Clifford, W. K.: analysis of the
axiom of free mobility, 199-203 ;
criticism of this analysis, 199-203;
popular exposition of metageo-
metry, 219-22; on elementary
flatness of surface, 220, 221;
analogous conception in relation
to space, 222; views criticized,

220-2.

Conant, Professor: on the nature of
the number-concept, 56-59.
Concept: definition of (Dictionary
of Philos. and Psych.), 46; and
meaning, 69.
Congruence: in Geometry, 158;
Euclid's alleged tacit assumption
of, 198-9; assumption of, relevant
to mensuration, meaningless in
relation to geometry, 199, 203;
criticism of explanation of, as
a geometrical assumption by (1)
Clifford, 199-203, (2) Russell,
203-5; verbal equivalents of the
notion of, 205; so-called axiom
of, and Euclid's Axiom 1, 205, 255.
Contingent Relations: in Geometry,

principle of, 128-30; same as
Poncelet's principle of continuity,
130.

Contradiction: real and nominal,
206-7.

Conventionality of language, 12.
Conventions: in Algebra, import
of, 81; validity of, 81.
Couturat, L.: on the foundations

of geometry, 154-7; views criti-
cized, 154-7; on the rôle of
intuition in geometry, 169.

D

Darwin and Max Müller's views on
:
language, 24.
Definition limit of process of, 4,

14; and meaning, 14, 15; real
nature of, 16, 17; nature and
function of, compared, 18; of
geometry, 152; technical sense
of, in geometry, 154; of space
useless, 154; Riemann on, in
geometry, 223.

Demonstration: nature and object
of, in geometry, 169-71; R. Sim-
son on, 169; the Epicureans and,
169, 170.

Direction: relation to notion of
straightness, 160, 162, 163; no-
tion of, in connexion with non-
Euclidean spaces, 248-9; notion
of, involved in the conception of
linear shape, 168, 254.

H
Hamilton, Sir William : on lan-
guage as an aid to thought, 27–9.
Hamilton, Sir W. R.: on derivation
of concept of number, 56.
Helmholtz: popular exposition of
metageometry, 213-19; fallacies
involved in this exposition, 216-
18; analogical value of Flat-
land' and Sphereland', 218,
219; on Riemann's conception
of manifoldness, 230.
Henrici, O.: on imaginary loci,
131-3; on Euclid's assumptions,
172; on Euclid's axioms of
magnitude, 187.

Homonymy: in relation to alge-
braic expression, 90.

I

Image: representative and sym-
bolic, functions of, in the develop-
ment of thought, 42-4, 250.

Imaginaries, mathematical: doc-
trine of, Chap. VI; nature of
enigma involved in, according to
(1) Cayley, 68, 69, 70, 85, (2)
Whitehead, 79, 85.
Imaginary: loci, 69, 71, geometrical
doctrine of, Chap. IX; point,
notion of, how arrived at analyti-
cally, 69, geometrically, 70; points,
introduction of, in geometrical
involution, 132-4; points, a
'factor' in the definition of real
geometrical relations, 135; points,
derivation of, in analytical geome-
try, 142-5; quantity, 69; ob-
jects (of geometry) in relation to
experience, 73; unit, a 'factor'
in the expression of algebraic
quantity, 126; quantity, two
senses of the 'interpretation' of,
in geometry, 146-7; quantity,
textbook derivation of the no-
tion, and sophisms involved in
this derivation, 113-15.
Indefinables: in mathematics, 154,
156.

Indices: see Power and Root.
Intercommunication: presupposi-
tion involved in, 4, 8, 10.
Intuition: rôle of, in geometry, 169,
170.

Involution: in geometry, 132–4.

K

Kantians and non-Euclidean geo-
metry, V.
Klein, F.: on nature of geometrical
axioms, 179; on Riemann's con-
ception of space, 184; connexion
of Cayley's Theory of Distance
with Metageometry, 241, 242.

L

Language: and thought, Chap. II;
as an instrument of reason, Chap.
III; essentially conventional na-
ture of, 9; the original and in-
stinctive in expression replaced
by the artificial and conventional,
10; learning of, mechanical by
comparison with origination and
development of, II; not a neces-
sary condition of forming abstract
ideas, 12, 46; as a potential source
of illusory beliefs, 13; subjective
and objective aspects of, 15, 18,
19, 49; functions of, as embodi-
ment of acquired knowledge and
as aid in reasoning, 29, 46–7.
Laws of Algebra, no difference be-

tween, and conventions of same,
81; are symbolic of processes of
thought, 87.

