return

Celebratio Mathematica

Eliakim Hastings Moore

Analysis  ·  U Chicago

Eliakim Hastings Moore: 1862–1932

by G. A. Bliss and L. E. Dickson

The great de­vel­op­ment which has taken place in our Amer­ic­an math­em­at­ic­al school dur­ing and since the last dec­ade of the last cen­tury has been in large part due to the activ­it­ies of a re­l­at­ively small group of men whose names and de­voted in­terest are well known to all of us. In our memor­ies and our his­tor­ies their achieve­ments will be in­delibly re­cor­ded with grate­ful ap­pre­ci­ation and es­teem. One of the lead­ers among these men, in en­thu­si­asm and schol­ar­ship and clear­ness of vis­ion for the fu­ture, was Eliakim Hast­ings Moore.

He was born in Mari­etta, Ohio, on Janu­ary 26, 1862, and died on Decem­ber 30, 1932, in Chica­go, where he was pro­fess­or and head of the de­part­ment of math­em­at­ics at the Uni­versity of Chica­go. It is in­ter­est­ing to note that the en­vir­on­ment in which Moore grew to man­hood was a most suit­able nurs­ery for the dis­tinc­tion which he af­ter­ward at­tained in so great a meas­ure. His grand­fath­er was an earli­er Eliakim Hast­ings Moore, a banker and treas­urer of Ohio Uni­versity at Athens, Ohio, a county of­ficer and col­lect­or of in­tern­al rev­en­ue, and a Con­gress­man. Eliakim Hast­ings the young­er served as mes­sen­ger in Con­gress dur­ing one sum­mer va­ca­tion while his grand­fath­er was there. His fath­er was a Meth­od­ist min­is­ter, Dav­id Hast­ings Moore, and his moth­er was Ju­lia Car­penter Moore of Athens. The fam­ily moved from place to place while E. H. Moore was young, as ne­ces­sit­ated by the pro­fes­sion of his fath­er, but a con­sid­er­able part of his child­hood was spent in Athens, where one of his play­mates was Martha Mor­ris Young, who was af­ter­ward to be­come his wife. His fath­er, D. H. Moore, be­sides be­ing a preach­er, was suc­cess­ively a cap­tain, ma­jor, and lieu­ten­ant col­on­el in the Civil War; pres­id­ent of Cin­cin­nati Wes­ley­an Col­lege; an or­gan­izer and first chan­cel­lor of the Uni­versity of Den­ver; ed­it­or of the West­ern Chris­ti­an Ad­voc­ate; and Bish­op of the Meth­od­ist Epis­copal Church in Shang­hai with jur­is­dic­tion in China, Ja­pan, and Korea. His was the dis­tin­guished ca­reer of a man much be­loved. Our E. H. Moore, the son of D. H. Moore, was mar­ried in Colum­bus, Ohio, on June 21, 1892, to Martha Mor­ris Young, who sur­vives him. She is a sis­ter of John Wes­ley Young, late pro­fess­or of math­em­at­ics at Dart­mouth Col­lege, the memor­ies of whose friend­ship and achieve­ments are cher­ished pos­ses­sions of math­em­aticians in this coun­try. Her fath­er, Wil­li­am Henry Young of Athens, was a pro­fess­or at Ohio Uni­versity, a col­on­el in the Civil War, and the son of a Con­gress­man. Mrs. Moore her­self was be­fore her mar­riage an in­struct­or of Ro­mance lan­guages at the Uni­versity of Ohio, and also at the Uni­versity of Den­ver dur­ing the chan­cel­lor­ship of D. H. Moore. Shortly after their mar­riage the young couple went to live in Chica­go where Mr. Moore had just been ap­poin­ted pro­fess­or and act­ing head of the de­part­ment of math­em­at­ics in the new Uni­versity of Chica­go which opened its doors in the au­tumn of 1892. Pro­fess­or and Mrs. Moore have one son, also named Eliakim Hast­ings Moore, who was gradu­ated from the Uni­versity of Chica­go and who now lives in Texas.

While E. H. Moore was still in high school, Or­mond Stone, dir­ect­or of the Cin­cin­nati Ob­ser­vat­ory, se­cured him one sum­mer in an emer­gency as an as­sist­ant. Pro­fess­or Stone was af­ter­ward dir­ect­or of the Leander Mc­Con­nick Ob­ser­vat­ory of the Uni­versity of Vir­gin­ia, and a founder of the An­nals of Math­em­at­ics which began its ca­reer at Vir­gin­ia, later moved to Har­vard, and fi­nally to Prin­ceton. Though primar­ily an as­tro­nomer, Pro­fess­or Stone had a high ap­pre­ci­ation for math­em­at­ics, and he in­spired his young as­sist­ant with a first in­terest in that sci­ence. This in­terest was later con­firmed at Yale Uni­versity which the stu­dent, Moore, had been per­suaded to enter by two of his friends, Hor­ace Taft and Sher­man Thatch­er. The former is the broth­er of the late Pres­id­ent Wil­li­am Howard Taft, and the lat­ter is a son of a pro­fess­or at Yale. These two men were Moore’s best friends in col­lege, and life-long friends there­after. Curi­ously enough both of them foun­ded fam­ous schools for boys, one in Wa­ter­town, Con­necti­c­ut, and the oth­er in Cali­for­nia. But the man at Yale who had the most pro­found in­flu­ence upon E. H. Moore, and who first in­spired in him the spir­it of re­search, was Hubert An­son New­ton, pro­fess­or of math­em­at­ics and a sci­ent­ist of dis­tinc­tion. That Moore re­spon­ded ably to the per­son­al en­cour­age­ment of New­ton, as well as that of Stone, is in­dic­ated by his ca­reer as an un­der­gradu­ate. Dur­ing his col­lege course he took three prizes in math­em­at­ics and one each in Lat­in, Eng­lish, and as­tro­nomy. In his ju­ni­or year he won the “philo­soph­ic­al ora­tion ap­point­ment” and second prize at “ju­ni­or ex­hib­i­tion,” and in his seni­or year he held the Foote Schol­ar­ship and was va­le­dictori­an of his class. His nick­name was “Plus” Moore. He took his A.B. at Yale in 1883 and his Ph.D. in 1885. Pro­fess­or New­ton, deeply im­pressed with the abil­ity of the young math­em­atician, fin­anced for him a year of study at Göt­tin­gen and Ber­lin in re­turn for a prom­ise to pay at some fu­ture time.

We have been able to find only mea­ger in­form­a­tion con­cern­ing the year which Pro­fess­or Moore spent in Ger­many. He went first to Göt­tin­gen, in the sum­mer of 1885, where he stud­ied the Ger­man lan­guage and pre­pared him­self for the winter of 1885–6 in Ber­lin. The pro­fess­ors of math­em­at­ics most prom­in­ent in Göt­tin­gen at that time were Weber, Schwarz, and Klein. At Ber­lin, Wei­er­strass and Kro­neck­er were lec­tur­ing. We know that Moore was re­ceived in friendly fash­ion and greatly in­flu­enced by these dis­tin­guished men. It seems that the work of Kro­neck­er made the most last­ing im­pres­sion upon him, but in his habits of math­em­at­ic­al thought and in his later work there are many in­dic­a­tions of in­flu­ences which might be traced to Wei­er­strass and Klein. There is no doubt that the year abroad af­fected greatly Pro­fess­or Moore’s ca­reer as a schol­ar. It es­tab­lished his con­fid­ence in his abil­ity to take an hon­or­able place in the in­ter­na­tion­al as well as our na­tion­al circle of math­em­aticians, ac­quain­ted him at first hand with the activ­it­ies of European sci­ent­ists, and es­tab­lished in him a re­spect and friendly in­terest for Ger­man schol­ar­ship which las­ted throughout his life.

