Celebratio Mathematica

Eliakim Hastings Moore

The scientific work of E. H. Moore

by Gilbert Ames Bliss


In a pre­vi­ous num­ber of this Bul­let­in (Vol. 39 [1933], pp. 831–838) the writer of these pages has pub­lished a bio­graph­ic­al sketch of the life 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. I have known 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, I should 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], [5], [30], [40], [42], [43], [46], [63]; 1885–1913.

  2. Groups, num­bers, al­gebra; [6], [7], [8], [9], [4], [13], [16], [14], [17], [18], [19], [21], [20], [22], [23], [25], [24], [26], [27], [31], [33], [41], [44], [47], [61], [59], [66], [68], [69]; 1892–1922.

  3. The­ory of func­tions; [5], [10], [12], [11], [15], [28], [29], [35], [36], [34], [39], [37], [38], [50], [57], [67], [70], [73]; 1890–1926.

  4. In­teg­ral equa­tions, gen­er­al ana­lys­is; [49], [52], [53], [55], [58], [60], [51], [62], [64], [65], [71]; 1906–1922.

  5. Mis­cel­laneous; [32], [45], [48], [54], [56], [72]; 1900–1922.

The table in­dic­ates fairly well, I 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 syn­opses of Moore’s pa­pers in the fol­low­ing pages are ne­ces­sar­ily brief, but I hope that they may in­dic­ate clearly the de­vel­op­ment and se­quence of his ma­jor in­terests. For ana­lyses and eval­u­ations of the pa­pers in the first, second, and fourth groups of the table I am greatly in­debted to my col­leagues, Pro­fess­ors Logs­don, Dick­son, and Barn­ard, re­spect­ively.


Moore’s pa­pers on geo­metry fall for the most part in­to two groups, an early one con­cerned with al­geb­ra­ic geo­metry, and a later series of three pa­pers on pos­tu­la­tion­al found­a­tions. In his doc­tor­al dis­ser­ta­tion [71] of 1885, writ­ten when he was 23 years old, Moore gen­er­al­ized to ir­re­du­cible \( r \)-spreads of or­der \( n \) three the­or­ems of Clif­ford con­cern­ing skew curves. Such an \( r \)-spread al­ways lies in a flat space of \( n+r - 1 \) di­men­sions or less. If it lies in no space of few­er di­men­sions its points are in one-to-one cor­res­pond­ence with those of an r-spread of or­der \( n-k+r+1 \) in a flat space of \( r + 1 \) di­men­sions, and it is uni­curs­al. Fur­ther­more every flat sec­tion of such an r-spread is also uni­curs­al. In a second part of the pa­per Moore dis­cusses 2-spreads of or­der \( m \). By pro­ject­ing such a sur­face on two planes he sets up a Cre­mona trans­form­a­tion between the planes, and uses this cor­res­pond­ence to ob­tain a ca­non­ic­al equa­tion of the sur­face in \( (m+l) \)-space and prop­er­ties of its os­cu­lat­ing flat spaces and the spreads gen­er­ated by them. The con­sid­er­a­tion of curves on the sur­face leads to a cur­vi­lin­ear gen­er­al­iz­a­tion for them of Pas­cal’s the­or­em con­cern­ing a hexagon, and to vari­ous gen­er­al­iz­a­tions of the the­ory of plane curves. In the lat­ter part of the pa­per he gen­er­al­izes to spreads of odd or­der on hy­per­quad­rics the­or­ems which Cay­ley had de­duced for su­per­lines of quad­ric sur­faces in five di­men­sions.

The second pa­per [2] was a note, with C. N. Little, prob­ably writ­ten be­fore his dis­ser­ta­tion but pub­lished in 1886, con­cern­ing the num­ber and char­ac­ter of the re­gions in­to which a plane is di­vided by n straight lines. Some spe­cial cases of the di­vi­sion of spaces of k di­men­sions by flat spaces of \( k - 1 \) di­men­sions were also dis­cussed.

The pa­per [3] of 1888 is con­cerned with Pi­card’s the­or­em stat­ing that the only al­geb­ra­ic sur­faces whose plane sec­tions are uni­curs­al are uni­curs­al ruled sur­faces and the sur­face of fourth or­der of Stein­er. It is proved that such a sur­face may be re­garded as the pro­jec­tion of a uni­curs­al sur­face of the same or­der \( N \) in a flat space of \( N + 1 \) di­men­sions. Moore also ar­rives in­cid­ent­ally at the res­ult, not for­mu­lated ex­pli­citly, that every com­plete lin­ear sys­tem of plane curves of de­gree n can be trans­formed, by a Cre­mona trans­form­a­tion, either in­to a sys­tem of curves of or­der n pos­sess­ing an \( (n - 1) \)-fold base point and \( n - 1 \) simple base points, or in­to the sys­tem of \( \infty^5 \) con­ics.