Length: Euclid's conception of,
as involved in his definition of the
line, 167; Russell's derivation of
the notion, 167, inadmissible,
167-8.

Light: rectilinear propagation of,
in relation to geometrical theory,
246-7.

Linear shape: analysis of the notion
of, 168.

Lobatschewsky: and the concep-
tion of the straight line, 181; his
hypothesis regarding parallels,
207; his geometry in relation to
the modifiability of the conception
of space, 211-12; on astronomical
observations as a test of geometri-
cal theory, 244.

Lotze ineffectiveness of his attack
on non-Euclidean geometry, 218;
on the futility of astronomical
observations as a test of geo-
metrical theory, 244; Russell's
criticism of, 245.
Love, A. E. H.:
quantity, 61-2.

M

on number and

Manifoldness: notion of (Riemann),
229, (Helmholtz), 230; con-
tinuous, positions and colours,
229; ambiguities in definition of,
229-31; relation of notion of, to
that of space, 257:
Mathematics: symbolism in rela-
tion to, 6; apparent sophistry
and paradox in, 6-7; the domain
of definite and stable concepts, 6.
Max Müller: doctrine of the identity
of thought and language, 20−6;
sense in which he uses the terms
'identity' and 'inseparableness',
21; names as an essential element
of thought, 21-2; words the signs
of concepts not of things, 23; his
doctrine in conflict with Darwin's
views, 24-5; and Whitney, con-
flicting views of language har-
monized, 48-9.

Meaning and thought, difference
between, 8; and symbol, mutu-
ally constituted by association,
8; stability of, in relation to
symbol, 13; and definition, 14;
of an idea, 72.
Measure of Curvature : a meta-
phorical expression in relation to

a manifold, and to space, 233-4;
as an inherent property of surface,
235-6; fallacious analogy between
surface and space, 237-8.
Mensuration: relation of, to me-
trical geometry, 243-8, 258.
Metaphor: expression of analogy,
18.
Metaphysics: value of, as an intel-
lectual exercise, 3; effect of con-
flict of systems of, 3.

Mill, J. S. on the nature of defini-
tion, 16; Taine's criticism of,
16; on the truths of geometry,
72; on the existence of, and the
conception of, geometrical enti-
ties, 163, note 2.
Muscular Adjustment: function of,
in perception of shape, 160-1.
Mystical Illusion: in geometry, as
in algebra, prompted by anterior
use of semi-paradoxical expres-
sions, 136.

:

Mystical Tendency in geometry
as compared with algebra, 131,

serial association of signs common
to all systems of expressing, 60;
and quantity in the abstract,
61-2; and quantity in algebraic
symbolism, 63; as such neither
positive nor negative, 116.
Numbers: as co-ordinates in geo-
metry, 242.

Numerals: and the use of the
fingers to express numbers, 59.

P

Paradox as a means of expressing
real relations, 130, 132, 134, 135-6;
inception of in the metaphorical
expression of real relations, 141;
sanction of, 147; test of valid use
of, 253.

Particular, the: and the general,
unthinkable save in relation to
one another, 155-6.

Perception: a rudimentary process
of reasoning (Binet), 42; of
shape, 160-1.
Perpendicularity: relation of, sym-

bolized by =1, 122; irrele-
vance of demonstration to this
symbolization, 125; Euclid's
alleged assumption of the exist-
ence of, 180.

Plane: name of an identity of
surface-shape, 158; extension of
the meaning of term, 211.
Plane, the empirical origin of con-
ception of, 163; as standard of
comparison, 159.

Poincaré, H. on the existence of
mathematical entities, 174; on
the nature of geometrical axioms,
175-7; the fundamental axioms
of geometry, 176; on nature of
synthetic judgements a priori,
176; on the validity of Euclidean
geometry, 245.

'Point at Infinity,' the: derivation
of the notion of, 137-8; mystical
element in the notion of, 138–9;
real value of the notion in the
theory of projection and corre-
spondence, 139–41.

Postulates nature of, in Euclid's
Elements, 182-5; Euclid's are
complementary definitions of the
straight line and circle, 182-3.
Power and Root: arithmetical
definition of, 106; implied alge-
braic definition of, 106-7; and
the employment of indices, 107-8;
and relations of algebraic quan-

tity, 108-9; inconsistency in the
use of indices of, 108-9, III, 118;
validation of this inconsistency,
108, 111, 112, 119.
Pragmatism:
mysticism, iv.
Properties of space, ambiguity of
the term, 151–2.
Pythagoreans, the mental charac-
teristics, iii.

as a solvent of

Q

Quantities: positive and negative,
alleged heterogeneity of, 93, 116;
conventional test of inequality,
93, 116; sophisms involved in
this test, 94-7.

