ll l], when I is the idempotent base of the fourth group ; 
 [11 2], when the fourth group is nilpotent. 
 
 [ll l]. The denning equation of this case is 
 
 l* = l. 
 There are two cases : 
 
 [ll l 2 ], when ?7i is in the second subsidiary group of the fourth group ; 
 [11 12], when m is in the fourth subsidiary group of the fourth group. 
 
 [ll l 2 ]. The defining equations of this case are 
 
 Im m , ml = ; 
 
PEIRCE : Linear Associative Algebra. 59 
 
 which give m 2 =jm* = b^jm = 6 25 =jm , 
 
 = m z k = c 53 mk =. c 53 = mk ; 
 
 and # 23 cannot vanish in a pure algebra, so that we may assume 
 
 jit = i , 
 which gives 
 
 Jg Ic = k = d 3Z c 3 7t , jkj = j = d 3z b z j , 1 = d 32 c 43 = d sz b u , 
 jl = jP = bujl , & 2 2 4 = &.J4 = 1 , Ik = Pk = c 43 lk , c| 3 = c 43 = 1 = c? 33 , 
 y/ j , 1k = 1c, Igl = Z = 7^ , 
 
 and there is no pure algebra in this case.* 
 
 [11/12]. The defining equations of this case are 
 
 lm =. ml = , 
 which give 
 
 = jlm = bujm = bubtoj = ^As. = Imk c 53 //^ = c 43 c 53 7^ = c 43 c 53 , 
 ^7 = <4g? = buJy = b^d^l + b^e az m , % (7 32 ? = c 4s Jfj = c^d^l + c 43 c 32 m , 
 A^m = e 32 ??i 2 = byjfj = b^d 3Z l + b K e&m , mkj = e 3Z m z = c 53 Zy = c 53 4 2 ? + c^ 3 e 3Z m , 
 <^32 :=:::: b Z4 d 3 % c 43 ot 32 , = t> 25 ct 32 = 6 53 a 32 = O 24 e 32 = c 43 e 32 . y 
 
 There are two cases : 
 
 [11 121], when m is ideinpotent ; 
 
 [11 12 2 ], when m is nilpotent. 
 [11 121]. The denning equation of this case is 
 
 m 2 = ?T? , 
 which gives 
 
 6 32 C 53^32 : ^25 e 32 j 
 
 and it may be assumed that 
 
 24 = 0. 
 
 But if the algebra is then regarded as having I for its idempotent basis, it is 
 evident from 50 that the bonds required for a pure algebra are wanting, so 
 that there is no pure algebra in this case.J 
 
 *In fact, i, j , fc, I form the algebra (gr 4 ), and Z, m , the algebra (6 2 ). [C. S. P.] 
 
 t The last equation holds by I 68. [C. S. P.] 
 
 % Namely, d 32 z=0, and either e 32 1, when Z forms the algebra (Oj), and i , y , k , TO the algebra 
 (gr 4 ), or else e 32 = 0, when by [13] of triple algebra a 23 =0, and j and A; each forms the algebra (6 2 ) with 
 each of the letters i, I, m. [C. S. P.] 
 
60 PEIRCE : Linear Associative Algebra. 
 
 [11 12 2 ]. The defining equation of this case is 
 
 m 2 = 0, 
 which gives 
 
 =jm* = b K jm = b&j = 6 25 =jm, = m*k = c^mk = c^k = c 53 = mk 
 
 and there is no pure algebra in this case. * 
 
 [11/2], The defining equation of this case is 
 
 ^=0, 
 
 in which n is 2 or 3. We must then have 
 
 = Im = ml = m z , 
 which give 
 
 =jl s = b. 2 J(* = bjjl = b u =jl =jm = Ik = mk, = kjk = a, 3 k = 23 = jk , 
 and there is no pure algebra in this case, f 
 [2]. The defining equation of this case is 
 
 i n = . 
 There are five cases : 
 
 [21], when n = 6 ; 
 [2 2 ], when n = 5; 
 [23], when n = 4; 
 [24], when n = 3; 
 [25], when 7i = 2. 
 
 [21]. The defining equation of this case is 
 
 i 6 = , 
 and by 60, 
 
 P=j, i 3 = k, i* = l, $ = m . 
 
 This gives a quintuple algebra which may be called fe), its multiplication table 
 being 
 
 Here, m forms the algebra (6j) , and the other letters form (c/.J . [C. S. P.] 
 
 t Namely, if n = 2 , j, l , fc , form the algebra (d 3 ) (second form), i, y, and i , ft . the algebra (6.), and 
 m the algebraic,). Batif=8,/, k. I and m form an algebra transformable into (, 4 ) or (fc 4 ), while i, 
 j , and z , k form, each pair, the algebra (6 a ). [C. S. P.] 
 
PEIRCB : Linear Associative Algebra. 
 fe) i j Jc I m 
 
 61 
 
 j 
 
 Jc 
 
 I 
 
 m 
 
 1 
 
 
 If 
 
 I 
 
 m 
 
 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [2 2 ]. The defining equation of this case is 
 
 i 5 = 0, 
 and by 59, 
 
 There are then by 64 two quintuple algebras which may be called (r 5 ) and 
 their multiplication tables being 
 
 j Jc I m 
 
 I m 
 
 Te 
 
 j 
 
 Jc I 
 
 
 
 
 
 Jc 
 
 I 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 I 
 
 Te 
 
 j 
 
 Jc 
 
 I 
 
 
 
 
 
 Jc 
 
 I 
 
 
 
 
 
 
 
 {/ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 [23]. The defining equation of this case is 
 
 t 4 = ; 
 
 and by 59, 
 
 t= 
 
62 PEIRCE : Linear Associative Algebra. 
 
 and it may be assumed, from the principle of 63, that 
 
 8=0, 
 
 which gives 
 
 =.jl = kl = iU = il z = il 
 li = c^k + dj. + 6> 41 m , I 2 = cjc + d A l + e^m , Im = c^k + d 45 Z + e 4 5 
 
 There are two cases : 
 
 [231], when im = l-j 
 
 [232], when im = . 
 [231]. The defining equation of this case is 
 
 im = I , 
 whence 
 
 = jm = km =jmi jml = jm? = e 41 = e 4 = e 45 , 
 & = </ 4 7i , = K* = ^ 41 ?i 3 = dy* = dlU = d* = 1j = Ik , 
 
 P = d i f, Q = l* = dJ 3 = dt, K = cjc, P = cjc, Im^cJf. + dJ, 
 imi = U = cdc , mi = c 4i y + c 51 ^ + cZ 51 ? , wy = c 41 (1 + cZ 51 ) 7f , ?n^ = , 
 iwi? = P =2 cjc , ml = cj H- c 54 /^ + e4t? , 
 
 = m 4 = d 45 , Kwi = ? 2 = c^&m = w?^ = 
 
 = mlm d 5 Jm = d u c^ , = w?l = d^ml = cZ 54 .* 
 
 There are two cases : 
 
 [231 2 ], when c 41 does not vanish ; 
 
 [2312], when c^ vanishes. 
 [2 3 1 2 ]. The defining equation of this case is reducible to 
 
 li = k. 
 There are two cases : 
 
 [2 3 1 3 ], when c 45 does not vanish ; 
 
 [231*2], when c 45 vanishes. 
 [231 s ]. The defining equation of this case can be reduced to 
 
 Im = k, 
 which gives 
 
 = k-\- d^k + d&k = k 
 
 * To these equations are to be added the following, which is taken for granted below : ml = mini 
 C 4g d 61 fc. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 and if r is one of the imaginary cube roots of 1 , there are two cases : 
 
 [2 3 1 4 ], when cZ 51 = r ; 
 [231 3 2], when<Z B1 = 1. 
 
 [2 3 1 4 ]. The defining equation of this case is 
 
 4,1 = t , 
 
 which gives 
 
 i (m c 51 Z) = /, l(m c 51 Z) = k , 
 (m c B1 Z) i = j + Vl , (m c B1 Z)/ = (1 + r) & , 
 (w c 51 ?) & = , (m c 51 ?) I = vie , 
 (m - cjf =j + [C B - c 51 (l + r)] k + (2r - 1) I ; 
 
 so that the substitution of m c 51 ? for wi is the same as to make 
 There are two cases : 
 
 c 51 = . 
 
 [231 5 ], when c 5 does not vanish ; 
 [231 4 2], when c 5 vanishes. 
 
 [231 5 ]. The defining equation of this case can be reduced to 
 
 <%==! 
 
 There is then a quintuple algebra which may be called ( 5 ), its multiplication 
 table being * 
 
 * The author has overlooked the circumstance that (t 5 ) and (u s ) are forms of the same algebra. If in 
 (t b ) weputij = i t 2 j , ji j 2r 2 &, k l fc, Zj =. v 2 k-\- I , m l = r 2 J + m , we get (1*5). The struc 
 ture of this algebra may be shown by putting i- L =. ri, j\ =^l 2 y, fcj = A;, Z x =r 2 J rZ, mj i= r m, 
 when we have this multiplication table (where the subscripts are dropped): 
 
 ( 5 ) i j k I m 
 
 i 
 
 i 
 
 k 
 I 
 
 m 
 
 j 
 
 fc 
 
 
 
 k 
 
 z 
 
 k 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 rk 
 
 
 
 
 
 
 
 
 
 rZ 
 
 r 2 fc 
 
 
 
 
 
 
 
 In relative form, i A : B + A : C+ B : E+ C: D + E: (?, j = A :D+ A : E+ B :G, k = A:G, 
 = rA:E+C:G, m^^A : B+ A : F+ rC: E + D : G- F: G. [C. S. P.] 
 
64 
 
 PEIRCE : Linear Associative Algebra. 
 i j k I 
 
 in 
 
 k 
 
 m 
 
 j 
 
 k 
 
 
 
 
 
 i 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ft 
 
 o 
 
 
 
 
 
 * 
 
 j + r/ 
 
 (i + r)& 
 
 
 
 r& 
 
 j+k+ 
 
 (2r 1)1 
 
 [231 4 2]. The defining equation of this case is 
 
 There is then a quintuple algebra which may be called (%), its multiplication 
 table being 
 
 (MB) i j k I m 
 
 k 
 
 m 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 & 
 
 
 
 
 
 
 
 k 
 
 y + rZ 
 
 (i+r)ft 
 
 
 
 rk 
 
 j + 
 (2t 1)1 
 
 [231 3 2]. The defining equation of this case is 
 
 fi i 
 
 T - , . ^51 * j 
 
 which gives 
 
 d 5 = Q, i(m c 51 ?) = / , 1(m c 51 ?) = k , 
 (m c 5l l)i =j - /, ( W __ CBI ?)/ = _ ^ , (m c 51 2 =y 
 so that the substitution of m c 51 Z * for m is the same as to make 
 
 CM = . 
 
 * The original text has m c 51 fc throughout these equations, but it is plain that m c,,l is meant. 
 
 [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 65 
 
 There are two cases : 
 
 [231 3 21], when c 5 does not vanish; 
 [23P2 2 ], when c 5 vanishes. 
 [231*21]. The defining equation of this case can be reduced to 
 
 There is a quintuple algebra which may be called (v 5 ), its multiplication table 
 being * 
 
 K) i j If I m 
 
 Jc 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 Jc 
 
 j-l 
 
 
 
 
 
 Jc 
 
 j+Jc 
 
 [231 3 2 2 ]. The defining equation of this case is 
 
 C B = . 
 
 This gives a quintuple algebra which may be called (w^), its multiplication table 
 being * 
 
 * The algebra (V B ) reduces to (w 5 ) on substituting i, = i + ^j+ Jz . j\ =j+ k, A; 1 =fc. l^-k + l. 
 
 in>1 ~ 8 <? ~*" 3- z + TO To exhibit the structure of this algebra, we may put p and p for imaginary cube 
 roots of 1, and substitute in (w*) i-, =i + //??i, /, (1 p} ?4- k~l-/^RJ j. or. / /-i ,/\ ,-_L i. 
 
 _ > Ji V-L KjJ~T Ot , K! >,<! (I P)J-\-K 
 
 v 3Z , TOJ = i+ pm . Then, dropping the subscripts, we have this multiplication table. 
 
 i j k I m 
 
 
 
 
 
 k 
 
 ) 
 
 k \ 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 I 
 
 k 
 
 
 
 
 
 
 
 In relative form, i - f > A :B + P C:F+ 3 P D : E, j BpA : C + 3p D :F,k = 3A-D 1= Sp A E 
 +3pB:F. m = pA:D + 3p B:C+pE:F. [C. S. P.] 
 
66 
 
 PEIRCE : Linear Associative Algebra, 
 (w ) i j k I m 
 
 i 
 
 j 
 Jc 
 I 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 fc 
 
 
 
 
 
 
 
 fc 
 
 )-l 
 
 
 
 
 
 Jf. 
 
 j+& 
 
 [23P2]. The defining equation of this case is 
 
 /ra = 0, 
 which gives 
 
 ml 0, m z = csk + dj, m?i djc = [1 +d 51 )&, d,= l+d^, 
 
 and c 51 may be made to vanish without loss of generality. 
 There are three cases : 
 
 [231 2 21], when neither cZ 51 nor d 51 + 1 vanishes ; 
 [23P2 2 ], when d 51 + 1 vanishes ; 
 [231 2 23], when d 5l vanishes. 
 
 [23P21]. The defining formulae of this case are 
 
 d 51 :0, ^i^- 1 - 
 There are two cases : 
 
 [231 2 21 2 ], when c 5 does not vanish ; 
 [231 2 212], when c 5 vanishes. 
 