When Moore re­turned to the United States from his so­journ in Europe he entered at once upon his ca­reer as teach­er, schol­ar, and in­de­pend­ent in­vest­ig­at­or. His first po­s­i­tion was an in­struct­or­ship in the Academy at North­west­ern Uni­versity in 1886–7. For the next two years he was a tu­tor at Yale Uni­versity. In 1889 he re­turned to North­west­ern as as­sist­ant pro­fess­or, and in 1891 he was ad­vanced to an as­so­ci­ate pro­fess­or­ship. Mean­while he had pub­lished four pa­pers in the field of geo­metry, and one con­cern­ing el­lipt­ic func­tions, and his ag­gress­ive geni­us as a rising young schol­ar was re­cog­nized by Pres­id­ent Wil­li­am R. Harp­er of the newly foun­ded Uni­versity of Chica­go. When the Uni­versity first opened in the au­tumn of 1892, Moore was ap­poin­ted pro­fess­or and act­ing head of the de­part­ment of math­em­at­ics. In 1896, after four years of un­usu­al suc­cess in or­gan­iz­ing the new de­part­ment, he was made its per­man­ent head, and he held this po­s­i­tion un­til his par­tial re­tire­ment from act­ive ser­vice in 1931.

As a lead­er in his de­part­ment, Pro­fess­or Moore was de­votedly un­spar­ing of his own en­er­gies and re­mark­ably suc­cess­ful. He per­suaded Pres­id­ent Harp­er to as­so­ci­ate with him two un­usu­ally fine schol­ars, Os­kar Bolza and Hein­rich Masch­ke, both former stu­dents at Ber­lin and Ph.D.s of the Uni­versity of Göt­tin­gen. The three of them sup­ple­men­ted each oth­er per­fectly. Moore was bril­liant and ag­gress­ive in his schol­ar­ship, Bolza rap­id and thor­ough, and Masch­ke more de­lib­er­ate but saga­cious and one of the most de­light­ful lec­tur­ers on geo­metry of all time. They early or­gan­ized the Math­em­at­ic­al Club of the Uni­versity of Chica­go whose meet­ings are de­voted to re­search pa­pers, and which con­tin­ues to meet bi-weekly to the present day. Those of us who were stu­dents in those early years re­mem­ber well the tensely alert in­terest of these three men in the pa­pers which they them­selves and oth­ers read be­fore the Club. They were en­thu­si­asts de­voted to the study of math­em­at­ics, and ag­gress­ively ac­quain­ted with the activ­it­ies of math­em­aticians in a wide vari­ety of do­mains. The speak­er be­fore the Club knew well that the ex­cel­len­cies of his pa­per would be fully ap­pre­ci­ated, but also that its weak­nesses would be dis­covered and thor­oughly dis­cussed. Math­em­at­ics, as ac­cur­ate as our powers of lo­gic per­mit us to make it, came first in the minds of these lead­ers in the youth­ful de­part­ment at Chica­go, but it was ac­com­pan­ied by a friend­ship for oth­ers hav­ing ser­i­ous math­em­at­ic­al in­terests which many who ex­per­i­enced their en­cour­age­ment will nev­er for­get. With no loc­al pre­ced­ent or his­tory of any sort to guide them, Moore and Bolza and Masch­ke, with Moore as the guid­ing spir­it, cre­ated a math­em­at­ic­al de­part­ment which promptly took its place among the group of act­ive cen­ters from which have flowed the in­flu­ences cre­at­ive of our present Amer­ic­an math­em­at­ic­al school. Their suc­cess is re­cor­ded not only in pub­lished pa­pers, but also in the activ­it­ies of their stu­dents, who are dis­trib­uted widely in the col­leges and uni­versit­ies of this coun­try.

In the lec­ture room Pro­fess­or Moore’s meth­ods de­fied most es­tab­lished rules of ped­agogy. Such rules, in­deed, meant noth­ing to him in the con­duct of his ad­vanced courses. He was ab­sorbed in the math­em­at­ics un­der dis­cus­sion to the ex­clu­sion of everything else, and neither clock time nor meal time brought the dis­cus­sion to a close. His dis­course ended when some in­stinct told him that his top­ic for the day was ex­hausted. Fre­quently he came to his class with ideas im­per­fectly de­veloped, and he and his stu­dents stud­ied through suc­cess­fully or failed to­geth­er in the study of some ques­tion in which he was at the mo­ment in­ter­ested. He was ap­pre­ci­at­ive of rap­id un­der­stand­ing, and some­times im­pa­tient when com­pre­hen­sion came more slowly. No one could have been more sur­prised than he, or more gentle in his ex­pres­sions of re­gret, when someone called to his at­ten­tion the fact that feel­ings had been hurt by such im­pa­tience. It is easy to un­der­stand un­der these cir­cum­stances, however, that poor stu­dents of­ten shunned his courses, and that good stu­dents whose prin­cip­al in­terests were in oth­er fields some­times could not af­ford the time to take them. But it was a proud mo­ment when one who was am­bi­tious and in­ter­ested found him­self in the re­l­at­ively small group of those who could stand the pace. It is no won­der that among the ablest math­em­aticians of our coun­try at the present time those who drew their chief in­spir­a­tion from Pro­fess­or Moore are nu­mer­ous. He was es­sen­tially a teach­er of those who teach teach­ers. Un­less we pause to make a com­pu­ta­tion we of­ten fail to com­pre­hend the rapid­ity of spread and the mag­nitude of the in­flu­ence of such a man.

There is not space here to trace the achieve­ments of the men who were in­flu­enced primar­ily dur­ing their stu­dent years by Pro­fess­or Moore. The list of those whose thes­is work for the doc­tor’s de­gree was done un­der his su­per­vi­sion is, however, a dis­tin­guished one. We give it here in the or­der in which the de­grees were taken: L. E. Dick­son, H. E. Slaught, D. N. Lehmer, W. Find­lay, O. Veblen, T. E. McKin­ney, R. L. Moore, G. D. Birk­hoff, N. J. Lennes, F. W. Owens, H. E. McNeish, R. P. Baker, T. H. Hildebrandt, Anna J. Pell (Mrs. A. L. Wheel­er), A. D. Pitch­er, R. E. Root, E. W. Chit­tenden, M. G. Gaba, C. R. Dines, Mary E. Wells, A. R. Sch­weitzer, V. D. Gokhale, E. B. Zeisler, J. P. Bal­lan­tine, C. E. Van Horn, R. E. Wilson, M. H. In­gra­ham, R. W. Barn­ard, H. L. Smith, F. D. Perez.

Pro­fess­or Moore’s suc­cess as an edu­cat­or was due to his pro­found in­terest in math­em­at­ics and his fac­ulty for in­spir­ing his col­leagues, and es­pe­cially the strongest gradu­ate stu­dents, with some of his own en­thu­si­asm. With stu­dents not so far ad­vanced he was less suc­cess­ful. His own com­pre­hen­sion was so rap­id, and his con­cen­tra­tion on the math­em­at­ics at hand was so ab­sorb­ing to him, that he found it hard to com­pre­hend or await the slower de­vel­op­ment of un­der­stand­ing in the less ex­per­i­enced. But he ap­pre­ci­ated the im­port­ance of the prob­lems of ele­ment­ary in­struc­tion and at times par­ti­cip­ated act­ively in their solu­tion. Not many people re­mem­ber that in 1897 he ed­ited an arith­met­ic for use in ele­ment­ary schools. In 1903–4 and fol­low­ing years he mod­i­fied rad­ic­ally the meth­ods of un­der­gradu­ate in­struc­tion in math­em­at­ics at the Uni­versity of Chica­go, and he him­self gave courses in be­gin­ning cal­cu­lus. With char­ac­ter­ist­ic in­de­pend­ence he cast aside the text books and con­cen­trated on fun­da­ment­als and their graph­ic­al in­ter­pret­a­tions. The courses were so-called labor­at­ory courses, meet­ing two hours each day, and re­quir­ing no out­side work from the stu­dents. It might be ad­ded par­en­thet­ic­ally that, as with many such new plans, the amount of work re­quired of the in­struct­or was ex­ceed­ingly great. The two hour peri­od was the fea­ture which later caused the aban­don­ment of the plan be­cause of the very prac­tic­al dif­fi­culty in find­ing hours on sched­ules which would not in­ter­fere with the of­fer­ings of oth­er de­part­ments. At the present time we are fa­cing ser­i­ous cri­ti­cisms of the teach­ing of math­em­at­ics in col­leges and high schools. If we are to find a sat­is­fact­ory an­swer we must per­haps con­sider again the de­le­tion of the ir­rel­ev­ant and con­cen­tra­tion on fun­da­ment­als. The labor­at­ory meth­od too has dis­tinct ad­vant­ages. It has ap­pealed to many, and has in one form or an­oth­er been made a part of nu­mer­ous new plans for the teach­ing of math­em­at­ics. In these edu­ca­tion­al ex­per­i­ments which Pro­fess­or Moore un­der­took, as at every oth­er stage of his lead­er­ship in his de­part­ment, he had one per­man­ent char­ac­ter­ist­ic. He be­lieved in the ex­er­cise of in­di­vidu­al­ity in class-room in­struc­tion, and he gave his col­leagues un­lim­ited free­dom in the de­vel­op­ment of their class-room meth­ods. He ex­pec­ted and in­sisted on suc­cess, and he was al­ways sym­path­et­ic­ally in­ter­ested in a new pro­pos­al or pro­ced­ure, but so far as is known to us he nev­er pre­scribed a text­book.

