The contributions of J. H. M. Wedderburn to the theory of algebras: 1900-10

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The contributions of J. H. M. Wedderburn to the theory of algebras: 1900-10

Table of contents :
ACKNOWLEDGMENTS .................................................. ii
I. JOSEPH HENRY MACLAGAN WEDDERBURN (1882-1948) ............. 1
III. ON FINITE ALGEBRAS ........................................ 117
IV. THE STRUCTURE THEORY OF ALGEBRAS ......................... 169
THEORY OF ALGEBRAS ....................................... 248
APPENDIX ........................................................... 261
BIBLIOGRAPHY ...................................................... 271
GLOSSARY ........................................................... 286

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I would very much like to thank my advisor, I. N. Herstein, for steering me into the study of the history of the theory of algebras and for giving me free reins to pursue my research as I saw fit. Many others deserve special thanks.

My other readers, Allen

G. Debus, Robert J. Richards, and Jonathan L. Alperin have all pored over the many pages of text and provided helpful suggestions. Saunders Mac Lane, Brian Parshall, and Irving Kaplansky spotted problems in several of the arguments and offered advice on how to correct them.

Kurt Meyberg and Klaus Roggenkamp helped me trans­

late a couple of very sticky passages of nineteenth century German text. Each of the above contributed in some way to my research in its later stages of development, but without the guidance of the following people, the project would have been inconceivable.


would like to express my sincerest gratitude to Jacques Roger for opening the eyes of a young undergraduate to the splendor of the his­ tory of science, to my parents, Mike and Maurice Hunger, for encour­ aging me from the very start to stop at nothing short of the desired goal, and to my husband, Brian Parshall, for giving all of those hours of reading, listening, discussing, and typing.


ACKNOWLEDGMENTS .................................................. Chapter I. JOSEPH HENRY MACLAGAN WEDDERBURN (1882-1948) .............




THE EARLY YEARS OF THE THEORY OF A L G E B R A S ..................7


ON FINITE A L G E B R A S ........................................117


THE STRUCTURE THEORY OF A L G E B R A S ......................... 169


WEDDERBURN'S PLACE IN THE HISTORY OF THE THEORY OF A L G E B R A S ....................................... 248

A P P E N D I X ........................................................... 261 BIBLIOGRAPHY ......................................................


G L O S S A R Y ........................................................... 286


To the memory of my grandmother, Angie Virginia Wroton




In 1882 Scotland had only recently lost one of her greatest scientific sons, James Clerk Maxwell (1831-79), but the spirit of orig­ inal scientific endeavor which he had embodied was carried on in the person of his friend, Peter Guthrie Tait (1831-1901), among others. On the twenty-sixth of February of this same year in the town of Forfar, however, another Scotsman was born who would later be raised in the proud applied tradition of Maxwell and Tait only to branch off and make seminal contributions to pure mathematics.

The present study deals

with the early work of this man, Joseph Henry Maclagan Wedderburn.* Born into a rather well-to-do family which emanated from a long line of ministers on the paternal side and from an equally long line of lawyers on his mother's side, Wedderburn received his early education at the local Forfar Academy.

In preparation for more advanced studies,

however, he went away to George Watson's College in Edinburgh and ob­ tained his leaving scholarship in 1898.

Still well before his seven­

teenth birthday, then, Wedderburn entered the University of Edinburgh where he studied primarily mathematics and physics, or natural philo­ sophy as it was then called.

*For the most part, the biographical information which follows may be found in the published biography of Wedderburn by H. S. Taylor entitled "Joseph Henry Maclagan Wedderburn (1882-1948)," Obituary Notices of Fellows of the Royal Society 6 (1948-49):619-25.


His hard work during the next five years earned him a Master of Arts degree with First

Class Honors in Mathematics in 1903,

virtually every serious mathematics

Then, like

student of the day, Wedderburn

traveled to two of the

great German centers of his chosen discipline,

Leipzig and Berlin, to

continue his studies during the academic year


This trip ahroad was immediately followed by a year as a

Carnegie Fellow at the University of Chicago on the other side of the Atlantic.