 [231 2 21 2 ]. The defining equation of this case can always be reduced to 
 
 This gives a quintuple algebra which may be called (x 5 ), its multiplication table 
 being* 
 
 *In relative form, i = A :B + A :E+ B : D + D : F, jA: D + B :F. k A:F, l A:D. 
 m=(l + a) A :B + A:C+A:E+B:D+C:D + D:F+E:F. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 i j k I m 
 
 k 
 
 m 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 j+al 
 
 (1 + a)k 
 
 
 
 
 
 k+ 
 
 (l+a)l 
 
 [23P212]. The defining equation of this case is 
 
 c 5 = . 
 
 This gives a quintuple algebra which may be called (// 5 ), its multiplication table 
 being * 
 
 i j k- I m 
 
 k 
 
 m 
 
 J 
 
 k 
 
 
 
 
 
 I 
 
 k 
 
 o. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 j+v* 
 
 (!+)* 
 
 
 
 
 
 (1 + a)l 
 
 [231 2 2 2 ]. The defining equation of this case is 
 
 which gives 
 
 There are two cases: 
 
 mi =j /, mj = , m 2 = c 5 k . 
 
 [231 2 2 2 1], when c 5 does not vanish ; 
 [231 2 2 3 ], when C B vanishes. 
 
 The relative form is the same as that of (# 5 ) ; omitting from m the terms A : E and E : F. [C. S. P.] 
 
68 
 
 PEIRCE : Linear Associative Algebra. 
 
 [231 2 2 2 1]. The defining equation of this case can be reduced to 
 
 m* = k . 
 This gives a quintuple algebra which may be called (z 5 ), its multiplication table 
 
 being * 
 
 (z 5 ) i j If I m 
 
 i 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 j 
 
 k 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 * 
 
 
 
 
 
 
 
 
 
 m 
 
 i-l 
 
 
 
 
 
 
 
 & 
 
 [231 2 2 3 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (a 5 ), its multiplication 
 
 table being f 
 
 (fin 5 ) i j k I m 
 
 i 
 
 j 
 Jc 
 I 
 m 
 
 j * 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 j-l 
 
 
 
 
 
 
 
 *In relative form, i A : B + B : C+ C:D , j= .A : C+ B :D . k=A:D. l = A:C, m = B:C 
 + A:E+E:D. [C. S. P.] 
 
 t In relative form, the same as (z 5 ), except that m B : C. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 69 
 
 [231 2 23]. The defining equation of this case is 
 
 mi = j , 
 which gives 
 
 = (/ j)i = ( m i)i- 
 
 so that, by the substitution of I j for / and m i for m, this case would 
 virtually be reduced to [232]. 
 
 [2312]. The defining equation of this case is 
 
 K = 0, 
 
 which gives 
 
 mj = , mim = ml = d^lm , d 45 = , c 54 = d 51 c 45 , 
 m\ = d 5l ml = c 45 k , c 45 = d 5l c M , m s = d 5 lm = d 5 ml , ^ 5 (c 54 c 45 ) = . 
 
 There are two cases : 
 
 [23121], when d 5 does not vanish ; 
 [2312 2 ], when d^ vanishes. 
 
 [23121]. The defining equation of this case can be reduced to 
 
 d = 1 , 
 which gives 
 
 C 45 C 54 5 
 
 and it may be assumed without loss of generality that 
 
 c 5 = 0.* 
 There are two cases : 
 
 [23121 2 ], when c 45 does not vanish ; 
 [231212], when c 45 vanishes. 
 
 [23121 2 ]. The defining equation of this case can be reduced to 
 
 lm = ml = k , 
 which gives 
 
 t==l. 
 
 There are two cases : 
 
 [23121 3 ], when c B1 does not vanish ; 
 [23121 2 2], when c 51 vanishes. 
 
 [23121 3 ]. The defining equation of this case can be reduced to 
 
 * Namely, by putting 1 1 rz c 5 k + 1 m i = i c 5i / . [C. S. P.] 
 
70 
 
 PEIRCB : Linear Associative Algebra. 
 
 This gives a quintuple algebra which may be called (a& 5 ), its multiplication 
 
 table being * 
 
 i j Tf I m 
 
 k 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 Jc+l 
 
 
 
 
 
 If 
 
 i + i 
 
 [23121 2 2]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (ac 5 ), its multiplication 
 table being f 
 
 *The structure of this algebra is best seen on making the following substitutions: Let ^ : and \) 2 
 represent the two roots of the equation a? 2 = a?+l. That is, fy l rr - (l-\-*/ 5) and fh ^ U */ 5 ) 
 
 i). Then, we have the multiplication table : 
 i j k I m 
 
 i 
 
 3 
 k 
 
 I 
 m 
 
 3 
 
 It 
 
 
 
 
 
 ilhfc 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 iM 
 
 
 
 0. 
 
 k 
 
 I 
 
 In relative form, i = A:B + B :C+ C:D+~ ^^ :G+ H:D, jA : C+B :D. k = A:D. l=A:F 
 
 D 
 
 ^ 2 A :H+G:D. [C. S. P.] 
 o 
 
 t On making the same substitutions for i and m as in the last note, this algebra falls apart into two 
 algebras of the form (6 3 ). [C. S. P.] 
 
PEIBCE : Linear Associative Algebra. 
 (ac 5 ) i j k I m 
 
 71 
 
 m 
 
 ] 
 
 .; ^ 
 
 i 
 
 
 
 
 
 I 
 
 7c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 I 
 
 1 
 
 
 
 k 
 
 /+! 
 
 [231212]. The defining equation of this case is 
 
 ml = Im = . 
 There are two cases : 
 
 [2312121], when c 51 does not vanish ; 
 [231212 2 ], when c 51 vanishes. 
 
 [2312121], The defining equation of this case can be reduced to 
 
 % = 1 
 
 This gives a quintuple algebra which may be called (ad^), its multiplication 
 table being* 
 
 i j 7c I m 
 
 m 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k + al 
 
 
 
 
 
 
 
 1 
 
 * In relative form, t A :B+B: C+C : D+E : F+aF: G. j A :C+ B :D + aE:G , k = A:D. 
 E:O, m = A:C+E:F+F:G. [C. S. P.] 
 
72 PEIRCE : Linear Associative Algebra. 
 
 [231 2 12 2 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (oe 5 ), its multiplication 
 table being * 
 
 (ae) i j ~k I m 
 
 j 
 
 k 
 
 
 
 
 
 I 
 
 Tt 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 al 
 
 
 
 
 
 
 
 J 
 
 [2312 2 ]. The denning equation of this case is 
 
 d, = Q. 
 There are two cases : 
 
 [2312 2 1], when c- 45 does not vanish; 
 [2312 3 ], when c 45 vanishes. 
 
 [2312 2 !]. The defining equation of this case can be reduced to 
 
 1m = k, 
 which gives 
 
 C 45 ^51 C 45 ) ^51 = 1 
 
 There are two cases : 
 
 [2312 2 1 2 J, whend B1 = l; 
 [2312 2 12], whend 81 = 1. 
 
 [231 2 2 1 2 ]. The defining equation of this case is 
 
 ... . <^ = i. 
 
 which gives 
 
 * In relative form, the same as (od 5 ) except that m = E: F+ F:G. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 73 
 
 There are two cases : 
 
 [2312 2 ! 3 ], when c 51 does not vanish ; 
 [2312 2 1 2 2], when c 51 vanishes. 
 
 [2312 2 ! 3 ]. The defining equation of this case can be reduced to 
 
 c 51 = 1 . 
 
 This gives a quintuple algebra which may be called (c// 5 ), its multiplication 
 table being* 
 
 fa/s) i j k I m 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 Tc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -k 
 
 Jc+l 
 
 
 
 
 
 ~k 
 
 j+cic 
 
 * To show the construction of this algebra, we may substitute i l = i -f m , j i 2/+ (a + 1) k + 21 , 
 4& , ?! = 2j + (a - 1) k 21 , m l i m. This gives the following multiplication table : 
 
 i 
 
 3 
 k 
 
 I 
 m 
 
 j 
 
 k 
 
 
 
 
 
 2=1.. 
 
 4 ; 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 - a+1 fc 
 
 ~r 
 
 
 
 
 
 k 
 
 I 
 
 This algebra thus strongly resembles (o6 5 ). In relative foim, t A :B + B : C+ C : D-f A : G 
 
 :D. [C. S. P.] 
 
74 PEIRCE : Linear Associative Algebra. 
 
 [2312 2 1 2 2]. The defining equation of this case is 
 
 c n = 0. 
 There are two cases : 
 
 [2312 2 1 2 21], when c 5 does not vanish; 
 [2312 2 P2 2 ], when C B vanishes. 
 
 [2312 2 1 2 21]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (a# 5 ), its multiplication table 
 being * 
 
 i J * I 
 
 m 
 
 % 
 
 j 
 
 Jc 
 
 I 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 I 
 
 
 
 
 
 Jc 
 
 j + Jc 
 
 [2312 2 1 2 2 2 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (a/i 5 ), its multiplication 
 table being f 
 
 * On substituting t 1 =i+^ t / + m, m 1 =i+jm, this algebra falls apart into two of the form 
 
 (6 3 ). [C. S.P.] 
 
 t On substituting i l =i+m, m 1 i m,j l j+l, l^^j l, this algebra falls apart into two of 
 the form (6 3 ). [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 (7? 5 ) i j Jc I m 
 
 75 
 
 i 
 
 j 
 Jc 
 I 
 m 
 
 j 
 
 K 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 I 
 
 
 
 
 
 Jc 
 
 j 
 
 [2312 2 12]. The defining equation of this case is 
 
 which gives 
 
 There are two cases : 
 
 [2312 2 121], when c 51 does not vanish; 
 [2312*12*], when c 51 vanishes. 
 [2312 2 121]. The defining equation of this case can be reduced to 
 
 c 51 = 1 . 
 
 This gives a quintuple algebra which may be called (<M S ), its multiplication 
 table being * 
 
 (ai 5 ) i j Jc I m 
 
 Jc 
 
 m 
 
 j 
 
 i 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 fC v 
 
 
 
 
 
 -Jc 
 
 j+cJc 
 
 *In relative form, t = A : C B : F+ C : E + D : G + E : G, j = A : E+ C : G . k = A:O. 
 A : F B:G. m = A:B + A:D + B:E+C: F+ aD : G + F : G . [C. S. P.] 
 
76 PEIRCE : Linear Associative Algebra. 
 
 [2312 2 12 2 ]. The defining equation of this case is 
 
 There are two cases : 
 
 mi = ?. 
 
 [2312 2 12 2 1], when c 5 does not vanish ; 
 [2312 2 12 3 ], when c 5 vanishes. 
 
 [2312*12*1]. The defining equation of this case can be reduced to 
 
 c 5 = 1 . - 
 
 This gives a quintuple algebra which may be called (o/ 5 ), its multiplication 
 table being* 
 
 (0/5) j & I m 
 
 j 
 
 It 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 ft 
 
 I 
 
 
 
 
 
 -& 
 
 j + 7c 
 
 [2312 2 12 3 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (a& B ), its multiplication 
 table being f 
 
 * In relative form, i = A : C+ C : E+ E : G B :F, j A :E+ C : G, k 
 = A:B + B:E+C:F+F:G + A :D + D:G. [C. S. P.] 
 t In relative form, t = A:C+ C : D-\- D :F B \ E , j A :D -\-C\F. k 
 :D+C:E+E:F. [C. S. P.] 
 
 A:G, l A:FB:G, 
 = A:F, l=A:EB:F, 
 
PEIRCE : Linear Associative Algebra. 
 (a7c 5 ) i j 7c I m 
 
 77 
 
 i 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 j 
 
 7c 
 
 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 k 
 
 m 
 
 I 
 
 
 