The found­a­tions of Pro­fess­or Moore’s lead­er­ship lay un­doubtedly in his schol­ar­ship. In this bio­graphy no ad­equate de­scrip­tion of his in­vest­ig­a­tions can be giv­en. The read­er will find an ana­lys­is of his more im­port­ant re­search activ­it­ies in the second of the pa­pers to which ref­er­ence was made above. In his earli­er years he was a pro­lif­ic writer, and his pub­lished pa­pers promptly es­tab­lished him as a math­em­atician of re­source­ful­ness and power. Two of the char­ac­ter­ist­ic qual­it­ies of his re­search were ac­cur­acy and gen­er­al­ity. He was a mas­ter of math­em­at­ic­al lo­gic, and his ori­gin­al­ity in mak­ing one or more the­or­ies ap­pear as spe­cial in­stances of a new and more gen­er­al one was re­mark­able. We re­mem­ber a num­ber of meet­ings of the Math­em­at­ic­al Club of the Uni­versity of Chica­go at which this in­terest in gen­er­al­iz­a­tion was char­ac­ter­ist­ic­ally ex­hib­ited. At one of them an eco­nom­ist was strug­gling with the old prob­lem of the se­lec­tion of a mean for the prop­er in­ter­pret­a­tion of cer­tain stat­ist­ic­al data. At the next meet­ing Pro­fess­or Moore sum­mar­ized the pos­tu­lates im­plied in the pa­per of the eco­nom­ist, and ex­hib­ited the in­fin­ite to­tal­ity of gen­er­al­ized means which they char­ac­ter­ized. At an­oth­er meet­ing Pro­fess­or Bolza de­scribed some of the prop­er­ties of a fam­ily of cyc­loid arches which are im­port­ant for the bra­chis­to­chrone prob­lem of the cal­cu­lus of vari­ations. At the fol­low­ing meet­ing Pro­fess­or Moore showed that the class of fam­il­ies of arches with the same prop­er­ties is in­deed a much more gen­er­al one, as is now well re­cog­nized. His suc­cess in these and much more im­port­ant gen­er­al­iz­a­tions, es­pe­cially in the do­main of in­teg­ral equa­tions, cul­min­ated in a the­ory which he called Gen­er­al Ana­lys­is and which be­came his prin­cip­al in­terest. In 1906 when he lec­tured on this the­ory at the New Haven Col­loqui­um of the Amer­ic­an Math­em­at­ic­al So­ci­ety, he was ahead of the times. In re­cent years, however, many math­em­aticians have con­tin­ued his ideas or have en­countered them in in­de­pend­ent ap­proaches from oth­er stand­points. Pro­fess­or Moore’s en­thu­si­asm for math­em­at­ic­al re­search nev­er waned, but in his later years his in­terest in form­al writ­ing de­clined. This was due primar­ily, we think, to two reas­ons. In 1899 he be­came one of the chief ed­it­ors of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety and for eight years there­after he de­voted him­self un­stint­ingly to the af­fairs of the journ­al and of the So­ci­ety. The value of this work to our math­em­at­ic­al com­munity, then still in its youth­ful and form­at­ive stage, can­not be over­es­tim­ated. But for Pro­fess­or Moore him­self it had the ef­fect of de­creas­ing markedly the num­ber of his pub­lished pa­pers. Later, after oth­ers had as­sumed the re­spons­ib­il­it­ies which he had so long cour­ageously shouldered, he ad­op­ted a lo­gic­al sym­bol­ism, largely of his own cre­ation, for the ex­pres­sion of his math­em­at­ic­al ideas to him­self and his stu­dents. It was not well un­der­stood by math­em­aticians in gen­er­al, and not well suited for pub­lic­a­tion in journ­als. In those days, when re­search as­sist­ants for math­em­aticians were al­most un­known, the trans­la­tion of his writ­ings from his con­veni­ent sym­bol­isms to con­ven­tion­al math­em­at­ic­al lan­guage was far less in­ter­est­ing to him than the con­tinu­ation of his own in­vest­ig­a­tions. The res­ult has been that he has left in sym­bol­ic form a great leg­acy of un­pub­lished re­search ma­ter­i­al con­cern­ing Gen­er­al Ana­lys­is.

It is too early to at­tempt a judg­ment of the sig­ni­fic­ance for math­em­aticians in gen­er­al of Pro­fess­or Moore’s nota­tions. He was a spe­cial­ist in sym­bol­isms, every de­tail of which meant something to him. In think­ing or lec­tur­ing about math­em­at­ics, oth­ers as well as him­self have found his nota­tions not only con­veni­ent but also a po­tent aid in the for­mu­la­tion and test­ing of se­quences of lo­gic­al steps. They are es­pe­cially ef­fect­ive in the de­vel­op­ment of the­or­ies in­volving lim­it­ing pro­cesses. It is true that the im­port­ant things in math­em­at­ics are ideas rather than the sym­bols by means of which we rep­res­ent them, but it is evid­ent also that the struc­ture of our sci­ence as we know it today would be im­possible without the in­creas­ingly con­veni­ent nota­tions which math­em­aticians through the ages have suc­cess­ively de­veloped. That Pro­fess­or Moore was fully con­scious of this, and that he re­garded nota­tion­al prob­lems as among the most im­port­ant and dif­fi­cult ones which math­em­aticians have to face, is clearly in­dic­ated by his cor­res­pond­ence with the late Pro­fess­or Flori­an Ca­jori in 1919. It was their ex­change of let­ters at that time which led to the pre­par­a­tion and pub­lic­a­tion in 1928 and 1929 of Ca­jori’s two volumes on The His­tory of Math­em­at­ic­al Nota­tions.