The period from the fall of 1904 through the spring and sum­

mer of 1905 marked Wedderburn's true initiation into the world of re­ search mathematics and resulted in his discovery of the abstract theory of algebras as an active field of endeavor.

His first major contribu­

tion to this area, namely, the theorem which stated that every finite division algebra was a field, dated from this extremely productive American sojourn and serves as the focal point of the third chapter below. Upon returning to Scotland during the summer of 1905, Wedderburn went back to his alma mater and assumed the dual position of Lecturer in Mathematics and Assistant to Professor George Chrystal (1851-1911). At twenty-three, he had to wait three more years before he would be eligible for a doctoral degree, and furthermore he had to prove himself worthy of such a degree by producing a thesis which testified to his ability as a research mathematician.

The work he presented late in 1907,

his paper entitled "On Hypercomplex Numbers," not only won him a Ph.D. the next year but also secured him a prominent place in the history of the theory of algebras,*

*Joseph H. M, Wedderburn, "On Hypercomplex Numbers," Proceedings

As we shall see in chapter four, in this one article Wedderburn suc­ ceeded in completely revamping the theory of algebras by discovering elegant yet simple techniques which applied to algebras not only over the real and complex numbers but most generally over arbitrary fields. Thus, he was the first person to realize that much of the structure of the algebra could be determined independently of the structure and of the properties of the underlying field.

Although his later work does

not concern us here, his 1907 paper, its theorems, its techniques, and its overall point of view

defined the course that virtually all of

his subsequent research would follow. After earning his doctoral degree, Wedderburn carried on with his duties at Edinburgh.

Sometime during the 1908-1909 school year,

though, he received an offer from Princeton University which lured him back across the ocean to the Ur.'.ted States.

Not an offer of an assistant

professorship, Wedderburn was asked to be one of the fifty new precep­ tors, an educational innovation of the president of Princeton, Woodrow Wilson (1856-1924).*

The preceptors were, for the most part, younger

men who had just finished taking their degrees themselves.

As Wilson

conceived it, each preceptor was to be both teacher and friend to a small group of undergraduates.

By partitioning up the student body in

this way, Wilson hoped to infuse his undergraduates with a zest for learning which they did not seem to get from the regular faculty.


of the London Mathematical Society, 2d ser., 6 (Nov. 1907);77-118. Throughout this study the terms "algebra," "hypercomplex number system," and "linear associative algebra" will be used interchangeably. See the glossary for the standard definition. *For a brief description of Wilson's preceptorial system, see Laurence R. Veysey, The Emergence of the American University (Chicago: University of Chicago Press, 1965), pp. 298-99.

having the young and eager "preceptor guys," as they were fondly called,, working on a much more intimate and personal basis with the students, Wilson believed that he could finally bridge the deep gulf which tradi­ tionally existed between the faculty and the student body.


in order to fulfill such a goal, the preceptors would have to be chosen carefully.

They would have to be men with a certain mental vigor and

enthusiasm for learning, and furthermore they would have to be able to communicate this enthusiasm dynamically.

The fact that Wedderburn was

asked to take part in the new program, then, reveals much about his personality. His pleasant years as a preceptor, however, were interrupted in 1914 by the outbreak of World War I.

An extremely loyal Scot, he signed

up as a private in the British Army at the very start of the conflict, being the first resident of Princeton to take up arms.

Although he

chose to enter as a private instead of immediately seeking the commis­ sion which his distinguished academic background would have insured him, on December 5, 1914 he actually assumed his military duties with the commissioned rank of Lieutenant, and by January 1915 he had made Captain of the 10th Batallion of the Seaforth Highlanders.


When Wedderburn

returned home to Princeton from the war in 1919, he had the .distinction of having served the longest in the war effort of any member of the university or resident of the town.

^Oswald Veblen, a young mathematician Wedderburn had met and worked with during his year at Chicago, was chosen to be a preceptor in 1905. It was probably as a result of his suggestion that Wedderburn was offered the position in 1909. 2

The published biography of Wedderburn by Taylor conflicts with the war records held in Wedderburn's faculty file in the Princeton Uni-

During the 1920's, Wedderburn's mathematical achievements started to win him the recognition he deserved.

In 1920 he was made

an assistant professor, and then the following year he was promoted with tenure to an associate professorship.

