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% The Project Gutenberg EBook of The Foundations of Mathematics, by Paul Carus
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% Title: The Foundations of Mathematics                                   %
%        A Contribution to the Philosophy of Geometry                     %
%                                                                         %
% Author: Paul Carus                                                      %
%                                                                         %
% Release Date: June 18, 2018 [EBook #57355]                              %
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\begin{PGtext}
The Project Gutenberg EBook of The Foundations of Mathematics, by Paul Carus

This eBook is for the use of anyone anywhere in the United States and most
other parts of the world at no cost and with almost no restrictions
whatsoever.  You may copy it, give it away or re-use it under the terms of
the Project Gutenberg License included with this eBook or online at
book.klll.cc.  If you are not located in the United States, you'll have
to check the laws of the country where you are located before using this ebook.

Title: The Foundations of Mathematics
       A Contribution to the Philosophy of Geometry

Author: Paul Carus

Release Date: June 18, 2018 [EBook #57355]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK THE FOUNDATIONS OF MATHEMATICS ***
\end{PGtext}
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Produced by Andrew D. Hwang
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%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%
\FrontMatter
\PageSep{i}
\begin{center}
  \Huge THE FOUNDATIONS OF
  MATHEMATICS
  \vfill

  \small A CONTRIBUTION TO
  \bigskip

  \Large THE PHILOSOPHY OF GEOMETRY
  \vfill

  \small BY
  \bigskip

  \large DR. PAUL CARUS
  \vfill

  \small \Grk{<o je`os >ae`i gewmetre~i}---PLATO.
  \vfill

  \large CHICAGO \\
  \Large THE OPEN COURT PUBLISHING CO.
  \smallskip

  \small LONDON AGENTS \\
  KEGAN PAUL, TRENCH TRÜBNER \& CO., LTD. \\
  1908
\end{center}
\newpage
\PageSep{ii}
\begin{center}
  \null\vfill
  \small COPYRIGHT BY \\
  \normalsize THE OPEN COURT PUB. CO. \\
  1908
  \vfill
\end{center}
\PageSep{iii}

\TableofContents

\iffalse
TABLE OF CONTENTS.


THE SEARCH FOR THE FOUNDATIONS OF GEOMETRY:
HISTORICAL SKETCH.

PAGE

Axioms and the Axiom of Parallels 1

Metageometry 5

Precursors 7

Gauss 11

Riemann 15

Lobatchevsky 20

Bolyai 22

Later Geometricians 24

Grassmann 27

Euclid Still Unimpaired 31


THE PHILOSOPHICAL BASIS OF MATHEMATICS

The Philosophical Problem 35

Transcendentalism and Empiricism 38

The A Priori and the Purely Formal 40

Anyness and its Universality 46

Apriority of Different Degrees 49

Space as a Spread of Motion 56

Uniqueness of Pure Space 61

Mathematical Space and Physiological Space 63

Homogeneity of Space Due to Abstraction 66

Even Boundaries as Standards of Measurement 69

The Straight Line Indispensable 72

The Superreal 76

Discrete Units and the Continuum 78


MATHEMATICS AND METAGEOMETRY

Different Geometrical Systems 82

Tridimensionality 84

Three a Concept of Boundary 88
\PageSep{iv}

PAGE

Space of Four Dimensions 90

The Apparent Arbitrariness of the A Priori 96

Definiteness of Construction 99

One Space, But Various Systems of Space Measurement 104

Fictitious Spaces and the Apriority of All Space Measurement 109

Infinitude 116

Geometry Remains A Priori 119

Sense-Experience and Space 122

The Teaching of Mathematics 127


EPILOGUE 132

INDEX 139
\fi

\PageSep{1}
\MainMatter

\Chapter[Historical Sketch]{The Search for the Foundations
of~Geometry: Historical Sketch}

\Section{Axioms and the Axiom of Parallels}
\index{Axioms|indexff}%

\First{Mathematics} as commonly taught in our
schools is based upon axioms. These axioms
so called are a few simple formulas which the beginner
must take on trust.

Axioms are defined to be self-evident propositions,
and are claimed to be neither demonstrable
nor in need of demonstration. They are statements
which are said to command the assent of every
one who comprehends their meaning.

The word axiom\footnote
  {\Grk{>ax'iwma}.}
means ``honor, reputation,
high rank, authority,'' and is used by Aristotle
almost in the modern sense of the term, as ``a self-evident
highest principle,'' or ``a truth so obvious
as to be in no need of proof.'' It is derived from
the verb \Grk{<axio"un}, ``to deem worthy, to think fit, to
maintain,'' and is cognate with  \Grk{>'axios}, ``worth'' or
``worthy.''

Euclid does not use the term ``axiom.'' He
\index{Axiom@``Axiom,''!Euclid avoided}%
\index{Euclid|(}%
\index{Euclid!avoided "axiom,"}%
starts with Definitions,\footnote
  {\Grk{<'oroi}.}
which describe the meanings
of point, line, surface, plane, angle, etc. He
\PageSep{2}
then proposes Postulates\footnote
  {\Grk{a'iti'hmata}.}
\index{Postulates}%
in which he takes for
granted that we can draw straight lines from any
point to any other point, and that we can prolong
any straight line in a straight direction. Finally,
he adds what he calls Common Notions\footnote
  {\Grk{koina`i <'ennoiai}.}
\index{Common notions}%
which embody
some general principles of logic (of pure reason)
specially needed in geometry, such as that
things which are equal to the same thing are equal
to one another; that if equals be added to equals,
the wholes are equal, etc.

I need not mention here perhaps, since it is a
fact of no consequence, that the readings of the
several manuscripts vary, and that some propositions
(\eg, that all right angles are equal to one
another) are now missing, now counted among the
postulates, and now adduced as common notions.

The commentators of Euclid who did not understand
\index{Definitions of Euclid}%
the difference between Postulates and Common
Notions, spoke of both as axioms, and even
to-day the term Common Notion is mostly so translated.

In our modern editions of Euclid we find a
statement concerning parallel lines added to either
the Postulates or Common Notions. Originally it
appeared in Proposition~29 where it is needed to
prop up the argument that would prove the equality
of alternate angles in case a third straight line falls
upon parallel straight lines. It is there enunciated
as follows:

\begin{Quote}
``But those straight lines which, with another straight
\PageSep{3}
line falling upon them, make the interior angles on the same
side less than two right angles, do meet if continually produced.''
\end{Quote}

Now this is exactly a point that calls for proof.
Proof was then, as ever since it has remained, altogether
lacking. So the proposition was formulated
dogmatically thus:

\begin{Quote}
``If a straight line meet two straight lines, so as to make
the two interior angles on the same side of it taken together
less than two right angles, these straight lines being continually
produced, shall at length meet upon that side on
which are the angles which are less than two right angles.''
\end{Quote}

And this proposition has been transferred by
the editors of Euclid to the introductory portion of
the book where it now appears either as the fifth
Postulate or the eleventh, twelfth, or thirteenth
Common Notion. The latter is obviously the less
\index{Axioms!not Common Notions}%
appropriate place, for the idea of parallelism is
assuredly not a Common Notion; it is not a rule
of pure reason such as would be an essential condition
of all thinking, reasoning, or logical argument.
And if we do not give it a place of its own,
it should either be classed among the postulates, or
recast so as to become a pure definition. It is usually
referred to as ``the axiom of parallels.''
\index{Parallels, Axiom of}%

It seems to me that no one can read the axiom of
parallels as it stands in Euclid without receiving
the impression that the statement was affixed by a
later redactor. Even in Proposition~29, the original
place of its insertion, it comes in as an afterthought;
and if Euclid himself had considered the difficulty
\PageSep{4}
of the parallel axiom, so called, he would have placed
\index{Parallel lines in spherical space!theorem|indexnote}%
it among the postulates in the first edition of his
\index{Monist@\textit{Monist}|indexnote}%
book, or formulated it as a definition.\footnote
  {For Professor Halsted's ingenious interpretation of the origin
\index{Halsted, George Bruce|indexnote}%
  of the parallel theorem see \Title{The Monist}, Vol.~IV, No.~4, p.~487. He
  believes that Euclid anticipated metageometry, but it is not probable
  that the man who wrote the argument in Proposition~29 had the fifth
  Postulate before him. He would have referred to it or stated it at
  least approximately in the same words. But the argument in Proposition~29
  differs considerably from the parallel axiom itself.}

Though the axiom of parallels must be an interpolation,
it is of classical origin, for it was known
even to Proclus (410--485~\AD), the oldest commentator
\index{Proclus}%
of Euclid.

By an irony of fate, the doctrine of the parallel
axiom has become more closely associated with
Euclid's name than anything he has actually written,
\index{Euclid|)}%
and when we now speak of Euclidean geometry
we mean a system based upon that determination of
parallelism.

We may state here at once that all the attempts
made to derive the axiom of parallels from pure
reason were necessarily futile, for no one can prove
the absolute straightness of lines, or the evenness of
space, by logical argument. Therefore these concepts,
including the theory concerning parallels,
cannot be derived from pure reason; they are not
Common Notions and possess a character of their
\index{Common notions}%
own. But the statement seemed thus to hang in the
air, and there appeared the possibility of a geometry,
and even of several geometries, in whose domains
the parallel axiom would not hold good. This
large field has been called metageometry, hypergeometry,
\PageSep{5}
or pangeometry, and may be regarded
as due to a generalization of the space-conception
involving what might be called a metaphysics of
mathematics.

\Section{Metageometry}
\index{Metageometry|indexff}%

Mathematics is a most conservative science. Its
system is so rigid and all the details of geometrical
demonstration are so complete, that the science was
commonly regarded as a model of perfection. Thus
the philosophy of mathematics remained undeveloped
almost two thousand years. Not that there
were not great mathematicians, giants of thought,
men like the Bernoullis, Leibnitz and Newton, Euler,
and others, worthy to be named in one breath with
Archimedes, Pythagoras and Euclid, but they abstained
from entering into philosophical speculations,
and the very idea of a pangeometry remained
foreign to them. They may privately have reflected
on the subject, but they did not give utterance to
their thoughts, at least they left no records of them
to posterity.

It would be wrong, however, to assume that the
mathematicians of former ages were not conscious
of the difficulty. They always felt that there was
a flaw in the Euclidean foundation of geometry,
but they were satisfied to supply any need of basic
principles in the shape of axioms, and it has become
quite customary (I might almost say orthodox) to
say that mathematics is based upon axioms. In fact,
people enjoyed the idea that mathematics, the most
\PageSep{6}
lucid of all the sciences, was at bottom as mysterious
as the most mystical dogmas of religious faith.

Metageometry has occupied a peculiar position
among mathematicians as well as with the public at
large. The mystic hailed the idea of ``$n$-dimensional
spaces,'' of ``space curvature'' and of other conceptions
of which we can form expressions in abstract
terms but which elude all our attempts to render
them concretely present to our intelligence. He
relished the idea that by such conceptions mathematics
gave promise to justify all his speculations
and to give ample room for a multitude of notions
that otherwise would be doomed to irrationality.
In a word, metageometry has always proved attractive
to erratic minds. Among the professional mathematicians,
however, those who were averse to philosophical
speculation looked upon it with deep distrust,
and therefore either avoided it altogether or
rewarded its labors with bitter sarcasm. Prominent
mathematicians did not care to risk their reputation,
and consequently many valuable thoughts remained
unpublished. Even Gauss did not care to speak
\index{Gauss!his letter to Taurinus|indexf}%
out boldly, but communicated his thoughts to his
most intimate friends under the seal of secrecy, not
unlike a religious teacher who fears the odor of
heresy. He did not mean to suppress his thoughts,
but he did not want to bring them before the public
unless in mature shape. A letter to Taurinus concludes
\index{Taurinus, Letter of Gauss to|indexf}%
with the remark:

\begin{Quote}
``Of a man who has proved himself a thinking mathematician,
I fear not that he will misunderstand what I say,
\PageSep{7}
but under all circumstances you have to regard it merely as
a private communication of which in no wise public use, or
one that may lead to it, is to be made. Perhaps I shall publish
them myself in the future if I should gain more leisure
than my circumstances at present permit.
\Signature{``C. F. Gauss.}{``Goettingen,}{8.~November, 1824.''}
\end{Quote}

But Gauss never did publish anything upon this
topic although the seeds of his thought thereupon
fell upon fertile ground and bore rich fruit in the
works of his disciples, foremost in those of Riemann.


\Section{Precursors}

The first attempt at improvement in the matter
of parallelism was made by Nasir Eddin (1201--1274)
\index{Nasir Eddin}%
whose work on Euclid was printed in Arabic
in 1594 in Rome. His labors were noticed by John
Wallis who in 1651 in a Latin translation communicated
\index{Wallis, John|indexf}%
Nasir Eddin's exposition of the fifth Postulate
to the mathematicians of the University of
Oxford, and then propounded his own views in a
lecture delivered on July~11, 1663. Nasir Eddin
takes his stand upon the postulate that two straight
lines which cut a third straight line, the one at
right angles, the other at some other angle, will
converge on the side where the angle is acute and
diverge where it is obtuse. Wallis, in his endeavor
to prove this postulate, starts with the auxiliary
theorem:

\begin{Quote}
``If a limited straight line which lies upon an unlimited
straight line be prolonged in a straight direction,
\PageSep{8}
its prolongation will fall upon the unlimited straight
line.''
\end{Quote}

There is no need of entering into the details of
his proof of this auxiliary theorem. We may call
his theorem the proposition of the straight line and
may grant to him that he proves the straightness of
the straight line. In his further argument Wallis
shows the close connection of the problem of parallels
with the notion of similitude.

Girolamo Saccheri, a learned Jesuit of the seventeenth
\index{Saccheri, Girolamo|indexf}%
century, attacked the problem in a new way.
Saccheri was born September~5, 1667, at San Remo.
Having received a good education, he became a
member of the Jesuit order March~24, 1685, and
served as a teacher of grammar at the Jesuit College
di~Brera, in Milan, his mathematical colleague being
Tommaso Ceva (a brother of the more famous
Giovanni Ceva). Later on he became Professor of
Philosophy and Polemic Theology at Turin and in
1697 at Pavia. He died in the College di~Brera
October~25, 1733.

Saccheri saw the close connection of parallelism
with the right angle, and in his work on Euclid\footnote
  {\Title{Euclides ab omni naevo vindicatus; sive conatus geometricus
  quo stabiliuntur prima ipsa universae geometriae principia}. Auctore
  Hier\-onymo Saccherio Societatis Jesu. Mediolani, 1773.}
he
examines three possibilities. Taking a quadrilateral
$ABCD$ with the angles at~$A$ and~$B$ right angles
and the sides $AC$ and $BD$ equal, the angles at~$C$ and~$D$
are without difficulty shown to be equal each to
the other. They are moreover right angles or else
they are either obtuse or acute. He undertakes to
\PageSep{9}
prove the absurdity of these two latter suppositions
so as to leave as the only solution the sole possibility
left, viz., that they must be right angles. But he
finds difficulty in pointing out the contradiction to
which these assumptions may lead and thus he opens
a path on which Lobatchevsky (1793--1856) and
Bolyai (1802--1860) followed, reaching a new view
which makes three geometries possible, viz., the
geometries of (1)~the acute angle, (2)~the obtuse
angle, and (3)~the right angle, the latter being the
Euclidean geometry, in which the theorem of parallels
holds.
\begin{center}
  \Input{page009}
\end{center}

While Saccheri seeks the solution of the problem
through the notion of the right angle, the German
mathematician Lambert starts from the notion of
\index{Lambert, Johann Heinrich|indexf}%
the angle-sum of the triangle.

Johann Heinrich Lambert was born August~26,
1728, in Mühlhausen, a city which at that time was
a part of Switzerland. He died in 1777. His \Title{Theory
of the Parallel Lines}, written in 1766, was not
published till 1786, nine years after his death, by
Bernoulli and Hindenburg in the \Title{Magazin für die
\index{Bernoulli}%
reine und angewandte Mathematik}.

Lambert points out that there are three possibilities:
the sum of the angles of a triangle may be
\PageSep{10}
exactly equal to, more than, or less than $180$~degrees.
The first will make the triangle a figure in a plane,
the second renders it spherical, and the third produces
a geometry on the surface of an imaginary
sphere. As to the last hypothesis Lambert said not
without humor:\footnote
  {P.~351, last line in the \Title{Magazin für die reine und angewandte
  Mathematik}, 1786.}
\begin{Quote}
``This result\footnote
  {Lambert refers to the proposition that the mooted angle might
  be less than $90$~degrees.}
possesses something attractive which easily
suggests the wish that the third hypothesis might be true.''
\end{Quote}

He then adds:\footnote
  {\Foreign{Ibid.}, p.~352.}
\begin{Quote}
``But I do not wish it in spite of these advantages, because
there would be innumerable other inconveniences.
The trigonometrical tables would become infinitely more
complicated, and the similitude as well as proportionality of
figures would cease altogether. No figure could be represented
except in its own absolute size; and astronomy would
be in a bad plight, etc.''
\end{Quote}

Lobatchevsky's geometry is an elaboration of
\index{Lobatchevsky}%
Lambert's third hypothesis, and it has been called
``imaginary geometry'' because its trigonometric
formulas are those of the spherical triangle if its
sides are imaginary, or, as Wolfgang Bolyai has
shown, if the radius of the sphere is assumed to be
imaginary $=(\sqrt{-1})r$.

France has contributed least to the literature on
the subject. Augustus De~Morgan records the following
\index{De Morgan, Augustus}%
story concerning the efforts of her greatest
mathematician to solve the Euclidean problem. Lagrange,
\index{Lagrange|indexf}%
\PageSep{11}
he says, composed at the close of his life
a discourse on parallel lines. He began to read it
in the Academy but suddenly stopped short and
said: ``II faut que j'y songe encore.'' With these
words he pocketed his papers and never recurred
to the subject.