It was to be ex­pec­ted that a man so highly re­garded as a sci­ent­ist should be­come a lead­er in his uni­versity and in the as­so­ci­ations of work­ers in his field. Pro­fess­or Moore was one of the young­est, but also one of the most spir­ited, of the not­able group of schol­ars who in the nineties of the last cen­tury first shaped the char­ac­ter of the new Uni­versity of Chica­go and gave it great dis­tinc­tion. From the open­ing day of the Uni­versity he de­voted him­self un­selfishly to its in­terests, and his coun­sel through the years had great in­flu­ence. At all times he stood un­equi­voc­ally for the highest ideals of schol­ar­ship. His ser­vices to the Uni­versity were sig­nal­ized in 1929 by the es­tab­lish­ment of the Eliakim Hast­ings Moore Dis­tin­guished Ser­vice Pro­fess­or­ship, one among the few of these pro­fess­or­ships which have been named in hon­or of mem­bers of the fac­ulty of the Uni­versity. The first and present in­cum­bent is Pro­fess­or Le­onard Eu­gene Dick­son. Pro­fess­or Moore was a mov­ing spir­it in the or­gan­iz­a­tion of the sci­entif­ic con­gress at the World’s Columbi­an Ex­pos­i­tion of 1893, and in the first col­loqui­um of Amer­ic­an math­em­aticians held shortly there­after in Evan­ston with Klein as the prin­cip­al speak­er. He was in­flu­en­tial in the trans­form­a­tion of the loc­al New York Math­em­at­ic­al So­ci­ety in­to the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1894, and in the found­a­tion of the first so-called sec­tion of the So­ci­ety whose meet­ings were held in or near Chica­go and of which he was the first presid­ing of­ficer in 1897. The form­a­tion of the Chica­go Sec­tion was an out­growth of the Evan­ston col­loqui­um. After that meet­ing a num­ber of math­em­aticians from uni­versit­ies in and near Chica­go oc­ca­sion­ally met in­form­ally for the ex­change of math­em­at­ic­al ideas. After the or­gan­iz­a­tion of the Amer­ic­an Math­em­at­ic­al So­ci­ety they ap­plied for and were gran­ted re­cog­ni­tion as a sec­tion of the So­ci­ety. It was the suc­cess of this first sec­tion which led to the es­tab­lish­ment, in vari­ous parts of the coun­try, of oth­er sim­il­ar meet­ing places which have ad­ded greatly to the in­flu­ence and value of the So­ci­ety. Pro­fess­or Moore was vice-pres­id­ent of the So­ci­ety from 1898 to 1900, and pres­id­ent from 1900 to 1902. In 1921 he was pres­id­ent of the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence. In 1899 he and oth­er ag­gress­ive mem­bers in­duced the So­ci­ety to found the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety, now our lead­ing math­em­at­ic­al journ­al. The first ed­it­ors were E. H. Moore, E. W. Brown of Yale, and T. S. Eiske of Columbia. These men set stand­ards of ed­it­or­i­al su­per­vi­sion which have en­dured to this day. Pro­fess­or Moore re­tired from his ed­it­or­ship in 1907. From 1908 to 1932 he was a non-res­id­ent mem­ber of the coun­cil of the Cir­colo Matem­atico di Palermo and of the ed­it­or­i­al board of its Ren­diconti. From 1914 to 1929 he was the chair­man of the ed­it­or­i­al board of the Uni­versity of Chica­go Sci­ence Series. Nine­teen volumes were pub­lished in the Series dur­ing that peri­od, two of them, by H. F. Blich­feldt and L. E. Dick­son, in the do­main of math­em­at­ics. From 1915 to 1920 Pro­fess­or Moore was a mem­ber of the ed­it­or­i­al board of the Pro­ceed­ings of the Na­tion­al Academy of Sci­ences. In 1916, by his ad­vice and en­cour­age­ment, he gave great as­sist­ance to Pro­fess­or H. E. Slaught, who was a mov­ing spir­it in the form­a­tion of the Math­em­at­ic­al As­so­ci­ation of Amer­ica. In the dec­ades pre­ced­ing 1890 re­search schol­ars in math­em­at­ics in Amer­ica were few and scattered, with lim­ited op­por­tun­it­ies for sci­entif­ic in­ter­course. At the present time we have a well-pop­u­lated and ag­gress­ive Amer­ic­an math­em­at­ic­al school, with fre­quent op­por­tun­it­ies for meet­ings, one of the world’s great cen­ters for the en­cour­age­ment of sci­entif­ic geni­us. From the re­cord of Pro­fess­or Moore’s activ­it­ies de­scribed above, it is clear that at every im­port­ant stage in the de­vel­op­ment of this school he was one of the pro­gress­ive and in­flu­en­tial lead­ers.

That the dis­tinc­tion of Pro­fess­or Moore’s ser­vices to sci­ence and edu­ca­tion was re­cog­nized in oth­er uni­versit­ies as well as his own is in­dic­ated by the hon­ors con­ferred upon him. He re­ceived an hon­or­ary Ph.D. from the Uni­versity of Göt­tin­gen in 1899, and an LL.D. from Wis­con­sin in 1904. Since that time he has been awar­ded hon­or­ary doc­tor­ates of sci­ence or math­em­at­ics by Yale, Clark, Toronto, Kan­sas, and North­west­ern. Be­sides his mem­ber­ships in Amer­ic­an, Eng­lish, Ger­man, and Itali­an math­em­at­ic­al so­ci­et­ies, he was a mem­ber of the Amer­ic­an Academy of Arts and Sci­ences, the Amer­ic­an Philo­soph­ic­al So­ci­ety, and the Na­tion­al Academy of Sci­ences. Two funds have been es­tab­lished in his hon­or. The first is held by the Amer­ic­an Math­em­at­ic­al So­ci­ety for the pur­pose of as­sist­ing in the pub­lic­a­tion of his re­search and for the es­tab­lish­ment of a per­man­ent me­mori­al to him in the activ­it­ies of the So­ci­ety. The second has been ex­pen­ded for a por­trait of him which hangs in Bern­ard Al­bert Eck­hart Hall for the math­em­at­ic­al sci­ences at the Uni­versity of Chica­go. The in­terest in these funds among the friends and ad­mirers of Pro­fess­or Moore was a re­mark­able trib­ute to him sci­en­tific­ally and per­son­ally.

The activ­it­ies too con­cisely enu­mer­ated in the pre­ced­ing para­graphs were the ex­tern­al evid­ences of a re­mark­able per­son­al­ity, a per­son­al­ity bey­ond the power of the writers of these pages ad­equately to de­scribe. Pro­fess­or Moore be­lieved in math­em­at­ics, and his life was an un­selfish and vig­or­ous ex­pres­sion of his con­fid­ence in the im­port­ance of the op­por­tun­ity of study­ing and teach­ing his chosen sci­ence, not only for him­self but also for oth­ers who might have the in­terest and abil­ity. He was some­times mis­un­der­stood when he was im­petu­ous or im­pa­tient, but his im­pa­tience was rarely per­son­al. It was due al­most al­ways to the fact that someone was not un­der­stand­ing math­em­at­ics, and that someone might be either an­oth­er per­son or him­self. In the lat­ter case he was likely to be for the mo­ment un­usu­ally rest­less and ir­rit­able. In all of his activ­it­ies he sought un­ceas­ingly for the truth, and for the words or sym­bols which might ex­press truth ac­cur­ately. He had at times a curi­ous hes­it­a­tion in his speech, char­ac­ter­ist­ic of him, but un­ac­count­able to those who re­cog­nized the un­usu­al agil­ity of his mind but who did not know him well. He would hes­it­ate or stop com­pletely in the midst of a sen­tence, search­ing among the wealth of words which presen­ted them­selves that par­tic­u­lar one which would pre­cisely ex­press his mean­ing, just as in his math­em­at­ics he sought al­ways the pre­cisely sug­gest­ive sym­bol. In times of stress his pa­tience with his col­leagues was re­mark­able, and his friend­ship for them at all times was im­mov­able. He be­lieved in in­di­vidu­al­ity and en­cour­aged in­de­pend­ence in their teach­ing, and he pro­tec­ted them in their re­search, of­ten at great cost to him­self. Out­side, as well as in his own de­part­ment, his en­thu­si­asm, his sci­entif­ic in­teg­rity, and his deep in­sight es­tab­lished an in­flu­ence which will ex­tend wherever math­em­at­ics is stud­ied and truth is honored, bey­ond the con­fines of his coun­try or his day.

The scientific work of Eliakim Hastings Moore

The pre­ced­ing pages of this mem­oir are de­voted to a bio­graph­ic­al sketch of Eliakim Hast­ings Moore. No ac­count of his life can ap­prox­im­ate com­plete­ness, however, without a more de­tailed de­scrip­tion of his sci­entif­ic activ­it­ies than was giv­en there. His en­thu­si­asm for math­em­at­ic­al re­search was a dom­in­ant one, more char­ac­ter­ist­ic of him than any oth­er, in spite of the fact that he had many ad­min­is­trat­ive and ed­it­or­i­al re­spons­ib­il­it­ies which of­ten in­terfered ser­i­ously with his sci­entif­ic work. He had a cath­ol­ic in­terest in all do­mains of math­em­at­ics and a breadth of know­ledge which was re­mark­able. There have been few men with so great an ap­pre­ci­ation of the math­em­at­ic­al ef­forts of oth­ers, or so well qual­i­fied to dis­cuss them in many dif­fer­ent fields, qual­it­ies which were an im­port­ant part of his in­signia of lead­er­ship. If there were two char­ac­ter­ist­ics of his re­search which could be dis­tin­guished above oth­ers, one could say that they would be rig­or and gen­er­al­ity. He strove for pre­ci­sion in thought and lan­guage at a time when vague­ness and un­cer­tainty were com­mon in math­em­at­ic­al lit­er­at­ure, and he pro­foundly in­flu­enced both stu­dents and col­leagues in this re­spect by his teach­ing and ex­ample. He was fur­ther­more among the very first to re­cog­nize the pos­sib­il­ity and im­port­ance of the great gen­er­al­ity in ana­lys­is which is now sought by many writers.