 
 
 ~~~ K 
 
 j 
 
 [2312 3 ]. The defining equations of this case are 
 
 ml Ira = , m z = c 5 k . 
 There are two cases : 
 
 [2312 3 1], when cZ 51 is not unity; 
 [2312 4 ], when d m is unity. 
 
 [2312 3 !]. The defining equation of this case is 
 
 . . *i*i. 
 
 which gives 
 
 * [(1 4i) m c B1 y ] = (1 d m ) I c B1 & , i[(l d 5l ) I c 51 7r ] = , 
 [(1 A) l ~ Cn&ji = , [(1 dn) m c 5l j]i = d 51 [(Id 5l )l c 5l k] , 
 [(1 4>i) I c 5l k] [(1 d,~] m c 5i y] = , 
 [(1 rf n ) m c 5 J] [(1 d m ) I c^] = , 
 [(1 - ^i) m -c 51 /] 2 rz (1 - dWV ; 
 
 so that the substitution of (1 d 5l ) m c 51 / for m, and of (1 d 5l )l c B1 & for 
 I, is the same as to make 
 
 051 = 
 
 There are now two cases : 
 
 [2312 3 ! 2 ], when c 5 does not vanish ; 
 [2312 3 12], when c 5 vanishes. 
 
 [2312 3 ! 2 ]. The defining equation of this case can be reduced to 
 
 m z = k . 
 
78 
 
 PEIRCE : Linear Associative Algebra. 
 
 This gives a quintuple algebra which may be called (a7 B ), its multiplication 
 table being* 
 
 i j It I m 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 dl 
 
 
 
 
 
 
 
 Jc 
 
 [2312 3 12]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (ow? 5 ), its multiplication 
 table being 
 
 (am 5 ) i j k I m 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 I 
 
 7c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 dl 
 
 
 
 
 
 
 
 
 
 *In relative form, i A : B + B : C+C:D + dE:F, j A:C+B:D. 
 mA:E-\-B:F+E:D. [C. S. P.] 
 
 A\D, l A:F, 
 
PEIRCE : Linear Associative Algebra. 79 
 
 [2312 4 ]. The defining equation of this case is 
 
 There are two cases : 
 
 [231 2 4 1], when c 51 does not vanish; 
 [2312 5 ], when c 51 vanishes. 
 
 [2312 4 !]. The denning equation of this case is easily reduced to 
 
 ___ -i 
 There are two cases : 
 
 [2312 4 ! 2 ], when c 5 does not vanish ; 
 [2312 4 12], when c 5 vanishes. 
 
 [2312 4 ! 2 ]. The denning equation of this case is easily reduced to 
 
 m 2 = k . 
 
 This gives a quintuple algebra which may be called (cm 5 ), its multiplication 
 table being * 
 
 (an 5 ) i j Tc I m 
 
 j 
 
 Jc 
 
 
 
 o 
 
 I 
 
 7c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l + Tc 
 
 
 
 
 
 
 
 7c 
 
 [2312 4 12]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (ao 5 ), its multiplication 
 table being f 
 
 *In relative form, i = A :E+A :B + B :C+ C:D+E:F, jA :C+B :D+A:F, 
 
 :F,m = A:C+A:E+E:D. [C. S. P.] 
 
 tin relative form, i A : B + B : C+ C : D + E : F, j A:C+B:D. k = A:D 
 A:C+A:E+B:F. [C. S. P.] 
 
 A F 
 
80 
 
 PEIRCE : Linear Associative Algebra. 
 (ao 5 ) i j It I m 
 
 Tf 
 
 I 
 
 m 
 
 j 
 
 Tf 
 
 
 
 
 
 I 
 
 If 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 l + Tf 
 
 
 
 
 
 
 
 
 
 [2312 5 ]. The defining equation of this case is 
 
 mi = I. 
 There are two cases : 
 
 [2312 5 2], when c 5 does not vanish ; 
 
 [2312 6 ], when c 5 vanishes. 
 
 [2312 5 !]. The defining equation of this case can be reduced to 
 
 m 2 = Jc. 
 
 This gives a quintuple algebra which may be called (op 5 ), its multiplication 
 
 table being* 
 
 i j 1^ I m 
 
 J 
 Jc 
 
 j 
 
 Tf 
 
 
 
 
 
 I 
 
 If 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 z 
 
 
 
 
 
 
 
 Tf 
 
 *In relative form, t A : B + B : C + C : D + E : F, j = A:C+B:D, k = A:D, l A:F, 
 :F+E:D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 [2312 6 ]. The defining equation of this case is 
 
 81 
 
 This gives a quintuple algebra which may be called (aq^), its multiplication 
 table being 
 
 (aq 5 ) i j 7c I m 
 
 J 
 
 m 
 
 j 
 
 7c 
 
 
 
 
 
 I 
 
 7c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 [232]. The defining equation of this case is 
 
 im = ,* 
 
 which gives 
 
 = j m = km , 
 and it may be assumed that 
 
 li= 0. 
 This gives 
 
 lj = Ik = = iP = l z i = Urn = iml = mli = im . 
 
 There are two cases : 
 
 [2321], when mi = l; 
 
 [232 s ], when ww = 0. 
 [2321]. The defining equation of this case is 
 
 mi I, 
 which gives 
 
 = mj = mlc , Im = c^Jc + d^l -\- e^m , 
 
 Imi = P = e^l , = / 4 = e,J? = e^ = 7 2 , m 2 = cjc + dj + e,m , 
 m*i = ml = ej, m 4 / = e^mH = e 5 = ml; = ?w 2 = d^frw = ^ 45 . 
 
 * What is meant is that every quantity not involving powers of i is nilfaciend with reference to i. 
 Hence, il = , also. [C. S. P.] 
 
82 
 
 PEIRCE : Linear Associative Algebra. 
 
 There are two cases : 
 
 [2321 2 ], when c 45 does not vanish; 
 
 [23212], when c 45 vanishes. 
 [2321 2 ]. The denning equation of this case can be reduced to 
 
 lm = /,* 
 which gives 
 
 m* = cjc, (m c 5 ?) 2 = , 
 
 so that the substitution of m c s l for m is the same as to make 
 
 This gives a quintuple algebra which may be called (ar 5 ), of which the multipli 
 cation table is 
 
 (r 5 ) i j k I m 
 
 j 
 
 Tf 
 
 
 
 
 
 
 
 lc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 Tc 
 
 I 
 
 
 
 
 
 
 
 
 
 [23212]. The denning equation of this case is 
 
 lm = Q. 
 There are two cases : 
 
 [232121], when d 5 does not vanish; 
 
 [23212 2 ], when d 5 vanishes. 
 [232121]. The denning equation of this case can be reduced to 
 
 d 5 = l. 
 There are two cases : 
 
 [2 32 12 1 2 ], when c 5 does not vanish ; 
 
 [2321212], when c 5 vanishes. 
 [232121*]. The defining equation of this case can be reduced to 
 
 c 5 = 1 . 
 
 * But := im r= mim =. lm . Thus, this case disappears, and the algebra (or,,} is incorrect. [C. S. P.] 
 
PEIECE : Linear Associative Algebra. 
 
 83 
 
 This gives a quintuple algebra which can be called (c/,? 3 ), its multiplication 
 table being* 
 
 (as 5 ) i j Jc I m 
 
 j 
 
 k 
 
 
 
 
 
 
 
 Jc 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 Jc + l 
 
 [2321212]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (W 5 ), its multiplication 
 table being 
 
 (at 5 ) i j Jf I m 
 
 m 
 
 j 
 
 Jc 
 
 
 
 
 
 
 
 7c 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 I 
 
 [23212 2 ]. The defining equation of this case is 
 
 m z = c 5 & . 
 There are two cases : 
 
 [2321 2*1], when c 5 does not vanish ; 
 
 [23212 3 ], when c 5 vanishes. 
 
 *In relative form, i A : B + B : C+ C : D + E : F, jA:C+B:D, k = A:D, 
 = A:E+E:F+E:D. Omitting the last term of m , we have (at.,). [C. S. P.] 
 
 = A:F, 
 
84 PEIRCE : Linear Associative Algebra. 
 
 [23212 2 !]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called 
 
 table being * 
 
 (au$) i j If I m 
 
 , its multiplication 
 
 It, 
 
 m 
 
 j 
 
 ft 
 
 
 
 
 
 
 
 If 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 If 
 
 [23212 3 ]. The defining equation of this case is 
 
 m 2 . 
 This gives a quintuple algebra which may be called (a 5 ), its multiplication 
 
 table being 
 
 (av$) i j If I m 
 
 m 
 
 j 
 
 If 
 
 
 
 
 
 
 
 If 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 *In relative form, i A :B+ B : C+ C:D, j A: C+*B:D, k = A:D, l E:D, m E\C 
 + A : F+ F : D . The omission of the last two terms of m gives (av 5 ). [C. S. P.] 
 
PEIRCB : Linear Associative Algebra. 85 
 
 [232 8 ]. The defining equation of this case is 
 
 mi = , 
 which gives 
 
 = mj mk = Imi = m*i , 
 
 and there is no pure algebra in this case. 
 
 [24]. The defining equation of this case is 
 
 i 3 = , 
 and by 59 , 
 
 **=/, *y=y*=y* = o. 
 
 There are three cases: 
 
 [241], when ik I, il m ; 
 
 [242], when ik = l, il = im = ; 
 [243], when ik = il = im = . 
 
 [241]. The defining equations of this case are 
 
 ik = I, il m , 
 which give 
 
 jk m , im =jl jm = , = iml = ml? e 54 ml* e 54 , jk = m , 
 tT* =ml= & M y , 2 = 6 54 2i -f 6 4 y + e 4 7n , = I 3 = b^m + e 4 ml = b u = ml , 
 im? = , m 2 = 6 5 y + e 5 m , = m 3 = e 5 m 2 = e 5 , 
 iwii = , mi = b^j + e^m , mj = e^mi , mi 3 = e B1 , 
 i7i = mi = & B1 y , ?i = 6 5 ii -f ft^y + e 41 m , 
 
 fo7 = lm=. b 51 m , = ? 3 i7z = 5 51 = lm = mi wi7 = -m 2 , (li) i Ij , 
 i& 2 = lk = a 3 j + c 3 ? + c^ 3 m , i?^ = mk = c 3 m , K/^ = l z = a 31 m , 
 =r 77z^ 3 = c|m c s = mk l$m , 
 ley ^ 2 = 31 y + d 31 li = o 31 (l -r ^si)y + ^li^ , 
 
 ^i/ = km = 31 (1 + c? 31 ) m , = A?m = 31 (1 + d 3l )km = a 31 (1 + d 31 ) = ^ , 
 kj = <^wi , = & 3 = a 3 ? -j- 6 3 w + dg?A; = a s = b 3 + c^ = b 3 kj + c^fc? , 
 Id = a 31 ? + (6 31 + cZ 3 cZ 31 ) w , = klk = a 31 lk = d 3 a 3l = lid = a 31 l z = a 3l = I 2 , 
 = ^ 3 a 31 = b 31 d 3 
 * = In? + iki + M = 
 = #i + /^ + ^ = 
 (k + pi) i = b 31 j H- 4^ + %m + />/ = (6 31 + p pd 3l )j -\- d 31 (I +pj) + e 31 m , 
 
 * This line and the first equation of the next can be derived from 0= (i+ jfc) 3 . [C. S. P.] 
 
86 PEIRCE : Linear Associative Algebra. 
 
 so that if p satisfies the equation 
 
 the substitution of k+pi for k and of I -\- pj for / is the same as to make 
 
 There are four cases : 
 
 [241 s ], when neither e sl nor e 3 vanishes ; 
 [2412], when e 3l does not vanish but e 3 vanishes ; 
 [2413], when e 31 vanishes and not e 3 ; 
 [2414], when e sl and e 3 both vanish. 
 
 [241 2 ]. The defining equations of this case can be reduced, without loss of 
 generality, to 
 
 We thus obtain a quintuple algebra which may be called (aw s ), its multiplication 
 table being* 
 
 (aw 5 ) i j k I m 
 
 It 
 
 j 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 H+w 
 
 fm 
 
 m 
 
 
 
 
 
 Km 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [2412], The defining equations of this case can be reduced to 
 
 _ 1 _ /~k 
 
 * In relative form i = A:B + B:D+vC:E+vE:F+G:F, j A:D-\-^C:F, k = * i:C+B:E 
 + D : F+ A :G+G:F, 1= A : E+ B :F, m A:F. To obtain (aa? 5 ), omit the last term of k. To 
 obtain (cw/ 5 ), omit, instead, the last term of i. To obtain (oz 5 ), omit both these last terms. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 87 
 
 We thus obtain a quintuple algebra which may be called (ax 5 ), its multiplication 
 table being 
 
 (oa? B ) i j 7c I m 
 
 j 
 
 
 
 i 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 vl+m 
 
 r 2 w 
 
 
 
 
 
 
 
 rm 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [2413], The defining equations of this case can be reduced to 
 
 % = , e 3 = 1 . 
 
 We thus obtain a quintuple algebra which may be called (ay 5 ), its multiplication 
 table being 
 
 i j Jc I m 
 
 k 
 
 m 
 
 j 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 tl 
 
 r 2 m 
 
 m 
 
 
 
 
 
 Km 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [2414]. The defining equations of this case are 
 
 We thus obtain a quintuple algebra which may be called (Z B ), its multiplication 
 table being 
 
88 
 
 PEIRCE : Linear Associative Algebra. 
 (azs) i j Jc I m 
 
 j 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 rZ 
 
 r 2 m 
 
 
 
 
 
 
 
 rm 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [242]. The defining equations of this case are 
 
 ik =. I , il = im = , 
 which give 
 
 li = iki = OSLJ + c 3l l , = U 3 = c sl = 1$ = Ij , 
 
 U Ct 3 i 1 r& IftiK I 1KI 0^54 C 34 , 
 
 iTem Im a m j + c 35 ? , = Im 3 = c 35 , 
 
 = imk = a 53 = c 53 , = mk 3 = e 53 , 
 
 There are two cases : 
 
 [2421], when % does not vanish ; 
 [242 2 ], when e 31 vanishes. 
 
 [2421], The defining equation of this case can be reduced to 
 
 Td =. m , 
 
 e 34 = 
 
PEIRCE : Linear Associative Algebra. 
 
 89 
 
 which, by the aid of the above equations, gives 
 
 = mi =. Idl ml 1dm = m? , a 3 j = ik* = kH = Ik = km , = Im , 
 b 53 j = kik = Id = mk , = iff + kik + l&i = 2 3 + b 53 , 
 = k 3 =. a 3 6 53 = Id km = mk = ml ; 
 
 and if ^ is determined by the equation 
 
 k -\- pi , I -\- pj , and ?n + pj can be respectively substituted for k, I and m, 
 
 which is the same thing as to make 
 
 63=0. 
 There are three cases : 
 
 [2421 2 ], when neither d 3 nor e 3 vanishes ; 
 [24212], when d 3 vanishes and not e 3 ; 
 [24213], when d 3 and e 3 both vanish. 
 
 [2421 2 ]. The denning equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (& 5 ), its multiplication 
 
 table being* 
 
 (ba 5 ) i j k I m 
 
 k 
 
 m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 I -\-ern 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 *In relative form, i = A : B + B : C+ A : E, j A : C, fc= D : B + E : F -f D : (? + eG : C+A :E, 
 1 = A :F, m D:C. By omitting the last term of fc and putting e = 1 we get (6b 5 ), and by omitting the 
 last two terms of k we get (&c 5 ). [C. S. P.] 
 