Legendre's treatment of the subject appears in
\index{Legendre}%
the third edition of his elements of Euclid, but he
omitted it from later editions as too difficult for beginners.
Like Lambert he takes his stand upon the
notion of the sum of the angles of a triangle, and
like Wallis he relies upon the idea of similitude,
saying that ``the length of the units of measurement
is indifferent for proving the theorems in question.''\footnote
  {\Title{Mémoires de l'Académie des Sciences de l'Institut de France}.
  Vol.~XII, 1833.}


\Section{Gauss}
\index{Gauss|indexff}%

A new epoch begins with Gauss, or rather with
his ingenious disciple Riemann. While Gauss was
rather timid about speaking openly on the subject,
he did not wish his ideas to be lost to posterity. In
a letter to Schumacher dated May~17, 1831, he
said:
\begin{Quote}
\index{Schumacher, Letter of Gauss to}%
``I have begun to jot down something of my own meditations,
which are partly older than forty years, but which
I have never written out, being obliged therefore to excogitate
many things three or four times over. I do not wish
them to pass away with me.''
\end{Quote}

The notes to which Gauss here refers have not
been found among his posthumous papers, and it
\PageSep{12}
therefore seems probable that they are lost, and our
knowledge of his thoughts remains limited to the
comments that are scattered through his correspondence
with mathematical friends.

Gauss wrote to Bessel (1784--1846) January~27,
1829:
\begin{Quote}
\index{Bessel, Letter of Gauss to|indexff}%
\index{Gauss!his letter to Bessel|indexff}%
``I have also in my leisure hours frequently reflected upon
another problem, now of nearly forty years' standing. I
refer to the foundations of geometry. I do not know
whether I have ever mentioned to you my views on this
matter. My meditations here also have taken more definite
shape, and my conviction that we cannot thoroughly demonstrate
geometry \Foreign{a~priori} is, if possible, more strongly confirmed
than ever. But it will take a long time for me to
bring myself to the point of working out and making public
my \emph{very extensive} investigations on this subject, and possibly
this will not be done during my life, inasmuch as I
stand in dread of the clamors of the B{\oe}otians, which would
be certain to arise, if I should ever give \emph{full} expression to
my views. It is curious that \emph{in addition to} the celebrated
flaw in Euclid's Geometry, which mathematicians have hitherto
endeavored in vain to patch and never will succeed,
there is still another blotch in its fabric to which, so far as
I know, attention has never yet been called and which it will
by no means be easy, if at all possible, to remove. This is
the definition of a plane as a surface in which a straight
line joining \emph{any two} points lies \emph{wholly} in that plane. This
definition contains \emph{more} than is requisite to the determination
of a surface, and tacitly involves a theorem which is in
need of prior proof.''
\end{Quote}

Bessel in his answer to Gauss makes a distinction
between Euclidean geometry as practical and
metageometry (the one that does not depend upon
\PageSep{13}
the theorem of parallel lines) as true geometry.
He writes under the date of February~10, 1829:
\begin{Quote}
``I should regard it as a great misfortune if you were to
allow yourself to be deterred by the `clamors of the B{\oe}otians'
from explaining your views of geometry. From what
Lambert has said and Schweikart orally communicated, it
has become clear to me that our geometry is incomplete and
stands in need of a correction which is hypothetical and
which vanishes when the sum of the angles of a plane triangle
is equal to~$180°$. This would be the \emph{true} geometry
and the Euclidean the \emph{practical}, at least for figures on the
earth.''
\end{Quote}

In another letter to Bessel, April~9, 1830, Gauss
sums up his views as follows:
\begin{Quote}
``The ease with which you have assimilated my notions of
geometry has been a source of genuine delight to me, especially
as so few possess a natural bent for them. I am profoundly
convinced that the theory of space occupies an entirely
different position with regard to our knowledge \Foreign{a~priori}
from that of the theory of numbers (\Foreign{Grössenlehre});
that perfect conviction of the necessity and therefore the
absolute truth which is characteristic of the latter is totally
wanting to our knowledge of the former. We must confess
in all humility that a number is \emph{solely} a product of our mind.
Space, on the other hand, possesses also a reality outside of
our mind, the laws of which we cannot fully prescribe \Foreign{a~priori}.''
\end{Quote}

Another letter of Gauss may be quoted here in
full. It is a reply to Taurinus and contains an appreciation
of his essay on the Parallel Lines. Gauss
writes from Göttingen, Nov.~8, 1824:
\begin{Quote}
\index{Gauss!his letter to Taurinus|indexf}%
\index{Taurinus, Letter of Gauss to|indexf}%
``Your esteemed communication of October~30th, with
\PageSep{14}
the accompanying little essay, I have read with considerable
pleasure, the more so as I usually find no trace whatever of
real geometrical talent in the majority of the people who
offer new contributions to the so-called theory of parallel
lines.

``With regard to your effort, I have nothing (or not
much) more to say, except that it is incomplete. Your presentation
of the demonstration that the sum of the three angles
of a plane triangle cannot be greater than~$180°$, does
indeed leave something to be desired in point of geometrical
precision. But this could be supplied, and there is no doubt
that the impossibility in question admits of the most rigorous
demonstration. But the case is quite different with the
second part, viz., that the sum of the angles cannot be
smaller than~$180°$; this is the real difficulty, the rock on
which all endeavors are wrecked. I surmise that you have
not employed yourself long with this subject. I have pondered
it for more than thirty years, and I do not believe
that any one could have concerned himself more exhaustively
with this second part than I, although I have not
published anything on this subject. The assumption that
the sum of the three angles is smaller than~$180°$ leads to a
new geometry entirely different from our Euclidean,---a
geometry which is throughout consistent with itself, and
which I have elaborated in a manner entirely satisfactory
to myself, so that I can solve every problem in it with the
exception of the determining of a constant, which is not
\Foreign{a~priori} obtainable. The larger this constant is taken, the
nearer we approach the Euclidean geometry, and an infinitely
large value will make the two coincident. The propositions
of this geometry appear partly paradoxical and absurd
to the uninitiated, but on closer and calmer consideration
it will be found that they contain in them absolutely
nothing that is impossible. Thus, the three angles of a
triangle, for example, can be made as small as we will,
provided the sides can be taken large enough; whilst the
\PageSep{15}
area of a triangle, however great the sides may be taken,
can never exceed a definite limit, nay, can never once reach
it. All my endeavors to discover contradictions or inconsistencies
in this non-Euclidean geometry have been in vain,
and the only thing in it that conflicts with our reason is the
fact that if it were true there would necessarily exist in space
a linear magnitude quite \emph{determinate in itself}, yet unknown
to us. But I opine that, despite the empty word-wisdom of
the metaphysicians, in reality we know little or nothing of
the true nature of space, so much so that we are not at liberty
to characterize as \emph{absolutely impossible} things that strike us
as unnatural. If the non-Euclidean geometry were the true
geometry, and the constant in a certain ratio to such magnitudes
as lie within the reach of our measurements on the
earth and in the heavens, it could be determined \Foreign{a~posteriori}.
I have, therefore, in jest frequently expressed the desire that
the Euclidean geometry should not be the true geometry,
because in that event we should have an absolute measure
\Foreign{a~priori}.''
\end{Quote}

Schweikart, a contemporary of Gauss, may incidentally
\index{Schweikart}%
\index{Astral geometry}%
\index{Geometry!Astral}%
be mentioned as having worked out a
geometry that would be independent of the Euclidean
axiom. He called it astral geometry.\footnote
  {\Title{Die Theorie der Parallellinien, nebst dem Vorschlag ihrer Verbannung
  aus der Geometrie}. Leipsic and Jena, 1807.}


\Section{Riemann}
\index{Riemann|indexff}%

Gauss's ideas fell upon good soil in his disciple
Riemann (1826--1866) whose Habilitation Lecture
on ``The Hypotheses which Constitute the Bases of
Geometry'' inaugurates a new epoch in the history
of the philosophy of mathematics.

Riemann states the situation as follows. I quote
\PageSep{16}
from Clifford's almost too literal translation (first
\index{Clifford}%
published in Nature, 1873):
\index{Nature@\textit{Nature}}%
\begin{Quote}
``It is known that geometry assumes, as things given,
both the notion of space and the first principles of constructions
in space. She gives definitions of them which are
merely nominal, while the true determinations appear in the
form of axioms. The relation of these assumptions remains
consequently in darkness; we neither perceive whether and
how far their connection is necessary, nor, \Foreign{a~priori}, whether
it is possible.

``From Euclid to Legendre (to name the most famous of
modern reforming geometers) this darkness was cleared up
neither by mathematicians nor by such philosophers as concerned
themselves with it.''
\end{Quote}

Riemann arrives at a conclusion which is negative.
He says:
\begin{Quote}
``The propositions of geometry cannot be derived from
general notions of magnitude, but the properties which distinguish
space from other conceivable triply extended magnitudes
are only to be deduced from experience.''
\end{Quote}

In the attempt at discovering the simplest matters
of fact from which the measure-relations of
space may be determined, Riemann declares that---
\begin{Quote}
``Like all matters of fact, they are not necessary, but
  only of empirical certainty: they are hypotheses.''
  \end{Quote}

Being a mathematician, Riemann is naturally
bent on deductive reasoning, and in trying to find
a foothold in the emptiness of pure abstraction he
starts with general notions. He argues that position
must be determined by measuring quantities,
and this necessitates the assumption that length of
\PageSep{17}
lines is independent of position. Then he starts with
the notion of manifoldness, which he undertakes to
specialize. This specialization, however, may be
done in various ways. It may be continuous, as is
geometrical space, or consist of discrete units, as
do arithmetical numbers. We may construct manifoldnesses
of one, two, three, or $n$~dimensions, and
the elements of which a system is constructed may
be functions which undergo an infinitesimal displacement
expressible by~$dx$. Thus spaces become
possible in which the directest linear functions (analogous
to the straight lines of Euclid) cease to be
straight and suffer a continuous deflection which
may be positive or negative, increasing or decreasing.

Riemann argues that the simplest case will be,
if the differential line-element~$ds$ is the square root
of an always positive integral homogeneous function
of the second order of the quantities~$dx$ in which
the coefficients are continuous functions of the quantities~$x$,
viz., $ds = \sqrt{\sum dx^{2}}$, but it is one instance only
of a whole class of possibilities. He says:
\begin{Quote}
``Manifoldnesses in which, as in the plane and in space,
the line-element may be reduced to the form $\sqrt{\sum dx^{2}}$, are
therefore only a particular case of the manifoldnesses to be
here investigated; they require a special name, and therefore
these manifoldnesses in which the square of the line-element
may be expressed as the sum of the squares of complete differentials
I will call \emph{flat}.''
\end{Quote}

The Euclidean plane is the best-known instance
of flat space being a manifold of a zero curvature.
\PageSep{18}

Flat or even space has also been called by the
new-fangled word \emph{homaloidal},\footnote
  {From the Greek \Grk{<omal'os}, level.}
\index{Homaloidal}%
which recommends
itself as a technical term in distinction from the
popular meaning of even and flat.

In applying his determination of the general
notion of a manifold to actual space, Riemann expresses
its properties thus:
\begin{Quote}
``In the extension of space-construction to the infinitely
great, we must distinguish between \emph{unboundedness} and \emph{infinite
extent}; the former belongs to the extent-relations, the
latter to the measure relations. That space is an unbounded
threefold manifoldness, is an assumption which is developed
by every conception of the outer world; according to which
every instant the region of real perception is completed and
the possible positions of a sought object are constructed, and
which by these applications is forever confirming itself. The
unboundedness of space possesses in this way a greater empirical
certainty than any external experience. But its infinite
extent by no means follows from this; on the other
hand, if we assume independence of bodies from position,
and therefore ascribe to space constant curvature, it must
necessarily be finite, provided this curvature has ever so
small a positive value. If we prolong all the geodetics
starting in a given surface-element, we should obtain an
unbounded surface of constant curvature, \ie, a surface
which in a \emph{flat} manifoldness of three dimensions would take
the form of a sphere, and consequently be finite.''
\end{Quote}

It is obvious from these quotations that Riemann
is a disciple of Kant. He is inspired by his
teacher Gauss and by Herbart. But while he starts
a transcendentalist, employing mainly the method
of deductive reasoning, he arrives at results which
\PageSep{19}
would stamp him an empiricist of the school of Mill.
He concludes that the nature of real space, which is
only one instance among many possibilities, must
be determined \Foreign{a~posteriori}. The problem of tridimensionality
and homaloidality are questions
which must be decided by experience, and while
upon the whole he seems inclined to grant that
Euclidean geometry is the most practical for a solution
of the coarsest investigations, he is inclined
to believe that real space is non-Euclidean. Though
the deviation from the Euclidean standard can only
be slight, there is a possibility of determining it by
exact measurement and observation.

Riemann has succeeded in impressing his view
upon \Chg{meta-geometricians}{metageometricians} down to the present day.
They have built higher and introduced new ideas,
yet the cornerstone of metageometry remained the
same. It will therefore be found recommendable
in a discussion of the problem to begin with a criticism
of his Habilitation Lecture.

It is regrettable that Riemann was not allowed to
work out his philosophy of mathematics. He died
at the premature age of forty, but the work which
he pursued with so much success had already been
taken up by two others, Lobatchevsky and Bolyai,
who, each in his own way, actually contrived a
geometry independent of the theorem of parallels.

It is perhaps no accident that the two independent
and almost simultaneous inventors of a non-Euclidean
geometry are original, not to say wayward,
characters living on the outskirts of European
\PageSep{20}
civilization, the one a Russian, the other a
Magyar.


\Section{Lobatchevsky}
\index{Lobatchevsky|indexff}%

Nicolai Ivanovich Lobatchevsky\footnote
  {The name is spelled differently according to the different
  methods of transcribing Russian characters.}
was born October~22
(Nov.~2 of our calendar), 1793, in the
town of Makariev, about $40$~miles above Nijni Novgorod
on the Volga. His father was an architect
who died in 1797, leaving behind a widow and two
small sons in poverty. At the gymnasium Lobatchevsky
was noted for obstinacy and disobedience,
and he escaped expulsion only through the protection
of his mathematical teacher, Professor Bartels,
who even then recognized the extraordinary talents
of the boy. Lobatchevsky graduated with distinction
and became in his further career professor of
mathematics and in 1827 Rector of the University
of Kasan. Two books of his offered for official
publication were rejected by the paternal government
of Russia, and the manuscripts may be considered
as lost for good. Of his several essays on
the theories of parallel lines we mention only the
one which made him famous throughout the whole
mathematical world, \Title{Geometrical Researches on the
Theory of Parallels}, published by the University of
Kasan in 1835.\footnote
  {For further details see Prof.\ G.~B. Halsted's article ``Lobachevski''
\index{Halsted, George Bruce|indexnote}%
\index{Open Court@\textit{Open Court}|indexnote}%
  in \Title{The Open Court}, 1898, pp.~411~ff.}

Lobatchevsky divides all lines, which in a plane
go out from a point~$A$ with reference to a given
\begin{figure}[hb]
  \centering
  \Input{page021}
\end{figure}
\PageSep{21}
straight line~$BC$ in the same plane, into two classes---cutting
and not cutting. In progressing from the
not-cutting lines, such as $EA$ and~$GA$, to the cutting
lines, such as~$FA$, we must come upon one~$HA$
that is the boundary between the two classes; and it
is this which he calls the parallel line. He designates
the parallel angle on the perpendicular ($p = AD$,
dropped from~$A$ upon~$BC$) by~$\Pi$. If $\Pi(p) < \frac{1}{2}\pi$
(viz., $90$~degrees) we shall have on the other side
of~$p$ another angle $DAK = \Pi(p)$ parallel to~$DB$,
so that on this assumption we
must make a distinction of
\emph{sides in parallelism}, and we
must allow two parallels, one
on the one and one on the
other side. If $\Pi(p) = \frac{1}{2}\pi$
we have only intersecting
lines and one parallel; but if
$\Pi(p) < \frac{1}{2}\pi$ we have two parallel
lines as boundaries between
the intersecting and
non-intersecting lines.

We need not further develop Lobatchevsky's
idea. Among other things, he proves that ``if in
any rectilinear triangle the sum of the three angles
is equal to two right angles, so is this also the case
for every other triangle,'' that is to say, each instance
is a sample of the whole, and if one case is
established, the nature of the whole system to which
it belongs is determined.

The importance of Lobatchevsky's discovery
\PageSep{22}
consists in the fact that the assumption of a geometry
from which the parallel axiom is rejected, does
not lead to self-contradictions but to the conception
of a general geometry of which the Euclidean is one
possibility. This general geometry was later on
most appropriately called by Lobatchevsky ``Pangeometry.''
\index{Pangeometry}%


\Section{Bolyai}
\index{Bolyai@Bolyai, János|indexff}%

John (or, as the Hungarians say, János) Bolyai
imbibed the love of mathematics in his father's
house. He was the son of Wolfgang (or Farkas)
Bolyai, a fellow student of Gauss at Göttingen when
the latter was nineteen years old. Farkas was professor
of mathematics at Maros Vásárhely and
wrote a two-volume book on the elements of mathematics\footnote
  {\Title{Tentamen juventutem studiosam in elementa matheos \emph{etc.}\ introducendi},
  printed in Maros Vásárhely. By Farkas Bolyai. Part~I.
  Maros Vásárhely, 1832. It contains the essay by János Bolyai as an
  Appendix.}
and in it he incidentally mentions his vain
attempts at proving the axiom of parallels. His
book was only partly completed when his son János
wrote him of his discovery of a mathematics of pure
space. He said:
\begin{Quote}
``As soon as I have put it into order, I intend to write
and if possible to publish a work on parallels. At this
moment, it is not yet finished, but the way which I have followed
promises me with certainty the attainment of my aim,
if it is at all attainable. It is not yet attained, but I have
discovered such magnificent things that I am myself astonished
at the result. It would forever be a pity, if they were
lost. When you see them, my father, you yourself will concede
\PageSep{23}
it. Now I cannot say more, only so much that \emph{from
nothing I have created another wholly new world}. All that
I have hitherto sent you compares to it as a house of cards
to a castle.''\footnote
  {See Halsted's introduction to the English translation of Bolyai's
\index{Halsted, George Bruce}%
  \Title{Science Absolute of Space}, p.~xxvii.}
\end{Quote}

János being convinced of the futility of proving
Euclid's axiom, constructed a geometry of absolute
space which would be independent of the axiom of
parallels. And he succeeded. He called it the \Title{Science
Absolute of Space},\footnote
  {\Title{Appendix scientiam spatii absolute veram exhibens: a veritate
  aut falsitate axiomatis XI. Euclidei (a~priori haud unquam decidenda)
  independentem; Adjecta ad casum falsitatis quadratura circuli
  geometrica.}}
an essay of twenty-four
pages which Bolyai's father incorporated in the
first volume of his \Title{Tentamen} as an appendix.
\index{Tentamen@\textit{Tentamen}}%

Bolyai was a thorough Magyar. He was wont
to dress in high boots, short wide Hungarian trousers,
and a white jacket. He loved the violin and
was a good shot. While serving as an officer in
the Austrian army, János was known for his hot
temper, which finally forced him to resign his commission
as a captain, and we learn from Professor
Halsted that for some provocation he was challenged
by thirteen cavalry officers at once. János
calmly accepted and proposed to fight them all, one
after the other, on condition that he be permitted
after each duel to play a piece on his violin. We
know not the nature of these duels nor the construction
of the pistols, but the fact remains assured that
he came out unhurt. As for the rest of the report
that ``he came out victor from the thirteen duels,
\PageSep{24}
leaving his thirteen adversaries on the square,''
we may be permitted to express a mild but deep-seated
doubt.

