#### by Gilbert Ames Bliss

#### Introduction

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:

Geometry; [1], [2], [3], [5], [30], [40], [42], [43], [46], [63]; 1885–1913.

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.

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

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

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.

#### Geometry

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.