90 PEIRCE : Linear Associative Algebra. 
 
 [2421 2 ]. The defining equation of this case can be reduced to 
 
 = m. 
 
 This gives a quintuple algebra which may be called (66 5 ), its multiplication 
 
 table being 
 
 (bb 5 ) i j Jc 
 
 I m 
 
 m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 m 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [24213]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (6c B ), its multiplication 
 table being 
 
 (bc 5 ) i j 7c I m 
 
 i 
 
 j 
 
 
 
 I 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 le 
 
 m 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
PEIRCE : Linear Associative Algebra. 91 
 
 [242 2 ]. The defining equation of this case is 
 
 % = 0. 
 
 There are two cases : 
 
 [242 2 1], when e B does not vanish ; 
 
 [242 3 ], when e s vanishes. 
 
 jf 
 [242 2 1]. The defining equation of this case can be reduced to 
 
 1& =. a 3 i -f- m , 
 which gives 
 
 Idk = kl a 3 d 3l j , iJ$ = Ik = a 3 j , J#i = a 3 j -\- mi = d 3l kl a^d^j , 
 = Wi + ik H- TdJc a 3 (d? 3l + d 3l +- 1) , mi = a 3 (d 3l l l)j , ml = mik = , 
 = 1$ =. a 3 l + mlt a 3 ki + ^^ , 
 
 mk = a%l , km =: a 3 b 3l j a 3 d 3l l , Im = . 
 
 There are two cases : 
 
 [242 2 1 2 ], when a 3 does not vanish ; 
 
 [242 2 12], when a 3 vanishes. 
 [242 2 1 2 ]. The defining equation of this case can be reduced to 
 
 1$ = i -\- m , 
 which gives 
 
 d 3l =. v 1 = t , lk=j, mlc = I , 
 
 ki = km = b 3l j + r?, mi = (t 2 1) j" , m 2 = Vj. 
 
 There are two cases : 
 
 [242 2 1 3 ], when b 3l does not vanish ; 
 
 [242 2 1 2 2], when b 3l vanishes. 
 [242 2 1 3 ]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (bd^), its multiplication 
 table being * 
 
 * In relative form, i A : D + D : F+ B : E+ C : F, j - A : F, k = vA : B + vB : C+ D : E -D :F 
 -\-E\F, i=A:E~- l -A:F-\-B:F.m v 2 A:C-A:D B:EC:F. [C. S. P.] 
 
92 
 
 PEIRCE : Linear Associative Algebra,. 
 (bd 5 ) i j k I m 
 
 k 
 
 m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j+ rz 
 
 
 
 i + m 
 
 tf 
 
 -j-vl 
 
 
 
 
 
 j 
 
 
 
 
 
 (r 2 -l)y 
 
 
 
 j 
 
 
 
 -fj 
 
 [242*1*2], The defining equation of this case is 
 
 Id = vl . 
 
 This gives a quintuple algebra which may be called (6e B ), its multiplication 
 table being * 
 
 (be s ) i j k I m 
 
 k 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 tl 
 
 
 
 i-\-m 
 
 tf 
 
 yl 
 
 
 
 
 
 j 
 
 
 
 
 
 (r 2 -l)j 
 
 
 
 i/ 
 
 
 
 -0 
 
 [242*12]. The defining equation of this case is 
 
 V = m , 
 which gives 
 
 = Id Ik km = mh = m z tfi mi . 
 
 There are two cases : 
 
 [242*121], when b 3l does not vanish; 
 
 [242*12*], when 6 31 vanishes. 
 
 * On adding to the expression for k in the last note the term A : C , we have this algebra in relative 
 form. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 [242*121]. The denning equation of this case can be reduced to 
 
 93 
 
 This gives a quintuple algebra which may be called (6/ 5 ), its multiplication 
 table being * 
 
 i j k I m 
 
 i 
 
 j 
 Jc 
 I 
 m 
 
 j j o 
 
 / 
 
 i 
 
 
 
 
 
 
 
 
 
 j+dl 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [242 2 12 2 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (bg^), its multiplication 
 
 table being f 
 
 i j k I m 
 
 i 
 
 j 
 
 
 
 / 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 dl 
 
 
 
 m 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 *In relative form, i = A : B+ B : C+ D : E, jA:C. k A: 
 l A:E,m = A:E-\-A:F. [C. S. P.] 
 
 tin relative form, i A : B + B :C+D : E, j=A:C, k = dA \D + B : E+ B : F . l A:E. 
 m=.A:F. The algebra (ar s ) is what this becomes when d . [C. S. P.] 
 
94 PEIRCE : Linear Associative Algebra. 
 
 [242 3 ]. The defining equation of this case is 
 
 e 3 = 0, 
 which gives * 
 
 * It is not easy to see how the author proves that a 3 = . But it can be proved thus. = k 3 = 
 (a s i + b.J+ d 3 l)k = a 3 l + a 3 d 3 j . 
 
 The algebras of the case [242 3 ] are those quintuple systems in which every product containing j or I 
 as a factor vanishes, while every product which does not vanish is a linear function of j and I. Any 
 multiplication table conforming to these conditions is self -consistent, but it is a matter of some trouble 
 to exclude every case of a mixed algebra. An algebra of the class in question is separable, if all 
 products are similar. But this case requires no special attention ; and the only other is when two 
 dissimilar expressions U and V can be found, such that both being linear functions of i , k and m , 
 UVVUQ. It will be convenient to consider separately, first, the conditions under which 
 UV FC7=0, and, secondly, those under which UV+VUQ . To bring the subjects under a 
 familiar form, we may conceive of i , k , m as three vectors not coplanar. so that, writing 
 
 U = xi -}- yk -\- zm , V x i-\- y k-{- z m , 
 
 we have x , y , z , and x , y , z , the Cartesian coordinates of two points in space. [We might 
 imagine the space to be of the hyperbolic kind, and take the coefficients of j and I as coordinates of a 
 point on the quadric surface at infinity. But this would not further the purpose with which we now 
 introduce geometric conceptions.] But since we are to consider only such properties of U and V as 
 belong equally to all their numerical multiples, we may assume that they always lie in any plane 
 
 Ax + By + Cz 1 , 
 
 not passing through the origin ; and then x , y , z , and x , y , z , will be the homogeneous coordinates 
 of the two points U and V in that plane. Let it be remembered that, although i . k. mare vectors, yet 
 their multiplication does not at all follow the rule of quaternions, but that 
 
 l. mk = b 35 j-\-d 35 l . m 2 
 The condition that UV VUQ is expressed by the equations 
 
 (b 1 1 b sl )(xy x y) + (b l5 b 5l )(xz x z} + (b 35 & 53 ) (yz y z) Q, 
 (d 13 d 3l )(xy x y) + (d l5 d^}(xz x z)-\- (d 35 d 53 )(yz y z) = Q. 
 
 The first equation evidently signifies that for every value of U. Fmust be on a straight line, that this 
 line passes through U. and that it also passes through the point 
 
 The second equation expresses that the line between U and V contains the point 
 
 Q=(d 35 -d 53 )i+ (d 51 d l5 )k + (d l3 d 31 }m. 
 
 The two equations together signify, therefore, that C7and Frnay be any two points on the line between 
 the fixed points P and Q. Linear transformations of j and I may shift P and Q to any other situations 
 on the line joining them, but cannot turn the line nor bring the two points into coincidence. 
 The condition that UV+ VU= is expressed by the equations 
 
 2b 1 a-a; + 2b 3 yy + 2b 5 zz + (&is + b 31 )(xy + x y] + (6 15 + b 5l }(xz + x z] + (b 35 + b sa )(y*+y z) = 0, 
 
 2d^xx + 2d 3 yy -{-2d- } zz -f (d ls -f d sl )(a# + afy) + (di 5 + d- ol }(xz + x z) + (d 35 + d 53 )(yz +y z) = . 
 
 The first of these evidently signifies that for any position of Fthe locus of U is aline ; that U" being fixed 
 
 at any point on that line, V may be carried to any position on a line passing through its original position : 
 
 and that further, if U is at one of the two points where its line cuts the conic 
 
 b 5 2 2 + (Z> 18 + b 81 ) xy+ (6 15 +6 51 ) xz+ (& 85 
 
PEIRCE : Linear Associative Algebra. 
 
 95 
 
 = li = a 3 j = a 3 = Z/b = wiZ = M 
 = Tcmk = a? = 
 
 then F may be at an infinitely neighboring point on the same conic, so that tangents to the conic from 
 V cut the locus of C7" at their points of tangency . The second equation shows that the points U and V 
 have the same relation to the conic 
 
 3 y 2 + d-,z 2 + (d 13 + d 31 ) xy+ (d ls + d- al ) xz+ (d 35 +d 5B ) yz~ . 
 
 These conies are the loci of points whose squares contain respectively no term in j and no term in I . 
 Their four intersections represent expressions whose squares vanish. Hence, linear transformations of 
 j and I will change these conies to any others of the sheaf passing through these four fixed points. The 
 two equations together, then, signify that through the four fixed points, two conies can be drawn 
 tangent at U and V to the line joining these last points. 
 
 Uniting the conditions of UV VU and UV-\- VU . they signify that U and V are on the 
 line joining P and Q at those points at which this line is tangent to conies through the four fixed points 
 whose squares vanish. But if the algebra is pure, it is impossible to find two such points ; so that the 
 line between P and Q must pass through one of the four fixed points. In other words, the necessary 
 condition of the algebra being pure is that one and only one nilpotent expression in i . k. m . should be 
 a linear function of P and Q . 
 
 The two points P and Q together with the two conies completely determine all the constants of the 
 multiplication table. Let S and T be the points at which the two conies separately intersect the line 
 between P and Q . A. linear transformation ofj will move P to the point pP-\- (1 p) Q and will move 
 S to the point pS-{- (1 p) T , and a linear transformation of I will move Q and T in a similar way. The 
 points P and S may thus be brought into coincidence, and the point Q may be brought to the common 
 point of intersection of the two conies with the line from P to. Q . The geometrical figure determining 
 the algebra is thus reduced to a first and a second conic and a straight line having one common intersec 
 tion. This figure will have special varieties due to the coincidence of different intersections, etc. 
 
 There are six cases : [1], there is a line of quantities whose squares vanish and one quantity out of 
 the line ; [2], there are four dissimilar quantities whose squares vanish ; [3], two of these four quantities 
 coincide ; [4], two pairs of the four quantities coincide ; [5]. three of the four quantities coincide : [6], all 
 the quantities coincide. 
 
 We may, in every case, suppose the equation of the plane tobex-\-y-{-z=.l. 
 
 [1]. In this case, the line common to the two conies may be taken as y == , and the separate lines of 
 the conies as z := and x , respectively. We may also assume 2P -=.x-\-y and 2Q x + z . We 
 thus obtain the following multiplication table, where the rows and columns having j and I as their 
 arguments are omitted : 
 
 i k m 
 
 
 
 3Z -j 
 
 I 
 
 Bj+l 
 
 J 
 
 i-j o 
 
 [2]. In this case, we may take k as the common intersection of the two conies and the line, i , m , 
 and i k -f- in as the other intersections of the conies. We have Q = k . and we may write 
 
 rp rq}k + rgm . 
 
 TrP-}- (\r}Q 
 We thus obtain the following multiplication table : 
 
PEIRCE : Linear Associative Algebra. 
 
 There are two cases : 
 
 [242 3 1], when d 5 does not vanish ; 
 [242 4 ], when d$ vanishes. 
 
 [242 3 1]. The defining equation of this case can be reduced to 
 
 which gives 
 and if 
 
 i (k + b s i +pm) = I + bj = m z ; 
 
 3) 
 
 m 
 
 
 
 q(q + l)j+rq(rq l)l 
 
 [-2-p(p-3) + q(q+l)]j 
 + [ 2 rp(rp l} + rq(rq \}}l 
 
 q(q $)j+rq(rq l)l 
 
 
 
 p(p B)jrp(rp 1)1 
 
 [2-p(p+l) + a(g-8)]y + 
 [2 rp ( rp 1) + rq (rq 1)J| 
 
 p (p+l)j rp (rp 1) I 
 
 
 
 [3]. Let k be the double point common to the two conies, and let i and m be their other intersections. 
 Then all expressions of the form ku + t/fc are similar. The line between P and Q cannot pass through 
 k , because in that case all products would be similar. We may therefore assume that it passes through 
 i . Then, we have Q = i . we may assume S z= P = i - Jc+m , and we may write T rP + (1 r) Q 
 i _ r fc -\- rm . The equation of the common tangent to the conies at k may be written hx+ (I h)z Q. 
 Then the equations of the two conies are 
 
 hxy -f xz + (1 h)yz = , 
 hxy + (h + f hr)xz + (1 h)yz = . 
 
 We thus obtain the following multiplication table : 
 
 i k m 
 
 m 
 
 [4]. In this case we may take i and m as the two points of contact of the conies, k as P, and 
 t k + m as 7 1 . Then writing the equations of the two tangents 
 
 gy + z . x + hy . 
 
 g^ + a?z + %2 . 
 h 1) ?/ 2 -f gxy + xz + hyz . 
 
 and the multiplication table is as follows : 
 
 the two conies become 
 
PEIRCE : Linear Associative Algebra. 
 
 97 
 
 the substitution of 1& -f ^ + pin for /; and Z + Z> 5 / for ? is the same as to make 
 
 & 5 = rfg = . 
 
 o o 
 
 This gives 
 