In the pa­per [5] of 1888 Moore stud­ied the con­di­tions for six points \( K_i \), \( (i = 1,\dots \), \( 6) \), in a plane to form two cur­vi­lin­early per­spect­ive tri­angles in a net of con­ics. If the points \( K_i \) and two of the base points of the net, \( H_1 \) and \( H_2 \), are se­lec­ted ar­bit­rar­ily, the third base point \( H_3 \) and the cen­ter \( I \) of cur­vi­lin­ear per­spectiv­ity are cor­res­pond­ing points of an in­vol­u­tion on a quart­ic curve which has \( H_1 \) and \( H_2 \) as double points and passes through the six points \( K_i \). If the points \( K_i \) form two per­spect­ive tri­angles in the or­din­ary straight line sense, then the above-men­tioned quart­ic re­duces to the straight line \( H_1H_2 \) and a cu­bic of the pen­cil of cu­bics through the eight points \( H_1 \), \( H_2 \), \( K_i \). If the points \( K_i \) form more than one pair of rec­ti­lin­early per­spect­ive tri­angles, they will have cor­res­pond­ing cur­vi­lin­ear per­spect­ive prop­er­ties in every net of con­ics hav­ing base points \( H_1 \), \( H_2 \), \( H_3 \) which with the points \( K_i \) are the nine base points of a pen­cil of cu­bics, and con­versely. By ap­ply­ing quad­rat­ic Cre­mona trans­form­a­tions to these con­fig­ur­a­tions Moore finds ne­ces­sary and suf­fi­cient con­di­tions for six points to have cur­vi­lin­ear per­spect­ive prop­er­ties ana­log­ous to those de­scribed above in nets of curves of or­der \( 2^n \).

The qual­it­ies ex­hib­ited by Moore in this early group of pa­pers were in many ways char­ac­ter­ist­ic of his re­search throughout his life. The the­ory of lin­ear sys­tems of plane curves, which he freely uses, 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 Moore 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.

Moore pub­lished in 1900 one fur­ther pa­per [30] on al­geb­ra­ic geo­metry which might also be classed in Group II of the above table. The an­har­mon­ic ra­tios formed from \( n \) vari­ables \( z_1,\dots \), \( z_n \) are all ex­press­ible ra­tion­ally in terms of a fun­da­ment­al sys­tem \[ r_i = (z_n, z_{n-1}, z_{n-2}, z_i) ,\] \( (i = 1,\dots \), \( n - 3) \), con­sist­ing of \( n- 3 \) of them. The \( n! \) fun­da­ment­al sys­tems formed by per­mut­ing the vari­ables \( z_i \) are there­fore all ra­tion­ally ex­press­ible in terms of \( (r_i, \dots, r_{n-3}) \), and a group of \( n! \) Cre­mona trans­form­a­tions of the \( (n- 3) \)-di­men­sion­al flat space of points \( (r_1, \dots \), \( r_{n-3}) \) is thus defined. The group has as a sub­group Klein’s group of \( (n - 1) ! \) col­lin­eations which per­mute a cer­tain set of \( n - 1 \) fixed points \( P_k \), \( (k = 1, \dots \), \( n - 1) \), among them­selves. The re­main­ing trans­form­a­tions of the group are \( n! - (n - 1)! \) Cre­mona trans­form­a­tions whose crit­ic­al fig­ures are in­cluded in the com­plete \( (n - l) \)-gon of points \( P_k \).1 For the Klein group men­tioned above Moore found a fun­da­ment­al re­gion in the pa­per [31]. The ap­pear­ance of Hil­bert’s book on the found­a­tions of geo­metry in 1899 at­trac­ted the at­ten­tion of Moore and his stu­dents to pos­tu­la­tion­al meth­ods, in­clud­ing the earli­er work of Pasch and Peano as well as that of Hil­bert. In his pa­per [40] of 1902, Moore gave a new for­mu­la­tion of a sys­tem of ax­ioms for \( n \)-di­men­sion­al pro­ject­ive geo­metry, us­ing points only as un­defined ele­ments in­stead of the points, lines, and planes of Hil­bert in the 3-di­men­sion­al case, and de­fin­ing \( k \)-spaces as classes of points with suit­ably pos­tu­lated prop­er­ties. Schur had as­ser­ted in 1901 that a cer­tain three of Hil­bert’s ax­ioms were prov­able from cer­tain oth­ers. This state­ment Moore showed to be in­cor­rect. He did, however, find that two of Hil­bert’s ax­ioms are re­dund­ant. Moore’s pa­pers [42], [43] were also con­cerned with found­a­tions, but were per­haps of less­er im­port­ance.

A fi­nal pa­per [46] which may be classed with the geo­met­ric­al group con­tains a proof that through every pair of points in the up­per half-plane there passes one and but one of a two-para­met­er fam­ily of very gen­er­al arches. This res­ult is a gen­er­al­iz­a­tion of a well known the­or­em con­cern­ing the cyc­loid arches of the bra­chis­to­chrone prob­lem in the cal­cu­lus of vari­ations.

Groups, numbers, algebra

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 [6] in this field ap­peared in 1892, but his first pub­lished pa­per was the pa­per [8], [14] of 1893. If \( q \) is a prime great­er than 3, the group of the mod­u­lar equa­tion for the trans­form­a­tion of el­lipt­ic func­tions of or­der \( q \) is known to con­sist of all lin­ear frac­tion­al trans­form­a­tions on one vari­able hav­ing in­teg­ral coef­fi­cients, taken mod­ulo \( q \), of de­term­in­ant unity. The gen­er­al­iz­a­tion in the pa­per [8] is the case in which the coef­fi­cients are Galois ima­gin­ar­ies, that is, poly­no­mi­als in x taken with re­spect to the mod­uli \( q \), \( f(x) \) where \( f(x) \) is an ir­re­du­cible poly­no­mi­al mod­ulo \( q \). Moore also proved in this pa­per the in­ter­est­ing and im­port­ant the­or­em that every fi­nite field is a Galois field. In the later pa­per [44] of 1903 he de­term­ined all of the sub­groups of his gen­er­al­ized mod­u­lar group.

