Celebratio Mathematica

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.

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

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

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

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

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.

E. H. Moore: “Concerning triple systems,” Math. Ann. 43 : 2–3 (1893), pp. 271–285. MR 1510812 JFM 25.0198.02 article

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

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

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

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

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

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

I present to the American Mathematical Society to-day a continuation of the paper [1894] presented last November, entitled The group of holoedric transformation into itself of a given group. To recall briefly: The given (abstract) group \( G_n \) of order \( n \) has the elements \( s_1 = \) identity, \( s_2,\dots,s_n \). The substitution-group \( \Gamma^n \) of transformation of \( G_n \) into itself is the substitution-group of the \( n \) letters \( s_1,\dots,s_n \) which leaves invariant the multiplication-table for \( G_n \). Letters \( s \) which are conjugate with one another under \( \Gamma^n \) must as elements of \( G_n \) have the same period. Thus, \( s_1= \) identity is invariant, and \( \Gamma^n \) is really \( \Gamma^{n-1} \) on the \( n-1 \) letters \( s_2,\dots,s_n \).

We are to consider to-day the case that \( \Gamma^{n-1} \) is transitive on the \( n-1 \) letters \( s_2,\dots,s_n \). Then the \( n-1 \) elements \( s_2,\dots,s_n \) of \( G_n \) have the \( \textit{same} \) period, which must then be a prime \( p \). Hence \( G_n \) has the order \( n=p^{n^{\prime}} \). Every group \( G_{n=p^{n^{\prime}}} \) has, in accordance with an important (Sylow’s) theorem [1872], at least one element different from identity commutative with every element of the group. This property of the element may be read out of the multiplication-table for \( G_{n=p^{n^{\prime}}} \), and is hence invariant under \( \Gamma^{n-1} \). But \( \Gamma^{n-1} \) is transitive on the \( n-1 \) letters \( s_2,\dots,s_n \). Hence every element of \( G_{n=p^{n^{\prime}}} \) is commutative with every other element. Our given group \( G_n \) is then the Abelian \( G_{p^{n^{\prime}}} \), or rather, omitting the \( ^{\prime} \), \( G_{p^n} \) with \( n \) generating elements, each of order \( p \), and commutative with one another.

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

Henry S. White	Related
Oskar Bolza	Related
Heinrich Maschke	Related

E. H. Moore: “Concerning transcendentally transcendental functions,” Math. Ann. 48 : 1–2 (1896), pp. 49–74. MR 1510923 JFM 27.0307.01 article

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

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

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

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

E. H. Moore: “Concerning regular triple systems,” Bull. Am. Math. Soc. 4 : 1 (1897), pp. 11–16. MR 1557545 JFM 28.0127.02 article

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

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

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

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

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.

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

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

E. H. Moore: “On certain crinkly curves,” Trans. Am. Math. Soc. 1 : 1 (1900), pp. 72–90. MR 1500526 JFM 31.0564.03 article

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

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

The binary \( n \)-ic form has as an absolute irrational invariant the cross-ratio of any four of its roots. These cross-ratios are expressible rationally in terms of any \( n-3 \) independent ones. If any particular system of \( n-3 \) independent ratios be associated with a particular order of the \( n \) roots, by varying the order of the \( n \) roots, we shall have in all \( n! \) conjugate systems; these systems are expressible rationally in terms of the original system, and exactly so in terms of any system of the set. Hence arises, to speak geometrically, a group of \( n! \) Cremona transformations in flat space of \( n-3 \) dimensions. The various Cremona groups so obtained from the various initial systems are Cremona transformations of one another. In this paper I study more closely one of the simplest of such Cremona groups.

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

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.

William Fogg Osgood	Related
Jacob William Albert Young	Related
James Harkness	Related
William Henry Maltbie	Related
Frank Morley	Related

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

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

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

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

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

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.

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

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

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

E. H. Moore: “‘The betweenness assumptions’,” Am. Math. Mon. 9 : 6–7 (1902), pp. 152–153. MR 1515615 JFM 33.0488.03 article

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

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

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

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

In his paper “The determination of the constants in the problem of the brachistochrone” (Bulletin, January, 1904, pages 185–188) Professor Bolza has proved analytically the statement of Weierstrass (lectures of 1882) that, of all cycloid arches having bases in a given line and lying on one side of that line, precisely one passes through any two points, lying on that side of the line or one or both lying on the line, and not lying in the same perpendicular to the line.

The special case in which one of the points is on the base line was handled geometrically by the brothers Bernoulli (cf. Ostwald’s “Klassiker der exacten Wissenschaften,” number 46, pages 12 and 18), use being made of the fact that all cycloid arches are similar. In reading his paper before the mathematical club of the University of Chicago, November 20, 1903, Professor Bolza, after indicating this solution, added a geometric solution for the case in which the segment is parallel to the base line, use being made furthermore of the fact that a cycloid arch is symmetric with respect to the perpendicular bisector of its base.

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

In a paper entitled “A definition of abstract groups” (Transactions, vol. 3, pp. 485–492, October, 1902), my second definition (l.c, p. 490) \[ (M^{\prime\prime}) = (1,2,3^{\prime},3^{\prime\prime},3^{\prime\prime}_r,4^{\prime\prime}_l) \] involves postulates not mutually independent. I shall prove here (as stated in October, 1904, Transactions, vol. 5, p. 549) that (\( 3_r^{\prime\prime} \)) is redundant, that in the new definition \[ (\overline{M}^{\prime\prime}):\,(1, 2, 3^{\prime\prime}, 3^{\prime\prime}_l, 4^{\prime\prime}_l) \] the postulates are mutually independent, and that this mutual independence remains even for the system \[ (\overline{M}^{\prime\prime}_A):\, (1, 2, 3^{\prime\prime}, 3^{\prime\prime}_l, 4^{\prime\prime}_l, A) \] defining an abelian group, obtained by adding to those of (\( M^{\prime\prime} \)) the postulate (\( A \)) that the multiplication or composition of two elements is commutative.

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

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.

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

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

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

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

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.

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

Ernst Julius Wilczynski	Related
Charles Max Mason	Related

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

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

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

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

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

Ernest William Hobson	Related
Augustus Edward Hough Love	Related

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

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

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

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

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.

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

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

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

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

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

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

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.

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

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Eliakim Hastings Moore

The scientific work of E. H. Moore

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