 There are two cases : 
 
 [242 3 1 2 ], when b 3 does not vanish ; 
 [242*12], when 1 3 vanishes. 
 
 [242 8 ! 1 ]. The denning equation of this case can be .reduced to 
 
 *=,-. 
 
 (g + h i) I 
 
 ffl +to+D* 
 
 2? 
 
 gj -\- (g 1) I 
 
 
 
 V+(+D. 
 
 V 
 
 V+(-. 
 
 
 
 [5]. In this case, we may take k as the point of osculation of the conies and i as their point of inter 
 section. The line between P and Q must either, [51], pass through fc, or, [52], pass through i . 
 [51]. We may, without loss of generality, take 
 
 and the equations of the two conies are 
 
 2 2 + rxz = , rxy + 2qxz + 2yz = . 
 Then, the multiplication table is as follows : 
 
 i k m 
 
 
 
 
 
 ql 
 
 rl 
 
 
 
 I 
 
 rj+ql 
 
 I 
 
 j 
 
 [52]. We have Q i, we may take T 7 m, and we may assume 
 Then, we may write the equations of the two conies, 
 
 2i m and & 13 -f- &si 1 
 
 2z z + xy + xz 
 ra-?/ -f- (2 r) xz -}- ? 2 ?/2 = . 
 
 We thus obtain the following multiplication table : 
 
98 PEIRCE : Linear Associative Algebra. 
 
 This gives a quintuple algebra which can be called (&7/ 5 ), its multiplication 
 (Mi s ) i j ^ I m 
 
 table being * 
 
 j 
 
 o / 
 
 
 
 
 
 
 
 
 
 r 
 
 
 
 
 
 n 
 
 o j 
 
 
 
 ej+*l 
 
 
 
 
 
 I 
 
 
 
 
 
 A 
 
 o L5 
 
 r/7 
 
 I 
 
 
 
 rt 
 
 j_(r-2)Z 
 
 2/-H 
 
 
 
 (r-2)j 
 + (r 2 + l)Z 
 
 j-(r-2)Z 
 
 (r-2)j 
 + (r*-l)Z 
 
 V 
 
 m j (r - 
 
 [6]. The conies have but one point in common. This may be taken at k . We have Q = k , we may 
 take Ti and 2P 2S i+k. We may also take b l l. Then the equations of the two conies 
 
 may be written 
 
 1 2xii -\- 4:Qxz -4- 2n/z 
 
 r) xz + 2ryz =. . 
 
 We thus find this multiplication table : 
 
 j 
 
 j+l 
 
 (2q l)j+2(q + r p)l 
 
 j+l 
 
 
 
 (r+l)j+fi 
 
 (2q+l)+2(q+r+p)l 
 
 (r-l)/+ri 
 
 l>J4-(4-fpr*)l 
 
 If this analysis is correct, only three indeterminate coefficients are required for the multiplication 
 tables of this class of algebras. [C. S. P.] 
 
 * See last note. I do not give relative forms for this class of algebras, owing to the extreme case 
 with which they may be found. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 99 
 
 [242 3 12]. The defining equation of this case is 
 
 I* = . 
 There are two cases : 
 
 [242 3 121], when & 31 does not vanish ; 
 [242 8 12], when 6 31 vanishes. 
 
 [242*121]. The defining equation of this case can be reduced to 
 
 
 
 b sl = 1 . 
 
 This gives a quintuple algebra which may be called (&/ 5 ), its multiplication 
 
 table being 
 
 (M 5 ) i j Jc I m 
 
 i 
 
 j 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 o 
 
 
 
 
 
 ~k 
 
 j + al 
 
 
 
 
 
 
 
 bj+cl 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 m 
 
 a j 
 + VI 
 
 
 
 <0 
 
 +d l 
 
 
 
 I 
 
 [242 3 12 8 ]. The defining equation of this case is 
 
 There are two cases : 
 
 [242 3 12 2 1], when & 51 does not vanish ; 
 [242 3 12 3 ], when 6 51 vanishes. 
 
 [242 3 12 2 1]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (lj\), its multiplication 
 
 table being 
 
100 
 
 PEIRCE : Linear Associative Algebra. 
 i j k I m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 (ll 
 
 
 
 
 
 
 
 fy +cZ 
 
 
 
 
 
 
 
 
 
 
 
 j+M 
 
 
 
 Vj+il 
 
 
 
 I 
 
 [242 2 12 3 ]. The defining equation of this case is 
 
 which can always, in the case of a pure algebra, be reduced to 
 
 mi =. I . 
 
 This gives a quintuple algebra which may be called (&& 5 ), its multiplication 
 table being 
 
 (bk s ) i j It I m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 al 
 
 
 
 
 
 
 
 # + cZ 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 0* 
 
 a j 
 
 + VI 
 
 
 
 I 
 
 [242 4 ]. The defining equation of this case is 
 
PEIRCE : Linear Associative Algebra. 
 and it can be reduced to [242 3 1] unless 
 
 whence it may be assumed that 
 
 and since 
 
 when 
 
 101 
 
 k + lif = , 
 tf = . 
 
 it may also be assumed that 
 
 There are two cases : 
 
 [242 4 1], when b 3l does not vanish; 
 [242 5 ], when b 3l vanishes. 
 
 [242 4 1]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (5? 5 ), its multiplication 
 
 table being 
 
 i j If I m 
 
 m 
 
 j 
 
 
 
 z 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 j-l 
 
 
 
 
 
 
 
 aj-\-bl 
 
 
 
 
 
 
 
 
 
 
 
 3 
 
 
 
 aj+U 
 
 
 
 y 
 
 [242 5 ]. The defining equation of this case is 
 
 ki = L 
 
 There are two cases : 
 
 [242 5 1], when 6 35 does not vanish ; 
 [242 6 ], when J 35 vanishes. 
 
102 
 
 PEIRCE : Linear Associative Algebra. 
 
 [242 5 1]. The defining equation of this case can be reduced to 
 
 &S5 = 1 
 
 This gives a quintuple algebra which may be called (6w? 5 ), its multiplication 
 table being * 
 
 If I in 
 
 m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 V 
 
 
 
 
 
 
 
 j+al 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 bj cd 
 
 
 
 <9 
 
 [242 6 ]. The defining equation of this case is 
 
 There are two cases : 
 
 [242 6 1], when 6 53 does not vanish; 
 [242 7 ], when & 53 vanishes. 
 
 [242 8 1]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (bn s ), its multiplication 
 table being t 
 
 *This algebra is mixed. Namely, if &=j=l, it separates on substituting t j (1 &)i + fc. 
 k 1 = (l b) i + [a (1 b) + 1] k (1 &) 2 m ; but if b := 1 , it separates on substituting i { ai (a 2 + a 
 + c) fc + m , &i ai + g/c + ?/i . [C. S. P.] 
 
 t Substitute ij_ i k , fcj = ak + ???, , and the algebra separates. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 103 
 
 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -I 
 
 
 
 
 
 
 
 al 
 
 o 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 jal 
 
 
 
 9 
 
 [242 7 ]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (bo 5 ), its multiplication 
 table being * 
 
 (bo 5 ) i j It I m 
 
 j 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 -I 
 
 
 
 
 
 
 
 al 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 al 
 
 
 
 <y 
 
 
 1 
 
 
 
 [243]. The defining equations of this case are 
 
 ik = II im , 
 
 which give 
 
 * Substitute for m , ai + m , and the algebra separates. [C. S. P.] 
 
104 
 
 PEIRCE : Linear Associative Algebra. 
 
 There are two cases : 
 
 [2431], when Id = I, U = in , mi = ; 
 [2432], when Id = 7, H = mi = 0. 
 
 [2431]. The defining equations of this case are 
 
 Id = 1, li = m , mi = , 
 which give 
 
 l-j =. m , Ij = mj = = lie = ink = ? 2 = 1m = ml 
 H& = ild = ikm a 3 = r/. 34 = 35 , 
 
 ^ = yfem = c s m , = lm =c 3 = Jem , 
 = = bm + dm = 
 
 There are two cases : 
 
 [2431 2 ], when e s does not vanish ; 
 [24312], when e 3 vanishes. 
 
 [2431 2 ]. The defining equation of this case can be reduced to 
 
 This gives a quintuple algebra which may be called (6/> 5 ), its multiplication 
 table being * 
 
 * The structure of this algebra may be exhibited by putting k l =: 
 a~*m , when the multiplication table becomes 
 
 i i k I m 
 
 =i + a \j a fc, 1^j a V. 
 
 i 
 
 3 
 
 
 
 3 
 
 
 
 
 
 3 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 I 
 
 m 
 
 
 
 
 
 
 
 I 
 
 m 
 
 
 
 m 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 In relative form, i = B:C+ C:D, jB:D, k = A:B + C:D, l = A:C, mA\D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 (bp 5 ) i j ^ I n 
 
 105 
 
 
 
 
 
 i 
 
 j o o 
 
 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 Jc 
 
 I 
 
 m 
 
 -*H- 
 
 al-\-m 
 
 am 
 
 
 
 I 
 
 m 
 
 
 
 
 
 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 [24312]. The defining equation of this case is 
 
 This gives a quintuple algebra which may be called (635), its multiplication 
 table being* 
 
 (bq 5 ) i j If I m 
 
 i 
 
 j 
 Tf 
 I 
 m 
 
 !]. The definin 
 
 J : 
 
 
 
 
 
 000 
 
 
 
 I 
 
 I m \ J 
 
 am 
 
 m 
 
 
 
 000 
 
 
 
 g equations of this case are 
 ki = l, li = mi = , 
 
 *On substituting k l :=i a~ l k, l l -=ij a~ l l, 
 form given in the last note. [C. S. P.] 
 
 ^a" 1 ?^, this algebra reduces to (&p 5 ). in the 
 
106 
 
 PEIRCE : Linear Associative Algebra. 
 
 which give 
 
 = kj = Ij =. mj = lk = 1? = 1m = itf = a s , 
 
 = ikm = a 35 = kmi = c 35 = lm = e 35 = imk =. a 53 , 
 ml = c^l , = m s l = c 53 = ml = ml<? = e 53 ; 
 
 and it may be assumed that 
 which gives 
 
 1 = m , 
 =1 k 3 = km = mk = 
 
 There is then a quintuple algebra which may be called (br$), its multiplication 
 being * 
 
 (br 5 ) i j k 1 m 
 
 k 
 
 m 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 , 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [25]. The denning equations of this case are 
 
 = i 2 =j 2 = 1& = Z 2 = 77i 2 ij -\-ji ==. ik -\- ki = il -\- li 
 =jk -\- Itj =jl + Ij =jm -f- mj = Id -\- Ik = km -f- 
 
 and it may be assumed that 
 
 ij =nk = ji , il = m = li] 
 
 = im + mi , 
 
 =. Im + ml ; 
 
 *In relative form. i 
 [C. S. P.] 
 
 = D: E+ E \F , j=D:F, k=. A :B + B : jP+ C: E. l = C:F, m = A:F 
 
PEIRCE : Linear Associative Algebra. 107 
 
 which gives 
 
 = ik =. lei = jk = kj = im mi km = mk = lm = ml , 
 ijk = kl =. b^k -f- d u m = ilj = mj =jm , 
 = fm = d M jm = d u = kP = b 2 Jd = 6 84 = M= Ik =jm = mj , 
 / 2 7 = a 24 / = 24 , 
 
 so that it is easy to see that there is no pure algebra in this case. 
 
 SEXTUPLE ALGEBRA. 
 There are two cases : 
 
 [1], when there is an idempotent basis; 
 [2], when the algebra is nilpotent. 
 
 [1], The defining equation of this case is 
 
 There are 19 cases : 
 
 [I 2 ], when all the other units but i are in the first group ; 
 
 [12], when/, k, I, m are in the first and n in the second group ; 
 
 [13], when/, k and / are in the first and m and n in the second group ; 
 
 [14], when/, k and / are in the first, m in the second and n in the third group; 
 
 [15], when/ and k are in the first and /, m and n in the second group ; 
 
 [16], when / and k are in the first, / and m in the second and n in the third 
 
 group ; 
 [17], when/ and k are in the first, / in the second, m in the third, and n in the 
 
 fourth group ; 
 
 [18], when/ is in the first, and k, I, m and n in the second group ; 
 [19], when/ is in the first, k, I and m in the second, and n in the third group ; 
 [10 ], when / is in the first, k and / in the second, and m and n in the third group ; 
 [11 ], when / is in the first, k and / in the second, m in the third and n in 
 
 the fourth group ; 
 [12 ], when / is in the first, k in the second, I in the third and m and n in the 
 
 fourth group ; . 
 
108 PEIRCE : Linear Associative Algebra. 
 
 [13 ], when/, k, I , m and n are in the second group ; 
 
 [14 ], when/, k, I and m are in the second and m in the third group ; 
 
 [15 ], when/, k and I are in the second and m and n are in the third group; 
 
 [16 ], when /, k and / are in the second, m in the third, and n in the fourth 
 
 group ; 
 [17 ], when/ and k are in the second, / and m in the third, and n in the fourth 
 
 group ; 
 [18 ], when/ and k are in the second, I in the third, and m and n in the fourth 
 
 group ; 
 [19 ], when/ is in the second, k in the third, and /, m and n in the fourth group. 
 
 [I 2 ] ; The defining equations of this case are 
 ij =ji =-j , ik = ki = k , il = li = I , im = mi = m , in = ni = n . 
 
 and the 54 algebras of this case deduced from (5-5) to (br^) may be called (a 6 ) to 
 
 (,).* 
 
 [12]. The defining equations of this case are 
 
 ij =ji / , ik ki = k , il = li = I , im = mi = m , in = n , ni , 
 
 which give 
 
 =jn /a/ = kn = nk = ln = nl = mn = nm ==. n z , 
 
 so that there is no pure algebra in this case. 
 
 [13]. The defining equations of this case are 
 
 ij =ji = j, ik =. ki = k , il = li I , im = m , in = n , 7m = m = . 
 
 There are four cases, which correspond to relations between the units of the 
 first group similar to those of the quadruple algebras (a 4 ) , (6 4 ) , (c 4 ) or (c 4 ) . 
 [131]. The defining equations of this case are 
 
 and, in the result, we obtain 
 
 jm = n , jn km = kn = Im =. In = . 
 
 * The multiplication tables of these algebras, formed from the nilpotent quintuple algebras, in the 
 same manner in which the first class of quintuple algebras are formed from the nilpotent quadruple 
 algebras, have been omitted. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra.. 
 
 109 
 
 This gives a sextuple algebra which may be called (bc 6 ), of which the multipli 