Also in 1921, the Royal

Society of Edinburgh honored him with its MakDougall-Brisbane Gold Medal and monetary prize of n» ? » • • • )

C ci j 3 »Y > a', 0', Y'» a", 0", y ",

to denote the system of equations


£ = ax + 0y + y z . n = a ’x + 0*y + Y ’2 £ = a"x + 0"y + y"2

We get from this the equation

a". 0", Y".

which denotes the system of equations which gives x, y, z , . in terms of £, r|, 5, . . ., and we find ourselves led to the notation

a , 0 , a 1, 0', a", 0",

Y >•• • 1 y 1,.. . y "»•• •

of the inverse matrix. The terms of this matrix are fractions, having the determinant formed by the terms of the original matrix as a common denominator; the numerators are the minors formed by the terms of this same matrix by suppressing one of the rows and one of the columns.1 He continued by describing matrix multiplication explicitly and by showing how his new notation could be used in expressing homogeneous polynomials of arbitrary degree in arbitrarily many-variables.


It is

important to note that in this initial memoir, Cayley made no mention of any special properties of matrices such as associativity and noncommutativity.

He also did not consider the addition of matrices.

As Thomas Hawkins pointed out in his paper, "Another Look at Cayley and the Theory of Matrices," the notation which Cayley intro3 duced in 1855 was not completely new.

In his Disquisitiones Arith-

^Cayley, "Remarques sur la Notation des Fonctions algebriques," pp. 282-83, or Math. Papers A.C., 2:185-86.

2 For a discussion of Cayley's work on these polynomials, see Thomas Hawkins, "Another Look at Cayley and the Theory of Matrices," Archives internationales d'Histoire des Sciences 26 (June 1977):87-91. Ibid., pp. 83-84.

25 meticae of 1801, Carl Friedrich Gauss (1777-1855) used a similar nota­ tion during the discussion of transformations or "forms", as he called them.

In article 268, he wrote

A ternary form f of determinant D and with unknowns x, x ’, x" (the first = x etc.) is transformed into a ternary form g of deter­ minant E and unknowns y, y * ,y" by a substitution such as this

x = ay + 3yf + yy" (2)

x' = a'y + 3'y' + Y'y"

x" = a"y + 3"y* + y'y" where the nine coefficients a, 3, etc. are all integers. For brevity we will neglect the unknowns and say simply that f is transformed into g by the substitution (S)

a , 3 ,Y a 1,3', Y' a", 3", Y"


and that f implies g or g is contained in f .* Later when he gave numerical examples of these substitutions (S), he used notation such as



13, 4, 0 3, 1, 0 o, 0, 1

Gauss, however, did not name arrays such as those in (3) and (4), and he did not treat them as mathematical entities in the sense that he did not speak of either adding or multiplying them.

Thus, Cayley was

the first mathematician to realize that the square arrays themselves actually had algebraic properties. 1 Gauss, p. 294,


- He exploited these properties 2

Ibid., 303,

3 Ibid.,p.306. Here Gauss, gave transformations applied successively, ing the arrays.

the successive arrays for the but he did not consider multiply­

4 Hawkins, in "Another Look at Cayley and the Theory of Matrices," pp.85-87, mentioned Gotthold Eisenstein and Charles Hermite (1822-1901) and their use of a single letter notation to denote linear transformations. Eisenstein, in particular, recognized that linear transformations could be

26 during the year 1855 in his work on the transformation of quadratic functions.* After devoting much of his energy to his famous series of papers on quantics or forms during the two years after 1855, Cayley returned to the subject of matrices and of their application to quad­ ratic functions in 1858 in his papers "A Memoir on the Theory of Ma­ trices' and "A Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function" both of which appeared in the Philosophical


Transactions of the Royal Society.

Unlike his earlier paper on ma­

trices, the first of these 1858 papers presented a rather complete exposition of the various properties of matrices.