János Bolyai starts with straight lines in the
same plane, which may or may not cut each other.
Now there are two possibilities: there may be a
system in which straight lines can be drawn which
do not cut one another, and another in which they all
cut one another. The former, the Euclidean he
calls~$\Sigma$, the latter~$S$. ``All theorems,'' he says,
``which are not expressly asserted for~$\Sigma$ or for~$S$
are enunciated absolutely, that is, they are true
whether $\Sigma$~or $S$ is reality.''\footnote
  {See Halsted's translation, p.~14.}
The system~$S$ can
be established without axioms and is actualized in
spherical trigonometry, (\Foreign{ibid.}\ p.~21). Now $S$~can
%[** TN: "\Sigma" missing in the original]
be changed to~$\Sigma$, viz., plane geometry, by reducing
the constant~$i$ to its limit (where the sect $y = 0$)
which is practically the same as the construction
of a circle with $r = \infty$, thus changing its periphery
into a straight line.


\Section{Later Geometricians}

The labors of Lobatchevsky and Bolyai are significant
in so far as they prove beyond the shadow
of a doubt that a construction of geometries other
than Euclidean is possible and that it involves us
in no absurdities or contradictions. This upset the
traditional trust in Euclidean geometry as absolute
truth, and it opened at the same time a vista of new
\PageSep{25}
problems, foremost among which was the question
as to the mutual relation of these three different
geometries.

It was Cayley who proposed an answer which
\index{Cayley}%
was further elaborated by Felix Klein. These two
\index{Klein, Felix}%
ingenious mathematicians succeeded in deriving by
projection all three systems from one common aboriginal
form called by Klein \Foreign{Grundgebild} or the
Absolute. In addition to the three geometries hitherto
\index{Absolute@Absolute, The}%
known to mathematicians, Klein added a fourth
one which he calls \emph{elliptic}.\footnote
  {``Ueber die sogenannte nicht-euklidische Geometrie'' in \Title{Math.\
  Annalen},~\textbf{4}, 6 (1871--1872). \Title{Vorlesungen über nicht-euklidische Geometrie},
  Göttingen, 1893.}

Thus we may now regard all the different geometries
as three species of one and the same genus
and we have at least the satisfaction of knowing that
there is \Foreign{terra firma} at the bottom of our mathematics,
though it lies deeper than was formerly supposed.

Prof.\ Simon Newcomb of Johns Hopkins University,
\index{Newcomb, Simon}%
although not familiar with Klein's essays,
worked along the same line and arrived at similar
results in his article on ``Elementary Theorems Relating
to the Geometry of a Space of Three Dimensions
\index{Elliptic geometry}%
\index{Geometry!Elliptic}%
\index{Fourth dimension}%
and of Uniform Positive Curvature in the
Fourth Dimension.\Add{''}\footnote
  {Crelle's \Title{Journal für die reine und angewandte Mathematik}.
  1877.}

In the meantime the problem of geometry became
interesting to outsiders also, for the theorem
of parallel lines is a problem of space. A most
\index{Parallel lines in spherical space!theorem|indexf}%
\PageSep{26}
excellent treatment of the subject came from the
pen of the great naturalist Helmholtz who wrote
\index{Helmholtz}%
two essays that are interesting even to outsiders
because written in a most popular style.\footnote
  {``Ueber die thatsächlichen Grundlagen der Geometrie,'' in \Title{Wissenschaftl.\
  Abh}., 1866, Vol.~II., p.~610~ff., and ``Ueber die Thatsachen,
  die der Geometrie zu Grunde liegen,'' \Foreign{ibid.}, 1868, p.~618~ff.}

A collection of all the materials from Euclid to
Gauss, compiled by Paul Stäckel and Friedrich
Engel under the title \Title{Die Theorie der Parallellinien
\index{Engel, Friedrich}%
\index{Stäckel, Paul}%
von Euklid bis auf Gauss, eine Urkundensammlung
zur Vorgeschichte der nicht-euklidischen Geometrie},
is perhaps the most useful and important publication
in this line of thought, a book which has become
indispensable to the student of metageometry and
\index{Metageometry!History of}%
its history.

A store of information may be derived from
\index{Russell, Bertrand A. W.}%
Bertrand A.~W. Russell's essay on the \Title{Foundations
of Geometry}. He divides the history of metageometry
into three periods: The synthetic, consisting
of suggestions made by Legendre and Gauss; the
metrical, inaugurated by Riemann and characterized
by Lobatchevsky and Bolyai; and the projective,
represented by Cayley and Klein, who reduce
metrical properties to projection and thus show that
Euclidean and non-Euclidean systems may result
from ``the absolute.''

Among American writers no one has contributed
more to the interests of metageometry than the
indefatigable Dr.\ George Bruce Halsted.\footnote
  {From among his various publications we mention only his
\index{Halsted, George Bruce}%
  translations: \Title{Geometrical Researches on the Theory of Parallels by
  Nicholaus Lobatchewsky}. Translated from the Original. And \Title{The
  Science Absolute of Space, Independent of the Truth or Falsity of
  Euclid's Axiom~XI. (which can never be decided \Foreign{a priori}). By John
  Bolyai}. Translated from the Latin, both published in Austin, Texas,
  the translator's former place of residence. Further, we refer the
  reader to Halsted's bibliography of the literature on hyperspace and
  non-Euclidean geometry in the \Title{American Journal of Mathematics},
  Vol.~I., pp.~261--276, 384, 385, and Vol.~II., pp.~65--70.}
He has
\PageSep{27}
\index{Halsted, George Bruce}%
not only translated Bolyai and Lobatchevsky, but
\index{Bolyai@Bolyai, János!translated}%
\index{Lobatchevsky!translated}%
in numerous articles and lectures advanced his own
theories toward the solution of the problem.

Prof.\ B.~J. Delb{\oe}uf and Prof.\ H.~Poincaré have
\index{Delboeuf@Delb{\oe}uf, B. J.}%
\index{Poincare@Poincaré, H.}%
expressed their conceptions as to the nature of the
bases of mathematics, in articles contributed to \Title{The
Monist}.\footnote
  {They are as follows: ``Are the Dimensions of the Physical
  World Absolute?'' by Prof.\ B.~J. Delb{\oe}uf, \Title{The Monist}, January,
  1894; ``On the Foundations of Geometry,'' by Prof.\ H.~Poincaré,
  \Title{The Monist}, October, 1898; also ``Relations Between Experimental
  and Mathematical Physics,'' \Title{The Monist}, July, 1902.}
\index{Monist@\textit{Monist}}%
The latter treats the subject from a purely
mathematical standpoint, while Dr.\ Ernst Mach in
\index{Mach, Ernst}%
his little book \Title{Space and Geometry},\footnote
  {Chicago: Open Court Pub.\ Co., 1906. This chapter also appeared
  in \Title{The Monist} for April, 1901.}
in the chapter
``On Physiological, as Distinguished from Geometrical,
Space,'' attacks the problem in a very original
manner and takes into consideration mainly the natural
growth of space conception. His exposition
might be called ``the physics of geometry.''


\Section{Grassmann}
\index{Grassmann|indexff}%

I cannot conclude this short sketch of the history
of metageometry without paying a tribute to the
memory of Hermann Grassmann of Stettin, a mathematician
of first degree whose highly important
results in this line of work have only of late found
\PageSep{28}
the recognition which they so fully deserve. I do
not hesitate to say that Hermann Grassmann's \Title{Lineare
Ausdehnungslehre} is the best work on the
\index{Ausdehnungslehre@\textit{Ausdehnungslehre}, Grassmann's}%
philosophical foundation of mathematics from the
standpoint of a mathematician.

Grassmann establishes first the idea of mathematics
as the science of pure form. He shows that
the mathematician starts from definitions and then
proceeds to show how the product of thought may
originate either by the single act of creation, or by
the double act of positing and combining. The
former is the continuous form, or magnitude, in the
narrower sense of the term, the latter the discrete
form or the method of combination. He distinguishes
between intensive and extensive magnitude
and chooses as the best example of the latter the
sect\footnote
  {Grassmann's term is \Foreign{Strecke}, a word connected with the Anglo-Saxon
\index{Halsted, George Bruce|indexnote}%
  ``Stretch,'' being that portion of a line that stretches between
  two points. The translation ``sect\Chg{,}{}'' was suggested by Prof.\ G.~B.
  Halsted.}
or limited straight line laid down in some
definite direction. Hence the name of the new science,
``theory of linear extension.''

Grassmann constructs linear formations of
which systems of one, two, three, and $n$~degrees
are possible. The Euclidean plane is a system of
second degree, and space a system of third degree.
He thus generalizes the idea of mathematics, and
having created a science of pure form, points out
that geometry is one of its applications which originates
under definite conditions.

Grassmann made the straight line the basis of
\PageSep{29}
his geometrical definitions. He defines the plane
as the totality of parallels which cut a straight line
and space as the totality of parallels which cut the
plane. Here is the limit to geometrical construction,
but abstract thought knows of no bounds. Having
generalized our mathematical notions as systems
of first, second, and third degree, we can continue
in the numeral series and construct systems of four,
five, and still higher degrees. Further, we can determine
any plane by any three points, given in the
figures $x_{1}$, $x_{2}$, $x_{3}$, not lying in a straight line. If the
equation between these three figures be homogeneous,
the totality of all points that correspond to it will
be a system of second degree. If this homogeneous
equation is of the first grade, this system of second
degree will be simple, viz., of a straight line; but
if the equation be of a higher grade, we shall have
curves for which not all the laws of plane geometry
hold good. The same considerations lead to a distinction
between homaloidal space and non-Euclidean
\index{Ausdehnungslehre@\textit{Ausdehnungslehre}, Grassmann's|indexnote}%
systems.\footnote
  {See Grassmann's \Title{Ausdehnungslehre} , 1844, Anhang~I, pp.~273--274.}

Being professor at a German gymnasium and
not a university, Grassmann's book remained neglected
and the newness of his methods prevented
superficial readers from appreciating the sweeping
significance of his propositions. Since there was
no call whatever for the book, the publishers returned
the whole edition to the paper mill, and the
complimentary copies which the author had sent
\PageSep{30}
out to his friends are perhaps the sole portion that
was saved from the general doom.

Grassmann, disappointed in his mathematical
labors, had in the meantime turned to other studies
and gained the honorary doctorate of the University
of Tübingen in recognition of his meritorious
work on the St.~Petersburg Sankrit Dictionary,
when Victor Schlegel called attention to the similarity
\index{Schlegel, Victor}%
of Hamilton's theory of vectors to Grassmann's
concept of \Foreign{Strecke}, both being limited straight
lines of definite direction. Suddenly a demand for
Grassmann's book was created in the market; but
alas!\ no copy could be had, and the publishers
deemed it advisable to reprint the destroyed edition
of 1844. The satisfaction of this late recognition
was the last joy that brightened the eve of Grassmann's
life. He wrote the introduction and an appendix
to the second edition of his \Title{Lineare Ausdehnungslehre},
\index{Ausdehnungslehre@\textit{Ausdehnungslehre}, Grassmann's}%
but died while the forms of his book
were on the press.

At the present day the literature on metageometrical
subjects has grown to such an extent that
we do not venture to enter into further details. We
will only mention the appearance of Professor
Schoute's work on more-dimensional geometry\footnote
  {\Title{Mehrdimensionale Geometrie} von Dr.\ P.~H. Schoute, Professor
\index{Schoute, P. H.|indexnote}%
  der Math.\ an d.~Reichs-Universität zu Groningen, Holland. Leipsic.
  Göschen. So far only the first volume, which treats of linear space,
  has appeared.}
which promises to be the elaboration of the pangeometrical
ideal.
\PageSep{31}


\Section{Euclid Still Unimpaired}
\index{Euclid|indexff}%
\index{Euclid!Halsted on|indexf}%
\index{Euclidean geometry, classical}%
\index{Halsted, George Bruce!on Euclid|indexf}%

Having briefly examined the chief innovations
of modern times in the field of elementary geometry,
it ought to be pointed out that in spite of the well-deserved
fame of the metageometricians from Wallis
to Halsted, Euclid's claim to classicism remains
unshaken. The metageometrical movement is not
a revolution against Euclid's authority but an attempt
at widening our mathematical horizon. Let
us hear what Halsted, one of the boldest and most
iconoclastic among the champions of metageometry
of the present day, has to say of Euclid. Halsted
begins the Introduction to his English translation
%[** TN: "Bolyai's" italicized in the original]
of Bolyai's \Title{Science Absolute of Space} with a terse
description of the history of Euclid's great book
\Title{The Elements of Geometry}, the rediscovery of which
is not the least factor that initiated a new epoch in
the development of Europe which may be called the
era of inventions, of discoveries, and of the appreciation
as well as growth of science. Halsted says:
\begin{Quote}
``The immortal \Title{Elements} of Euclid was already in dim
antiquity a classic, regarded as absolutely perfect, valid
without restriction.

``Elementary geometry was for two thousand years as
stationary, as fixed, as peculiarly Greek as the Parthenon.
On this foundation pure science rose in Archimedes, in
\index{Archimedes}%
Apollonius, in Pappus; struggled in Theon, in Hypatia;
\index{Apollonius}%
\index{Hypatia}%
\index{Pappus}%
\index{Theon}%
declined in Proclus; fell into the long decadence of the Dark
\index{Proclus}%
Ages.

``The book that monkish Europe could no longer understand
\PageSep{32}
was then taught in Arabic by Saracen and Moor in
the Universities of Bagdad and Cordova.

``To bring the light, after weary, stupid centuries, to
Western Christendom, an Englishman, Adelhard of Bath,
journeys, to learn Arabic, through Asia Minor, through
Egypt, back to Spain. Disguised as a Mohammedan student,
he got into Cordova about 1120, obtained a Moorish
copy of Euclid's \Title{Elements}, and made a translation from the
Arabic into Latin.

``The first printed edition of Euclid, published in Venice
in 1482, was a Latin version from the Arabic. The translation
into Latin from the Greek, made by Zamberti from a
\index{Zamberti}%
manuscript of Theon's revision, was first published at Venice
\index{Theon}%
in 1505.

``Twenty-eight years later appeared the \Foreign{editio princeps} in
Greek, published at Basle in 1533 by John Hervagius,
edited by Simon Grynaeus. This was for a century and
three-quarters the only printed Greek text of all the books,
and from it the first English translation (1570) was made
by `Henricus Billingsley,' afterward Sir Henry Billingsley,
\index{Billingsley, Sir H.}%
Lord Mayor of London in 1591.

``And even to-day, 1895, in the vast system of examinations
carried out by the British Government, by Oxford,
and by Cambridge, no proof of a theorem in geometry will
be accepted which infringes Euclid's sequence of propositions.

``Nor is the work unworthy of this extraordinary immortality.

``Says Clifford: `This book has been for nearly twenty-two
\index{Clifford}%
centuries the encouragement and guide of that scientific
thought which is one thing with the progress of man from
a worse to a better state.

``\,`The encouragement; for it contained a body of knowledge
that was really known and could be relied on.

``\,`The guide; for the aim of every student of every subject
\PageSep{33}
was to bring his knowledge of that subject into a form
as perfect as that which geometry had attained.'\,''
\end{Quote}

Euclid's \Title{Elements of Geometry} is not counted
among the books of divine revelation, but truly it
deserves to be held in religious veneration. There
is a real sanctity in mathematical truth which is
not sufficiently appreciated, and certainly if truth,
helpfulness, and directness and simplicity of presentation,
give a title to rank as divinely inspired literature,
Euclid's great work should be counted among
the canonical books of mankind.

\Thoughtbreak

Is there any need of warning our readers that
the foregoing sketch of the history of metageometry
is both brief and popular? We have purposely
avoided the discussion of technical details, limiting
our exposition to the most essential points and trying
to show them in a light that will render them
interesting even to the non-mathematical reader.
It is meant to serve as an introduction to the real
matter in hand, viz., an examination of the foundations
upon which geometrical truth is to be rationally
justified.

The author has purposely introduced what
might be called a biographical element in these expositions
of a subject which is commonly regarded
as dry and abstruse, and endeavored to give something
of the lives of the men who have struggled
and labored in this line of thought and have sacrificed
their time and energy on the altar of one of the
\PageSep{34}
noblest aspirations of man, the delineation of a
philosophy of mathematics. He hopes thereby to
relieve the dryness of the subject and to create an
interest in the labor of these pioneers of intellectual
progress.
\PageSep{35}


\Chapter[The Philosophical Basis]{The Philosophical Basis of Mathematics}

\Section{The Philosophical Problem}

\First{Having} thus reviewed the history of non-Euclidean
geometry, which, rightly considered,
is but a search for the philosophy of mathematics,
I now turn to the problem itself and, in the
conviction that I can offer some hints which contain
its solution, I will formulate my own views in
as popular language as would seem compatible with
exactness. Not being a mathematician by profession
I have only one excuse to offer, which is this:
that I have more and more come to the conclusion
that the problem is not mathematical but philosophical;
and I hope that those who are competent
to judge will correct me where I am mistaken.