 cation table is * 
 
 (bc e ) i j Tf I m n 
 
 m 
 
 i j 
 
 If I 
 
 j 
 
 m n 
 
 j * 
 
 I 
 
 , 
 
 n 
 
 Jc I 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 i 
 
 
 
 
 
 
 
 
 
 
 
 [132]. The defining equations of this case are 
 z = Jc = l\ Ijak, jk=jl=fy = 
 
 which give 
 
 There are two cases : 
 
 km = kn . 
 
 [1321], when <? 24 does not vanish ; 
 [132 2 ], when e 24 vanishes. 
 
 [1321]. The defining equation of this case can be reduced to 
 
 jn = m , 
 which gives 
 
 =jm = Im . 
 
 This gives a sextuple algebra which may be called (&c 6 ), of which the multipli 
 cation table is f 
 
 * In relative form, i A: A + B :B + C : (7+ D:D,j = A:B+B: C+ C:D. k = A: C + B:D. 
 l A:D,m B:E,n A:E. [C. S. P.] 
 
 t This algebra is distinguishable into two. in the same manner as (c 3 ). Namely, if a= 2 . on sub 
 stituting l^lj. we have Z 2 = . jl k. lj= k, and the multiplication table is otherwise un 
 changed. Otherwise, on substituting j\ Z-f cj, l l =zk + c~ l j . where 2e adr^/a 2 4. we have 
 J 2 =l 2 = 0.jl=(l c 2 )k. lj=(\ c--)k,jn=(b-\-c)k. In (b-f c- 1 ) k. and otherwise the multi 
 plication table is unchanged. The following is a relative form for the first variety: i=.A :A + B :B 
 
 For the second variety, i = A : A + B : B+ C : C -f D : D, j =. A : B+ (1 c 3 ) C:D. k = A :D, l = A:C 
 + (1 c~*)B:D. m = A:E. n= (b + c) B :E+ (b + c- 1 ) C : E. [C. S. P.] 
 
MO 
 
 PEIRCE : Linear Associative Algebra. 
 (bd 6 ) i j k I m 
 
 i 
 
 j 
 * 
 I 
 m 
 n 
 
 i 
 
 1 
 j 
 
 & 
 
 I 
 
 m 
 
 n 
 
 j 
 
 Tf 
 
 
 
 
 
 
 
 m 
 
 Tt 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 ok 
 
 
 
 A- 
 
 
 
 bm 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [132 2 ]. The defining equation of this case is 
 
 jn = > 
 
 and there is no pure algebra in this case. 
 
 [13 2 ]. The defining equations of this case are 
 
 which give 
 
 km = kn = 
 
 There is a sextuple algebra in this case which may be called (&e e ), of which the 
 multiplication table is * 
 
 *This algebra may be a little simplified by substituting jlforj. In relative form, i A: A 
 + B:B+ C:C+D:Z>, j = A:D+B:C, k A:C, l = A:B, m = A:E, n bB:E+aD:E. 
 [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 (be 6 ) i j k I m n 
 
 111 
 
 i 
 
 i 
 
 j 
 
 If 
 
 / 
 
 m 
 
 n 
 
 j 
 
 j 
 
 If 
 
 
 
 
 
 
 
 am 
 
 Jc 
 
 If, 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 I 
 
 Tf 
 
 
 
 
 
 
 
 bm 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [134], The defining equations of this case are 
 
 jk = kj= I, /= tf=jJc. = kj=kl=lk= / 2 = 0. 
 
 There is a sextuple algebra in this case which may be called (fe/ e ), of which the 
 multiplication table is * 
 
 m n 
 
 i 
 
 i 
 
 j 
 
 Tf 
 
 I 
 
 m 
 
 n 
 
 j 
 
 j 
 
 
 
 
 
 Tf 
 
 
 
 m 
 
 Jc 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 I 
 
 Jc 
 
 
 
 
 
 
 
 am 
 
 m 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * In relative form, t = 4 :4 + J5:B+ C:C+ D:D, j=A:BC:D, k A:D, l = A:C+B:D, 
 = A:E, n = B :E+aC:E . [C. S. P.] 
 
112 PEIRCE : Linear Associative Algebra. 
 
 [14]. The defining equations of this case are 
 
 ij ji j , ik = M = k , il = li = I , im . =. m , ni = n , mi = in = , 
 which give 
 =jm =jn = km = kn = lm = ln = mj = nj = mk = nk = ml=.nl = m z =nm =n z 
 
 There are four cases denned as in [13]. 
 
 [141]. The defining equations of this case are 
 
 which give 
 
 mn = 
 
 There is a sextuple algebra which may be called 
 cation table is* 
 
 i Jf I 
 
 m 
 
 n 
 
 of which the multipli- 
 
 m n 
 
 i 
 
 j 
 
 k 
 
 
 
 m 
 
 
 
 j 
 
 k 
 
 I 
 
 
 
 
 
 
 
 Jf 
 
 I 
 
 
 
 
 
 
 
 
 
 I 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 [142]. The denning equations of this case are the same as in [132], 
 
 which give 
 
 mn = 
 
 *In relative form, i = A:A + B:B+C:C+D:D,j= A :B+ B :C+ C :D . k = A:C+ B :D 
 A:D.m A:E,n = E:D. [C. S. P.] 
 
PEIRCE : Linear Associative Algebra. 
 
 113 
 
 There is a sextuple algebra which may be called (bh 9 ), of which the multipli 
 cation table is * 
 
 i j k I m n 
 
 k 
 
 m 
 
 n 
 
 i 
 
 . 
 .1 
 
 k 
 
 I 
 
 m 
 
 
 
 j 
 
 Tf 
 
 
 
 
 
 
 
 
 
 ~k 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 ok 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 k 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 [143]. The denning equations of this case are the same as in [13 2 ]. There 
 is a sextuple algebra which may be called (U,\ of which the multiplication 
 table isf 
 
 (fo e) i j If I m n 
 
 i 
 
 j 
 
 * 
 
 / 
 
 m 
 
 1 
 
 
 j 
 
 k 
 
 
 
 
 
 
 
 
 
 A- 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 * 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 * This algebra has two varieties, analogous to those of (c 3 ). The first is, in relative form, i = A-A 
 
 + B:B+C:C+D:D,j=A:B + B:C+A:D,lc=A:C.l=-A:B + D:C,m = A:E.n = E : .C. 
 m relative form is the same except that./ A:B-\- 6- Z> : C, l = A:D bB C [CSP] 
 
 g ? ra maj be Sligh% sim P lified b y P uttiQ g /- 1 forj. Then, in relative form. i=A A 
 :C. j = B : C . k-A:C, l-A:B. m = A:D. n = D:C. [C. S. P.] 
 
114 
 
 PEIRCE : Linear Associative Algebra. 
 
 [14 3 ]. The defining equations of this case are the same as in [134]. There 
 is a sextuple algebra which may be called (6/ 6 ), of which the multiplication 
 table is * 
 
 m n 
 
 i 
 
 j 
 
 It I 
 
 m 
 
 
 
 j 
 
 
 
 A- 
 
 
 
 
 
 k 
 
 
 
 
 
 o 
 
 
 
 ^ 
 
 fc 
 
 () 
 
 
 
 
 
 
 
 
 
 
 
 1 
 
 
 
 If 
 
 
 
 
 
 
 
 
 
 
 
 [15]. The defining equations of this case are 
 
 ij = ji = j , ik = ki = k, il=J, im = m , wi = w , /i = wii = ni = , 
 which give 
 
 / = k, =jk = kj = tf = /; = // = /* = lm = 1n= mj mk = ml = in 1 
 
 =. mn = nj = nk = nl = nm = ?i 2 . 
 
 There is a sextuple algebra which may be called (/>7,- 6 ), of which the multipli 
 cation table isf 
 
 *t = A\A + B:B+ C\C+D:D, jA:B C:D, k A:D. l A: 
 E:D. [C. S. P.] 
 
 tin relative form, t = A : A + B : B+ C : C, j=A:B + B:C, k A : C. 
 A:D. [C. S. P.] 
 
 , m A:E. 
 = C : D, m = B :D. 
 
PEIRCE : Linear Associative Algebra. 
 (bk 6 ) i j k I m n 
 
 115 
 
 k 
 
 tn 
 
 n 
 
 i 
 
 j 
 
 k 
 
 I 
 
 m 
 
 ft 
 
 j 
 
 k 
 
 
 
 m 
 
 n 
 
 
 
 k 
 
 
 
 
 
 it 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [16]. The defining equations of this case are 
 
 ij =ji J , ik = ki = k , ^7 = / , im = m , ni n , U = mi =. in = , 
 which give 
 
 y* = k , =-jk =jn = kj = k* = kn = Ij = lk = I* = lm = mj = mk =. nl = m 
 
 =. nj = nk = nl = nm = ri*. 
 
 There is a sextuple algebra which may be called (&/ 6 ), of which the multipli 
 cation table is * 
 
 (bl 6 ) i j k I m n 
 
 k 
 
 n 
 
 i 
 
 j 
 
 K 
 
 I 
 
 m 
 
 
 
 j 
 
 k 
 
 
 
 m 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 o 
 
 
 
 
 
 k 
 
 
 
 
 
 
 
 
 
 
 
 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 *In relative form. i = A: A + B : B + C : C, jA:B + B:C. k A:C, l = B: 
 = A:D, n E :C. [C. S. P.] 
 
116 PEIRCE : Linear Associative Algebra. 
 
 [17]. The defining equations of this case are 
 ij =ji ==j , ik Id = k , il =. I , mi := m , /i = iwi = in = m = . 
 
 There is no pure algebra in this case. 
 
 [18]. The defining equations of this case are 
 
 ij = ji = j, ik = k , il = I , im = m , in = n , ki = li= mi = ni = . 
 
 There is no pure algebra in this case. 
 
 [19]. The defining equations of this case are 
 
 ij =ji = j, ik = k , il = I , im =. m , ni = n , in = kl = li = ni = n . 
 
 There is no pure algebra in this case. 
 
 [10 ]. The defining equations of this case are 
 
 ij =-ji = j, ik lt, il = I , mi = m , ni = n , i/n = ^w = & = li = . 
 
 There is no pure algebra in this case. 
 
 [11 ]. The defining equations of this case are 
 
 ij =-ji =-j , ik = k, il =. I , mi = m , im = Zt = m = m = 0. 
 
 There is no pure algebra in this case. 
 
 [12 ]. The defining equations of this case are 
 
 ij =ji j, ik = k, li / , il =. im = in =.ld =- mi = ni = . 
 
 There is no pure algebra in this case. 
 
 [13 ]. The defining equations of this case are 
 
 ij j , ik == k , il = I , im = m , in = n , ji = ki li = mi = ni = . 
 
 There is no pure algebra in this case. 
 
 [14 ]. The defining equations of this case are 
 
 ij =j , ik = k , il = I , im = m , ni = n , /& = /M = li = mi = in = Q . 
 
 There is no pure algebra in this case. 
 
 [15 ]. The defining equations of this case are 
 
 ij = j , ik = k , il = I, mi =. m , ni = n , im = in = ji = ki = ?i =r . 
 There is no pure algebra in this case. 
 
PBIRCE : Linear Associative Algebra. 117 
 
 [16 ]. The defining equations of this case are 
 
 *} = y ik = k, il = I , mi = m , im = in = jk = kl = li = ni = . 
 There is no pure algebra in this case. 
 
 [17 ]. The defining equations of this case are 
 
 ij =y , ik = k , U=l, mi = m , ?7 = iwi = m = ji =. ki = ni = . 
 There is no pure algebra in this case. 
 
 [18 ]. The defining equations of this case are 
 
 if =y , ik = k , li l, ji = ki = il = im = in = mi = m = . 
 There are six cases : 
 
 [18 1], when m* = m , mn = n , nm = , 
 [18 2], when m* == m , mn = , nm = n , 
 [18 3], when w 2 = n , mn = nm = , ri* = w , 
 [18 4], when m z = m , mn = T^TZ = yi 2 1= , 
 [18 5], when m 2 = n , m 3 = , 
 [18 6], when w 2 = ^ 2 = 0. 
 
 [18 1]. The defining equations of this case are 
 
 m z =. m , mn = n , nm . 
 There are two cases : 
 
 [18 1 2 ], wheniZ = 0; 
 [1812], when ml=l. 
 [18 1 2 ]. The defining equation of this case is 
 
 mf=0. 
 There is no pure algebra in this case. 
 
 [18 12]. The defining equation of this case is 
 
 ml I. 
 There are two cases : 
 
 [18 121], when >!=./; 
 [1812 2 ], whenym = 0. 
 
 [18 121]. The defining equation of this case is 
 
 jm =j. 
 
 There is a sextuple algebra which may be called (6m 6 ), of which the multipli 
 cation table is * 
 
 *In relative form, i = A:A,j=A:B, k = A:C, l=.B:A, m = B:B, n = B:C. [C. S. P.] 
 
118 
 
 PEIRCE : Linear Associative Algebra. 
 (bm 6 ) i j k I m n 
 
 m 
 
 n 
 
 i 
 
 
 > 
 
 
 
 
 
 
 
 
 
 
 
 
 
 $ 
 
 J 
 
 * 
 
 
 
 
 
 
 
 o 
 
 o 
 
 
 
 I 
 
 m 
 
 n 
 
 
 
 
 
 
 
 
 
 
 
 
 
 I 
 
 m 
 
 w 
 
 
 
 
 
 
 
 
 
 
 
 
 
 [18 12 2 ]. The defining equation of this case is 
 
 jm =. . 
 
 There is no pure algebra in this case. 
 
 [18 2]. The defining equations of this case are 
 
 m? =. m , mn = , nm = n . 
 There are two cases : 
 
 [18 21], when ml = I; 
 
 [18 2 3 ], whenml=Q. 
 
 [18 21]. The defining equation of this case is 
 
 ml I . 
 There are two cases : 
 
 [18 2P], whenyw=y; 
 
 [18 212], whenym=0. 
 
 [18 21 2 ]. The defining equation of this case is 
 
 jm=j. 
 
 There is no pure algebra in this case. 
 
 [18 212]. The defining equation of this case is 
 
 jm = . 
 There is no pure algebra in this case. 
 
PEIRCE : Linear Associative Algebra. 119 
 
 [18 2 2 ]. The defining equation of this case is 
 
 ml = 0. 
 
 There is no pure algebra in this case. 
 
 [18 3]. The defining equations of this case are 
 
 m 2 = m , mn = nm =. , n 2 = n . 
 
 There is no pure algebra in this case. 
 
 [18 4], The defining equations of this case are 
 
 m 3 = m , mn = nm = n z = . 
 There are two cases : 
 
 [18 41], when/wi =j; 
 
 [18 42], when/m = 0. 
 
 [18 41]. The defining equation of this case is 
 
 jm ./. 
 
 There is no pure algebra in this case. 
 
 [18 42]. The defining equation of this case is 
 
 jm =. . 
 
 There is no pure algebra in this case. 
 
 [18 5]. The defining equations of this case are 
 
 m 2 n , m 3 . 
 
 There is no pure algebra in this case. 
 
 [18 6]. The defining equations of this case are 
 
 m 2 n* = . 
 
 There is no pure algebra in this case. 
 
 [19 ]. The defining equations of this case are 
 
 ij =j , ki =. k , ji = ik = il = im =: in = li = mi = ni = . 
 
 There is no pure algebra in this case. 
 
 [2]. The algebras belonging to this case are not investigated, because it is 
 evident from 69 that they are rarely of use unless combined with an idempo- 
 tent basis, so as to give septuple algebras. 
 
 NATURAL CLASSIFICATION. 
 
 There are many cases of these algebras which may obviously be combined 
 into natural classes, but the consideration of this portion of the subject will be 
 reserved to subsequent researches. 
 
120 PEIRCE : Linear Associative Algebra. 
 
 ADDENDA. 