An iso­morph­ism of an ab­stract group \( G \) was defined by Moore in the pa­per [9] of 1894 to be a sub­sti­tu­tion on the ele­ments of \( G \) which pre­serves the mul­ti­plic­a­tion table of \( G \). The res­ult­ing group of iso­morph­isms had been defined in­de­pend­ently by O. Hold­er.2 Later, in the pa­per [13] of 1895, Moore took \( G \) to be the Abeli­an group of or­der \( p^n \) of type \( (1, 1,\dots \), \( 1) \), and proved that its group of iso­morph­isms is Jordan’s group of all lin­ear ho­mo­gen­eous trans­form­a­tions on \( n \) vari­ables whose coef­fi­cients are in­tegers taken mod­ulo a prime \( p \). He defined three re­lated tac­tic­al con­fig­ur­a­tions. Else­where, in the pa­per [21] of 1897, he defined ab­stract groups which are simply iso­morph­ic with the gen­er­al sym­met­ric and al­tern­at­ing sub­sti­tu­tion groups. In the pa­per [22] of 1896 and 1898 Moore an­nounced his dis­cov­ery of the im­port­ant fact that every fi­nite group \( G \) of lin­ear trans­form­a­tions on \( n \) vari­ables has a Her­mitian in­vari­ant. If one starts with any pos­it­ive Her­mitian form, ap­plies to it all of the trans­form­a­tions of \( G \), and adds the res­ult­ing forms, the sum is evid­ently in­vari­ant un­der \( G \). This the­or­em was an­nounced in­de­pend­ently by A. Loewy and Fuchs.

One should men­tion here the in­ter­est­ing group of Cre­mona trans­form­a­tions in Moore’s pa­per [30], [31] of 1900, de­scribed in an earli­er para­graph above. His defin­i­tions of an ab­stract group, in the pa­pers [41], [47] of 1902 and 1905, are well known.

Sev­er­al of his pa­pers [7], [4], [20], [23] em­ploy the no­tion of a triple sys­tem, that is, an ar­range­ment of n let­ters in triples (the or­der of the let­ters in a triple be­ing im­ma­ter­i­al) such that every pair ap­pears ex­actly once in some triple. For \( n = 7 \) the unique triple sys­tem of the ordered let­ters \( abcdrst \) is \( abc \), \( adr \), \( ast \), \( bds \), \( brt \), \( cdt \), \( crs \). As an ap­plic­a­tion Moore stud­ied in the pa­per of 1899 the re­solvent equa­tion of de­gree 15 of the gen­er­al equa­tion of de­gree 7 by us­ing the 30 equi­val­ent triple sys­tems in 7 let­ters (each in­vari­ant un­der a group of 168 even sub­sti­tu­tions whose in­dex un­der the al­tern­at­ing group is 15). Next he used a quad­ruple sys­tem in 8 let­ters such that every triple ap­pears in one and only one quad­ruple. There are 14 quad­ruples in the sys­tem. Each quad­ruple sys­tem is in­vari­ant un­der a group of \( 8{\times}168 \) even sub­sti­tu­tions whose in­dex un­der the al­tern­at­ing group of or­der \( 8!/2 \) is 15. Hence the gen­er­al equa­tion of de­gree 8 has a re­solvent of de­gree 15.

Fi­nally Moore ap­plied groups, in the pa­per [18] of 1896, to a highly ab­stract the­ory which in­cludes as spe­cial cases whist tour­na­ment ar­range­ments, and vari­ous gen­er­al­iz­a­tions of the prob­lem of 15 school girls in­volving a triple sys­tem with \( n = 15 \).

Moore pub­lished also a series of pa­pers con­cern­ing the the­ory of num­bers and mod­u­lar sys­tems. In ho­mo­gen­eous form Fer­mat’s the­or­em states that \[ x^py-y^px \] is identic­ally con­gru­ent to a product of lin­ear func­tions mod­ulo a prime p. Moore proved, in the pa­per [17] of 1896, the cor­res­pond­ing the­or­em for the de­term­in­ant \[ \begin{vmatrix} \mkern2mu x&y&z\\ \mkern2mu x^p&y^p&z^p\\ \mkern2mu x^{p^2} &y^{p^2} &z^{p^2} \end{vmatrix} \] and for the ana­log­ous de­term­in­ant of or­der \( n \). In the pa­per [61] of 1907 the mod­u­lar sys­tem com­posed of the coef­fi­cients of the dif­fer­ence between the de­term­in­ant and the product of lin­ear func­tions mod­ulo \( p \) men­tioned above was de­com­posed in­to prime mod­u­lar sys­tems. Else­where, in [19] of 1897, Moore filled a gap in a proof by Molk con­cern­ing gen­er­al mod­u­lar sys­tems.

The pa­pers [27], [16] of 1900 and 1896 were de­voted to al­geb­ra­ic ques­tions. In the former Moore dis­cussed a de­term­in­ant \( D \) each of whose ele­ments is a product of two factors, in par­tic­u­lar, the fac­tor­iz­a­tion of \( D \) in­to de­term­in­ants. In the lat­ter he de­notes by \( f(x, m) \) the func­tion \[ a + h(m + x) + bmx + g(m^2 + x^2) + f(m^2x + x^2m) + cm^2x^2 \] sym­met­ric in \( m \), \( x \) and of de­gree 2 in each. Let \( x_i \), be the roots of \( f = 0 \). Then \( f(x_1, x_2) \) is zero identic­ally in \( m \) if and only if \[ (b - g)g+ac-fh=0 ,\] ex­cept when \( f \) is \( g(x - m)^2 \).