The problem of the philosophical foundation of
mathematics is closely connected with the topics
of Kant's \Title{Critique of Pure Reason}. It is the old
\index{Kant}%
\index{Kant!his term \textit{Anschauung}}%
quarrel between Empiricism and Transcendentalism.
\index{Transcendentalism@Transcendentalism and Empiricism}%
Hence our method of dealing with it will naturally
be philosophical, not typically mathematical.

The proper solution can be attained only by
analysing the fundamental concepts of mathematics
\PageSep{36}
and by tracing them to their origin. Thus alone can
we know their nature as well as the field of their
applicability.

We shall see that the data of mathematics are
not without their premises; they are not, as the
Germans say, \Foreign{voraussetzungslos}; and though mathematics
is built up from nothing, the mathematician
does not start with nothing. He uses mental implements,
and it is they that give character to his
science.

Obviously the theorem of parallel lines is one
instance only of a difficulty that betrays itself everywhere
in various forms; it is not the disease of
geometry, but a symptom of the disease. The theorem
that the sum of the angles in a triangle is
equal to $180$~degrees; the ideas of the evenness or
homaloidality of space, of the rectangularity of the
square, and more remotely even the irrationality of~$\pi$
and of~$e$, are all interconnected. It is not the
author's intention to show their interconnection,
nor to prove their interdependence. That task is
the work of the mathematician. The present investigation
shall be limited to the philosophical side
of the problem for the sake of determining the nature
of our notions of evenness, which determines
both parallelism and rectangularity.

At the bottom of the difficulty there lurks the old
\index{Apriority of different degrees!Problem of}%
problem of apriority, proposed by Kant and decided
\index{Kant!and the \textit{a priori}}%
by him in a way which promised to give to mathematics
a solid foundation in the realm of transcendental
thought. And yet the transcendental method
\PageSep{37}
finally sent geometry away from home in search of
a new domicile in the wide domain of empiricism.

Riemann, a disciple of Kant, is a transcendentalist.
\index{Riemann}%
He starts with general notions and his arguments
are deductive, leading him from the abstract down
to concrete instances; but when stepping from the
ethereal height of the absolute into the region of
definite space-relations, he fails to find the necessary
connection that characterizes all \Foreign{a~priori} reasoning;
and so he swerves into the domain of the \Foreign{a~posteriori}
and declares that the nature of the specific features
of space must be determined by experience.

The very idea seems strange to those who have
been reared in traditions of the old school. An unsophisticated
man, when he speaks of a straight
line, means that straightness is implied thereby;
and if he is told that space may be such as to render
all straightest lines crooked, he will naturally be
bewildered. If his metageometrical friend, with
much learnedness and in sober earnest, tells him that
when he sends out a ray as a straight line in a forward
direction it will imperceptibly deviate and
finally turn back upon his occiput, he will naturally
become suspicious of the mental soundness of his
company. Would not many of us dismiss such ideas
with a shrug if there were not geniuses of the very
first rank who subscribe to the same? So in all
modesty we have to defer our judgment until competent
study and mature reflection have enabled us
to understand the difficulty which they encounter
and then judge their solution. One thing is sure,
\PageSep{38}
however: if there is anything wrong with metageometry,
the fault lies not in its mathematical
portion but must be sought for in its philosophical
foundation, and it is this problem to which the
present treatise is devoted.

While we propose to attack the problem as a
philosophical question, we hope that the solution
will prove acceptable to mathematicians.


\Section{Transcendentalism and Empiricism}
\index{Empiricism, Transcendentalism and|indexff}%
\index{Transcendentalism@Transcendentalism and Empiricism|indexff}%

In philosophy we have the old contrast between
the empiricist and transcendentalist school. The
former derive everything from experience, the latter
insist that experience depends upon notions not derived
from experience, called transcendental, and
these notions are \Foreign{a~priori}. The former found their
\index{Apriori@\textit{A priori}}%
representative thinkers in Locke, Hume, and John
Stuart Mill, the latter was perfected by Kant. Kant
\index{Kant!and the \textit{a priori}}%
\index{Mill, John Stuart}%
establishes the existence of notions of the \Foreign{a~priori}
on a solid basis asserting their universality and
necessity, but he no longer identified the \Foreign{a~priori}
with innate ideas. He granted that much to empiricism,
stating that all knowledge begins with experience
and that experience rouses in our mind
the \Foreign{a~priori} which is characteristic of mind. Mill
went so far as to deny altogether necessity and universality,
claiming that on some other planet $2 × 2$
might be~$5$. French positivism, represented by
Comte and Littré, follows the lead of Mill and thus
\index{Comte}%
\index{Littre@Littré}%
they end in agnosticism, and the same result was
\PageSep{39}
reached in England on grounds somewhat different
by Herbert Spencer.

The way which we propose to take may be characterized
as the New Positivism. We take our stand
upon the facts of experience and establish upon the
systematized formal features of our experience a
new conception of the \Foreign{a~priori}, recognizing the universality
and necessity of formal laws but rejecting
Kant's transcendental idealism. The \Foreign{a~priori} is
not deducible from the sensory elements of our sensations,
but we trace it in the formal features of
experience. It is the result of abstraction and systematization.
Thus we establish a method of dealing
with experience (commonly called Pure Reason)
which is possessed of universal validity, implying
logical necessity.

The New Positivism is a further development of
philosophic thought which combines the merits of
both schools, the Transcendentalists and Empiricists,
in a higher unity, discarding at the same time
their aberrations. In this way it becomes possible
to gain a firm basis upon the secure ground of facts,
according to the principle of positivism, and yet to
preserve a method established by a study of the
purely formal, which will not end in nescience (the
ideal of agnosticism) but justify science, and thus
establishes the philosophy of science.\footnote
  {We have treated the philosophical problem of the \Foreign{a~priori} at
  full length in a discussion of Kant's \Title{Prolegomena}. See the author's
\index{Kant@\textit{Kant's Prolegomena}|indexnote}%
\index{Carus, Paul!Fundamental Problems@\textit{Fundamental Problems}|indexnote}%
\index{Carus, Paul!Kant's Prolegomena@\textit{Kant's Prolegomena}|indexnote}%
\index{Carus, Paul!Primer of Philosophy@\textit{Primer of Philosophy}|indexnote}%
  \Title{Kant's Prolegomena}, edited in English, with an essay on Kant's Philosophy
  and other Supplementary Material for the study of Kant,
  pp.~167--240. Cf.~\Title{Fundamental Problems}, the chapters ``Form and
  Formal Thought,'' pp.~26--60, and ``The Old and the New Mathematics,''
  pp.~61--73; and \Title{Primer of Philosophy}, pp.~51--103.}
\PageSep{40}

It is from this standpoint of the philosophy of
science that we propose to investigate the problem
of the foundation of geometry.


\Section{The A Priori and the Purely Formal}
\index{Apriori@\textit{A priori}!and the purely formal|indexff}%

The bulk of our knowledge is from experience,
\ie, we know things after having become acquainted
with them. Our knowledge of things is \Foreign{a~posteriori}.
If we want to know whether sugar is sweet, we must
taste it. If we had not done so, and if no one had
tasted it, we could not know it. However, there is
another kind of knowledge which we do not find out
by experience, but by reflection. If I want to know
how much is $3 × 3$, or $(a + b)^{2}$ or the angles in a
regular polygon, I must compute the answer in
my own mind. I need make no experiments but
must perform the calculation in my own thoughts.
This knowledge which is the result of pure thought
is \Foreign{a~priori}; viz., it is generally applicable and holds
good even \emph{before} we tried it. When we begin to
make experiments, we presuppose that all our \Foreign{a~priori}
arguments, logic, arithmetic, and mathematics,
will hold good.

Kant declared that the law of causation is of the
\index{Causation!Kant on}%
\index{Kant}%
same nature as arithmetical and logical truths, and
that, accordingly, it will have to be regarded as
\Foreign{a~priori}. Before we make experiments, we know
that every cause has its effects, and wherever there
\PageSep{41}
is an effect we look for its cause. Causation is not
proved by, but justified through, experience.

The doctrine of the \Foreign{a~priori} has been much misinterpreted,
especially in England. Kant calls that
\index{Kant!his use of ``transcendental,''}%
\index{Transcendental@`Transcendental,'' Kant's use of}%
which transcends or goes beyond experience in the
sense that it is the condition of experience ``transcendental,''
and comes to the conclusion that the
\Foreign{a~priori} is transcendental. Our \Foreign{a~priori} notions
are not derived from experience but are products
of pure reflection and they constitute the conditions
of experience. By experience Kant understands
sense-impressions, and the sense-impressions of the
outer world (which of course are \Foreign{a~posteriori}) are
reduced to system by our transcendental notions;
and thus knowledge is the product of the \Foreign{a~priori}
and the \Foreign{a~posteriori}.

A sense-impression becomes a perception by being
regarded as the effect of a cause. The idea of
causation is a transcendental notion. Without it
experience would be impossible. An astronomer
measures angles and determines the distance of the
moon and of the sun. Experience furnishes the
data, they are \Foreign{a~posteriori}; but his mathematical
methods, the number system, and all arithmetical
functions are \Foreign{a~priori}. He knows them before he
collects the details of his investigation; and in so
far as they are the condition without which his
sense-impressions could not be transformed into
knowledge, they are called transcendental.

Note here Kant's use of the word transcendental
which denotes the clearest and most reliable knowledge
\PageSep{42}
in our economy of thought, pure logic, arithmetic,
geometry, etc. But transcendental is frequently
(though erroneously) identified with ``transcendent,''
which denotes that which transcends our
knowledge and accordingly means ``unknowable.''
Whatever is transcendental is, in Kantian terminology,
never transcendent.

That much will suffice for an explanation of the
historical meaning of the word transcendental. We
must now explain the nature of the \Foreign{a~priori} and its
source.

The \Foreign{a~priori} is identical with the purely formal
which originates in our mind by abstraction. When
we limit our attention to the purely relational, dropping
all other features out of sight, we produce a
field of abstraction in which we can construct purely
formal combinations, such as numbers, or the ideas
of types and species. Thus we create a world of
pure thought which has the advantage of being
applicable to any purely formal consideration of
conditions, and we work out systems of numbers
which, when counting, we can use as standards of
reference for our experiences in practical life.

But if the sciences of pure form are built upon
an abstraction from which all concrete features are
omitted, are they not empty and useless verbiage?

Empty they are, that is true enough, but for all
that they are of paramount significance, because
they introduce us into the \Foreign{sanctum sanctissimum} of
the world, the intrinsic necessity of relations, and
thus they become the key to all the riddles of the
\PageSep{43}
universe. They are in need of being supplemented
by observation, by experience, by experiment; but
while the mind of the investigator builds up purely
formal systems of reference (such as numbers) and
purely formal space-relations (such as geometry),
the essential features of facts (of the objective
world) are in their turn, too, purely formal, and
they make things such as they are. The suchness
of the world is purely formal, and its suchness alone
is of importance.

In studying the processes of nature we watch
transformations, and all we can do is to trace the
changes of form. Matter and energy are words
which in their abstract significance have little value;
they merely denote actuality in general, the one of
being, the other of doing. What interests us most
are the \emph{forms} of matter and energy, how they
change by transformation; and it is obvious that
the famous law of the conservation of matter and
energy is merely the reverse of the truth that causation
is transformation. In its elements which in
their totality are called matter and energy, the elements
of existence remain the same, but the forms in
which they combine change. The sum-total of the
mass and the sum-total of the forces of the world
can be neither increased nor diminished; they remain
the same to-day that they have always been
and as they will remain forever.

All \Foreign{a~posteriori} cognition is concrete and particular,
\index{Aposteriori@\textit{A posteriori}}%
while all \Foreign{a~priori} cognition is abstract and
general. The concrete is (at least in its relation
\PageSep{44}
to the thinking subject) incidental, casual, and individual,
but the abstract is universal and can be used
as a general rule under which all special cases may
be subsumed.

The \Foreign{a~priori} is a mental construction, or, as Kant
\index{Apriori@\textit{A priori}!is ideal}%
\index{Ideal@``Ideal'' and ``subjective,'' Kant's identification of|indexf}%
\index{Kant!his identification of ``ideal'' and ``subjective,''|indexf}%
\index{Subjective and ideal!Kant's identification of|indexf}%
says, it is ideal, viz., it consists of the stuff that
ideas are made of, it is mind-made. While we grant
that the purely formal is ideal we insist that it is
made in the domain of abstract thought, and its
fundamental notions have been abstracted from experience
by concentrating our attention upon the
purely formal. It is, not directly but indirectly and
ultimately, derived from experience. It is not derived
from sense-experience but from a consideration
of the relational (the purely formal) of experience.
Thus it is a subjective reconstruction of
certain objective features of experience and this
reconstruction is made in such a way as to drop
every thing incidental and particular and retain
only the general and essential features; and we gain
the unspeakable advantage of creating rules or formulas
which, though abstract and mind-made, apply
to \emph{any} case that can be classified in the same category.

Kant made the mistake of identifying the term
``ideal'' with ``subjective,'' and thus his transcendental
idealism was warped by the conclusion that
our purely formal laws were not objective, but were
imposed by our mind upon the objective world. Our
mind (Kant said) is so constituted as to interpret
all facts of experience in terms of form, as appearing
\PageSep{45}
in space and time, and as being subject to the law
of cause and effect; but what things are in themselves
we cannot know. We object to Kant's subjectivizing
the purely formal and look upon form
as an essential and inalienable feature of objective
existence. The thinking subject is to other thinking
subjects an object moving about in the objective
world. Even when contemplating our own existence
we must grant (to speak with Schopenhauer)
that our bodily actualization is our own object; \ie,
we (each one of us as a real living creature) are as
much objects as are all the other objects in the
world. It is the objectified part of our self that in
its inner experience abstracts from sense-experience
the interrelational features of things, such as right
and left, top and bottom, shape and figure and structure,
succession, connection, etc. The formal adheres
to the object and not to the subject, and every
object (as soon as it develops in the natural way of
evolution first into a feeling creature and then into
a thinking being) will be able to build up \Foreign{a~priori}
from the abstract notion of form in general the several
systems of formal thought: arithmetic, geometry,
algebra, logic, and the conceptions of time,
space, and causality.

Accordingly, all formal thought, although we
grant its ideality, is fashioned from materials abstracted
from the objective world, and it is therefore
a matter of course that they are applicable to the
objective world. They belong to the object and,
when we thinking subjects beget them from our own
\PageSep{46}
minds, we are able to do so only because we are objects
that live and move and have our being in the
objective world.


\Section{Anyness and its Universality}
\index{Anyness|indexff}%

We know that facts are incidental and haphazard,
and appear to be arbitrary; but we must not
rest satisfied with single incidents. We must gather
enough single cases to make abstractions. Abstractions
are products of the mind; they are subjective;
but they have been derived from experience, and
they are built up of elements that have objective
significance.

The most important abstractions ever made by
man are those that are purely relational. Everything
from which the sensory element is entirely
omitted, where the material is disregarded, is called
``pure form,'' and the relational being a consideration
neither of matter nor of force or energy, but
of number, of position, of shape, of size, of form,
of relation, is called ``the purely formal.'' The notion
of the purely formal has been gained by abstraction,
viz., by abstracting, \ie, singling out and
retaining, the formal, and by thinking away, by cancelling,
by omitting, by leaving out, all the features
which have anything to do with the concrete sensory
element of experience.

And what is the result?

We retain the formal element alone which is
void of all concreteness, void of all materiality, void
\PageSep{47}
of all particularity. It is a mere nothing and a
non-entity. It is emptiness. But one thing is left,---position
or relation. Actuality is replaced by
mere potentiality, viz., the possible conditions of
\emph{any} kind of being that is possessed of form.

The word ``any'' denotes a simple idea, and yet
it contains a good deal of thought. Mathematics
builds up its constructions to suit any condition.
``Any'' implies universality, and universality includes
necessity in the Kantian sense of the term.

In every concrete instance of an experience the
subject-matter is the main thing with which we are
concerned; but the purely formal aspect is after all
the essential feature, because form determines the
character of things, and thus the formal (on account
of its anyness) is the key to their comprehension.

The rise of man above the animal is due to his
ability to utilize the purely formal, as it revealed
itself to him especially in types for classifying things,
as genera and species, in tracing transformations
which present themselves as effects of causes and reducing
them to shapes of measurable relations. The
abstraction of the formal is made through the instrumentality
of language and the result is reason,---the
faculty of abstract thought. Man can see the
universal in the particular; in the single experiences
he can trace the laws that are generally applicable
to cases of the same class; he observes some instances
and can describe them in a general formula so
as to cover any other instance of the same kind, and
thus he becomes master of the situation; he learns
\PageSep{48}
to separate in thought the essential from the accidental,
and so instead of remaining the prey of circumstance
he gains the power to adapt circumstances
to himself.

Form pervades all nature as an essential constituent
\index{Form!and reason}%
thereof. If form were not an objective feature
of the world in which we live, formal thought would
never without a miracle, or, at least, not without
the mystery of mysticism, have originated according
to natural law, and man could never have arisen.
But form being an objective feature of all existence,
it impresses itself in such a way upon living creatures
that rational beings will naturally develop
among animals whose organs of speech are perfected
as soon as social conditions produce that demand
for communication that will result in the creation
of language.

The marvelous advantages of reason dawned
\index{Reason, Form and}%
upon man like a revelation from on high, for he did
not invent reason,\Pagelabel{48} he discovered it; and the sentiment
that its blessings came to him from above,
from heaven, from that power which sways the destiny
of the whole universe, from the gods or from
God, is as natural as it is true. The anthropoid did
not seek reason: reason came to him and so he became
man. Man became man by the grace of God,
by gradually imbibing the Logos that was with God
in the beginning; and in the dawn of human evolution
we can plainly see the landmark of mathematics,
for the first grand step in the development of
\PageSep{49}
man as distinguished from the transitional forms of
the anthropoid is the ability to count.