Quantity: not, as such, either
positive or negative, 116, 120,
122; vicious reaction of alge-
braic symbolism on the notion of,
252.

R

Reasoning subjects of, distribu-

table between two extremes, 5;
process of, in relation to typical
imagery and to symbolic ima-
gery, 5; of animals, 30, 32–3, 35,
compared with man, 35; with
and without the aid of words,
39-41.

Reid on definition, 14.
Riemann: conception of space as

finite but unbounded, 184; out-
line of dissertation on foundations
of geometry, 223-4, and criticism
of the filiation of ideas in, 224-6;
his alternative to Euclid's Axiom
10, 226; the notion of manifold-
ness and its obscurities, 227–32;
relation of
of his mathematical
analysis to this notion, 232, and
of space to this notion, 232.
Right Angle, the: see Perpendicu-
larity.

Russell, B. A. W.: on algebraic
imaginaries, 74, 83, note; on the
derivation of the notion of length,
167; on intuition in geometry,
170; on the axiom of parallels,
192; on the axiom of congruence,
203-5; on the irrelevance of
motion to the foundations of
geometry, 204; on congruence
and rigid bodies, 204; on Helm-
holtz's Flatland' and 'Sphere-
land', 218; on the relation be-
tween different space-constants,
237-8; on the reduction of

Sounds: articulate, primary associa-
tion of, with persons rather that
with things, 10.

:

Space as a particular case of a
more general notion, 65; kinds
of, and systems of geometry, 152;
infinitude and unboundedness of,
184; elementary flatness of, 220,
222; as a manifold, 229, 232;
measure of curvature of, 234;
non-Euclidean, as conceivable
but not imaginable, 234 and
note 2; conflicting notions on
curvature of, 235.
Space-constants: relation between
different, 237-8; see also Measure
of Curvature.

Spaces equality of, notion derived
from that of congruence of figure,
199.

Stallo, J. B. on the use of the

term 'quantity' in Algebra, 62.
Stout, G. F. on distinction between
words and substitute signs, 37;
the distinction too trenchant, 37;
on the relation of language to
conception, 45-7; limitations of
natural signs, 47; general agree-
ment with his views on relations
of thought and language, 47.
Straight Line, the derivation and
analysis of the notion, Chap. XI;
Euclid's treatment of, 137; con-
sidered as given as an infinite
whole' a confusion of thought,
137; is a particular shape, 158
and note, and the standard of
linear shape, 159; sense in which
it is said to be indefinable, 159;
alleged a priority of the notion,
163 and note, 254; Cayley and
J. S. Mill on the existence of, 163
and note, 164 and note; genesis
of notion in experience, 164-6;
various definitions of, 181.
Straightness: name of an identity
of linear shapes, 158.

Symbol: technical sense in which

the term is employed, 5, note; of
number, dual use of, 60; of
equality, real meaning of in
Algebra, 95; and meaning, con-
fusion of thought from neglect of
distinguishing between, 117.
Symbolism: originates in purpo-
sive adaptation of physical ad-
juncts of mental states, 10;
mathematical, distinction be-
tween and ordinary language, 37;
transition from representation to,
59; retro-active effect of, on pro-
cess of conception, 103.
Symbols: conditions of valid rea-
soning by the aid of, 77-8; alge-
braic, in relation to the notions of
number and quantity, 62-3, 251.

T

Taine on the nature of definition,
16.

Truth: relevance of, to geometrical
systems, 178.

W

Whitehead, A. N.: on philosophy

of mathematical imaginaries, 75-
83; his view of the part played
by substitutive signs in reasoning,
74-5; criticism of this view, 75-
7, which apparently derives from
Boole, 77; his elucidation of the
enigma involved in algebraic
imaginaries, 79; an elucidation
after the manner of Boole, who
begs the question, 80; influence
of his views on contemporary
philosophy of mathematics, 83;
his conception of Algebra, 82, 109,
252.

Whitney, W. D.: on the difference
of mental action in men and other
animals, 30, 32, 33, 35; on the
aid afforded by language to
thought, 34, 35; criticism of his
views, 31, 32, 33, 35-6.
Words: the fortresses of thought
(Hamilton), 27; and substitute
signs, functions of, 37, 38; as
unessential adjuncts in the pro-
cess of reasoning, 41, 45; as real
instruments in this process, 46.
(See also Language and Names.)