Theory of functions

Moore’s in­terest in the the­ory of func­tions was first in­dic­ated in a pa­per [5] of 1890 con­cern­ing el­lipt­ic func­tions, and a second pa­per [12] of 1895 con­cern­ing the char­ac­ter­ist­ics of theta-func­tions. They were a com­ple­tion and re­vi­sion of proofs of the­or­ems by Hal­phen and Prym, re­spect­ively. A much more im­port­ant con­tri­bu­tion, show­ing per­haps for the first time his full power in ana­lys­is, was his mem­oir [15] of 1896 con­cern­ing tran­scend­ent­ally tran­scend­ent­al func­tions, in­spired by Hölder’s proof that the func­tion \( \Gamma(x) \) sat­is­fies no al­geb­ra­ic dif­fer­en­tial equa­tion with coef­fi­cients ra­tion­al in the vari­able \( x \). Moore defines a “realm of ra­tion­al­ity” as the to­tal­ity of ra­tion­al func­tions of \( n \) ana­lyt­ic func­tions \( f_i(x) \) \( (i - 1, \dots \), \( n) \) hav­ing a com­mon do­main of ex­ist­ence, the coef­fi­cients in the ra­tion­al func­tions be­ing com­plex con­stants. A func­tion \( \phi(x) \) is tran­scend­ent­ally tran­scend­ent­al with re­spect to such a realm if it sat­is­fies no al­geb­ra­ic dif­fer­en­tial equa­tion with coef­fi­cients in the realm. Moore es­tab­lished an in­geni­ous suf­fi­cient con­di­tion that a func­tion sat­is­fy­ing a func­tion­al equa­tion of very gen­er­al type shall be tran­scend­ent­ally tran­scend­ent­al, and ap­plied it to show that this prop­erty is pos­sessed by the two func­tions \[ \phi(x) = \sum^{\infty}_{\nu=0}x^{a^\nu}, \quad \psi(y)= \sum^{\infty}_{\nu=0} e^{e^{\nu}y} \qquad (a = \text{integer} > 1), \] defined for \( | x | < 1 \) and real part of \( y \) neg­at­ive and sat­is­fy­ing the func­tion­al equa­tions \[ \phi(x^a) = \phi(x) -x \quad\text{and}\quad \psi(ay) =\psi(y) -e^y .\] In the lat­ter part of the pa­per he ap­plies his meth­ods to prove that every solu­tion of the func­tion­al equa­tion \[ \Gamma (x+1) =x\Gamma(x) \] is tran­scend­ent­ally tran­scend­ent­al in the realm of ra­tion­al­ity of the single func­tion \( f_1(x) =x \).

In 1900 math­em­aticians in this coun­try were greatly in­ter­ested in Goursat’s re­capit­u­la­tion, in the first volume of the Trans­ac­tions of the Amer­ic­an Math­em­at­ic­al So­ci­ety, of his proof of Cauchy’s the­or­em for a func­tion \( f(z) \) without the as­sump­tion of the con­tinu­ity of the de­riv­at­ive \( f^{\prime}(2) \). In a pa­per [29] of the same year Moore ex­hib­its a proof of this im­port­ant res­ult by an in­dir­ect meth­od which seems un­usu­ally use­ful and ef­fect­ive.

Wei­er­strass de­scribed a con­tinu­ous plane curve pos­sess­ing nowhere a tan­gent, and Peano and Hil­bert have giv­en ex­amples of con­tinu­ous plane curves \[ x = \phi(t), \quad y = \psi(t) \] which com­pletely cov­er por­tions of the plane. In his pa­per [28] of 1900 Moore re­stud­ies in il­lu­min­at­ing geo­met­ric­al-ana­lyt­ic­al fash­ion the curves of Peano and Hil­bert. He re­defines geo­met­ric­ally the \( ty \)-curve \( y =\psi(t) \) of Peano and shows that it also is an ex­ample of a con­tinu­ous curve pos­sess­ing nowhere a well-defined tan­gent. It is in some ways a more in­ter­est­ing ex­ample than that of Wei­er­strass since it is pos­sible to show that the curve has nowhere a pro­gress­ive or re­gress­ive non­ver­tic­al tan­gent, and to char­ac­ter­ize the points, every­where dense on the curve, at which it has pro­gress­ive or re­gress­ive ver­tic­al tan­gents.