Moore was a pro­lif­ic thinker, though not throughout his life­time a pro­lif­ic writer. His pa­pers, as giv­en in the bib­li­o­graphy at the end of this art­icle, fall roughly in­to the groups in­dic­ated in the fol­low­ing table which lists the num­bers of the items in the bib­li­o­graphy be­long­ing to each field and the dates of the first and last pa­pers in each group:

  1. Geo­metry; [1], [2], [3], [4], [35], [42], [44], [45], [48], [63]; 1885–1913.

  2. Groups, num­bers, al­gebra; [6], [7], [8], [9], [11], [14], [20], [17], [19], [16], [15], [22], [21], [23], [25], [24], [26], [27], [28], [29], [30], [33], [43], [47], [49], [52], [64], [68], [70], [73]; 1892–1922.

  3. The­ory of func­tions; [5], [10], [12], [13], [18], [31], [34], [40], [41], [36], [39], [37], [38], [53], [59], [69], [71], [75]; 1890–1926.

  4. In­teg­ral equa­tions, gen­er­al ana­lys­is; [51], [54], [55], [57], [60], [65], [61], [62], [66], [67], [72], [76], [77]; 1906–1922.

  5. Mis­cel­laneous; [32], [46], [50], [56], [58], [74]; 1900–1922.

The table in­dic­ates fairly well, we think, the se­quence of his ma­jor in­terests, though it does not rep­res­ent ad­equately the re­l­at­ive en­thu­si­asms with which he pur­sued them. The do­mains sug­ges­ted in the second and fourth entries were the ones to which he gave most thought. His stud­ies in al­gebra and the the­ory of groups fell in the peri­od of his greatest activ­ity as a writer, while in­teg­ral equa­tions and gen­er­al ana­lys­is were his ab­sorb­ing in­terest dur­ing the lat­ter part of his life when he pub­lished least. For gen­er­al ana­lys­is, in par­tic­u­lar, he nev­er lost his en­thu­si­asm. He con­tin­ued his spec­u­la­tions in that field in­to the last year of his life, as long as his strength per­mit­ted.

The com­ments on Moore’s pa­pers in these pages must ne­ces­sar­ily be brief. For a more com­plete syn­op­sis the read­er is re­ferred again to the second of the pa­pers to which ref­er­ence is made on the first page of this mem­oir. The pa­pers on geo­metry fall in­to two groups, an early one con­cerned with al­geb­ra­ic geo­metry, which was Moore’s first math­em­at­ic­al in­terest, and a later series of three pa­pers on pos­tu­la­tion­al found­a­tions. The qual­it­ies ex­hib­ited by Moore in the earli­er of these two groups of pa­pers were in many ways char­ac­ter­ist­ic of his re­search throughout his life. The the­ory of lin­ear sys­tems of plane curves, which he freely used, was at that time a cent­ral in­terest in al­geb­ra­ic geo­metry, as in­dic­ated, for ex­ample, in nu­mer­ous pa­pers which ap­peared between 1884 and 1887 in the Palermo Ren­diconti. The skill which he showed in hand­ling such sys­tems, and the el­eg­ance of his res­ults, are in­dic­at­ive of un­usu­al power in so young a man, and the prob­lems which he stud­ied were fun­da­ment­al ones for the al­geb­ra­ic geo­metry of that peri­od. The pa­pers on pos­tu­la­tion­al found­a­tions were in­spired by Hil­bert’s book on the found­a­tions of geo­metry of 1899 which at­trac­ted the at­ten­tion of Moore and his stu­dents to pos­tu­la­tion­al meth­ods, in­clud­ing earli­er work of Pasch and Peano as well as that of Hil­bert. Moore ana­lysed skill­fully ques­tions which had aris­en con­cern­ing the in­de­pend­ence of Hil­bert’s ax­ioms, and gave a new for­mu­la­tion of a sys­tem of ax­ioms for \( n \)-di­men­sion­al geo­metry, us­ing points only as un­defined ele­ments in­stead of the points, lines, and planes of Hil­bert in the 3-di­men­sion­al case.

Moore early be­came in­ter­ested in the the­ory of ab­stract groups, one of the fields of re­search in which he was at vari­ous times most deeply en­gaged. An ab­stract of his first pa­per in this field, num­ber [6] in the bib­li­o­graphy be­low, ap­peared in 1892, but his first pub­lished pa­per was the pa­per [8], [17] which he presen­ted at the math­em­at­ic­al con­gress held at the World’s Columbi­an Ex­pos­i­tion in Chica­go in 1893. It con­tained a gen­er­al­iz­a­tion of the mod­u­lar group, a state­ment and proof for the first time of the in­ter­est­ing and im­port­ant the­or­em which says that every fi­nite al­geb­ra­ic field is a Galois field, and a char­ac­ter­iz­a­tion of a doubly in­fin­ite sys­tem of so-called simple groups, only a few of which had been known be­fore. Al­though Moore’s al­geb­ra­ic in­terests centered largely in groups and their ap­plic­a­tions to the the­ory of equa­tions, he was nev­er­the­less act­ively in­ter­ested at times in a vari­ety of oth­er al­geb­ra­ic ques­tions, not­ably the the­ory of num­bers and mod­u­lar sys­tems. In each of these do­mains he made in­ter­est­ing and im­port­ant con­tri­bu­tions.

Moore’s early in­terest in the the­ory of func­tions was in­dic­ated by the re­l­at­ively simple pa­pers [5] of 1890 and [12] of 1895. A much more im­port­ant con­tri­bu­tion, show­ing per­haps for the first time his full power in ana­lys­is, was his mem­oir [18] of 1897 con­cern­ing tran­scend­ent­ally tran­scend­ent­al func­tions. It is a mod­el of clar­ity and el­eg­ance, and gives evid­ence of his in­creas­ing in­terest and in­genu­ity in math­em­at­ic­al gen­er­al­iz­a­tions. In 1900 math­em­aticians in this coun­try were greatly in­ter­ested in an ex­ten­sion by Goursat of a fun­da­ment­al the­or­em in the the­ory of func­tions, and in the space-filling curves of Peano and Hil­bert. Moore’s pa­pers [34] and [31] of that year il­lu­min­ated these top­ics with an ac­cur­acy and clar­ity char­ac­ter­ist­ic of him. His pa­pers [40], [41], [36] of 1901 on im­prop­er def­in­ite in­teg­rals may with justice be re­garded as the cli­max of the lit­er­at­ure on this sub­ject pre­ced­ing the de­vel­op­ment of the later and more ef­fect­ive in­teg­ra­tion the­or­ies of Borel and Le­besgue.

But the do­mains which cap­tured Moore’s in­terest most ef­fect­ively were the the­or­ies of in­teg­ral equa­tions and gen­er­al ana­lys­is. In the years fol­low­ing 1900 the fun­da­ment­al pa­pers of Fred­holm and Hil­bert on in­teg­ral equa­tions at­trac­ted wide at­ten­tion. Moore saw that the equa­tions which they stud­ied, as well as cor­res­pond­ing and more ele­ment­ary ones well known in al­gebra, must be spe­cial in­stances of a much more gen­er­al lin­ear equa­tion, and he set about the con­struc­tion of a gen­er­al the­ory which should in­clude them all. His guid­ing prin­ciple, as of­ten stated, was that “the ex­ist­ence of ana­lo­gies between cent­ral fea­tures of vari­ous the­or­ies im­plies the ex­ist­ence of a gen­er­al ab­stract the­ory which in­cludes the par­tic­u­lar the­or­ies and uni­fies them with re­spect to these cent­ral fea­tures.” This prin­ciple was the dom­in­ant note of his Col­loqui­um lec­tures [57] de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety at Yale Uni­versity in 1906. The lec­tures were pub­lished in 1910, after the ap­pear­ance of his re­lated pa­per [55] presen­ted at the In­ter­na­tion­al Con­gress of Math­em­aticians in Rome in 1909. In these pa­pers and two later ones [60], [65] Moore gave in es­sen­tial out­line his first the­ory of gen­er­al ana­lys­is and his gen­er­al­iz­a­tion of pre­ced­ing the­or­ies of lin­ear equa­tions. His at­tack was highly pos­tu­la­tion­al, and ori­gin­al es­pe­cially in the fact that the pos­tu­lates ap­plied to classes of func­tions rather than to in­di­vidu­al ones. He was able to se­cure for his gen­er­al the­ory most of the res­ults which are of in­terest in the more spe­cial cases, but some of them eluded him. The at­tempts which he made in or­der to com­plete the the­ory in these re­spects led to such com­plex­it­ies that he fi­nally turned from his first meth­od to a second more con­struct­ive the­ory of sim­il­ar char­ac­ter but with a much sim­pler basis. The res­ults which he at­tained have ap­peared in part in his mem­oirs. They are now be­ing re­vised by Pro­fess­or R. W. Barn­ard and Dr. Max Cor­al and are be­ing pub­lished by the Amer­ic­an Philo­soph­ic­al So­ci­ety.