Man's distinctive characteristic remains, even
to-day, reason, the faculty of purely formal thought;
and the characteristic of reason is its general application.
All its verdicts are universal and involve
apriority or beforehand knowledge so that man can
\index{Apriority of different degrees|indexff}%
foresee events and adapt means to ends.


\Section{Apriority of Different Degrees}

Kant has pointed out the kinship of all purely
formal notions. The validity of mathematics and
logic assures us of the validity of the categories
including the conception of causation; and yet geometry
cannot be derived from pure reason alone,
but contains an additional element which imparts
to its fundamental conceptions an arbitrary appearance
if we attempt to treat its deductions as rigidly
\Foreign{a~priori}. Why should there be straight lines at all?
Why is it possible that by quartering the circle we
should have right angles with all their peculiarities?
All these and similar notions can not be subsumed
under a general formula of pure reason from which
we could derive it with logical necessity.

When dealing with lines we observe their \Typo{extention}{extension}
in one direction, when dealing with planes we
have two dimensions, when measuring solids we
have three. Why can we not continue and construct
bodies that extend in four dimensions? The limit
set us by space as it positively presents itself to us
\PageSep{50}
seems arbitrary, and while transcendental truths
are undeniable and obvious, the fundamental notions
of geometry seem as stubborn as the facts of
our concrete existence. Space, generally granted
to be elbow-room for motion in all directions, after
all appears to be a definite magnitude as much as
a stone wall which shuts us in like a prison, allowing
us to proceed in such a way only as is permissible
by those co-ordinates and no more. We can by
no resort break through this limitation. Verily we
might more easily shatter a rock that impedes our
progress than break into the fourth dimension. The
boundary line is inexorable in its adamantine rigor.

Considering all these unquestionable statements,
is there not a great probability that space is a concrete
fact as positive as the existence of material
things, and not a mere form, not a mere potentiality
of a general nature? Certainly Euclidean geometry
contains some such arbitrary elements as we should
expect to meet in the realm of the \Foreign{a~posteriori}. No
wonder that Gauss expressed ``the desire that the
Euclidean geometry should not be the true geometry,''
because ``in that event we should have an absolute
measure \Foreign{a~priori}.''

Are we thus driven to the conclusion that our
space-conception is not \Foreign{a~priori}; and if, indeed, it is
not \Foreign{a~priori}, it must be \Foreign{a~posteriori}! What else can
it be? \Foreign{Tertium non datur.}

If we enter more deeply into the nature of the
\Foreign{a~priori}, we shall learn that there are different kinds
\PageSep{51}
of apriority, and there is a difference between the
logical \Foreign{a~priori} and the geometrical \Foreign{a~priori}.
\index{Apriori@\textit{A priori}!Source of the}%

Kant never investigated the source of the \Foreign{a~priori}.
He discovered it in the mind and seemed satisfied
\index{Mind!Origin of}%
with the notion that it is the nature of the mind
to be possessed of time and space and the categories.
He went no further. He never asked, how did mind
originate?

Had Kant inquired into the origin of mind, he
would have found that the \Foreign{a~priori} is woven into
the texture of mind by the uniformities of experience.
The uniformities of experience teach us the
laws of form, and the purely formal applies not to
one case only but to any case of the same kind, and
so it involves ``anyness,'' that is to say, it is \Foreign{a~priori}.

Mind is the product of memory, and we may
briefly describe its origin as follows:

Contact with the outer world produces impressions
in sentient substance. The traces of these
impressions are preserved (a condition which is
called ``memory'') and they can be revived (which
state is called ``recollection''). Sense-impressions
are different in kind and leave different traces, but
those which are the same in kind, or similar, leave
traces the forms of which are the same or similar;
and sense-impressions of the same kind are registered
in the traces having the same form. As a note
of a definite pitch makes chords of the same pitch
vibrate while it passes all others by; so new sense-impressions
revive those traces only into which they
fit, and thereby announce themselves as being the
\PageSep{52}
same in kind. Thus all sense-impressions are systematized
according to their forms, and the result
is an orderly arrangement of memories which is
called ``mind.''\footnote
  {For a more detailed exposition see the author's \Title{Soul of Man};
  also his \Title{Whence and Whither}.}

Thus mind develops through uniformities in sensation
\index{Mind!develops through uniformities}%
\index{Uniformities!Mind develops through}%
according to the laws of form. Whenever
a new sense-perception registers itself mechanically
and automatically in the trace to which it belongs,
the event is tantamount to a logical judgment which
declares that the object represented by the sense-impression
belongs to the same class of objects
which produced the memory traces with which it is
registered.

If we abstract the interrelation of all memory-traces,
omitting their contents, we have a pure system
of genera and species, or the \Foreign{a~priori} idea of
``classes and subclasses.''

The \Foreign{a~priori}, though mind-made, is constructed
of chips taken from the objective world, but our
several \Foreign{a~priori} notions are by no means of one and
the same nature and rigidity. On the contrary,
there are different degrees of apriority. The emptiest
forms of pure thought are the categories, and
the most rigid truths are the logical theorems, which
can be represented diagrammatically so as to be a \Foreign{demonstratio
ad oculos}.

If all $b$s are~$B$ and if $\beta$~is a~$b$, then $\beta$~is a~$B$. If
all dogs are quadrupeds and if all terriers are dogs,
then terriers are quadrupeds. It is the most rigid
\PageSep{53}
kind of argument, and its statements are practically
tautologies.

The case is different with causation. The class
\index{Causation!apriori@\textit{a priori}}%
of abstract notions of which causation is an instance
is much more complicated. No one doubts that
every effect must have had its cause, but one of the
keenest thinkers was in deep earnest when he
doubted the possibility of proving this obvious statement.
And Kant, seeing its kinship with geometry
and algebra, accepted it as \Foreign{a~priori} and treated it
as being on equal terms with mathematical axioms.
Yet there is an additional element in the formula
of causation which somehow disguises its \Foreign{a~priori}
origin, and the reason is that it is not as rigidly
\Foreign{a~priori} as are the norms of pure logic.

What is this additional element that somehow
savors of the \Foreign{a~posteriori}?

If we contemplate the interrelation of genera
and species and subspecies, we find that the categories
with which we operate are at rest. They
stand before us like a well arranged cabinet with
several divisions and drawers, and these drawers
have subdivisions and in these subdivisions we keep
boxes. The cabinet is our \Foreign{a~priori} system of classification
and we store in it our \Foreign{a~posteriori} impressions.
If a thing is in box~$\beta$, we seek for it in
drawer~$b$ which is a subdivision of the department~$B$.

How different is causation! While in logic
everything is at rest, causation is not conceivable
without motion. The norms of pure reason are
static, the law of cause and effect is dynamic; and
\index{Logic is static}%
\PageSep{54}
thus we have in the conception of cause and effect
an additional element which is mobility.

Causation is the law of transformation. We have
\index{Causation!transformation@{and transformation}}%
\index{Transformation, Causation and}%
a definite system of interrelated items in which we
observe a change of place. The original situation
and all detailed circumstances are the conditions;
the motion that produces the change is the cause;
the result or new arrangement of the parts of the
whole system is the effect. Thus it appears that causation
is only another version of the law of the conservation
of matter and energy. The concrete items
of the whole remain, in their constitutional elements,
the same. No energy is lost; no particle of matter
is annihilated; and the change that takes place is
mere transformation.\footnote
  {See the author's \Title{Ursache, Grund und Zweck}, Dresden, 1883;
  also his \Title{Fundamental Problems}.}

The law of causation is otherwise in the same
predicament as the norms of logic. It can never be
satisfactorily proved by experience. Experience
justifies the \Foreign{a~priori} and verifies its tenets in single
\index{Apriori@\textit{A priori}!The logical}%
\index{Apriori@\textit{A priori}!The rigidly}%
instances which prove true, but single instances can
never demonstrate the universal and necessary validity
of any \Foreign{a~priori} statement.

The logical \Foreign{a~priori} is rigidly \Foreign{a~priori}; it is the
\Foreign{a~priori} of pure reason. But there is another kind
of \Foreign{a~priori} which admits the use of that other abstract
notion, mobility, and mobility as much as
form is part and parcel of the thinking mind. Our
conception of cause and effect is just as ideal as our
conception of genera and species. It is just as much
\PageSep{55}
mind-made as they are, and its intrinsic necessity
and universal validity are the same. Its apriority
cannot be doubted; but it is not rigidly \Foreign{a~priori}, and
\index{Apriori@\textit{A priori}!The purely}%
\index{Apriori@\textit{A priori}!The rigidly}%
\index{Purely \textit{a priori}, The}%
we will call it purely \Foreign{a~priori}.

We may classify all \Foreign{a~priori} notions under two
headings and both are transcendental (viz., conditions
of knowledge in their special fields): one is
the \Foreign{a~priori} of being, the other of doing. The rigid
\Foreign{a~priori} is passive anyness, the less rigid \Foreign{a~priori}
is active anyness. Geometry belongs to the latter.
Its fundamental concept of space is a product of active
apriority; and thus we cannot derive its laws
from pure logic alone.

The main difficulty of the parallel theorem and
the straight line consists in our space-conception
\index{Space-conception!product of pure activity}%
which is not derived from rationality in general,
but results from our contemplation of motion. Our
space-conception accordingly is not an idea of pure
reason, but the product of pure activity.

Kant felt the difference and distinguished between
pure reason and pure intuition or \Foreign{Anschauung}.
He did not expressly say so, but his treatment
suggests the idea that we ought to distinguish between
two different kinds of \Foreign{a~priori}. Transcendental
logic, and with it all common notions of
Euclid, are mere applications of the law of consistency;
they are ``rigidly \Foreign{a~priori}.'' But our pure
space-conception presupposes, in addition to pure
reason, our own activity, the potentiality of moving
about in any kind of a field, and thus it admits
another factor which cannot be derived from pure
\PageSep{56}
reason alone. Hence all attempts at proving the
theorem on rigidly \Foreign{a~priori} grounds have proved
failures.


\Section{Space as a Spread of Motion}
\index{Space!a spread of motion|indexff}%

Mathematicians mean to start from nothingness,
so they think away everything, but they retain their
own mentality. Though even their mind is stripped
of all particular notions, they retain their principles
of reasoning and the privilege of moving about,
and from these two sources geometry can be constructed.

The idea of causation goes one step further: it
admits the notions of matter and energy, emptied
of all particularity, in their form of pure generalizations.
It is still \Foreign{a~priori}, but considerably more
complicated than pure reason.

The field in which the geometrician starts is pure
nothingness; but we shall learn later on that nothingness
is possessed of positive qualifications. We
must therefore be on our guard, and we had better
inquire into the nature and origin of our nothingness.

The geometrician cancels in thought all positive
existence except his own mental activity and starts
moving about as a mere nothing. In other words,
we establish by abstraction a domain of monotonous
sameness, which possesses the advantage of ``anyness,''
\ie, an absence of particularity involving
universal validity. In this field of motion we proceed
to produce geometrical constructions.
\PageSep{57}

The geometrician's activity is pure motion,
which means that it is mere progression; the ideas
of a force exerted in moving and also of resistance
to be overcome are absolutely excluded.

We start moving, but whither? Before us are
infinite possibilities of direction. The inexhaustibility
of chances is part of the indifference as
to definiteness of determining the mode of motion
(be it straight or curved). Let us start at once
in all possible directions which are infinite, (a proposition
which, in a way, is realized by the light),
and having proceeded an infinitesimal way from the
starting-point~$A$ to the points $B$, $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$,~\dots\Add{,}
$B_{\infty}$; we continue to move in infinite directions
at each of these stations, reaching from~$B$ the points
$C$, $C_{1}$, $C_{2}$, $C_{3}$, $C_{4}$,~\dots\Add{,} $C_{\infty}$. From~$B_{1}$ we would switch
off to the points $C_{1}^{B_{1}}$, $C_{2}^{B_{1}}$, $C_{3}^{B_{1}}$, $C_{4}^{B_{1}}$,\dots\Add{,} $C_{\infty}^{B_{1}}$, etc.\ until
we reach from~$B_{\infty}$ the points $C_{1}^{B_{\infty}}$, $C_{2}^{B_{\infty}}$, $C_{3}^{B_{\infty}}$, $C_{4}^{B_{\infty}}$,\dots\Add{,}
$C_{\infty}^{B_{\infty}}$, thus exhausting all the points which cluster
around every $B_{1}$, $B_{2}$, $B_{3}$, $B_{4}$,~\dots\Add{,} $B_{\infty}$. Thus, by
moving after the fashion of the light, spreading
again and again from each new point in all directions,
in a medium that offers no resistance whatever,
we obtain a uniform spread of light whose
intensity in every point is in the inverse square of
its distance from its source. Every lighted spot becomes
a center of its own from which light travels
on in all directions. But among these infinite directions
there are rays, $A$, $B$, $C$,~\dots, $A_{1}$, $B_{1}$, $C_{1}$,~\dots,
$A_{2}$, $B_{2}$, $C_{2}$,~\dots, etc., that is to say, lines of motion
that pursue the original direction and are paths of
\PageSep{58}
maximum intensity. Each of these rays, thus ideally
constructed, is a representation of the straight
line which being the shortest path between the starting-point~$A$
\index{Path of highest intensity a straight line}%
\index{Ray a final boundary}%
and any other point, is the climax of
directness: it is the upper limit of effectiveness and
its final boundary, a \Foreign{non plus ultra}. It is a maximum
because there is no loss of efficacy. The
straight line represents a climax of economy, viz.,
the greatest intensity on the shortest path that is
reached among infinite possibilities of progression
by uniformly following up all. In every ray the
maximum of intensity is attained by a minimum
of progression.

Our construction of motion in all directions after
the fashion of light is practically pure space; but
to avoid the forestalling of further implications we
will call it simply the spread of motion in all directions.

The path of highest intensity in a spread of
motion in all directions corresponds to the ray in an
ideal conception of a spread of light, and it is equivalent
to the straight line in geometry.

We purposely modify our reference to light in
our construction of straight lines, for we are well
aware of the fact that the notion of a ray of light
as a straight line is an ideal which describes the
progression of light only as it appears, not as it is.
The physicist represents light as rays only when
measuring its effects in reflection, etc., but when
considering the nature of light, he looks upon rays
as transversal oscillations of the ether. The notion
\PageSep{59}
\index{Straight line!a path of highest intensity}%
of light as rays is at bottom as much an \Foreign{a~priori}
construction as is Newton's formula of gravitation.

The construction of space as a spread of motion
\index{Space!the possibility of motion}%
in all directions after the analogy of light is a summary
creation of the scope of motion, and we call it
``ideal space.'' Everything that moves about, if it
develops into a thinking subject, when it forms the
abstract idea of mobility, will inevitably create out
of the data of its own existence the ideal ``scope of
motion,'' which is space.

When the geometrician starts to construct his
figures, drawing lines and determining the position
of points, etc., he tacitly presupposes the existence
of a spread of motion, such as we have described.
Motility is part of his equipment, and motility presupposes
a field of motion, viz., space.

Space is the possibility of motion, and by ideally
moving about in all possible directions the number
of which is inexhaustible, we construct our notion
of pure space. If we speak of space we mean this
construction of our mobility. It is an \Foreign{a~priori} construction
and is as unique as logic or arithmetic.
There is but one space, and all spaces are but portions
of this one construction. The problem of tridimensionality
will be considered later on. Here
we insist only on the objective validity of our \Foreign{a~priori}
construction, which is the same as the objective
validity of all our \Foreign{a~priori} constructions---of
logic and arithmetic and causality, and it rests
upon the same foundation. Our mathematical space
\index{Space-conception!how far \textit{a priori?}|indexf}%
omits all particularity and serves our purpose of
\PageSep{60}
universal application: it is founded on ``anyness,''
\index{Anyness!Space founded on}%
\index{Space!founded on ``anyness,''}%
and thus, within the limits of its abstraction, it holds
good everywhere and under all conditions.

There is no need to find out by experience in the
domain of the \Foreign{a~posteriori} whether pure space is
\index{Aposteriori@\textit{A posteriori}}%
curved. Anyness has no particular qualities; we
create this anyness by abstraction, and it is a matter
of course that in the field of our abstraction, space
will be the same throughout, unless by another act
of our creative imagination we appropriate particular
qualities to different regions of space.

The fabric of which the purely formal is woven
\index{Purely \textit{a priori}, The!formal, absence of concreteness}%
is an absence of concreteness. It is (so far as matter
\index{Concreteness, Purely formal, absence of}%
is concerned) nothing. Yet this airy nothing
is a pretty tough material, just on account of its
indifferent ``any''-ness. Being void of particularity,
it is universal; it is the same throughout, and if we
proceed to build our air-castles in the domain of
anyness, we shall find that considering the absence
of all particularity the same construction will be the
same, wherever and whenever it may be conceived.

Professor Clifford says:\footnote
  {\Foreign{Loc.\ cit.}, p.~53.}
\index{Clifford}%
``We \emph{assume} that two
lengths which are equal to the same length are
equal to each other.'' But there is no ``assumption''
about it. The atmosphere in which our mathematical
creations are begotten is sameness. Therefore
the same construction is the same wherever and
whenever it may be made. We consider form only;
\index{Form}%
we think away all other concrete properties, both
\PageSep{61}
of matter and energy, mass, weight, intensity, and
qualities of any kind.


\Section{Uniqueness of Pure Space}
\index{Space!the potentiality of measuring}%
\index{Space!Uniqueness of pure|indexff}%

Our thought-forms, constructed in the realm of
\index{Thought-forms, systems of reference}%
empty abstraction, serve as models or as systems of
reference for any of our observations in the real
world of sense-experience. The laws of form are
as well illustrated in our models as in real things,
and can be derived from either; but the models of
our thought-forms are always ready at hand while
the real things are mostly inaccessible. The anyness
of pure form explains the parallelism that obtains
\index{Pure form!space, Uniqueness of|indexff}%
between our models and actual experience,
which was puzzling to Kant. And truly at first
\index{Kant}%
sight it is mystifying that a pure thought-construction
can reveal to us some of the most important
and deepest secrets of objective nature; but the simple
solution of the mystery consists in this, that the
actions of nature are determined by the same conditions
of possible motions with which pure thought
is confronted in its efforts to construct its models.
Here as well as there we have consistency, that is
to say, a thing done is uniquely determined, and,
in pure thought as well as in reality, it is such as it
has been made by construction.