In 1901 Moore pub­lished three pa­pers [35], [36], [34] on im­prop­er def­in­ite in­teg­rals. They were con­cerned es­pe­cially with the­or­ies of Har­nack, Stolz, Jordan, Hölder, Schoen­fliess [sic], and de la Vallée Poussin, which oc­cu­pied a dom­in­ant po­s­i­tion in the the­ory of in­teg­ra­tion up to the time of the de­vel­op­ment of the new­er the­or­ies of Borel, Le­besgue, and their fol­low­ers. As the basis of his the­ory Moore ad­op­ted and im­proved no­tions of Har­nack. Let \( \Xi \) be a closed point set of Jordan meas­ure zero on a fi­nite in­ter­val \( a\leq x\leq b \), and let \( F(x) \) be a real single-val­ued func­tion at each point of ab not in \( \Xi \). Let \( I \) be a set of a fi­nite num­ber of dis­crete in­ter­vals con­tain­ing the points of \( S \) as in­teri­or points, and let \( F_I(x) \) be a func­tion equal to \( F(x) \) out­side of \( I \) and equal to zero on \( I \). Then the S-in­teg­ral of \( F(x) \) on \( ab \) \[ \int^b_{a\Xi}F(x)\,dx \] is said to ex­ist if the Riemann in­teg­ral of \( F_I(x) \) ex­ists on \( ab \) for every \( I \) and ap­proaches a fi­nite lim­it as the meas­ure of \( I \) ap­proaches zero. There are two kinds of \( \Xi \)-in­teg­rals, the “nar­row” and the “broad”, ac­cord­ing as every in­ter­val of the set \( I \) is re­quired to con­tain at least one point of \( \Xi \) or not. The two types have dif­fer­ences ana­log­ous to those of con­di­tion­al and ab­so­lute con­ver­gence. In his first pa­per [35] Moore ana­lyzed these dif­fer­ences and gave a mas­terly dis­cus­sion of Har­nack in­teg­rals and their re­la­tions to the im­prop­er in­teg­rals of oth­er writers. Pre­ced­ing mem­oirs con­tained a num­ber of im­port­ant the­or­ems whose truth had been doubted or falsely as­ser­ted. In char­ac­ter­ist­ic fash­ion Moore es­tab­lished by proofs the ones which were true and con­struc­ted ex­amples show­ing the fail­ure of the oth­ers. His second pa­per [36] in this field was an out­growth of the first. It con­tains a clas­si­fic­a­tion of the im­prop­er in­teg­rals of pre­ced­ing writers and a defin­i­tion by pos­tu­la­tion­al meth­ods of a new and more gen­er­al type in­clud­ing the oth­ers as spe­cial cases. The pa­per [34] is con­cerned with two in­teg­rabil­ity the­or­ems of Du Bois-Rey­mond. The first states that a con­tinu­ous func­tion of in­teg­rable func­tions is in­teg­rable, and the second that an in­teg­rable func­tion of in­teg­rable func­tions is in­teg­rable. Moore gen­er­al­izes the former of these the­or­ems and gives an ex­ample to show that the lat­ter is in­cor­rect. Pro­fess­or Os­wald Veblen has re­cently re­marked to me that the re­l­at­ive ef­fect­ive­ness of the in­teg­ra­tion the­ory of Borel and Le­besgue, and oth­er equi­val­ent the­or­ies, is un­doubtedly due to the pres­ence of the two lim­it­ing pro­cesses which they util­ize, a first in the defin­i­tion of meas­ure and a second in the sub­sequent defin­i­tion of the in­teg­ral. It is evid­ent, even from the brief de­scrip­tion above giv­en, that the former of these lim­it­ing pro­cesses is ab­sent in the older the­or­ies in which Moore and many oth­ers were in­ter­ested.

The pa­per [50] is a note ex­plain­ing in in­ter­est­ing fash­ion how the Four­i­er con­stants of a product \( f(x)g(x) \) can be de­term­ined from those of \( f(x) \) and \( g(x) \) by a cal­cu­la­tion in­volving as its prin­cip­al step the form­al mul­ti­plic­a­tion of two Laurent series.

The pa­per [70] on a gen­er­al the­ory of lim­its by E. H. Moore and H. L. Smith, pub­lished in 1922, should per­haps be classed with the pa­pers on gen­er­al ana­lys­is, but it has great in­terest for stu­dents of the the­ory of func­tions in gen­er­al. In his gen­er­al ana­lys­is the­or­ies Moore had in­tro­duced the fol­low­ing no­tion of a lim­it. Let \( Q \) be a class of ele­ments \( q \) and \( S \) the class of all fi­nite classes \( s \) of ele­ments \( q \). A nu­mer­ic­ally val­ued func­tion \( a(s) \) on the range \( S \) is said to con­verge to a num­ber \( a \) as a lim­it if for every \( e > 0 \) there ex­ists a class \( s_e \) such that \( |\alpha(s) - a| < e \) for every class \( s \) con­tain­ing \( s_e \). The pa­per here un­der dis­cus­sion gives a defin­i­tion of a gen­er­al­ized lim­it which in­cludes the or­din­ary lim­it of a se­quence and the lim­it just de­scribed, as well as many oth­er con­ceiv­able cases, as spe­cial in­stances. Let \( P \) be a class of ele­ments \( p \) and \( R \) a re­la­tion such that for every pair of ele­ments \( p_1, p_2 \) of \( P \) the ele­ment \( p_1 \) is either in the re­la­tion \( R \) to \( p_2 \) or not. Let \( \alpha(p) \) be a nu­mer­ic­ally val­ued func­tion on the range \( P \). Then \( \alpha(p) \) has by defin­i­tion the lim­it \( a \) provided that for every \( e > 0 \) there ex­ists an ele­ment \( p_e \) such that \( |\alpha(p)- a| < e \) for every \( p \) in the re­la­tion \( R \) to \( p_e \), in nota­tion \( p R p_e \). The re­la­tion \( R \) is sup­posed to be trans­it­ive, so that \( p_1 R p_2 \) and \( p_2 R p_3 \) im­ply \( p_1 R p_3 \) and it has the com­pos­i­tion prop­erty that for every pair of ele­ments \( p_1,p_2 \) there ex­ists an ele­ment \( p_3 \) such that \( p_3 R p_1 \) and \( p_3 R p_2 \). The pa­per con­tains a thor­ough ana­lys­is of the prop­er­ties of these lim­its in­clud­ing such top­ics as uni­form con­ver­gence, double lim­its, and a re­vised for­mu­la­tion of the­or­ems of Fréchet con­cern­ing com­pact sets and cov­er­ing the­or­ems.