The titles of the pa­pers in the group des­ig­nated as “mis­cel­laneous” in the table are for the most part self-ex­plan­at­ory. Three of these should be men­tioned more ex­pli­citly, however, Moore’s ad­dresses [46], [56], [74] as re­tir­ing pres­id­ent of the Amer­ic­an Math­em­at­ic­al So­ci­ety in 1902, at the 20th an­niversary of Clark Uni­versity in 1909, and as re­tir­ing pres­id­ent of the Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence in 1922. The first con­tains in its earli­er pages an il­lu­min­at­ing de­scrip­tion of Moore’s con­cep­tion of the lo­gic­al struc­tures of pure and ap­plied math­em­at­ic­al sci­ences, the lat­ter part be­ing de­voted to a dis­cus­sion of the ped­ago­gic­al meth­ods by means of which one might hope to es­tab­lish such con­cepts clearly in the minds of stu­dents in our schools, col­leges and uni­versit­ies. It was writ­ten at a time when Moore him­self was greatly in­ter­ested in a labor­at­ory meth­od of in­struc­tion for col­lege stu­dents of math­em­at­ics, and at the height of the so-called Perry move­ment in Eng­land which aroused great in­terest and dis­cus­sion among those re­spons­ible for in­struc­tion in the math­em­at­ic­al sci­ences in our own coun­try.

The second pa­per [56] was ap­par­ently un­pub­lished, and we have as yet found no manuscript. But there is a some­what in­form­ally writ­ten pa­per with nearly the same title and date in the archives of the De­part­ment of Math­em­at­ics at the Uni­versity of Chica­go. There seems little doubt that it con­tains the ma­ter­i­al of the Clark ad­dress. It con­tains a non-tech­nic­al de­scrip­tion of the work of Pasch, Peano, and Hil­bert on found­a­tions of geo­metry, and of the con­tri­bu­tions of Can­tor, Rus­sell, and Zer­melo to the the­ory of classes.

His ad­dress [74] as re­tir­ing pres­id­ent of the Amer­ic­an As­so­ci­ation was also un­pub­lished, but a type­writ­ten copy is ex­tant. It is a de­scrip­tion of the his­tor­ic­al de­vel­op­ment of the num­ber sys­tems of math­em­at­ics with the pur­pose of es­tab­lish­ing the in­ter­est­ing thes­is that math­em­at­ic­al the­or­ies, though well re­cog­nized as highly de­duct­ive in their ul­ti­mately soph­ist­ic­ated forms, are nev­er­the­less the products of in­duct­ive de­vel­op­ments sim­il­ar to those well known in the labor­at­ory sci­ences.

Works

[1]E. H. Moore: “Ex­ten­sions of cer­tain the­or­ems of Clif­ford and Cay­ley in the geo­metry of \( n \) di­men­sions,” Trans. Conn. Acad. Arts Sci. 7 (1885), pp. 1–​18.

[2]E. H. Moore, Jr. and C. N. Little, Jr.: “Note on space di­vi­sions,” Am. J. Math. 8 : 2 (February 1886), pp. 127–​131. MR 1505415 JFM 18.​0576.​01 article

[3]E. H. Moore, Jr.: “Al­geb­ra­ic sur­faces of which every plane-sec­tion is uni­curs­al in the light of \( n \)-di­men­sion­al geo­metry,” Am. J. Math. 10 : 1 (October 1887), pp. 17–​28. MR 1505461 JFM 19.​0787.​02 article

[4]E. H. Moore, Jr.: “A prob­lem sug­ges­ted in the geo­metry of nets of curves and ap­plied to the the­ory of six points hav­ing mul­tiply per­spect­ive re­la­tions,” Am. J. Math. 10 : 3 (1888), pp. 243–​257. MR 1505481 JFM 20.​0609.​01 article

[5]E. H. Moore: “Note con­cern­ing a fun­da­ment­al the­or­em of el­lipt­ic func­tions, as treated in Hal­phen’s Traité, Vol. I, pages 39–41,” Red. Circ. Mat. Palermo 4 (1890), pp. 186–​194. JFM 22.​0448.​01 article

[6]E. H. Moore: “Con­cern­ing a con­gru­ence group of or­der 360 con­tained in the group of lin­ear frac­tion­al sub­sti­tu­tions,” Proc. Am. As­soc. Adv. Sci. 41 (1892), pp. 62.

[7]E. H. Moore: “Con­cern­ing triple sys­tems,” Math. Ann. 43 : 2–​3 (1893), pp. 271–​285. MR 1510812 JFM 25.​0198.​02 article

[8]E. H. Moore: “Ab­stract for ‘A doubly-in­fin­ite sys­tem of simple groups’,” New York M. S. Bull. III 3 : 3 (1893), pp. 73–​78. Ab­stract only; pub­lished in Chica­go Con­gress, Math­em­at­ic­al pa­pers (1896), pp. 208–242. MR 1557275 JFM 25.​0198.​01 article

[9]E. H. Moore: “The group of holoedric trans­form­a­tion in­to it­self of a giv­en group,” Bull. Am. Math. Soc. 1 : 3 (1894), pp. 61–​66. MR 1557292 JFM 25.​0198.​03 article

[10]E. H. Moore: “Con­cern­ing the defin­i­tion by a sys­tem of func­tion­al prop­er­ties of the func­tion \( f(z)=(\sin\pi z)/\pi \),” Ann. of Math. 9 : 1–​6 (1894–1895), pp. 43–​49. MR 1502180 JFM 26.​0470.​04 article

[11]E. H. Moore: “Con­cern­ing triple sys­tems,” Red. Circ. Mat. Palermo 9 : 1 (1895), pp. 86. JFM 26.​0181.​05 article

[12]E. H. Moore: “On a the­or­em con­cern­ing \( p \)-rowed char­ac­ter­ist­ics with de­nom­in­at­or 2,” Bull. Am. Math. Soc. 1 : 10 (1895), pp. 252–​255. MR 1557392 JFM 26.​0516.​02 article

[13]E. H. Moore: “A note on mean val­ues,” Am. Math. Mon. 2 : 11 (1895), pp. 303–​304. MR 1513931 article

[14]E. H. Moore: “Con­cern­ing Jordan’s lin­ear groups,” Bull. Am. Math. Soc. 2 : 2 (1895), pp. 33–​43. MR 1557406 JFM 26.​0173.​02 article

[15]E. H. Moore: “Tac­tic­al memor­anda I–III (con­tin­ued),” Am. J. Math. 18 : 4 (1896), pp. 291–​303. MR 1505717 article

[16]E. H. Moore: “Tac­tic­al memor­anda I–III,” Am. J. Math. 18 : 3 (1896), pp. 264–​290. MR 1505714 JFM 27.​0472.​02 article

[17]E. H. Moore: “A doubly-in­fin­ite sys­tem of simple groups,” pp. 208–​242 in Math­em­at­ic­al pa­pers read at the in­ter­na­tion­al math­em­at­ic­al con­gress held in con­nec­tion with the world’s Columbi­an ex­pos­i­tion (Chica­go, 1893). Edi­ted by E. H. Moore, O. Bolza, H. Masch­ke, and H. S. White. Mac­mil­lan and Co. (New York), 1896. JFM 28.​0147.​01 incollection

[18]E. H. Moore: “Con­cern­ing tran­scend­ent­ally tran­scend­ent­al func­tions,” Math. Ann. 48 : 1–​2 (1896), pp. 49–​74. MR 1510923 JFM 27.​0307.​01 article

[19]E. H. Moore: “A two-fold gen­er­al­iz­a­tion of Fer­mat’s the­or­em,” Bull. Am. Math. Soc. 2 : 7 (1896), pp. 189–​199. MR 1557441 JFM 27.​0139.​05 article