 I. 
 
 On the Uses and Transformations of Linear Algebra. 
 BY BENJAMIN PEIRCE. 
 
 [Presented to the American Academy of Arts and Sciences, May 11, 1875.] 
 
 Some definite interpretation of a linear algebra would, at first sight, appear 
 indispensable to its successful application. But on the contrary, it is a singular 
 fact, and one quite consonant with the principles of sound logic, that its first and 
 general use is mostly to be expected from its want of significance. The interpre 
 tation is a trammel to the use. Symbols are essential to comprehensive argument. 
 The familiar proposition that all A is B, and all B is G, and therefore all A is C, 
 is contracted in its domain by the substitution of significant words for the 
 symbolic letters. The A, B, and G, are subject to no limitation for the purposes 
 and validity of the proposition ; they may represent not merely the actual, but 
 also the ideal, the impossible as well as the possible. In Algebra, likewise, the 
 letters are symbols which, passed through a machinery of argument in accord 
 ance with given laws, are developed into symbolic results under the name of 
 formulas. When the formulas admit of intelligible interpretation, they are 
 accessions to knowledge ; but independently of their interpretation they are 
 invaluable as symbolical expressions of thought. But the most noted instance 
 is the symbol called the impossible or imaginary, known also as the square root 
 of minus one, and which, from a shadow of meaning attached to it, may be 
 more definitely distinguished as the symbol of semi-inversion. This symbol is 
 restricted to a precise signification as the representative of perpendicularity in 
 quaternions, and this wonderful algebra of space is intimately dependent upon 
 the special use of the symbol for its symmetry, elegance, and power. The 
 immortal author of quaternions has shown that there are other significations 
 which may attach to the symbol in other cases. But the strongest use of the 
 symbol is to be found in its magical power of doubling the actual universe, and 
 
PEIRCE : Linear Associative Algebra. 121 
 
 placing by its side an ideal universe, its exact counterpart, with which it can be 
 compared and contrasted, and, by means of curiously connecting fibres, form 
 with it an organic whole, from which modern analysis has developed her 
 surpassing geometry. The letters or units of the linear algebras, or to use the 
 better term proposed by Mr. Charles S. Peirce, the vids of these algebras, are 
 fitted to perform a similar function each in its peculiar way. This is their 
 primitive -and perhaps will always be their principal use. It does not exclude 
 the possibility of some special modes of interpretation, but, on the contrary, a 
 higher philosophy, which believes in the capacity of the material universe for 
 all expressions of human thought, will find, in the utility of the vids, an indica 
 tion of their probable reality of interpretation. Doctor Hermann Hankel s 
 alternate numbers, with Professor Clifford s applications to determinants, are a 
 curious and interesting example of the possible advantage to be obtained from 
 the new algebras. Doctor Spottiswoode in his fine, generous, and complete 
 analysis of my own treatise before the London Mathematical Society in Novem 
 ber of 1872, has regarded these numbers as quite different from the algebras 
 discussed in my treatise, because they are neither linear nor limited. But there 
 is no difficulty in reducing them to a linear form, and, indeed, my algebra (<? 3 ) is 
 the simplest case of Hankel s alternate numbers ; and in any other case, in which 
 n is the number of the Hankel elements employed, the complete number of vids 
 of the corresponding linear algebra is ( 2 n 1 . The limited character of the 
 algebras which I have investigated may be regarded as an accident of the mode 
 of discussion. There is, however, a large number of unlimited algebras 
 suggested by the investigations, and Hankel s numbers themselves would have 
 been a natural generalization from the proposition of 65 of my algebra. * 
 Another class of unlimited algebras, which would readily occur from the 
 inspection of those which are given, is that in which all the powers of a vid are 
 adopted as independent vids, and the highest power may either be zero, or unity, 
 or the vid itself, and the zero power of the fundamental vid, i. e. unity itself, 
 may also be retained as a vid. But I desire to draw especial attention to that 
 class, which is also unlimited, and for which, when it was laid before the math 
 ematical society of London in January of 1870, Professor Clifford proposed the 
 appropriate name of quadrates. 
 
 * This remark is not intended as a foundation for a claim upon the Hankel numbers, which were 
 published in 1867. three years prior to the publication of my own treatise. B. P. [They were given 
 much earlier under the name of clefs by Cauchy, and (substantially) at a still earlier date by Grassmann. 
 C. S. P.] 
 
122 PEIRCE: Linear Associative Algebra. 
 
 Quadrates. 
 
 The best definition of quadrates is that proposed by Mr. Charles S. Peirce. 
 If the letters A, B, C, etc., represent absolute quantities, differing in quality, 
 the vids may represent the relations of these quantities, and may be written in 
 the form 
 
 (A:A)(A:B)(A:O) . . . (B : A) (B : B) . . . (C:A), etc. 
 
 subject to the equations 
 
 (A :B)(B:G) = (A: C) 
 
 (A:B)(C:D) = Q. 
 
 In other words, every product vanishes, in which the second letter of the multi 
 plier differs from the first letter of the multiplicand ; and when these two letters 
 are identical, both are omitted, and the product is the vid which is compounded 
 of the remaining letters, which retain their relative position. 
 
 Mr. Peirce has shown by a simple logical argument that the quadrate is the 
 legitimate form of a complete linear algebra, and that all the forms of the 
 algebras given by me must be imperfect quadrates, and has confirmed this 
 conclusion by actual investigation and reduction. His investigations do not 
 however dispense with the analysis by which the independent forms have 
 been deduced in my treatise, though they seem to throw much light upon their 
 probable use. 
 
 Unity. 
 
 The sum of the vids (A : A), (B :B), (C : G), etc., extended so as to include 
 all the letters which represent absolute quantities in a given algebra, whether it 
 be a complete or an incomplete quadrate, has the peculiar character of being 
 idempotent, and of leaving any factor unchanged with which it is combined as 
 multiplier or multiplicand. This is the distinguishing property of unity, so that 
 this combination of the vids can be regarded as unity, and may be introduced 
 as such and called the vid of unity. There is no other combination which 
 possesses this property. 
 
 But any one of the vids (A: A), (B : B), etc., or the sum of any of these 
 vids is idempotent. There are many other idempotent combinations, such as 
 
 (A:A) + x(A:B), y (A : B) + (B : B), 
 ^(A:A)-\-^(A:B) + ^(B:A) + ^(B:B), 
 
 which may deserve consideration in making transformations of an algebra 
 preparatory to its application. 
 
PEIRCE : Linear Associative Algebra,. 123 
 
 Inversion. 
 
 A vid which differs from unity, but of which the square is equal to unity, 
 may be called a vid of inversion. For such a vid when applied to some other 
 combination transforms it ; but, whatever the transformation, a repetition of the 
 application restores the combination to its primitive form. A very general form 
 of a vid of inversion is 
 
 (A : A) (B : B) (G : C) etc., 
 
 in which each doubtful sign corresponds to two cases, except that at least one of 
 the signs must be negative. The negative of unity might also be regarded as a 
 symbol of inversion, but cannot take the place of an independent vid. Besides 
 the above vids of inversion, others may be formed by adding to either of them 
 a vid consisting of two different letters, which correspond to two of the one- 
 lettered vids of different signs ; and this additional vid may have any numerical 
 coefficient whatever. Thus 
 
 (A:A) + (B:B) (C:C) + x(A: C) + y (B : C) 
 
 is a vid of inversion. 
 
 The new vid which Professor Clifford has introduced into his biquaternions 
 is a vid of inversion. 
 
 Sem i- Inversion. 
 
 A vid of which the square is a vid of inversion, is a vid of semi-inversion. 
 A very general form of a vid of semi-inversion is 
 
 (A:A) (B:B)J(C: C) etc. 
 
 in which one or more of the terms (A: A), (B : B), etc., have J for a coeffi 
 cient. The combination 
 
 (A:A) J(B : B) + x(A : B) + etc. 
 
 is also a vid of semi-inversion. With the exception of unity, all the vids of 
 Hamilton s quaternions are vids of semi-inversion. 
 
 The Use of Commutative Algebras. 
 
 Commutative algebras are especially applicable to the integration of 
 differential equations of the first degree with constant coefficients. If i, j, k, 
 
124 
 
 PEIRCE : Linear Associative Algebra. 
 
 etc., are the vids of such an algebra, while x, y, z, etc., are independent 
 variables, it is easy to show that a solution may have the form F (xi + yj -\- zk 
 -fete.), in which .Pis an arbitrary function, and i, ./, k, etc., are connected by 
 some simple equation. This solution can be developed into the form 
 
 F(xi + yj + zk + etc.) = Mi + Nj + Pk + etc. 
 
 in which M, N, P, etc., will be functions of x, y, z, etc., and each of them is a 
 solution of the given equation. Thus in the case of Laplace s equation for the 
 potential of attracting masses, the vids must satisfy the equation 
 
 +# = 0. 
 
 The algebra ( 3 ) of which the multiplication table is 
 
 i j k 
 
 i 
 
 j 
 
 n 
 
 j 
 
 * 
 
 
 
 k 
 
 
 
 
 
 may be used for this case. Combinations i l7 j\, ^ of these vids can be found 
 which satisfy the equation 
 
 *!+/! + *! = o, 
 
 and if the functional solution 
 
 F(vii+yji + *ki) 
 
 is developed into the form of the original vids 
 
 M+ Nj+Pk, 
 
 M, N, and P will be independent solutions, of such a kind that the surfaces for 
 which N and P are constant will be perpendicular to that for which M is 
 constant, which is of great importance in the problems of electricity. 
 
 The Use of Mixed Algebras. 
 
 It is quite important to know the various kinds of pure algebra in making 
 a selection for special use, but mixed algebras can also be used with advantage 
 
PEIRCE: Linear Associative Algebra. 125 
 
 in certain cases. Thus, in Professor Clifford s biquaternions, of which he has 
 demonstrated the great value, other vids can be substituted for unity and his 
 new vid, namely their half sum and half difference, and each of the original 
 vids of the quaternions can be multiplied by these, giving us two sets of vids, 
 each of which will constitute an independent quadruple algebra of the same 
 form with quaternions. Thus if i,j, k, are the primitive quaternion vids and 
 ID the new vid, let 
 
 tt! = iz (1 + w) . a 8 = 2 (1 w). 
 
 i^ rz: a-ii . 4 = a z i . 
 
 Then since 
 
 a i a i a l = 
 
 
 
 ^ = 
 = = 
 
 in which M l denotes any combination of the vids of the first algebra, and N z any 
 combination of those of the second algebra. It may perhaps be claimed that 
 these algebras are not independent, because the sum of the vids a x and a 2 is 
 absolute unity. This, however, should be regarded as a fact of interpretation 
 which is not apparent in the defining equations of the algebras. 
 
 II. 
 
 On the Relative Forms of the Algebras. 
 BY C. S. PEIRCE. 
 
 Griven an associative algebra whose letters are i, /,./<", /, etc., and whose 
 
 multiplication table is 
 
 i* = a u i + b u j -\- c n k + etc.* 
 
 ij = a i& + &12/ + c lz k + etc. 
 ji = a^i + b 2l j + c^k + etc., 
 etc., etc. 
 
 I proceed to explain what I call the relative form of this algebra. 
 
 I have used a l l , etc., in place of the a l , etc.. used by my father in his text. 
 
126 PEIRCE : Linear Associative Algebra. 
 
 Let us assume a number of new units, A, /, J, K, L, etc., one more in 
 number than the letters of the algebra, and every one except the first, A , 