Cayley summarized

his new results in this way: It will be seen that matrices (attending only to those of the same order) comport themselves as single quantities; they may be added, multiplied or compounded together §c.: the law of addition of matrices is precisely similar to that for the addition of ordinary algebraical quantities; as regards their multiplication (or compo­ sition) , there is the peculiarity that matrices are not in general

added and multiplied and so could be treated as separate algebraic enti­ ties. As we pointed out above, however, at this time, the intimate relationship between matrices and linear transformations was not appre­ ciated. Thus, the realization that linear transformations could be treated as algebraic entities and the identical realization about matrices were two separate discoveries. Eisenstein was not thinking of matrices in the same way Cayley was, and so his pronouncements on linear transformations should not detract from the importance of Cayley's development of matrices. ■^Arthur Cayley, "Sur la Transformation d'une Fonction quadratique en elle-merae par des Substitutions lineaires," Crelle 50 (1855): 288-99, or Math. Papers A.C., 2:192-201. 2

Arthur Cayley, "A Memoir on the Theory of Matrices," Phil­ osophical Transactions of the Royal Society of London 148 (Jan. 1858): 17-37, or Math. Papers A.C., 2:475-96; and "A Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function," Philosophical Transactions of the Royal Society of London 148 (Jan. 1858):39-46, or Math. Papers A.C., 2:497-505.

27 convertible; it is nevertheless possible to form the powers (posi­ tive or negative, integral or fractional) of a matrix, and thence to arrive at the notion of a rational and integral function, or generally of any algebraical function, of a matrix. I obtain the remarkable theorem that any matrix whatever satisfies an algebraical equation of its own order, the coefficient of the highest power being unity, and those of the other powers functions of the terms of the matrix, the last coefficient being in fact the determir. nant . . . * Thus, Cayley had realized by 1858 that matrices could be treated as algebraical

quantities satisfying all of the properties which would

later constitute an algebra.

He wrote down the additive and multipli­

cative identities, the rules for addition, scalar multiplication and matrix multiplication, and he proved associativity and noncommutativity. He also dealt with many algebra-theoretic properties of matrices, such as the existence of zero divisors, the conditions for commutativity and skew commutativity, and the properties of the transpose.


In addition to systematically going through these fundamentals, Cayley analyzed deeper results such as the so-called Hamilton-Cayley theorem.

In Cayley's words, this theorem

. . . will be best understood by a complete development of a partic­ ular case. Imagine a matrix

and form the determinant la - M, b I I c d - Mj * the developed expression of this determinant is M 2 - (a + d)M^ + (ad - bc)M^; the values of M 2, M 1, M° are ( a2 + be., bfa + d) ) I c(a + d), d + be (,

f a, b ] c, d


f 1, 0 ) f, | 0 , 1 [,

Cayley, "A Memoir on the Theory of Matrices," p. 17, or Math. Papers A.C., 2:475-76.

2 Ibid., pp. 27-31, or pp. 486-91.

28 and substituting these values the determinant becomes equal to the matrix zero, viz. we have la - M, b I_ . (0,0) | c , d - M|“ ' ' ' “ I0,0 [


. . . where the matrix of the determinant is

■iSW-MKI"-1 After running through the above verification for a 3 x 3 matrix M, Cayley concluded that it was unnecessary ". . . t o undertake the la­ bour of a formal proof of the theorem in the general case of a matrix of any degree."


Cayley’s exposition of this theorem illustrated quite well one aspect of his mathematical personality, namely, the tendency to let mere verification of small examples suffice for proof in general.


his various papers on the history of algebra during the nineteenth century, Thomas Hawkins referred to this character trait as "generic ■ reasoning" and noted that at the time Cayley was working in England, mathematicians like Karl Weierstrass (1815-97) in Germany and Cauchy in France labored to introduce a much greater degree of rigor into mathematical argument.


They demanded proofs in general for general state-

Ibid., p. 23, or p. 482. As Thomas Hawkins pointed out in "Another Look at Cayley and the Theory of Matrices," p. 92, Cayley had actually written out this "proof" to Sylvester in a letter dated 19 November 1857.