Integral equations, general analysis

Moore’s in­terest in the pos­tu­la­tion­al found­a­tions of vari­ous math­em­at­ic­al the­or­ies was in­dic­ated as early as 1893, in his pa­per [8], [14] in which he set down simple pos­tu­lates for an ab­stract field and showed that every such field is the ab­stract form of a Galois field. It was an in­terest which ap­peared fre­quently in his pa­pers, however, not­ably in his char­ac­ter­iz­a­tion [36] of a gen­er­al­ized type of im­prop­er in­teg­ral, in his pa­pers [41], [42], [43] on the found­a­tions of geo­metry, in his defin­i­tions [40], [47] of an ab­stract group, and in his pres­id­en­tial ad­dress [45] on the found­a­tions of math­em­at­ics. In the early years of this cen­tury, when vari­ous the­or­ies of in­teg­ral equa­tions fol­lowed one an­oth­er rap­idly, his at­ten­tion was led quite nat­ur­ally, in ac­cord with this in­terest in found­a­tions, to the for­mu­la­tion of pos­tu­la­tion­al the­or­ies which should in­clude as spe­cial in­stances nu­mer­ous known the­or­ies of lin­ear equa­tions. 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 those cent­ral fea­tures.”

Moore found that the spe­cial the­or­ies of lin­ear equa­tions which he de­sired to uni­fy could be re­garded as spe­cial in­stances of a gen­er­al the­ory of lin­ear func­tion­al equa­tions in which the func­tions \( \mu(p) \) in­volved are defined on an en­tirely un­res­tric­ted range \( \boldsymbol{P} \) of ele­ments \( p \). In his ef­forts to pre­serve the gen­er­al­ity ne­ces­sary in this range he soon dis­covered that it was im­prac­tic­al to try to gen­er­al­ize such prop­er­ties as con­tinu­ity or dif­fer­en­ti­ab­il­ity of func­tions. He was able to leave the range quite un­res­tric­ted, however, if he pre­sup­posed that the func­tions \( p \), with which he was work­ing be­longed to a class \( \mathfrak{M} \) of such func­tions, and if he pos­tu­lated suit­able prop­er­ties for the class \( \mathfrak{M} \) as a whole. This idea was ef­fect­ive in gen­er­al­iz­ing the the­ory of lin­ear equa­tions, and it seemed to prom­ise sim­il­ar ex­ten­sions in many oth­er do­mains of math­em­at­ics. Thus Moore was led to the com­pre­hens­ive de­vel­op­ment of the the­ory of classes of func­tions on a gen­er­al range which he called “gen­er­al ana­lys­is”.

The prin­ciple of gen­er­al­iz­a­tion quoted above was the dom­in­ant note of Moore’s col­loqui­um lec­tures at Yale Uni­versity in 1906, but his pa­per [53], presen­ted at the In­ter­na­tion­al Con­gress of Math­em­aticians in Rome in 1909, was the first de­tailed pub­lic­a­tion in­dic­at­ing the form which the the­ory of gen­er­al ana­lys­is was tak­ing in his mind. The func­tions p which he stud­ied in this pa­per have nu­mer­ic­al val­ues \( \mu(p) \) defined for an en­tirely ar­bit­rary range \( \boldsymbol{P} \) of ele­ments \( p \). The class \( \mathfrak{M} \) to which they be­long pos­sesses, be­sides the usu­al lin­ear­ity prop­er­ties, the so-called prop­er­ties of dom­in­ance and self-clos­ure, and a com­pos­i­tion prop­erty. Func­tion­al trans­form­a­tions between classes of func­tions are con­sidered, and vari­ous prop­er­ties of these trans­form­a­tions, such as lin­ear­ity, bounded­ness, and norm prop­er­ties, are in­tro­duced. The res­ult was a gen­er­al the­ory of trans­form­a­tions which in­cluded the trans­form­a­tions of the spe­cial cases of lin­ear equa­tion the­ory from which Moore star­ted, and of course much more be­sides. In­stances of the gen­er­al the­ory are the class \( \mathfrak{M} \) of Hil­bert se­quences re­garded as func­tions of \( p \) on the range of ele­ments \( p=0, 1,2 \), \( \dots \), the class of con­tinu­ous func­tions on the range \( 0\leq p\leq 1 \), and their trans­form­a­tions. The suc­cess of Moore’s the­ory in in­clud­ing such het­ero­gen­eous spe­cial cases was de­pend­ent largely upon his no­tion of uni­form con­ver­gence with re­spect to a scale-func­tion, which was fre­quently ef­fect­ive in his work, and which will be de­scribed some­what fur­ther in a later para­graph of this pa­per. The pa­per closes with a study of the lin­ear func­tion­al dif­fer­en­tial equa­tion \[ \frac{\partial \rho(t,p)}{\partial t} = K\rho, \qquad (t_1\leqq t \leqq t_2; p \text{ in } \mathfrak{B}) \] where \( K \) is a lin­ear func­tion­al trans­form­a­tion tak­ing p(t, p) in­to a func­tion of the same ar­gu­ments. The dif­fer­en­tial equa­tion with suit­able ini­tial con­di­tions is equi­val­ent to a lin­ear in­teg­ral equa­tion which has a unique solu­tion.