 corresponding to a particular letter of the algebra. These new units are sus 
 ceptible of being multiplied by numerical coefficients and of being added 
 together ; * but they cannot be multiplied together, and hence are called non- 
 relative units. 
 
 Next, let us assume a number of operations each denoted by bracketing 
 together two non-relative units separated by a colon. These operations, equal in 
 number to the square of the number of non-relative units, may be arranged as 
 follows : 
 
 (A :A) (A: /) (A :J) (A: K\ etc. 
 
 (I: A) (1:1) (I:J) (I : K), etc. 
 
 (J:A) (/:/) (J:J) (J : K\ etc. 
 
 Any one of these operations performed upon a polynomial in non-relative units, 
 of which one term is a numerical multiple of the letter folio wing the colon, gives the 
 same multiple of the letter preceding the colon. Thus, (I:J) (/+ bJ+ cK) = bI.-f 
 These operations are also taken to be susceptible of associative combination. 
 Hence (/ : J) (J: K) = (/ : K) for (J : K) K = J and (I:J)J= I, so that 
 (I:J)(J:K)K=I. And (I:J)(K:L) = Q-, for (K: L) L = K and (/: J) K 
 = (/:/) (Q.J+K) = 0./= 0. We further assume the application of the 
 distributive principle to these operations ; so that, for example, 
 
 \(I:J) + (K-.J) + (K:L)\(aJ+ bL) = aJ+ (a + b)K. 
 
 Finally, let us assume a number of complex operations denoted by i , /, #, 
 I , etc., corresponding to the letters of the algebra and determined by its multi 
 plication table in the following manner : 
 
 i = (I: A) + flll (/ : /) + b n (J: I) + c n (K : I) + etc. 
 
 .+ au(I:J) + b n (J: J) + Cu(K: J) + etc. 
 
 + a ls (/ : K) + b w (J: K) + Cl8 (JT: K) + etc. + etc. 
 /= (J:A)+ a n (I:l) + b^(J : I) + % (JT: /) + etc. 
 
 + 0^(1 :J) + bn(J:J) H- c^(K:J) + etc. 
 
 -f a 23 (/: K) + bn(J:K) + c zs (K: K) + etc. + etc. 
 A/=: etc. 
 
 * Any one of them multiplied by gives . t If b = , of course the result is . 
 
PEIRCB : Linear Associative Algebra. 127 
 
 Any two operations are equal which, being performed on the same operand, 
 invariably give the same result. The ultimate operands in this case are the non- 
 relative units. But any operations compounded by addition or multiplication 
 of the operations i f , /, #, etc., if they give the same result when performed 
 upon A, will give the same result when performed upon any one of the non- 
 relative units. For suppose i j A = VI! A . We have 
 
 i j A = i J = a n l + b lz J + c n K + etc. 
 
 etc. 
 
 so that a^ = a M , b 12 = b 34 , c 12 = c 34 , etc., and in our original algebra ij = Id . 
 Hence, multiplying both sides of the equation into any letter, say m, ijm = Urn 
 But 
 
 ijm = i (a^i + b m j + c 25 7c + etc.) = (a n a 2 , + a 12 6 25 + 13 c 25 + etc.)* 
 
 + (&1125 + ^12^25 H- ^13^25 + etC.)y + GtC. 
 
 But we have equally 
 
 i j m A = ( n a 25 H- a 12 & 25 + c/ 13 c 25 + etc.)/-f (A n 25 + /, 12 6 25 + i lst . gB + etc.) J+ etc. 
 
 So that i j m A = Mm A. Hence, t yjf = MM. It follows, then, that if i j A 
 = &W, then * / into any non-relative unit equals M into the same unit, so that 
 i f= Ml . We thus see that whatever equality subsists between compounds of 
 the accented letters i , /, //, etc., subsists between the same compounds of the 
 corresponding unaccented letters i,j, k, so that the multiplication tables of the 
 two algebras are the same.* Thus, what has been proved is that any associ- 
 tive algebra can be put into relative form, *. e. (see my brochure entitled 
 A brief Desertion of the Algebra of Relatives) that every such algebra may be 
 represented by a matrix. 
 
 Take, for example, the algebra (bd< 5 ). It takes the relative form 
 
 j = (J : A) , 
 
 l=(L:A) + (J:K), m = ( M : A) + (r 2 - - 1) (J : I) - (L : K) - r 2 (J : M) . 
 
 * A brief proof of this theorem, perhaps essentially the same as the above, was published by me in 
 the Proceedings of the American Academy of Arts and Sciences, for May 11. 1875. 
 
128 
 
 PEIRCE: Linear Associative Algebra. 
 
 This is the same as to say that the general expression xi + yj + zk 
 of this algebra has the same laws of multiplication as the matrix 
 
 o, 
 
 0, 
 
 o, 
 
 o, 
 
 o, 
 
 o, 
 
 X, 
 
 o, 
 
 o, 
 
 2, 
 
 o, 
 
 o, 
 
 
 X -|- 2 
 
 
 
 
 
 lh 
 
 _i_ /2 |\,y 
 
 o, 
 
 u, 
 
 V2, 
 
 2 
 
 2, 
 
 ^ o, 
 
 o, 
 
 0. 
 
 o, 
 
 o, 
 
 U, 
 
 rz, 
 
 o, 
 
 cc v , 
 
 o, 
 
 1 
 
 V , 
 
 0, 
 
 o, 
 
 2, 
 
 o, 
 
 0. 
 
 Of course, every algebra may be put into relative form in an infinity of 
 ways ; and simpler ways than that which the rule affords can often be found. 
 Thus, for the above algebra, the form given in the foot-note is simpler, and so is 
 the following : 
 
 i=(B:A) + (C:B) + (F:D) + (CiE), j=(C:A), 
 k = (2):A) + (E:D)+ (C:B) + r(F:B) + r(C : F), 
 l = (]T.A) + (C:D), m = (E:A) + (V*l)(C:E)(B\A) (F:D}(C:E). 
 
 These different forms will suggest transformations of the algebra. Thus, the 
 relative form in the foot-note to (H) suggests putting 
 
 m m j 
 
 when we get the following multiplication table, where p is put for r 
 
 i j Jc I m 
 
 
 
 
 
 
 
 
 
 j 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 i 
 
 i 
 
 1 
 
 
 
 
 
 9J 
 
 
 
 
 
 fr 
 
 
 
 9 l 
 
 
 
 .; 
 
PEIRCE : Linear Associative Algebra. 129 
 
 Ordinary algebra with imaginaries, considered as a double algebra, is, in 
 relative form, 
 
 1=(X:X) + (Y: F), J = (X: Y)-(Y:X). 
 
 This shows how the operation J turns a vector through a right angle in the 
 plane of X, Y. Quaternions in relative form is 
 
 l=.(W: W) + (X:X) + (Y: Y] + (Z : Z) , 
 i=(X: W) --(W:X) + (Z: Y)-(Y:Z), 
 j=(Y: W) --(Z:X) - - (W : Y) + (X:Z], 
 
 We see that we have here a reference to a space of four dimensions corres 
 ponding to X, F, Z, W. 
 
 III. 
 
 On the Algebras in which Division is Unambiguous. 
 BY C. S. PEIRCE. 
 
 1. In the Linear Associative Algebra, the coefficients are permitted to be 
 imaginary. In this note they are restricted to being real. It is assumed that 
 we have to deal with an algebra such that from AB = A C we can infer that 
 A = or B = G . It is required to find what forms such an algebra may take. 
 
 2. If AB = 0, then either A = or 5 = 0. For if not, AC=A(B+C), 
 although A does not vanish and G is unequal to B + C . 
 
 3. The reasoning of 40 holds, although the coefficients are restricted to 
 being real. It is true, then, that since there is no expression (in the 
 algebra under consideration) whose square vanishes, there must be an expression, 
 i, such that i 2 = i. 
 
 4. By 41, it appears that for every expression in the algebra we have 
 
 iA = Ai = A . 
 
 5. By the reasoning of 53, it appears that for every expression A there is 
 
 an equation of the form 
 
 Z m (a m A m )+bi = 0. 
 
 But i is virtually arithmetical unity, since iA = Ai = A ; and this equation may 
 be treated by the ordinary theory of equations. Suppose it has a real root, a ; 
 then it will be divisible by (A a) , and calling the quotient B we shall have 
 
 (A ai)B = 0. 
 
130 P EI ROE : Linear Associative Algebra. 
 
 But A ai is not zero, for A was supposed dissimilar to i . Hence a product of 
 finites vanishes, which is impossible. Hence the equation cannot have a real 
 root. But the whole equation can be resolved into quadratic factors, and some 
 one of these must vanish. Let the irresoluble vanishing factor be 
 
 (A -sf + * 2 0. 
 Then 
 
 or, every expression, upon subtraction of a real number (i. e. a real multiple of i). 
 can be converted, in one way only, into a quantity whose square is a negative 
 number. We may express this by saying that every quantity consists of a scalar 
 and a vector part. A quantity whose square is a negative number we here call 
 a vector. 
 
 6. Our next step is to show that the vector part of the product of two 
 vectors is linearly independent of these vectors and of unity. That is, i and j 
 being any two vectors.* if 
 
 ij s -\- v 
 
 where s is a scalar and v a vector, we cannot determine three real scalars 
 a, b, c, such that 
 
 v = a -f- hi -f- cj . 
 
 This is proved, if we prove that no scalar subtracted from ij leaves a remainder 
 bi -|- cj . If this be true when i and j are any unit vectors whatever, it is true 
 when these are multiplied by real scalars, and so is true of every pair of vectors. 
 We will, then, suppose i and j to be unit vectors. Now, 
 
 ij z = i. 
 If therefore we had 
 
 ij = a + bi + cj , 
 we should have 
 
 - i =. if = aj -\- bij c = ab c + b z i -\- (a + bc)j ; 
 whence, i andy being dissimilar, 
 
 -i = Vi, b* = l, 
 
 and b could not be real. 
 
 * The idempotent basis having been shown to be arithmetical unity, we are free to use the letter i to 
 denote another unit. 
 
PEIRCE : Linear Associative Algebra. 131 
 
 7. Our next step is to show that, i and/ being any two vectors, and 
 
 ij s + v, 
 s being a scalar and v a vector, we have 
 
 ji = r(s v), 
 
 where r is a real scalar. It will be obviously sufficient to prove this for the case 
 in which i and/ are unit vectors. Assuming them such, let us write 
 
 ji = s + v 1 , vv s"+v" , 
 where s and s" are scalars, while v and v" are vectors. Then 
 
 ij.ji (-S- + V) (.S- -f V ) = 6V/ + SV+ S V + V"+ S". 
 
 But we have 
 
 ij .ji =. ij*i p 1 . 
 
 Hence, 
 
 v"= 1 ss 1 s" *v s v. 
 
 But v" is the vector of vv , so that by the last paragraph such an equation cannot 
 subsist unless v" vanishes. Thus we get 
 
 0=1 ss s" sv s v , 
 or 
 
 8V = 1 - SS 1 - s" - s v . 
 
 But a quantity can only be separated in one way into a scalar and a vector part ; 
 so that 
 
 That is, 
 
 ji=~(s-v). Q.E.D. 
 
 S 
 
 8. Our next step is to prove that s = s ; so that if ij = s H- v then ji = 
 s v. It is obviously sufficient to prove this when i and / are unit vectors. Now 
 from any quantity a scalar may be subtracted so as to leave a remainder whose 
 square is a scalar. We do not yet know whether the sum of two vectors is a 
 vector or not (though we do know that it is not a scalar). Let us then take such 
 a sum as ai + bj and suppose x to be the scalar which subtracted from it makes 
 the square of the remainder a scalar. Then, C being a scalar, 
 
 + + = C. 
 
132 PEIRCE : Linear Associative Algebra. 
 
 But developing the square we have 
 
 ( x + ai + bjj = x z a* I? + abs + abs 1 2axi + 2bxj + ab(l - \= C; 
 
 V 
 i. e. 
 
 ab(l -}v = Cx 2 + a 2 + tf abs abs + 2cm + 2bxj . 
 
 \ 9 / 
 
 But v being the vector of i/, by the last paragraph but one the equation must 
 vanish. Either then v = or 1 - -- = . But if v = , ij = s, and multiply 
 ing into/, 
 
 ~~" v * Oj * 
 
 at 
 
 which is absurd, i and / being dissimilar. Hence 1 = and 
 
 s 
 
 ji s v. Q.E.D. 
 
 9. The number of independent vectors in the algebra cannot be two. For 
 the vector of ij is independent of i and/. There may be no vector, and in that 
 case we have the ordinary algebra of reals ; or there may be only one vector, 
 and in that case we have the ordinary algebra of imaginaries. 
 
 Let i and/ be two independent vectors such that 
 
 ij = s -\- v . 
 Let us substitute for / 
 
 /! = si + / . 
 Then we have 
 
 Jv = ii = = , vj = = , 
 iv = <fy\ = j\ , vi = ij\i = j$ = /! . 
 
 Thus we have the algebra of real quaternions. Suppose we have a fourth unit 
 vector, Jc , linearly independent of all the others, and let us write 
 
 jjc = s 1 + V*, 
 
 ld = s"+ v". 
 Let us substitute for k 
 
 IfcSSjft + < + *! 
 
 and we get 
 
 JA = s"v + tf, k^\ = s"v tf, 
 kfi = s v + v", ik^ = s v v". 
 
PEIRCE : Linear Associative Algebra. 133 
 
 Let us further suppose 
 
 (itik, = s> + " . 
 Then, because ij\ is a vector, 
 
 k l (ij\) = S f"-vf ff . 
 But 
 
 hji = JA , kj< iki , 
 because both products are vectors. 
 
 Hence 
 
 * -JA = i. &i/i = MI -ji = % -ji = &i ij\ 
 Hence 
 
 s" + v" = s" v" 
 
 or v" = , and the product of the two unit vectors is a scalar. These vectors 
 cannot, then, be independent^ or k cannot be independent of if = v . Thus it is 
 proved that a fourth independent vector is impossible, and that ordinary real 
 algebra, ordinary algebra with irnaginaries, and real quaternions are the only 
 associative algebras in which division by finites always yields an unambiguous 
 quotient. 
 

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