2 Cayley, "A Memoir on the Theory of Matrices," p. 24, or Math. Papers A.C., 2:483. 3 See, Thomas Hawkins, "The Theory of Matrices in the 19th Centur ,," Proceedings of the International Congress of Mathematicians: Vancouver, 1974, 2 vols, (n.p,; Canadian Mathematical Congress, 1975), 2:561-70; and "Weierstrass and the Theory of Matrices," Archive for History of Exact Sciences 17 (July 1977):119-63, in addition to "Another Look at the Theory of Matrices."

raents as well as a careful analysis of possible exceptions.


however, a product of the English school as influenced by the push toward abstraction of Peacock, DeMorgan, and others, paid little atten­ tion to such detail.*

The general proof of the Hamilton-Cayley theorem

would have to wait until 1878 when Weierstrass’s famous student, Georg Frobenius, published his seminal paper entitled "Ueber lineare Substitutionen und bilineare Formen,"


In that paper, as Hawkins showed, Frobenius developed the singl letter notation for linear transformations to a great extent.



Cayley was already using the single letter notation in a highly sug­ gestive way for matrices in both of his 1858 papers on that topic and especially in the one oh matrix applications mentioned above.

In this

latter paper, we see chains of matrix computations like:

S2(fi - Y)_1(fi +

- Y)(ft + Y)”1^ = A,

or what is the same thing, - Y ) - 1 (fi + Y)ft- 1 (ft - y ) (£2 + Y ) - 1 ^ =


+ Y)^”1(fi - y ) = cn - y )^-1(^ + Y)> + Y)Si-1Cfl - Y) =

Cl + fi_1Y)Cl - ^_1Y) = 4 which is a mere identity, . . .

" Y)G-1(B + Y) j (1 - ^-1Y H 1 + n-1Y),

*See, Hawkins, "Another Look at Cayley and the Theory of Matrices," pp. 94-95 for examples of Cayley’s generic reasoning.


Ferdinand Georg Frobenius, "Ueber lineare Substitutionen und bilineare Formen," Ferdinand Georg Frobenius: Gesammelte Abhandlungen, ed. Jean-Pierre Serre, 3 vols. (New York: Springer-Verlag, 1968), 1: 343-405 (hereafter cited as F.G.F.:Ges. Abh.). 3 Hawkins, "Another Look at Cayley and the Theory of Matrices," pp. 98-101. 4

Cayley, "A Memoir on the Automorphic Linear Transformation of a Bipartite Quadric Function," p. 43, or Math. Papers A.C., 2:502.

30 (Here 0 and y are arbitrary 3 x 3


This kind of shorthand

definitely stripped much of the obscurity from matrix manipulations by highlighting the essential features of the computations. During the course of Cayley's early work on invariant theory and on matrices, he and Sylvester met and talked mathematics regularly. As we have already mentioned, Sylvester attributed his renewed interest in research mathematics to his friendship with Cayley.

Thus, it comes

as no surprise that Sylvester also becamecdeeply interested in the theory of invariants while it was still in its infancy and nourished it greatly with his own ideas.

As for the theory of matrices, however, he did not

seem to have been attracted right away by it even though Cayley had corresponded with him on his new matrix-theoretic results at least as 1 early as 1854.

Whatever the reason for this initial lack of interest,

by 1882 Sylvester had rediscovered matrices and had been devoting a substantial part of his time to their development, but we have gotten a bit ahead of our story. When we left off, Sylvester and Cayley had just met while study­ ing to become lawyers.

This was in 1846.

Sylvester was called to the

bar in 1850, but like Cayley, he persisted doggedly in publishing his mathematical researches, producing seven papers in 1850 and thirteen in 1851.

Virtually all of this work dealt in one way or another with the

theory of invariants.

The life of a lawyer apparently did not appeal

to him either, though, because twice in 1854 he applied for and was denied academic positions.

Finally, in 1855, the professorship in math­

ematics, which he had sought the previous year, at the Royal Military 1_ See n. 1 on p.

31 Academy in Woolwich became vacant, and Sylvester was elected to the post he would keep until 1870,. In volume two of his collected works, 106 of the 110 articles attested to the productivity he enjoyed during this time.


he began to reap the benefits of the recognition his work so greatly deserved.

Within a period of three years from 1861 to 1864, he won a

Royal Medal, the position of correspondent in mathematics in the French Academy of Sciences, and the presidency of the London Mathematical Society.

All looked well until once again his academic career was

marred by his premature retirement from Woolwich in 1870 at the age of fifty-six.