The mem­oir [55], pub­lished in 1910, is a sys­tem­at­ic in­tro­duc­tion to the form of gen­er­al ana­lys­is, which, as in­dic­ated above, was in­aug­ur­ated in Moore’s col­loqui­um lec­tures of 1906. The first part of the lec­tures is de­voted to a study of the clos­ure and dom­in­ance prop­er­ties of classes \( \mathfrak{M} \) of real single-val­ued func­tions \( \mu \) on a gen­er­al range \( \boldsymbol{P} \) of ele­ments \( p \). Much use is made of the im­port­ant no­tion of uni­form con­ver­gence of a se­quence \( \mu_n \), \( (n = 1, 2 \), \( \dots) \), to a func­tion \( \mu \) re­l­at­ive to a scale func­tion \( \sigma \). Such con­ver­gence means that for every \( e > 0 \) there ex­ists an in­teger \( n_e \) such that \( |\mu_n(p)-\mu(p)| < e_{\sigma}(p) \) for every \( n > n_e \) and \( p \) in \( \boldsymbol{P} \). The second part of the pa­per is con­cerned with prop­er­ties of a class of func­tions of two vari­ables on two gen­er­al ranges, the class be­ing ob­tained by vari­ous ex­ten­sions of the class of products of pairs of func­tions from two of the classes con­sidered in the first part of the pa­per. De­sir­able prop­er­ties of the product class may be ob­tained by im­pos­ing suit­able prop­er­ties on one or both of the com­pon­ent classes.

In two pa­pers [58], [60] of the years 1912 and 1913, re­spect­ively, Moore gave in es­sen­tial out­line the Fred­holm the­ory of lin­ear in­teg­ral equa­tions, and the Hil­bert–Schmidt the­ory of in­teg­ral equa­tions with Her­mitian ker­nels, from the point of view of his gen­er­al ana­lys­is. In the former of these pa­pers he notes that an ob­vi­ous basis for the Fred­holm the­ory, in nota­tion \( (\mathfrak{U};\boldsymbol{P};\mathfrak{R};J) \), con­sists of the class \( \mathfrak{U} \) of real num­bers, a range \( \boldsymbol{P} \) of un­defined ele­ments \( p \), a class \( \mathfrak{M} \) of func­tions \( \mu \) on \( \boldsymbol{P} \), a class \( \mathfrak{R} \) of ker­nel func­tions \( \kappa \) on pairs \( (p^{\prime}, p^{\prime\prime}) \) to \( \mathfrak{U} \), and a lin­ear func­tion­al op­er­a­tion \( J \). The func­tion­al equa­tion stud­ied has the form \[ \xi = \eta - z J \kappa \eta ,\] where \( z \) is a real para­met­er, \( \xi \) and \( \kappa \) are giv­en func­tions in their re­spect­ive classes \( \mathfrak{M} \) and \( \mathfrak{R} \), and a solu­tion \( \eta \) is to be found in \( \mathfrak{M} \). By means of the de­vel­op­ments in his pre­vi­ous pa­per [55] he suc­cess­ively finds mod­i­fied or sim­pli­fied bases, al­to­geth­er six in num­ber, with re­spect to which the the­ory can be car­ried through. It is in­ter­est­ing and im­port­ant to note that on ac­count of the flex­ib­il­ity of the range \( \boldsymbol{P} \) the the­ory of a sys­tem of lin­ear equa­tions for func­tions \( \eta \) of sev­er­al vari­ables is in­cluded in the the­ory of a single equa­tion for a single func­tion \( \eta \) of a single vari­able \( p \). The pa­per [60] of 1912 is con­cerned es­pe­cially with res­ults ana­log­ous to those of the Hil­bert–Schmidt the­ory of lin­ear in­teg­ral equa­tions with sym­met­ric or Her­mitian ker­nel func­tions \( \kappa \). Very gen­er­al res­ults are found by means of a the­ory of lin­ear Her­mitian pos­it­ive def­in­ite func­tion­al op­er­a­tions. As has been re­marked by Pro­fess­or T. H. Hildebrandt,3 the meth­ods of Moore in the the­ory of lin­ear func­tion­al equa­tions are epoch-mak­ing in that they shift the at­ten­tion from the prop­er­ties of in­di­vidu­al func­tions to prop­er­ties of classes of func­tions, and from the form of the op­er­at­or \( J \) to its prop­er­ties, thus at­tain­ing far-reach­ing gen­er­al­ity.

The char­ac­ter­ist­ic value prob­lem in the the­ory of lin­ear equa­tions in­volving a de­nu­mer­able in­fin­ity of vari­ables, as presen­ted for ex­ample by Hil­bert, gave rise to res­ults which Moore failed to at­tain in the gen­er­al the­ory which he had de­veloped in the pa­pers de­scribed in the pre­ced­ing para­graphs. At­tempts to modi­fy his the­ory led to such com­plex­it­ies that he fi­nally aban­doned the highly pos­tu­la­tion­al meth­od of at­tack in fa­vor of a con­struct­ive the­ory. The res­ults which he at­tained are in their fi­nal for­mu­la­tion con­cerned only with a pos­it­ive Her­mitian mat­rix func­tion \( \epsilon \) of pairs of ar­gu­ments \( (p^{\prime}, p^{\prime\prime}) \) on the same range \( \boldsymbol{P} \), and with suit­able defin­i­tions of an in­teg­ra­tion pro­cess \( J \) and a class of in­teg­rable func­tions. In terms of these no­tions he de­veloped in the years fol­low­ing 1915 a the­ory of in­teg­ral equa­tions, char­ac­ter­ist­ic val­ues, and ex­pan­sions, which is a gen­er­al­iz­a­tion of the ana­log­ous the­ory for the lim­ited matrices of Hil­bert. The de­tails of this work of Moore have re­mained for the most part un­pub­lished, but are now be­ing as­sembled and will ap­pear in print. The lim­it­ing pro­cess which was used in de­fin­ing his in­teg­ral was first de­scribed in the pa­per [64] of 1915, but was af­ter­ward presen­ted in more de­tail in the joint pa­per [70] with H. L. Smith pub­lished in 1922 and de­scribed in a pre­ced­ing para­graph. The pa­per [71] of 1922, on power series in gen­er­al ana­lys­is, gives the most im­port­ant res­ults of a chapter in the new the­ory which may be roughly de­scribed as a gen­er­al­ized Four­i­er series the­ory. The first part of the pa­per gives an il­lus­tra­tion of these res­ults by means of an ap­plic­a­tion to cer­tain types of power series in which the num­ber of vari­ables is not ne­ces­sar­ily de­nu­mer­ably in­fin­ite.