Our constructions are made in anyness and apply
to all possible instances of the kind; and thus we
may as well define space as the potentiality of measuring,
which presupposes moving about. Mobility
\PageSep{62}
granted, we can construct space as the scope of our
\index{Space!Mathematical and actual}%
motion in anyness. Of course we must bear in mind
that our motion is in thought only and we have
dropped all notions of particularity so as to leave
an utter absence of force and resistance. The motor
element, \Foreign{qua} energy, is not taken into consideration,
but we contemplate only the products of progression.

Since in the realm of pure form, thus created
by abstraction, we move in a domain void of particularity,
it is not an assumption (as Riemann
\index{Riemann}%
declares in his famous inaugural dissertation), but
a matter of course which follows with logical necessity,
that lines are independent of position; they
\index{Position}%
are the same anywhere.

In actual space, position is by no means a negligible
quantity. A real pyramid consisting of actual
material is possessed of different qualities according
to position, and the line~$AB$, representing a path
\index{Line created by construction!independent of position}%
from the top of a mountain to the valley is very
different from the line~$BA$, which is the path from
the valley to the top of the mountain. In Euclidean
geometry $AB = BA$.

Riemann attempts to identify the mathematical
space of a triple manifold with actual space and expects
a proof from experience, but, properly speaking,
they are radically different. In real space position
is not a negligible factor, and would necessitate
a fourth co-ordinate which has a definite relation
to the plumb-line; and this fourth co-ordinate
(which we may call a fourth dimension) suffers a
\PageSep{63}
constant modification of increase in inverse proportion
to the square of the distance from the center of
this planet of ours. It is rectilinear, yet all the plumb-lines
are converging toward an inaccessible center;
accordingly, they are by no means of equal value in
their different parts. How different is mathematical
space! It is homogeneous throughout. And it
\index{Mathematical space|indexff}%
\index{Space!Mathematical|indexff}%
\index{Space!Physiological|indexff}%
is so because we made it so by abstraction.

Pure form is a feature which is by no means
\index{Pure form}%
a mere nonentity. Having emptied existence of all
concrete actuality, and having thought away everything,
we are confronted by an absolute vacancy---a
zero of existence: but the zero has positive characteristics
and there is this peculiarity about the
zero that it is the mother of infinitude. The thought
is so true in mathematics that it is trite. Let any
number be divided by nought, the result is the infinitely
great; and let nought be divided by any
number, the result is the infinitely small. In thinking
away everything concrete we retain with our
nothingness potentiality. Potentiality is the empire
\index{Potentiality}%
of purely formal constructions, in the dim background
of which lurks the phantom of infinitude.


\Section{Mathematical Space and Physiological Space}
\index{Physiological space|indexff}%

If we admit to our conception of space the qualities
of bodies such as mass, our conception of real
space will become more complicated still. What
we gain in concrete definiteness we lose from universality,
and we can return to the general applicability
\PageSep{64}
of \Foreign{a~priori} conditions only by dropping
\index{Apriori@\textit{A priori}}%
all concrete features and limiting our geometrical
constructions to the abstract domain of pure form.

Mathematical space with its straight lines,
planes, and right angles is an ideal construction.
It exists in our mind only just as much as do logic
and arithmetic. In the external world there are
no numbers, no mathematical lines, no logarithms,
no sines, tangents, nor secants. The same is true
of all the formal sciences. There are no genera and
species, no syllogisms, neither inductions nor deductions,
running about in the world, but only concrete
individuals and a concatenation of events. There
are no laws that govern the motions of stars or molecules;
yet there are things acting in a definite way,
and their actions depend on changes in relational
conditions which can be expressed in formulas. All
the generalized notions of the formal sciences are
mental contrivances which comprise relational features
in general rules. The formulas as such are
purely ideal, but the relational features which they
describe are objectively real.

Thus, the space-conception of the mathematician
\index{Ideal@``Ideal'' and ``subjective,'' Kant's identification of!not synonyms|indexf}%
\index{Subjective and ideal!not synonyms}%
is an ideal construction; but the ideal has objective
significance. Ideal and subjective are by no means
synonyms. With the help of an ideal space-conception
we can acquire knowledge concerning the
real space of the objective world. Here the Newtonian
law may be cited as a conspicuous example.

How can the thinking subject know \Foreign{a~priori} anything
about the object? Simply because the subject
\PageSep{65}
is an object moving about among other objects.
Mobility is a qualification of the object, and I, the
thinking subject, become conscious of the general
rules of motion only, because I also am an object
endowed with mobility. My ``scope of motion'' cannot
be derived from the abstract idea of myself as
a thinking subject, but is the product of a consideration
of my mobility, generalized from my activities
as an object by omitting all particularities.

Mathematical space is \Foreign{a~priori} in the Kantian
\index{Mathematical space!a priori@\textit{a priori}}%
sense. However, it is not ready-made in our mind,
it is not an innate idea, but the product of much
toil and careful thought. Nor will its construction
be possible, except at a maturer age after a long
development.

Physiological space is the direct and unsophisticated
\index{Physiological space!originates through experience}%
space-conception of our senses. It originates
through experience, and is, in its way, a truer picture
\index{Experience, Physiological space originates through}%
of actual or physical space than mathematical
space. The latter is more general, the former more
concrete. In physiological space position is not indifferent,
for high and low, right and left, and up
and down are of great importance. Geometrically
congruent figures produce (as Mach has shown)
\index{Mach, Ernst}%
remarkably different impressions if they present
themselves to the eyes in different positions.

In a geometrical plane the figures can be shoved
about without suffering a change of form. If they
are flopped, their inner relations remain the same,
as, \eg, helices of opposite directions are, mathematically
considered, congruent, while in actual life
\PageSep{66}
they would always remain mere symmetrical counterparts.
So the right and the left hands considered
as mere mathematical bodies are congruent, while
in reality neither can take the place of the other.
A glove which we may treat as a two-dimensional
thing can be turned inside out, but we would need
a fourth dimension to flop the hand, a three-dimensional
body, into its inverted counterpart. So long
as we have no fourth dimension, the latter being a
mere logical fiction, this cannot be done. Yet mathematically
considered, the two hands are congruent.
Why? Not because they are actually of the same
shape, but because in our mathematics the qualifications
of position are excluded; the relational
alone counts, and the relational is the same in both
cases.

Mathematical space being an ideal construction,
\index{Space!Homogeneity of|indexff}%
it is a matter of course that all mathematical problems
must be settled by \Foreign{a~priori} operations of pure
thought, and cannot be decided by external experiment
or by reference to \Foreign{a~posteriori} information.


\Section{Homogeneity of Space Due to Abstraction}
\index{Homogeneity of space|indexff}%

When moving about, we change our place and
pass by different objects. These objects too are
moving; and thus our scope of motion tallies so
exactly with theirs that one can be used for the computation
of the other. All scopes of motion are possessed
of the same anyness.

Space as we find it in experience is best defined
\PageSep{67}
as the juxtaposition of things. If there is need of
distinguishing it from our ideal space-conception
\index{Space!the juxtaposition of things}%
which is the scope of our mobility, we may call the
former pure objective space, the latter pure subjective
space, but, our subjective ideas being rooted
in our mobility, which is a constitutional feature
of our objective existence, for many practical purposes
the two are the same.

But though pure space, whether its conception
be established objectively or subjectively, must be
accepted as the same, are we not driven to the conclusion
that there are after all two different kinds
of space: mathematical space, which is ideal, and
\index{Mathematical space}%
\index{Space!Mathematical}%
physiological space, which is real? And if they are
\index{Physiological space}%
\index{Space!Physiological}%
different, must we not assume them to be independent
of each other? What is their mutual relation?

The two spaces, the ideal construction of mathematical
space and the reconstruction in our senses
of the juxtaposition of things surrounding us, are
different solely because they have been built up upon
two different planes of abstraction; physiological
space includes, and mathematical space excludes,
the sensory data of juxtaposition. Physiological
space admits concrete facts,---man's own upright
position, gravity, perspective, etc. Mathematical
space is purely formal, and to lay its foundation we
have dug down to the bed-rock of our formal knowledge,
which is ``anyness.'' Mathematical space is
\Foreign{a~priori}, albeit the \Foreign{a~priori} of motion.

At present it is sufficient to state that the homogeneity
of a mathematical space is its anyness, and
\PageSep{68}
its anyness is due to our construction of it in the
domain of pure form, involving universality and
excluding everything concrete and particular.

The idea of homogeneity in our space-conception
is the tacit condition for the theorems of similarity
and proportion, and also of free mobility without
change, viz., that figures can be shifted about without
suffering distortion either by shrinkage or by
expanse. The principle of homogeneity being admitted,
we can shove figures about on any surface
the curvature of which is either constant or zero.
This produces either the non-Euclidean geometries
of spherical, pseudo-spherical, and elliptic surfaces,
or the plane geometry of Euclid---all of them \Foreign{a~priori}
constructions made without reference to reality.

Our \Foreign{a~priori} constructions serve an important
purpose. We use them as systems of reference.
We construct \Foreign{a~priori} a number system, making a
simple progression through a series of units which
we denominate from the starting-point~$0$, as $1$, $2$,
$3$, $4$, $5$, $6$, etc. These numbers are purely ideal constructions,
but with their help we can count and
measure and weigh the several objects of reality
that confront experience; and in all cases we fall
back upon our ideal number system, saying, the
table has four legs; it is two and a half feet high, it
weighs fifty pounds, etc. We call these modes of
determination quantitative.

The element of quantitative measurement is the
ideal construction of units, all of which are assumed
\PageSep{69}
to be discrete and equivalent. The equivalence of
numbers as much as the homogeneity of space, is
due to abstraction. In reality equivalent units do
not exist any more than different parts of real space
may be regarded as homogeneous. Both constructions
have been made to create a domain of anyness,
for the purpose of standards of reference.


\Section{Even Boundaries as Standards of Measurement}
\index{Even boundaries!as standards of measurement|indexff}%
\index{Measurement!Even@Even boundaries as standards of|indexff}%
\index{Standards of measurement!Even@Even boundaries as|indexff}%

Standards of reference are useful only when
they are unique, and thus we cannot use any path
of our spread of motion in all directions, but must
select one that admits of no equivocation. The only
line that possesses this quality is the ray, viz., the
straight line or the path of greatest intensity.
\index{Straight line}%

The straight line is one instance only of a whole
class of similar constructions which with one name
may be called ``even boundaries,'' and by even I
mean congruent with itself. They remain the same
in any position and no change originates however
they may be turned.

Clifford, starting from objective space, constructs
\index{Clifford!Plane constructed by}%
\index{Plane@Plane, a zero of curvature!constructed by Clifford}%
the plane by polishing three surfaces, $A$,~$B$,
and~$C$, until they fit one another, which means until
they are congruent.\footnote
  {\Title{Common Sense of the Exact Sciences}, Appleton \&~Co., p.~66.}
His proposition leads to the
same result as ours, but the essential thing is not
so much (as Clifford has it) that the three planes
are congruent, each to the two others, but that each
plane is congruent with its own inversion. Thus,
\PageSep{70}
under all conditions, each one is congruent with
itself. Each plane partitions the whole infinite space
into two congruent halves.

Having divided space so as to make the boundary
surface congruent with itself (viz., a plane),
we now divide the plane (we will call it~$P$) in the
same way,---a process best exemplified in the folding
of a sheet of paper stretched flat on the table.
The crease represents a boundary congruent with
itself. In contrast to curved lines, which cannot be
flopped or shoved or turned without involving a
change in our construction, we speak of a straight
line as an even boundary.

A circle can be flopped upon itself, but it is not
an even boundary congruent with itself, because
the inside contents and the outside surroundings
are different.

If we take a plane, represented by a piece of
paper that has been evenly divided by the crease~$AB$,
and divide it again crosswise, say in the point~$O$,
by another crease~$CD$, into two equal parts, we
establish in the four angles round~$O$ a new kind
of even boundary.

The bipartition results in a division of each half
plane into two portions which again are congruent
the one to the other; and the line in the crease~$CD$,
constituting, together with the first crease,~$AB$, two
angles, is (like the straight line and the plane)
nothing more nor less than an even boundary construction.
The right angle originates by the process
\PageSep{71}
of halving the straight line conceived as an
\index{Straight line}%
angle.

Let us now consider the significance of even
boundaries.

A point being a mere locus in space, has no extension
whatever; it is congruent with itself on
\index{Point congruent with itself}%
account of its want of any discriminating parts. If
it rotates in any direction, it makes no difference.

There is no mystery about a point's being congruent
with itself in any position. It results from
our conception of a point in agreement with the abstraction
we have made; but when we are confronted
with lines or surfaces that are congruent
with themselves we believe ourselves nonplussed;
yet the mystery of a straight line is not greater than
that of a point.

A line which when flopped or turned in its direction
remains congruent with itself is called straight,
and a surface which when flopped or turned round
on itself remains congruent with itself is called
plane or flat.

The straight lines and the flat surfaces are,
among all possible boundaries, of special importance,
for a similar reason that the abstraction of
pure form is so useful. In the domain of pure form
we get rid of all particularity and thus establish
a norm fit for universal application. In geometry
straight lines and plane surfaces are the climax
of simplicity; they are void of any particularity
that needs further description, or would complicate
the situation, and this absence of complications in
\PageSep{72}
their construction is their greatest recommendation.
The most important point, however, is their quality
of being unique by being even. It renders them
specially available for purposes of reference.

We can construct \Foreign{a~priori} different surfaces that
are homogeneous, yielding as many different systems
of geometry. Euclidean geometry is neither
more nor less true than spherical or elliptic geometry;
all of them are purely formal constructions,
they are \Foreign{a~priori}, being each one on its own premises
irrefutable by experience; but plane geometry is
more practical for general purposes.

The question in geometry is not, as some metageometricians
\index{Geometry!Question in}%
\index{Question in geometry}%
would have it, ``Is objective space
flat or curved?'' but, ``Is it possible to make constructions
that shall be unique so as to be serviceable
as standards of reference?'' The former question
is due to a misconception of the nature of mathematics;
the latter must be answered in the affirmative.
All even boundaries are unique and can therefore
be used as standards of reference.


\Section{The Straight Line Indispensable}
\index{Straight line!does not exist}%
\index{Straight line!indispensable|indexff}%

Straight lines do not exist in reality. How
rough are the edges of the straightest rulers, and
how rugged are the straightest lines drawn with
instruments of precision, if measured by the standard
of mathematical straightness! And if we consider
the paths of motion, be they of chemical atoms
or terrestrial or celestial bodies, we shall always
\PageSep{73}
find them to be curves of high complexity. Nevertheless
the idea of the straight line is justified by
\index{Straight line!Nature of}%
experience in so far as it helps us to analyze the
complex curves into their elementary factors, no
one of which is truly straight; but each one of
which, when we go to the end of our analysis, can
be represented as a straight line. Judging from
the experience we have of moving bodies, we cannot
doubt that if the sun's attraction of the earth (as
well as that of all other celestial bodies) could be
annihilated, the earth would fly off into space in
a straight line. Thus the mud on carriage wheels,
when spurting off, and the pebbles that are thrown
with a sling, are flying in a tangential direction
which would be absolutely straight were it not for
the interference of the gravity of the earth, which
is constantly asserting itself and modifies the
straightest line into a curve.

Our idea of a straight line is suggested to us by
experience when we attempt to resolve compound
forces into their constituents, but it is not traceable
in experience. It is a product of our method of
measurement. It is a creation of our own doings,
yet it is justified by the success which attends its
employment.

The great question in geometry is not, whether
\index{Geometry!Question in}%
\index{Question in geometry}%
straight lines are real but whether their construction
is not an indispensable requisite for any possible
system of space measurement, and further,
what is the nature of straight lines and planes and
right angles; how does their conception originate
\index{Plane@Plane, a zero of curvature!Nature of}%
\index{Right angle!Nature of}%
\PageSep{74}
and why are they of paramount importance in geometry.

We can of course posit that space should be
\index{Space!homaloidal}%
filled up with a medium such as would deflect every
ray of light so that straight rays would be impossible.
For all we know ether may in an extremely
slight degree operate in that way. But there would
be nothing in that that could dispose of the thinkability
of a line absolutely straight in the Euclidean
sense with all that the same involves, so that Euclidean
geometry would not thereby be invalidated.

Now the fact that the straight line (as a purely
\index{Straight line!possible}%
mental construction) is possible cannot be denied:
we use it and that should be sufficient for all practical
purposes. That we can construct curves also
does not invalidate the existence of straight lines.

So again while a geometry based upon the idea
of homaloidal space will remain what it has ever
\index{Homaloidal}%
been, the other geometries are not made thereby illegitimate.
Euclid disposes as little of Lobachevsky
and Bolyai as they do of Euclid.

As to the nature of the straight line and all the
\index{Standards of measurement}%
other notions connected therewith, we shall always
be able to determine them as concepts of boundary,
either reaching the utmost limit of a certain function,
be it of the highest (such as~$\infty$) or lowest measure
(such as~$0$); or dividing a whole into two congruent
parts.

The utility of such boundary concepts becomes
\index{Boundary concepts, Utility of}%
apparent when we are in need of standards for
measurement. An even boundary being the utmost
\index{Measurement!Standards for}%
\PageSep{75}
limit is unique. There are innumerable curves, but
there is only one kind of straight line. Accordingly,
\index{Line created by construction!Straightest}%
\index{Straight line!One kind of}%
if we need a standard for measuring curves, we
must naturally fall back upon the straight line and
determine its curvature by its deviation from the
straight line which represents a zero of curvature.

The straight line is the simplest of all boundary
concepts. Hence its indispensableness.

If we measure a curvature we resolve the curve
into infinitesimal pieces of straight lines, and then
determine their change of direction. Thus we use
the straight line as a reference in our measurement
of curves. The simplest curve is the circle, and its
\index{Circle!the simplest curve}%
curvature is expressed by the reciprocal of the radius;
but the radius is a straight line. It seems
that we cannot escape straightness anywhere in
geometry; for it is the simplest instrument for measuring
distance. We may replace metric geometry
by projective geometry, but what could projective
geometricians do if they had not straight lines for
their projections? Without them they would be
in a strait indeed!