This came as quite a blow to Sylvester, and the subsequent

haggling over his rightful pension left him bitter and unsure . 1


five mathematical papers came from his prolific pen in the five years between 1870 and 1875,


but at long last in 1875 academe came knocking

at Sylvester's door and begging for yet another chance. In 1875 the Johns Hopkins University was founded at Baltimore. A letter to Sylvester from the celebrated Joseph Henry, of date 25 August 1875, seems to indicate that Sylvester had expressed at least a willingness to share in forming the tone of the young university; the authorities seem to have felt that a Professor of Mathematics and a Professor of Classics could inaugurate the work of a Univer­ sity without expensive buildings or elaborate apparatus. It was finally agreed that Sylvester should go, securing, besides his travelling expenses, an annual stipend of 5000 dollars '* paid in

■^Sylvester, Math. Papers J.J.S., 4:xxix.


In 1870, however, his book The Laws of Verse; or. Principles of Versification Exemplified in Metrical Translations; Together With an Annotated Reprint of the Inaugural Presidential Address to the Math­ ematical and Physical Sections of the British Association at Exeter (London: Longmans, Green , 8 Co., 1870) was published. In this work, he set a life-long interest in versification down on paper and included a number of his own original poems. These five years of his life seem to have been occupied primarily by this sort of literary pursuit.

32 gold. By all accounts, Sylvester’s eight years at Johns Hopkins were some of the happiestyears of his life because for once he was wellreceived and well-liked by both his colleagues and his students.


expressed his sincere gratitude to the people at his new university in his Commemoration Day address on 22 February, 1877: It is always a satisfaction to meet those from whom we have received marks of regard, and whom we know to be favorably disposed towards us; and I should be heartless, indeed, and more callous than an oyster, who, twin-soul to the mathematician, working in silence and seclusion between the folding-doors of his mansion, elaborates the pearl that may, hereafter, deck an empress's brow, could I be insensible to many proofs of kind and generous feeling which both within and without the walls of this University, have been so wisely and unequivocably accorded to me. I scruple not to say (for it is strictly the truth) that I have experienced from the authorities of the University a degree of delicate consideration and forebearance from all claims that might be supposed to interfere, in any respect, with my comfort or ease of mind, that, as long as I live will endear to me the name of the Johns Hopkins University . 2 Sylvester repaid his new friends for this goodwill continually during his second stay in America with his endearing eccentricity, his un­ bounded enthusiasm, and his multifarious contributions to mathematics. Like E. H. Moore (1862-1932) almost twenty years later at the University of Chicago, Sylvester did more than his own mathematical research while at Johns Hopkins.

In 1878 the first volume of the Amer­

ican Journal of Mathematics appeared under his joint editorship with William Story (1850-1930), Simon Newcomb (1835-1909), H. A. Rowland (1848-1901), and Benjamin Peirce (1809-80).

This new publication, like

Sylvester, Math. Papers J.J.S., 4:xxix-xxx. 2

James Joseph Sylvester, "Address on Commemoration Day at Johns Hopkins University, 22 February, 1877," Math. Papers J.J.S., 3:73.

33 Moore's Transactions of the American Mathematical Society later, helped to fill a great void in the American mathematical community by provide ing another receptacle for American mathematical contributions and by bringing the work of many great European mathematicians to American shores.

Sylvester, like Moore, also built

a strong, research-oriented

department of mathematics at Hopkins which produced young mathematicians interested in furthering their subject in their native land. Two of them, Fabian Franklin and Thomas Craig, remained at Hopkins; others introduced modern mathematical teaching to many American universities: for instance, George B. Halstead at the University of Texas; Washington Irving Stringham at the University of Cali^ fornia at Berkeley; C. A. van Velzer at the University of Wisconsin. Finally, like Moore, Sylvester brought a famous European mathematician to the New World to lecture on his current research.

As we shall see,

in chapter three, in Moore's case, this mathematician was Felix Klein (1849-1925); in Sylvester's case, it was Arthur Cayley. Cayley came to Johns Hopkins during the spring semester of 1882 and gave a series of lectures on abelian and theta functions. Whether as a result of this visit and a chance conversation with his friend, whether as a result of a subconscious recollection of a certain letter from 1854, or whether as a result of his own Sylvester

research, by 1882

had turned from a concerted attack on invariant theory to a

full-scale assault on the theory of matrices.