[20]E. H. Moore and E. C. Ack­er­mann: “On an in­ter­est­ing sys­tem of quad­rat­ic equa­tions,” Am. Math. Mon. 3 : 2 (1896), pp. 38–​41. MR 1514009 article

[21]E. H. Moore: “Con­cern­ing the ab­stract groups of or­der \( k! \) and \( (1/2)k! \) holo­hedric­ally iso­morph­ic with the sym­met­ric and the al­tern­at­ing sub­sti­tu­tion-groups on \( k \) let­ters,” Proc. Lon. Math. Soc. S1-28 : 1 (1897), pp. 357–​366. MR 1576641 JFM 28.​0121.​03 article

[22]E. H. Moore: “The de­com­pos­i­tion of mod­u­lar sys­tems of rank \( n \) in \( n \) vari­ables,” Bull. Am. Math. Soc. 3 : 10 (1897), pp. 372–​380. MR 1557534 JFM 28.​0087.​03 article

[23]E. H. Moore: “Con­cern­ing reg­u­lar triple sys­tems,” Bull. Am. Math. Soc. 4 : 1 (1897), pp. 11–​16. MR 1557545 JFM 28.​0127.​02 article

[24]E. H. Moore: “Con­cern­ing Abeli­an-reg­u­lar trans­it­ive triple sys­tems,” Math. Ann. 50 : 2–​3 (1898), pp. 225–​240. MR 1510993 JFM 29.​0120.​01 article

[25]E. H. Moore: “An uni­ver­sal in­vari­ant for fi­nite groups of lin­ear sub­sti­tu­tions: with ap­plic­a­tion in the the­ory of the ca­non­ic­al form of a lin­ear sub­sti­tu­tion of fi­nite peri­od,” Math. Ann. 50 : 2–​3 (1898), pp. 213–​219. MR 1510991 JFM 29.​0114.​04 article

[26]E. H. Moore: “Ab­stract for ‘A two-para­met­er class of solv­able quintics, in which the ra­tion­al re­la­tions amongst the roots, by threes, do not con­tain the para­met­ers’,” Bull. Am. Math. Soc. 4 (1898), pp. 364. Un­pub­lished ad­dress.

[27]E. H. Moore: “Con­cern­ing the gen­er­al equa­tions of the sev­enth and eighth de­grees,” Math. Ann. 51 : 3 (1898), pp. 417–​444. MR 1511033 JFM 29.​0080.​01 article

[28]E. H. Moore: “Ab­stract for ‘On the sub­groups of abeli­an groups’,” Bull. Am. Math. Soc. 5 (1899), pp. 382–​383. Un­pub­lished ad­dress.

[29]E. H. Moore: “A fun­da­ment­al re­mark con­cern­ing de­term­in­antal nota­tions with the eval­u­ation of an im­port­ant de­term­in­ant of spe­cial form,” Ann. of Math. (2) 1 : 1–​4 (1899–1900), pp. 177–​188. MR 1502269 JFM 31.​0155.​01 article

[30]E. H. Moore: “Con­cern­ing Klein’s group of \( (n+1)! \) \( n \)-ary col­lin­eations,” Am. J. Math. 22 : 4 (1900), pp. 336–​342. MR 1507869 JFM 31.​0146.​02 article

[31]E. H. Moore: “On cer­tain crinkly curves,” Trans. Am. Math. Soc. 1 : 1 (1900), pp. 72–​90. MR 1500526 JFM 31.​0564.​03 article

[32]W. H. Malt­bie: “The un­der­gradu­ate cur­riculum,” Bull. Am. Math. Soc. 7 (1900), pp. 14–​24. Re­port of a dis­cus­sion at the sev­enth sum­mer meet­ing of the Amer­ic­an Math­em­at­ic­al So­ci­ety, in­volving con­tri­bu­tions from Profs. Moore, Hark­ness, Os­good, Mor­ley and Young.

[33]E. H. Moore: “Ab­stract for ‘On the gen­er­a­tion­al de­term­in­a­tion of ab­stract groups’,” Bull. Am. Math. Soc. 6 (1900), pp. 379–​380. Un­pub­lished ad­dress.

[34]E. H. Moore: “A simple proof of the fun­da­ment­al Cauchy–Goursat the­or­em,” Trans. Am. Math. Soc. 1 : 4 (1900), pp. 499–​506. MR 1500551 JFM 31.​0398.​02 article

[35]E. H. Moore: “The cross-ra­tio group of \( n! \) Cre­mona trans­form­a­tions of or­der \( n-3 \) in flat space of \( n-3 \) di­men­sions,” Am. J. Math. 22 : 3 (1900), pp. 279–​291. MR 1505837 JFM 31.​0655.​01 article

[36]E. H. Moore: “Con­cern­ing du Bois-Rey­mond’s two re­l­at­ive in­teg­rabil­ity the­or­ems,” Ann. of Math. (2) 2 : 1–​4 (1900–1901), pp. 153–​158. MR 1503491 JFM 32.​0392.​02 article

[37]E. H. Moore: “Ab­stract for ‘On double lim­its’,” Bull. Am. Math. Soc. 7 (1901), pp. 257. Un­pub­lished ad­dress.

[38]E. H. Moore: “Ab­stract for ‘Con­cern­ing the second mean value the­or­em of the in­teg­ral cal­cu­lus’,” Bull. Am. Math. Soc. 8 (1901), pp. 19–​20. Un­pub­lished ad­dress.

[39]E. H. Moore: “Ab­stract for ‘On the uni­form­ity of con­tinu­ity’,” Bull. Am. Math. Soc. 7 (1901), pp. 245. Un­pub­lished ad­dress.

[40]E. H. Moore: “Con­cern­ing Har­nack’s the­ory of im­prop­er def­in­ite in­teg­rals,” Trans. Am. Math. Soc. 2 : 3 (1901), pp. 296–​330. MR 1500570 JFM 32.​0299.​02 article

[41]E. H. Moore: “On the the­ory of im­prop­er def­in­ite in­teg­rals,” Trans. Am. Math. Soc. 2 : 4 (1901), pp. 459–​475. MR 1500580 JFM 32.​0300.​01 article

[42]E. H. Moore: “On the pro­ject­ive ax­ioms of geo­metry,” Trans. Am. Math. Soc. 3 : 1 (1902), pp. 142–​158. MR 1500592 JFM 33.​0487.​01 article

[43]E. H. Moore: “A defin­i­tion of ab­stract groups,” Trans. Am. Math. Soc. 3 : 4 (1902), pp. 485–​492. MR 1500616 JFM 33.​0142.​01 article

[44]E. H. Moore: “‘The between­ness as­sump­tions’,” Am. Math. Mon. 9 : 6–​7 (1902), pp. 152–​153. MR 1515615 JFM 33.​0488.​03 article

[45]E. H. Moore: “Ab­stract for ‘On Hil­bert’s plane ar­guesian geo­metry’,” Bull. Am. Math. Soc. 8 (1902), pp. 202. Un­pub­lished ad­dress.

[46]E. H. Moore: “On the found­a­tions of math­em­at­ics. Pres­id­en­tial ad­dress de­livered be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety at its ninth an­nu­al meet­ing, Decem­ber 19, 1902,” Bull. Am. Math. Soc. 9 : 8 (1903), pp. 402–​424. MR 1558011 JFM 34.​0068.​03 article

[47]E. H. Moore: “The sub­groups of the gen­er­al­ized fi­nite mod­u­lar group,” pp. 141–​190 in The decen­ni­al pub­lic­a­tions of the Uni­versity of Chica­go. Uni­versity of Chica­go Press, 1903. JFM 34.​0172.​02 incollection

[48]E. H. Moore: “On doubly in­fin­ite sys­tems of dir­ectly sim­il­ar con­vex arches with com­mon base line,” Bull. Am. Math. Soc. 10 : 7 (1904), pp. 337–​341. MR 1558119 JFM 35.​0730.​04 article

[49]E. H. Moore: “On a defin­i­tion of ab­stract groups,” Trans. Am. Math. Soc. 6 : 2 (1905), pp. 179–​180. MR 1500704 JFM 36.​0193.​01 article

[50]E. H. Moore: “The cross-sec­tion pa­per as a math­em­at­ic­al in­stru­ment,” Sch. Rev. 14 : 5 (May 1906), pp. 317–​338.