But suppose we renounced with Lobatchevsky
\index{Lobatchevsky}%
the conventional method of even boundary conceptions,
especially straightness of line, and were satisfied
with straightest lines, what would be the result?
\index{Straightest line}%
He does not at the same time, surrender either the
principle of consistency or the assumption of the
homogeneity of space, and thus he builds up a geometry
independent of the theorem of parallel lines,
which would be applicable to two systems, the Euclidean
\PageSep{76}
of straight lines and the non-Euclidean of
curved space. But the latter needs the straight line
as much as the former and finds its natural limit
in a sphere whose radius is infinite and whose curvature
is zero. He can measure no spheric curvature
without the radius, and after all he reaches the
straight line in the limit of curvature. Yet it is
noteworthy that in the Euclidean system the straight
line is definite and $\pi$~irrational, while in the non-Euclidean,
$\pi$~is a definite number according to the
measure of curvature and the straight line becomes
irrational.


\Section{The Superreal}
\index{Superreal, The|indexff}%

We said in a former chapter (\Chg{p.~48}{\Pageref{48}}), ``man did
not invent reason, he discovered it,'' which means
\index{Reason, Form and!Nature of}%
that the nature of reason is definite, unalterable,
and therefore valid. The same is true of all anyness
\index{Anyness}%
of all formal thought, of pure logic, of mathematics,
and generally of anything that with truth
can be stated \Foreign{a~priori}. Though the norms of anyness
are woven of pure nothingness, the flimsiest
material imaginable, they are the factors which determine
the course of events in the entire sweep
of actual existence and in this sense they are real.
They are not real in the sense of materiality; they
are real only in being efficient and in distinction
to the reality of corporeal things we may call them
superreal.

On the one hand it is true that mathematics is
\PageSep{77}
a mental construction; it is purely ideal, which means
it is woven of thought. On the other hand we must
grant that the nature of this construction is foredetermined
in its minutest detail and in this sense
all its theorems must be discovered. We grant
that there are no sines, and cosines, no tangents
and cotangents, no logarithms, no number~$\pi$, nor
even lines, in nature, but there are relations in
nature which correspond to these notions and suggest
the invention of symbols for the sake of determining
them with exactness. These relations
possess a normative value. Stones are real in the
sense of offering resistance in a special place, but
these norms are superreal because they are efficient
factors everywhere.

The reality of mathematics is well set forth in
\index{Mathematics!Reality of}%
these words of Prof.\ Cassius Jackson Keyser, of
Columbia University:
\begin{Quote}
\index{Keyser, Cassius Jackson}%
``Phrase it as you will, there is a world that is peopled
with ideas, ensembles, propositions, relations, and implications,
in endless variety and multiplicity, in structure ranging
from the very simple to the endlessly intricate and complicate.
That world is not the product but the object, not
the creature but the quarry of thought, the entities composing
it---propositions, for example,---being no more identical
with thinking them than wine is identical with the drinking
of it. Mind or no mind, that world exists as an extra-personal
affair,---pragmatism to the contrary notwithstanding.''
\end{Quote}

While the relational possesses objective significance,
the method of describing it is subjective and
of course the symbols are arbitrary.
\PageSep{78}

In this connection we wish to call attention to a
most important point, which is the necessity of creating
fixed units for counting. As there are no
\index{Units!Discrete|indexff}%
logarithms in nature, so there are no numbers;
\index{Nature@\textit{Nature}!a continuum}%
there are only objects or things sufficiently equal
which for a certain purpose may be considered
equivalent, so that we can ignore these differences,
and assuming them to be the same, count them.


\Section{Discrete Units and the Continuum}
\index{Continuum@Continuum|indexff}%
\index{Discrete units|indexff}%

Nature is a continuum; there are no boundaries
\index{Boundaries}%
among things, and all events that happen proceed
in an uninterrupted flow of continuous transformations.
For the sake of creating order in this flux
which would seem to be a chaos to us, we must
distinguish and mark off individual objects with
definite boundaries. This method may be seen in all
branches of knowledge, and is most in evidence in
arithmetic. When counting we start in the domain
of nothingness and build up the entire structure of
arithmetic with the products of our own making.

We ought to know that whatever we do, we must
first of all take a definite stand for ourselves. When
we start doing anything, we must have a starting-point,
and even though the world may be a constant
flux we must for the sake of definiteness regard our
starting-point as fixed. It need not be fixed in reality,
but if it is to serve as a point of reference we
must regard it as fixed and look upon all the rest
as movable; otherwise the world would be an indeterminable
\PageSep{79}
tangle. Here we have the first rule
of mental activity. There may be no rest in the
\index{Mental activity, First rule of}%
world yet we must create the fiction of a rest as a
\Grk{d'os moi po~u st~w} and whenever we take any step
we must repeat this fictitious process of laying
down definite points.

All the things which are observed around us are
compounds of qualities which are only temporarily
combined. To call them things as if they were
\index{As if}%
separate beings existing without reference to the
rest is a fiction, but it is part of our method of
classification, and without this fictitious comprehension
\index{Classification}%
of certain groups of qualities under definite
names and treating them as units, we could make
no headway in this world of constant flux, and all
events of life would swim before our mental eye.

Our method in arithmetic is similar. We count
as if units existed, yet the idea of a unit is a fiction.
We count our fingers or the beads of an abacus or
any other set of things as if they were equal. We
count the feet which we measure off in a certain
line as if each one were equivalent to all the rest.
For all we know they may be different, but for our
purpose of measuring they possess the same significance.
This is neither an hypothesis nor an assumption
nor a fiction, but a postulate needed for a
definite purpose. For our purpose and according
to the method employed they are the same. We
postulate their sameness. We have made them the
same, we treat them as equal. Their sameness depends
\PageSep{80}
upon the conditions from which we start and
on the purpose which we have in view.

There are theorems which are true in arithmetic
\index{Order in life and arithmetic}%
but which do not hold true in practical life. I will
only mention the theorem $2 + 3 + 4 = 4 + 3 + 2 = 4 + 2 + 3$
etc. In real life the order in which things
are pieced together is sometimes very essential, but
in pure arithmetic, when we have started in the
domain of nothingness and build with the products
of our own counting which are ciphers absolutely
equivalent to each other, the rule holds good and
it will be serviceable for us to know it and to utilize
its significance.

The positing of units which appears to be an
\index{Units!Positing of}%
indispensable step in the construction of arithmetic
is also of great importance in actual psychology
and becomes most apparent in the mechanism of
vision.

Consider the fact that the kinematoscope has
\index{Kinematoscope}%
become possible only through an artificial separation
of the successive pictures which are again fused
together into a new continuum. The film which
passes before the lens consists of a series of little
pictures, and each one is singly presented, halting
a moment and being separated from the next by a
rotating fan which covers it at the moment when it
is exchanged for the succeeding picture. If the
moving figures on the screen did not consist of a
definite number of pictures fused into one by our
eye which is incapable of distinguishing their quick
succession the whole sight would be blurred and we
\PageSep{81}
could see nothing but an indiscriminate and unanalyzable
perpetual flux.

This method of our mind which produces units
in a continuum may possess a still deeper significance,
for it may mark the very beginning of the
real world. For all we know the formation of the
chemical atoms in the evolution of stellar nebulas
may be nothing but an analogy to this process. The
manifestation of life too begins with the creation
of individuals---of definite living creatures which
develop differently under different conditions and
again the soul becomes possible by the definiteness
of single sense-impressions which can be distinguished
as units from others of a different type.

Thus the contrast between the continuum and
the atomic formation appears to be fundamental
and gives rise to many problems which have become
especially troublesome in mathematics. But
if we bear in mind that the method, so to speak, of
atomic division is indispensable to change a world
of continuous flux into a system that can be computed
and determined with at least approximate
accuracy, we will be apt to appreciate that the
atomic fiction in arithmetic is an indispensable part
\index{Atomic fiction}%
of the method by which the whole science is created.
\PageSep{82}


\Chapter{Mathematics and Metageometry}
\index{Metageometry!Mathematics and|indexff}%

\Section{Different Geometrical Systems}

\First{Straightness}, flatness, and rectangularity
are qualities which cannot (like numbers) be
determined in purely quantitative terms; but they
are determined nevertheless by the conditions under
which our constructions must be made. A right
angle is not an arbitrary amount of ninety degrees,
but a quarter of a circle, and even the nature of
angles and degrees is not derivable either from
arithmetic or from pure reason. They are not purely
quantitative magnitudes. They contain a qualitative
element which cannot be expressed in numbers
alone. A plane is not zero, but a zero of curvature
\index{Plane@Plane, a zero of curvature}%
in a boundary between two solids; and its qualitative
element is determined, as Kant would express
it, by \Foreign{Anschauung}, or as we prefer to say, by pure
\index{Anschauung@\textit{Anschauung}}%
motility, \ie, it belongs to the domain of the \Foreign{a~priori}
of doing. For Kant's term \Foreign{Anschauung} has
the disadvantage of suggesting the passive sense
denoted by the word ``contemplation,'' while it is
important to bear in mind that the thinking subject
by its own activities creates the conditions that determine
the qualities above mentioned.
\PageSep{83}

Our method of creating by construction the
\index{Line created by construction}%
\index{Straight line!created by construction}%
straight line, the plane, and the right angle, does
\index{Plane@Plane, a zero of curvature!created by construction}%
\index{Right angle!created by construction}%
not exclude the possibility of other methods of
space-measurement, the standards of which would
not be even boundaries, such as straight lines, but
lines possessed of either a positive curvature like
the sphere or a negative curvature rendering their
surface pseudo-spherical.

Spheres are well known and do not stand in
need of description. Their curvature which is positive
is determined by the reciprocal of their radius.
\begin{center}
  \Input{page083}
\end{center}

Pseudo-spheres are surfaces of negative curvature,
\index{Helmholtz}%
\index{Pseudo-spheres}%
and pseudo-spherical surfaces are saddle-shaped.
Only limited pieces can be connectedly
represented, and we reproduce from Helmholtz,\footnote
  {\Foreign{Loc.\ cit.}, p.~42.}
two instances. If arc~$ab$ in figure~1 revolves round
an axis~$AB$, it will describe a concave-convex surface
like that of the inside of a wedding-ring; and
in the same way, if either of the curves of figure~2
revolve round their axis of symmetry, it will describe
one half of a pseudospherical surface resembling
\PageSep{84}
the shape of a morning-glory whose tapering
stem is infinitely prolonged. Helmholtz compares
the former to an anchor-ring, the latter to a champagne
glass of the old style.

The sum of the angles of triangles on spheres
always exceeds~$180°$, and the larger the sphere the
more will their triangles resemble the triangle in
the plane. On the other hand, the sum of the angles
of triangles on the pseudosphere will always be
somewhat less than~$180°$. If we define the right
angle as the fourth part of a whole circuit, it will
be seen that analogously the right angle in the
plane differs from the right angles on the sphere as
well as the pseudosphere.

We may add that while in spherical space several
shortest lines are possible, in pseudospherical
space we can draw one shortest line only. Both surfaces,
\index{Line created by construction!Shortest}%
\index{Shortest line}%
however, are homogeneous (\ie, figures can
be moved in it without suffering a change in dimensions),
but the parallel lines which do not meet are
\index{Parallel lines in spherical space}%
impossible in either.

We may further construct surfaces in which
changes of place involve either expansion or contraction,
but it is obvious that they would be less
serviceable as systems of space-measurement the
more irregular they grow.


\Section{Tridimensionality}
\index{Tridimensionality|indexff}%

Space is usually regarded as tridimensional, but
there are some people who, following Kant, express
\index{Kant}%
\PageSep{85}
themselves with reserve, saying that the mind of
man may be built up in such a way as to conceive
of objects in terms of three dimensions. Others
think that the actual and real thing that is called
space may be quite different from our tridimensional
conception of it and may in point of fact be four,
or five, or $n$-dimensional.

Let us ask first what ``dimension'' means.
\index{Dimension, Definition of}%

Does dimension mean direction? Obviously not,
for we have seen that the possibilities of direction in
space are infinite.

Dimension is only a popular term for co-ordinate.
In space there are no dimensions laid down,
but in a space of infinite directions three co-ordinates
are needed to determine from a given point of reference
the position of any other point.

In a former section on ``Even Boundaries as
\index{Boundaries!produced by halving, Even}%
\index{Even boundaries!as standards of measurement}%
\index{Even boundaries!produced by halving}%
Standards of Measurement,'' we have halved space
\index{Measurement!Even@Even boundaries as standards of}%
\index{Standards of measurement!Even@Even boundaries as}%
and produced a plane~$P_{1}$ as an even boundary between
the two halves; we have halved the plane~$P_{1}$
by turning the plane so upon itself, that like a crease
in a folded sheet of paper the straight line~$AB$
was produced on the plane. We then halved the
straight line, the even boundary between the two
half-planes, by again turning the plane upon itself
so that the line~$AB$ covered its own prolongation.
It is as if our folded sheet of paper were folded a
second time upon itself so that the crease would
be folded upon itself and one part of the same fall
exactly upon and cover the other part. On opening
the sheet we have a second crease crossing the first
\PageSep{86}
one making the perpendicular~$CD$, in the point~$O$,
thus producing right angles on the straight line~$AB$,
represented in the cross-creases of the twice
folded sheet of paper. Here the method of producing
even boundaries by halving comes to a natural end.
\index{Boundaries!produced by halving, Even}%
\index{Even boundaries!as standards of measurement}%
\index{Even boundaries!produced by halving}%
So far our products are the plane, the straight line,
the point and as an incidental but valuable by-product,
the right angle.

We may now venture on a synthesis of our materials.
\index{Measurement!Even@Even boundaries as standards of}%
\index{Standards of measurement!Even@Even boundaries as}%
We lay two planes, $P_{2}$ and~$P_{3}$, through the
two creases at right angles on the original plane~$P_{1}$,
represented by the sheet of paper, and it becomes
apparent that the two new planes $P_{2}$ and $P_{3}$ will
intersect at~$O$, producing a line~$EF$ common to both
planes $P_{2}$ and~$P_{3}$, and they will bear the same relation
to each as each one does to the original plane~$P_{1}$,
that is to say: the whole system is congruent
with itself. If we make the planes change places, $P_{1}$~may
as well take the place of~$P_{2}$ and $P_{2}$ of~$P_{3}$ and
$P_{3}$ of~$P_{2}$ or $P_{1}$ of~$P_{3}$, etc., or \Foreign{vice versa},
and all the
internal relations would remain absolutely the same.
Accordingly we have here in this system of the
three planes at right angles (the result of repeated
halving), a composition of even boundaries which,
as the simplest and least complicated construction
of its kind, \Typo{recomends}{recommends} itself for a standard of measurement
of the whole spread of motility.

The most significant feature of our construction
consists in this, that we thereby produce a convenient
system of reference for determining every
\PageSep{87}
possible point in co-ordinates of straight lines standing
at right angles to the three planes.

If we start from the ready conception of objective
space (the juxtaposition of things) we can
\index{Space!the juxtaposition of things}%
refer the several distances to analogous \Foreign{loci} in our
system of the three planes, mutually perpendicular,
each to the others. We cut space in two equal halves
by the horizontal plane~$P_{1}$. We repeat the cutting so
as to let the two halves of the first cut in their angular
relation to the new cut (in~$P_{2}$) be congruent
with each other, a procedure which is possible only
if we make use of the even boundary concept with
which we have become acquainted. Accordingly,
the second cut should stand at right angles on the
first cut. The two planes $P_{1}$ and $P_{2}$ have one line
in common,~$EF$, and any plane placed at right angles
to~$EF$ (in the point~$O$) will again satisfy the
demand of dividing space, including the two planes
$P_{1}$ and~$P_{2}$, into two congruent halves. The two
new lines, produced by the cut of the third plane~$P_{3}$
through the two former planes $P_{1}$ and~$P_{2}$, stand
both at right angles to~$EF$. Should we continue
our method of cutting space at right angles in~$O$
on either of these lines, we would produce a plane
coincident with~$P_{1}$, which is to say, that the possibilities
of the system are exhausted.

This implies that in any system of pure space
\emph{three co-ordinates are sufficient} for the determination
of any place from a given reference point.
\PageSep{88}


\Section{Three a Concept of Boundary}
\index{Three, The number|indexf}%

The number three is a concept of boundary as
much as the straight line. Under specially complicated
conditions we might need more than three
co-ordinates to calculate the place of a point, but in
empty space the number three, the lowest number
that is really and truly a number, is sufficient. If
space is to be empty space from which the notion
of all concrete things is excluded, a kind of model
constructed for the purpose of determining juxtaposition,
three co-ordinates are sufficient, because
our system of reference consists of three planes,
and we have seen above that there is no possibility
of introducing a fourth plane without destroying
its character of being congruent with itself, which
imparts to it the simplicity and uniqueness that
render it available for a standard of measurement.

Three is a peculiar number which is of great
significance. It is the first real number, being the
simplest multiplex. One and two and also zero
are of course numbers if we consider them as members
of the number-system in its entirety, but singly
regarded they are not yet numbers in the full sense
of the word. One is the unit, two is a couple or a
pair, but three is the smallest amount of a genuine
plurality. Savages who can distinguish only between
one and two have not yet evolved the notion
of number; and the transition to the next higher
stage involving the knowledge of ``three'' passes
through a mental condition in which there exists
\PageSep{89}
only the notion one, two, and plurality of any kind.
When the idea of three is once definitely recognized,
the naming of all other numbers can follow in rapid
succession.

In this connection we may incidentally call attention
\index{Dual number}%
to the significance of the grammatical dual
number as seen in the Semitic and Greek languages.
It is a surviving relic and token of a period during
which the unit, the pair, and the uncounted plurality
constituted the entire gamut of human arithmetic.
The dual form of grammatical number by
the development of the number-system became redundant
and cumbersome, being retained only for
a while to express the idea of a couple, a pair that
naturally belong together.