He detailed his arrival

on this new battleground as follows: Much as I owe in the way of fruitful suggestion to Cayley's immortal memoir ["On the Theory of Matrices"], the idea of subjecting matrices to the additive process and of their consequent amenability to the laws of functional operation was not taken from it, but occurred

Judith V. Grabiner, "Mathematics in America: The First Hundred Years," in The Bicentennial Tribute to American Mathematics: 1776-1976, ed. Dalton Tarwater (n.p.: Mathematical Association of America, 1977), p. 2 0 .

34 to me independently before 1 had seen the memoir or was acquainted with its contents; . . . My memoir on Tchebysheff's method concerning the totality of prime numbers within certain limits [1881], was the indirect cause of turning my attention to the subject, as (through the systems of difference equations therein employed to contract Tchebysheff1s limits) I was led to the discovery of the properties of the latent [characteristic] roots of matrices, and had made considerable progress in developing the theory of matrices considered as quantities, when on writing to Prof. Cayley upon the subject he referred me to the memoir in question: all this only proves how far from being artificial or factitious, but, on the contrary, was bound to be evolved, in the fullness of time, as a necessary sequel previously acquired cognitions.* In the fall of 1881, Sylvester's new course on multinomial quan­ tities, or what he would later call universal algebra, must have played some role in firing up his interest in the whole subject of matrices. From testimonies of those who knew him, we know that once the spark had been ignited in him there was no stopping him until some other topic caught his fancy.

The theory of matrices apparently firmly held his

attention in the years 1882 to 1884.

One of his students recalled

Sylvester's single-minded pursuit of this topic this way: Sylvester's methods! He had none. "Three lectures will be delivered on a New Universal Algebra," he would say; then "the course must be extended to twelve." It did last all the rest of that year. The following year the course was to be "Substitution Theory, by Netto." We all got the text. He lectured about three times, following the text closely, but stopping sharp at the end of the h o u r . Then he began to think about Matrices again. "I must give one lecture a week on these," he said. He could not confine himself to the hour nor to the one lecture a week. Two weeks passed and Netto was for­ gotten and never mentioned aga-in. 2 Although not his most important mathematical work, Sylvester's

James Joseph Sylvester, "Lectures on the Principles of Universal Algebra," American Journal of Mathematics 6 (1884):271, or Math. Papers J.J.S., 4:209.

2 George E. Andrews, Partitions: Yesterday and Today (Wellington, N.Z.: New Zealand Mathematical Society, 1979), p. 3.

35 investigations on matrices and matrix algebras figure prominently in our present historical development for two main reasons.

Aside from

the fact that his research on such topics as the characteristic and minimum polynomials was very useful in analyzing the internal structure of algebras, Sylvester published many of his new ideas and theorems from 1882 to 1884 in the Comptes rendus, one of the more widely read journals for mathematical research of the day.

Thus, his work had the

potential for reaching a relatively large audience.

Combined with this

potential recognition, actual recognition came to his papers when Henri Poincare (1854-1912) singled them out in a short but influential note, also in the Comptes rendus, entitled "Sur les nombres complexes.1' * Through Poincare's note, Sylvester's work became known to European math­ ematicians who worked in

the same area but who looked at the subject

from a different point of view.

This heightened awareness of the re­

search being done on the other side of the Atlantic marked the first step toward a unification of the various approaches to the theory of algebras.

We shall return to this paper by Poincare shortly, but for

the moment we must look at Sylvester's actual work on matrix algebra. As Sylvester said in the passage cited above, his interest in matrix questions was aroused by his research on Tchebycheff's inequality on prime numbers.

In an 1852 paper entitled "Sur les nombres premiers"

which appeared in the Journal de Mathematiques, the Russian mathemati­ cian Pafnuti Tchebycheff (1821-94) produced upper and lower bounds on the number of primes less than a given number x.

If ir(x) equaled the

number of primes less than x, he showed that

^Henri Poincar£, "Sur les nombres complexes," Comptes; rendus 99 (Nov. 1884):740-42.

36 A

A1 where 0.922