644 
ubiquitous than either of them, disclosing 
things that curiously match the things that 
they disclose and countless things besides, 
namely, the world of ideas and the rela- 
tions that bind them: a cosmic world, in 
the center whereof is the home of the mer- 
man. There remains to be named a fourth 
kind of light. I mean the light of emotion, 
the radiance and glory of things that, save 
by gleams and intimations, are not re- 
vealed in perception or in imagination or 
in thought: the light of the seraph’s world, 
the world of the good, the true and the 
beautiful, of the spirit of art, of aspiration 
and of religion. 
Such, in brief, is the cluster of worlds 
wherein dwell the spiritual interests of the 
human beings to whom it is our mission to 
teach mathematics. My thesis is that it is 
our privilege to show, in the way of our 
teaching it, that its human significance is 
not confined to one of the worlds but, like 
a subtle and ubiquitous ether, penetrates 
them all. Objectively viewed, conceptually 
taken, these worlds, unlike the spheres of 
the geometrician, do not intersect—a thing 
in one of them is not in another; but the 
things in one of them and the things in 
‘another may own a fine resemblance sery- 
ing for mutual recall and illustration, ef- 
fecting transfer of attention—transforma- 
tion as the mathematicians call it—from 
world to world; for whilst these worlds of 
interest, objectively viewed, have naught 
in common, yet subjectively they are 
united, united as differing mansions of the 
house of the human spirit. A relation, for 
example, between three independent vari- 
ables exists only in the gray light of 
thought, only in the world of the merman; 
the habitation of the geometric locus of the 
relation is the world of imagination; if a 
model of the locus be made or a drawing of 
it, this will be a thing in the world of per- 
ception; finally, the wondrous correlation 
SCIENCE 
[N.S. Vou. XXXV. No. 904 
of the three things, or the spiritual quali- 
ties of them—the sensuous beauty of the 
model or the drawing, the unfailing valid- 
ity of the given relation holding as it does 
throughout ‘‘the cycle of the eternal 
year,’’ the immobile presence of the locus 
or image poised there in eternal calm like 
a figure of justice—these may serve, in 
contemplating them, to evoke the radiance 
of the seraph’s world: and thus the circuit 
and interplay, ranging through the world 
of imagination and the world of thought 
from what is sensuous to what is supernal, 
is complete. It would not have seemed to 
Plato, as it may seem to us, a far ery from 
the prayer of a poet to the theorem of 
Pythagoras, for example, or to that of 
Archimedes respecting a sphere and its 
circumscribing cylinder. Yet I venture to 
say, that calm reflection upon the existence . 
and nature of such a theorem—cloistral 
contemplation, I mean, of the fact that it 
is really true, of its serene beauty, of its 
silent omnipresence throughout the infinite 
universe of space, of the absolute exacti- 
tude and invariance of its truth from ever- 
lasting to everlasting—will not fail to yield 
a sense of reverence and awe akin to the 
feeling that, for example, pervades this 
choral prayer by Sophocles: 
““ Oh! that my lot may lead me in the 
path of holy innocence of word and deed, 
the path which august laws ordain, laws 
that in the highest empyrean had their 
birth, of which Heaven is the father alone, 
nor did the race of mortal men beget them, 
nor shall oblivion put them to sleep. The 
god is mighty in them and he groweth not 
old.”’ 
But why should we think it strange that 
interests, though they seem to cluster about 
opposite poles, are yet united by a com- 
mon mood? Of the great world of human 
interests, mathematics is indeed but a part; 
but is a central part, and, in a profound 