In the bib­li­o­graphy at the end of this pa­per a num­ber of ab­stracts [49], [52], [51], [62], [65] are lis­ted which were con­cerned with gen­er­al ana­lys­is. Of these the first four have to do with Moore’s earli­er the­ory, and the con­tents of the pa­pers they de­scribe are either in­cluded in his more ex­tens­ive mem­oirs, or are as­so­ci­ated with parts of the the­ory in a read­ily un­der­stand­able way. The last [65] is the title only of an ad­dress by Moore in 1915 as chair­man of the Chica­go Sec­tion of the Amer­ic­an Math­em­at­ic­al So­ci­ety. It un­doubtedly con­cerned the second the­ory of gen­er­al ana­lys­is which was then tak­ing form in his mind, but we have not been able so far to find a re­cord of it oth­er than what may be con­tained in the manuscripts of the the­ory now be­ing pre­pared for pub­lic­a­tion.

In con­clud­ing these re­marks con­cern­ing Moore’s the­or­ies of gen­er­al ana­lys­is I wish to call at­ten­tion to a sen­tence from the ad­dress of Pro­fess­or E. W. Chit­tenden at the me­mori­al meet­ing re­ferred to on a pre­ced­ing page: “The jus­ti­fic­a­tion for gen­er­al ana­lys­is and sim­il­ar gen­er­al the­or­ies will not be found in the con­tri­bu­tions which are made to the spe­cial the­or­ies which sug­gest the gen­er­al­iz­a­tion. Nev­er­the­less, minor con­tri­bu­tions do res­ult from the meth­ods of ap­proach re­quired by the gen­er­al point of view. The de­sired jus­ti­fic­a­tion lies in the con­tri­bu­tion of the gen­er­al the­ory to a more per­fect com­pre­hen­sion of the nature and sig­ni­fic­ance of the un­der­ly­ing math­em­at­ic­al ele­ments, in the res­ult­ing cla­ri­fic­a­tion and con­dens­a­tion of proof, and in the ex­ten­sion of the range of ap­plic­a­tion for a sig­ni­fic­ant group of ideas”.

Miscellaneous papers

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 [45], [54], [72] 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 [54] 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 non­tech­nic­al 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.

The ad­dress [9] as re­tir­ing pres­id­ent of the Amer­ic­an As­so­ci­ation was also un­pub­lished, but a type-writ­ten 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.

Moore presen­ted nu­mer­ous pa­pers be­fore the Amer­ic­an Math­em­at­ic­al So­ci­ety whose con­tents did not af­ter­ward ap­pear in print or which ap­peared un­der dif­fer­ent titles. The very in­ter­est­ing no­tions which he had con­cern­ing double lim­its in the pa­per [39], for ex­ample, were de­veloped in Chapter IX of his lec­tures on “Ad­vanced In­teg­ral Cal­cu­lus” in 1900, a hand-writ­ten ac­count of which by Pro­fess­or Os­wald Veblen is in the Uni­versity of Chica­go lib­rary. The ma­ter­i­al de­scribed in the ab­stract [67] seems to be in­cluded in some­what mod­i­fied form in the pub­lished pa­per [70] and [73] is a part of Moore’s second the­ory of gen­er­al ana­lys­is now in pre­par­a­tion for pub­lic­a­tion.


[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: “A note on mean val­ues,” Am. Math. Mon. 2 : 11 (1895), pp. 303–​304. MR 1513931 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: “Con­cern­ing Jordan’s lin­ear groups,” Bull. Am. Math. Soc. 2 : 2 (1895), pp. 33–​43. MR 1557406 JFM 26.​0173.​02 article

[14]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

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

[16]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

[17]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

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

[19]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

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

[23]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

[24]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

[25]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.

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

[27]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

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

[29]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

[30]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

[31]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

[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: “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

[35]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

[36]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

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

[41]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

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

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

[44]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

[45]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

[46]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

[47]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

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

[49]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.

[50]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

[51]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

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

[53]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

[54]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.

[55]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

[56]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

[57]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

[58]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

[59]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

[60]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

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

[65]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.

[66]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

[67]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

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

[69]E. H. Moore: “Ab­stract for ‘On the de­term­in­ant of an Her­mitian mat­rix with qua­ternion­ic ele­ments: Defin­i­tion and ele­ment­ary prop­er­ties with ap­plic­a­tions’,” Bull. Am. Math. Soc. 28 : 6 (1922), pp. 296. Art­icle un­pub­lished. JFM 48.​1150.​08 article

[70]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

[71]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

[72]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.

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