[51]E. H. Moore: “Ab­stract for ‘On the the­ory of sys­tems of in­teg­ral equa­tions of the second kind’,” Bull. Am. Math. Soc. 12 (1906), pp. 280. Un­pub­lished ad­dress.

[52]E. H. Moore: “The de­com­pos­i­tion of mod­u­lar sys­tems con­nec­ted with the doubly gen­er­al­ized Fer­mat the­or­em,” Bull. Am. Math. Soc. 13 : 6 (1907), pp. 280–​288. MR 1558462 JFM 38.​0238.​02 article

[53]E. H. Moore: “Note on Four­i­er’s con­stants,” Bull. Am. Math. Soc. 13 : 5 (1907), pp. 232–​234. MR 1558448 JFM 38.​0307.​01 article

[54]E. H. Moore: “Ab­stract for ‘Ho­mo­gen­eous dis­tributive func­tion­al op­er­a­tions of de­gree \( n \),” Bull. Am. Math. Soc. 13 (1907), pp. 217–​219. Un­pub­lished ad­dress.

[55]E. H. Moore: “On a form of gen­er­al ana­lys­is with ap­plic­a­tion to lin­ear dif­fer­en­tial and in­teg­ral equa­tions,” pp. 98–​114 in Atti del IV Con­gresso In­ternazionale dei Matem­atici (Rome, 6–11 April 1908), vol. 2. Edi­ted by G. Castel­nuovo. 1909. JFM 40.​0396.​01 incollection

[56]E. H. Moore: “Ab­stract for ‘The role of pos­tu­la­tion­al meth­ods in math­em­at­ics’,” Bull. Am. Math. Soc. 16 (1909), pp. 41. Un­pub­lished ad­dress giv­en at Clark Uni­versity, 20th an­niversary.

[57]E. H. Moore: “In­tro­duc­tion to a form of gen­er­al ana­lys­is,” pp. 1–​150 in E. H. Moore, E. J. Wil­czyn­ski, and M. Ma­son: The New Haven Math­em­at­ic­al Col­loqui­um (New Haven, CT, 5–8 Septem­ber 1906). Yale Uni­versity Press (New Haven, CT), 1910. JFM 41.​0376.​01 incollection

[58]E. H. Moore: “A gen­er­al­iz­a­tion of the game called Nim,” Ann. of Math. (2) 11 : 3 (1910), pp. 93–​94. MR 1502397 JFM 41.​0263.​02 article

[59]E. H. Moore: “Ab­stract for ‘Mul­ti­plic­at­ive in­ter­re­la­tions of cer­tain classes of se­quences of pos­it­ive terms’,” Bull. Am. Math. Soc. 18 : 9 (1912), pp. 444–​445. Un­pub­lished ad­dress. JFM 43.​0334.​01 article

[60]E. H. Moore: “On the found­a­tions of the the­ory of lin­ear in­teg­ral equa­tions,” Bull. Am. Math. Soc. 18 : 7 (1912), pp. 334–​362. MR 1559219 JFM 43.​0424.​02 article

[61]E. H. Moore: “Ab­stract for ‘On nowhere neg­at­ive ker­nels’,” Bull. Am. Math. Soc. 19 : 6 (1913), pp. 287–​288. Un­pub­lished ad­dress. JFM 44.​0420.​08 article

[62]E. H. Moore: “Ab­stract for ‘On a class of con­tinu­ous func­tion­al op­er­a­tions as­so­ci­ated with the class of con­tinu­ous func­tions on a fi­nite lin­ear in­ter­val’,” Bull. Am. Math. Soc. 20 : 2 (1913), pp. 70–​71. Un­pub­lished ad­dress. JFM 44.​0420.​09 article

[63]E. H. Moore: “Ab­stract for ‘On the geo­metry of lin­ear ho­mo­gen­eous trans­form­a­tions of \( m \) vari­ables’,” Bull. Am. Math. Soc. 19 : 9 (1913), pp. 457–​458. Un­pub­lished ad­dress. JFM 44.​0755.​04 article

[64]E. H. Moore: “A mode of com­pos­i­tion of pos­it­ive quad­rat­ic forms,” pp. 413 in 82nd Meet­ing of the Brit­ish As­so­ci­ation for the Ad­vance­ment of Sci­ence (Dun­dee, Scot­land, Septem­ber 1912). Brit­ish As­so­ci­ation (Lon­don), 1913. JFM 44.​0248.​11 incollection

[65]E. H. Moore: “On the fun­da­ment­al func­tion­al op­er­a­tion of a gen­er­al the­ory of lin­ear in­teg­ral equa­tions,” pp. 230–​255 in Pro­ceed­ings of the fifth In­ter­na­tion­al Math­em­at­ics Con­fer­ence (Cam­bridge, UK, 22–28 Au­gust 1912), vol. 1. Edi­ted by E. W. Hob­son and A. E. H. Love. 1913. JFM 44.​0405.​03 incollection

[66]E. H. Moore: “Defin­i­tion of lim­it in gen­er­al in­teg­ral ana­lys­is,” Proc. Natl. Acad. Sci. USA 1 : 12 (December 1915), pp. 628–​632. JFM 45.​0426.​03 article

[67]E. H. Moore: “Ab­stract for ‘Re­port on in­teg­ral equa­tions in gen­er­al ana­lys­is’,” Bull. Am. Math. Soc. 21 (1915), pp. 430. Un­pub­lished ad­dress by chair­man of AMS Chica­go Sec­tion.

[68]E. H. Moore: “Ab­stract for ‘On prop­erly pos­it­ive Her­mitian matrices’,” Bull. Am. Math. Soc. 23 : 2 (1916), pp. 66–​67. Un­pub­lished ad­dress. JFM 46.​0165.​03 article

[69]E. H. Moore: “Ab­stract for ‘On a defin­i­tion of the concept: lim­it of a func­tion’,” Bull. Am. Math. Soc. 22 : 9 (1916), pp. 439–​440. Un­pub­lished ad­dress. JFM 46.​0364.​03 article

[70]E. H. Moore: “Ab­stract for ‘On the re­cip­roc­al of the gen­er­al al­geb­ra­ic mat­rix’,” Bull. Am. Math. Soc. 26 (1920), pp. 394–​395. Un­pub­lished ad­dress.

[71]E. H. Moore and H. L. Smith: “A gen­er­al the­ory of lim­its,” Am. J. Math. 44 : 2 (1922), pp. 102–​121. MR 1506463 JFM 48.​1254.​01 article

[72]E. H. Moore: “On power series in gen­er­al ana­lys­is,” Math. Ann. 86 : 1–​2 (1922), pp. 30–​39. MR 1512076 JFM 48.​0488.​02 article

[73]E. H. Moore: “Ab­stract for ‘On the de­term­in­ant of an Her­mitian mat­rix of qua­ternion­ic ele­ments’,” Bull. Am. Math. Soc. 28 : 4 (1922), pp. 161. Art­icle un­pub­lished. JFM 48.​0128.​07 article

[74]E. H. Moore: “Ab­stract for ‘What is a num­ber sys­tem?’,” Bull. Am. Math. Soc. 29 (1923), pp. 91. Un­pub­lished re­tir­ing pres­id­ent’s ad­dress to Amer­ic­an As­so­ci­ation for the Ad­vance­ment of Sci­ence, Cam­bridge, MA, Decem­ber 1922.

[75]E. H. Moore: “Ab­stract for ‘In­tro­duc­tion to a the­ory of gen­er­al­ized Hellinger in­teg­rals’,” Bull. Am. Math. Soc. 32 (1926), pp. 224. Un­pub­lished ad­dress.

[76]E. H. Moore: Gen­er­al ana­lys­is, part 1. Edi­ted by R. W. Barn­ard. Mem­oirs of the Amer­ic­an Philo­soph­ic­al So­ci­ety 1. Amer­ic­an Philo­soph­ic­al So­ci­ety (Phil­adelphia), 1935. JFM 61.​0433.​06 Zbl 0013.​11605 book

[77]E. H. Moore: Gen­er­al ana­lys­is, part 2: The fun­da­ment­al no­tions of gen­er­al ana­lys­is. Edi­ted by R. W. Barn­ard. Mem­oirs of the Amer­ic­an Philo­soph­ic­al So­ci­ety 1. Amer­ic­an Philo­soph­ic­al So­ci­ety (Phil­adelphia), 1939. JFM 65.​0497.​05 Zbl 0020.​36601 book