Certainly, the origin of the notion three has its
germ in the nature of abstract anyness. Nor is
it an accident that in order to construct the simplest
figure which is a real figure, at least three lines
are needed. The importance of the triangle, which
becomes most prominent in trigonometry, is due to
its being the simplest possible figure which accordingly
possesses the intrinsic worth of economy.

The number three plays also a significant part
in logic, and in the branches of the applied sciences,
and thus we need not be astonished at finding the
very idea, three, held in religious reverence, for the
doctrine of the Trinity has its basis in the constitution
\index{Trinity, Doctrine of the}%
of the universe and can be fully justified by the
laws of pure form.
\PageSep{90}


\Section{Space of Four Dimensions}
\index{Dimensions, Space of four|indexff}%
\index{Four dimensions!Space of|indexff}%
\index{Space!of four dimensions|indexff}%

The several conceptions of space of more than
three dimensions are of a purely abstract nature,
yet they are by no means vague, but definitely determined
by the conditions of their construction.
Therefore we can determine their properties even
in their details with perfect exactness and formulate
in abstract thought the laws of four-, five-, six-,
and $n$-dimensional space. The difficulty with which
we are beset in constructing $n$-dimensional spaces
consists in our inability to make them representable
to our senses. Here we are confronted with what
may be called the limitations of our mental constitution.
These limitations, if such they be, are conditioned
by the nature of our mode of motion, which,
if reduced to a mathematical system, needs three
co-ordinates, and this means that our space-conception
is tridimensional.

We ourselves are tridimensional; we can measure
the space in which we move with three co-ordinates,
yet we can definitely say that if space were
four-dimensional, a body constructed of two factors,
so as to have a four-dimensional solidity, would
be expressed in the formula:
\[
(a + b)^{4} = a^{4} + 4a^{3}b + 6a^{2} b^{2} + 4ab^{3} + b^{4}.
\]

We can calculate, compute, excogitate, and describe
all the characteristics of four-dimensional
space, so long as we remain in the realm of abstract
thought and do not venture to make use of our
motility and execute our plan in an actualized construction
\PageSep{91}
of motion. From the standpoint of pure
logic, there is nothing irrational about the assumption;
but as soon as we make an \Foreign{a~priori} construction
of the scope of our motility, we find out the
incompatibility of the whole scheme.

In order to make the idea of a space of more than
\index{Tridimensional beings!space, Two-dimensional beings and}%
\index{Two-dimensional beings and tridimensional space}%
three dimensions plausible or intelligible, we resort
to the relation between two-dimensional beings and
tridimensional space. The nature of tridimensional
space may be indicated yet not fully represented in
\begin{figure}[hb]
  \centering
  \Input[1.25in]{page091}
\end{figure}
two-dimensional space. If we construct a square
upon the line~$AB$ one inch long, it will be bounded
by four lines each an inch in length. In order to
construct upon the square $ABCD$ a cube of the
same measure, we must raise the square by one inch
into the third dimension in a direction at right
angles to its surface, the result being a figure
bounded by six surfaces, each of which is a one-inch
square. If two-dimensional beings who could not
rise into the third dimension wished to gain an idea
of space of a higher dimensionality and picture in
\PageSep{92}
their own two-dimensional mathematics the results
of three dimensions, they might push out the square
in any direction within their own plane to a distance
of one inch, and then connect all the corners of the
image of the square in its new position with the
corresponding points of the old square. The result
would be what is to us tridimensional beings the
picture of a cube.

When we count the plane quadrilateral figures
produced by this combination we find that there are
six, corresponding to the boundaries of a cube. We
must bear in mind that only the original and the new
square will be real squares, the four intermediary
figures which have originated incidentally through
our construction of moving the square to a distance,
exhibit a slant and to our two-dimensional beings
they appear as distortions of a rectangular relation,
which faultiness has been caused by the insufficiency
of their methods of representation. Moreover
all squares count in full and where their surfaces
overlap they count double.

Two-dimensional beings having made such a
construction must however bear in mind that the
field covered by the sides\Chg{,}{} $GEFH$ and $BFDH$ does
not take up any room in their own plane, for it is
only a picture of the extension which reaches out
either above or below their own plane; and if they
venture out of this field covered by their construction,
they have to remember that it is as empty and
unoccupied as the space beyond the boundaries $AC$
and~$AB$.
\PageSep{93}

Now if we tridimensional beings wish to do the
\index{Four dimensional@Four-dimensional space and tridimensional beings}%
\index{Tridimensional beings!Four-dimensional space and}%
same, how shall we proceed?

We must move a tridimensional body in a rectangular
\index{Fourth dimension!illustrated by mirrors|indexff}%
\index{Mirrors, Fourth dimension illustrated by|indexff}%
direction into a new (\ie, the fourth)
dimension, and being unable to accomplish this we
may represent the operation by mirrors. Having
three dimensions we need three mirrors standing
at right angles. We know by \Foreign{a~priori} considerations
according to the principle of our construction
that the boundaries of a four-dimensional body must
be solids, \ie, tridimensional bodies, and while the
sides of a cube (algebraically represented by~$a^{3}$)
must be six surfaces (\ie, two-dimensional figures,
one at each end of the dimensional line) the boundaries
of an analogous four-dimensional body built
up like the cube and the square on a rectangular
plan, must be eight solids, \ie, cubes. If we build
up three mirrors at right angles and place any
object in the intersecting corner we shall see the
object not once, but eight times. The body is reflected
below, and the object thus doubled is mirrored
not only on both upright sides but in addition
in the corner beyond, appearing in either of the upright
mirrors coincidingly in the same place. Thus
the total multiplication of our tridimensional boundaries
of a four-dimensional complex is rendered
eightfold.

We must now bear in mind that this representation
of a fourth dimension suffers from all the faults
of the analogous figure of a cube in two-dimensional
space. The several figures are not eight independent
\PageSep{94}
bodies but they are mere boundaries and the
four dimensional space is conditioned by their interrelation.
It is that unrepresentable something which
they enclose, or in other words, of which they are
assumed to be boundaries. If we were four-dimensional
beings we could naturally and easily enter
into the mirrored space and transfer tridimensional
bodies or parts of them into those other objects reflected
here in the mirrors representing the boundaries
of the four-dimensional object. While thus
on the one hand the mirrored pictures would be as
real as the original object, they would not take up
the space of our three dimensions, and in this respect
our method of representing the fourth dimension
by mirrors would be quite analogous to the
cube pictured on a plane surface, for the space to
which we (being limited by our tridimensional
space-conception) would naturally relegate the seven
additional mirrored images is unoccupied, and if
we should make the trial, we would find it empty.

Further experimenting in this line would render
constructions of a more complicated character more
and more difficult although not quite impossible.
Thus we might represent the formula $(a + b^{4}$ by
placing a wire model of a cube, representing the
proportions $(a + b)^{3}$, in the corner of our three mirrors,
and we would then verify by ocular inspection
the truth of the formula
\[
(a + b)^{4} = a^{4} + 4a^{3}b + 6a^{2} b^{2} + 4ab^{3} + b^{4}.
\]

However, we must bear in mind that all the
solids here seen are merely the boundaries of four-dimensional
\PageSep{95}
bodies. All of them with the exception
of the ones in the inner corner are scattered around
and yet the analogous figures would have to be regarded
as being most intimately interconnected,
each set of them forming one four-dimensional complex.
Their separation is in appearance only, being
due to the insufficiency of our method of presentation.

We might obviate this fault by parceling our
wire cube and instead of using three large mirrors
for reflecting the entire cube at once, we might insert
in its dividing planes double mirrors, \ie, mirrors
which would reflect on the one side the magnitude~$a$
and on the other the magnitude~$b$. In this
way we would come somewhat nearer to a faithful
representation of the nature of four-dimensional
space, but the model being divided up into a number
of mirror-walled rooms, would become extremely
complicated and it would be difficult for us to bear
always in mind that the mirrored spaces count on
both sides at once, although they overlap and (tridimensionally
considered) seem to fall the one into
the other, thus presenting to our eyes a real labyrinth
of spaces that exist within each other without
interfering with one another. They thus render
new depths visible in all three dimensions, and in
order to represent the whole scheme of a four-dimensional
complex in its full completeness, we ought
to have three mirrors at right angles placed at
every point in our tridimensional space. The scheme
itself is impossible, but the idea will render the
\PageSep{96}
nature of four-dimensional space approximately
clear. If we were four-dimensional beings we would
be possessed of the mirror-eye which in every direction
could look straightway round every corner
of the third dimension. This seems incredible, but it
can not be denied that tridimensional space lies
open to an inspection from the domain of the fourth
dimension, just as every point of a Euclidean plane
is open to inspection from above to tridimensional
vision. Of course we may demur (as we actually
do) to believing in the reality of a space of four
dimensions, but that being granted, the inferences
can not be doubted.


\Section{The Apparent Arbitrariness of the A Priori}
\index{Apriori@\textit{A priori}!Apparent arbitrariness of the|indexff}%

Since Riemann has generalized the conception of
\index{Riemann}%
space, the tridimensionality of space seems very
arbitrary.

Why are three co-ordinates sufficient for pure
space determinations? The obvious answer is, Because
we have three planes in our construction of
space-boundaries. We might as well ask, why do
the three planes cut the entire space into $8$~equal
parts? The simple answer is that we have halved
space three times, and $2^{3} = 8$. The reason is practically
the same as that for the simpler question,
why have we two halves if we divide an apple into
two equal parts?

These answers are simple enough, but there is
another aspect of the question which here seems in
\PageSep{97}
order: Why not continue the method of halving?
And there is no other answer than that it is impossible.
The two superadded planes $P_{2}$ and $P_{3}$ both
standing at right angles to the original plane, necessarily
halve each other, and thus the four right angles
of each plane $P_{2}$ and $P_{3}$ on the center of intersection,
form a complete plane for the same reason
that four quarters are one whole. We have in each
case four quarters, and $\frac{4}{4} = 1$.

Purely logical arguments (\ie, all modes of
reasoning that are rigorously \Foreign{a~priori}, the \Foreign{a~priori}
of abstract being) break down and we must resort to
the methods of the \Foreign{a~priori} of doing. We cannot
understand or grant the argument without admitting
the conception of space, previously created by
a spread of pure motion. Kant would say that we
\index{Anschauung@\textit{Anschauung}}%
\index{Kant!his term \textit{Anschauung}}%
need here the data of \Foreign{reine Anschauung}, and Kant's
\Foreign{reine Anschauung} is a product of our motility. As
soon as we admit that there is an \Foreign{a~priori} of doing
(of free motility) and that our conception of pure
space and time (Kant's \Foreign{reine Anschauung}) is its
product, we understand that our mathematical conceptions
cannot be derived from pure logic alone
but must finally depend upon our motility, viz., the
function that begets our notion of space.

If we divide an apple by a vertical cut through
its center, we have two halves. If we cut it again
by a horizontal cut through its center, we have
four quarters. If we cut it again with a cut that
is at right angles to both prior cuts, we have eight
eighths. It is obviously impossible to insert among
\PageSep{98}
these three cuts a fourth cut that would stand at
right angles to these others. The fourth cut through
the center, if we needs would have to make it, will
fall into one of the prior cuts and be a mere repetition
of it, producing no new result; or if we made
it slanting, it would not cut all eight parts but only
four of them; it would not produce sixteen equal
parts, but twelve unequal parts, viz., eight sixteenths
plus four quarters.

If we do not resort to a contemplation of the
scope of motion, if we neglect to represent in our
imagination the figure of the three planes and rely
on pure reason alone (\ie, the rigid \Foreign{a~priori}), we
have no means of refuting the assumption that we
ought to be able to continue halving the planes by
other planes at right angles. Yet is the proposition
as inconsistent as to expect that there should be
regular pentagons, or hexagons, or triangles, the
angles of which are all ninety degrees.

From the standpoint of pure reason alone we
cannot disprove the incompatibility of the idea of a
rectangular pentagon. If we insist on constructing
\index{Rectangular pentagon}%
by hook or crook a rectangular pentagon, we will
succeed, but we must break away from the straight
line or the plane. A rectangular pentagon is not
absolutely impossible; it is absolutely impossible in
the plane; and if we produce one, it will be twisted.

Such was the result of Lobatchevsky's and Bolyai's
\index{Bolyai@Bolyai, János}%
\index{Lobatchevsky}%
\index{Parallel lines in spherical space!theorem}%
construction of a system of geometry in which
the theorem of parallels does not hold. Their geometries
cease to be even; they are no longer Euclidean
\PageSep{99}
and render the even boundary conceptions unavailable
as standards of measurement.

If by logic we understand consistency, and if
anything that is self-contradictory and incompatible
with its own nature be called illogical, we would
say that it is not the logic of pure reason that renders
certain things impossible in our geometric constructions,
but the logic of our scope of motion. The
latter introduces a factor which determines the nature
of geometry, and if this factor is neglected or
misunderstood, the fundamental notions of geometry
must appear arbitrary.


\Section{Definiteness of Construction}
\index{Geometrical construction, Definiteness of|indexff}%

The problems which puzzle some of the metaphysicians
of geometry seem to have one common
foundation, which is the definiteness of geometrical
construction. Geometry starts from empty nothingness,
and we are confronted with rigid conditions
which it does not lie in our power to change. We
make a construction, and the result is something
new, perhaps something which we have not intended,
something at which we are surprised. The
synthesis is a product of our own making, yet there
is an objective element in it over which we have no
command, and this objective element is rigid, uncompromising,
an irrefragable necessity, a stubborn
fact, immovable, inflexible, immutable. What is it?

Our metageometricians overlook the fact that
their nothingness is not an absolute nothing, but
\PageSep{100}
only an absence of concreteness. If they make definite
constructions, they must (if they only remain
consistent) expect definite results. This definiteness
is the logic that dominates their operations.
Sometimes the results seem arbitrary, but they never
\index{Why@Why?}%
are; for they are necessary, and all questions \emph{why?}\
can elicit only answers that turn in a circle and are
mere tautologies.

Why, we may ask, do two straight lines, if they
intersect, produce four angles? Perhaps we did
not mean to construct angles, but here we have
them in spite of ourselves.

And why is the sum of these four angles equal
to~$360°$? Why, if two are acute, will the other two
be found obtuse? Why, if one angle be a right
angle, will all four be right angles? Why will the
sum of any two adjacent angles be equal to two
right angles?\ etc. Perhaps we should have preferred
three angles only, or four acute angles, but
we cannot have them, at least not by this construction.

We have seen that the tridimensionality of space
is arbitrary only if we judge of it as a notion of
pure reason, without taking into consideration the
method of its construction as a scope of mobility.
Tridimensionality is only one instance of apparent
arbitrariness among many others of the same kind.

We cannot enclose a space in a plane by any
figure of two straight lines, and we cannot construct
a solid of three even surfaces.

There are only definite forms of polyhedra possible,
\PageSep{101}
and the surfaces of every one are definitely
determined. To the mind uninitiated into the secrets
of mathematics it would seem arbitrary that
there are two hexahedra (viz., the cube and the
duplicated tetrahedron), while there is no heptahedron.
And why can we not have an octahedron
with quadrilateral surfaces? We might as well ask,
why is the square not round!

Prof.\ G.~B. Halsted says in the Translator's
\index{Halsted, George Bruce}%
Appendix to his English edition of Lobatchevsky's
\Title{Theory of Parallels}, p.~48:
\begin{Quote}
\index{Theory of Parallels@\textit{Theory of Parallels}, Lobatchevsky's}%
\index{Lobatchevsky's \textit{Theory of Parallels}}%
``But is it not absurd to speak of space as interfering
\index{Space!Interference of|indexff}%
with anything? If you think so, take a knife and a raw
potato and try to cut it into a seven-edged solid.''
\end{Quote}

Truly Professor Halsted's contention, that the
laws of space interfere with our operations, is true.
Yet it is not space that squeezes us, but the laws of
construction determine the shape of the figures
which we make.

A simple instance that illustrates the way in
\index{Chessboard, Problem of}%
which space interferes with our plans and movements
is the impossible demand on the chessboard
to start a rook in one corner (A1) and pass it with
the rook motion over all the fields once, but only
once, and let it end its journey on the opposite corner
(H8). Rightly considered it is not space that interferes
with our mode of action, but the law of consistency.
The proposition does not contain anything
illogical; the words are quite rational and the
sentences grammatically correct. Yet is the task
\PageSep{102}
impossible, because we cannot turn to the right and
left at once, nor can we be in two places at once,
neither can we undo an act once done or for the
\begin{figure}[hbt]
  \centering
  \Input{page102a}
\end{figure}
nonce change the rook into a bishop; but something
of that kind would have to be done, if we start from
\begin{figure}
  \centering
  \Input{page102b}
\end{figure}
A1 and pass with the rook motion through A2
and B1 over to B2. In other words:
Though the demand is not in conflict
with the logic of abstract being
or the grammar of thinking, it is
impossible because it collides with
the logic of doing; the logic of moving
about, the \Foreign{a~priori} of motility.

The famous problem of crossing seven bridges
leading to the two Königsberg Isles, is of the same
\begin{figure}[hb]
  \centering
  \Input{page103a} \\
  \footnotesize THE SEVEN BRIDGES OF KÖNIGSBERG.
\end{figure}
\index{Bridges of Königsberg|indexf}%
\index{Konigsberg@Königsberg, Seven bridges of|indexf}%
\PageSep{103}
kind. Near the mouth of the Pregel River there
is an island called Kneiphof, and the situation of
the seven bridges is shown as in the adjoined diagram.
A discussion arose as to whether it was possible
to cross all the bridges in a single promenade
\begin{figure}
  \centering
  \Input{page103b} \\
  \footnotesize EULER'S DIAGRAM.
\end{figure}
without crossing any one a second time. Finally
Euler solved the problem in a memoir presented to
\PageSep{104}
the Academy of Sciences of St.~Petersburg in 1736,
pointing out why the task could not be done. He
reproduced the situation in a diagram and proved
that if the number of lines meeting at the point~$K$
(representing the island Kneiphof as~$K$) were even
the task was possible, but if the number is odd it
can not be accomplished.

The squaring of the circle is similarly an impossibility.
\index{Circle!Squaring of the}%
\index{Squaring of the circle}%

We cannot venture on self-contradictory enterprises
\index{Determinism in mathematics}%
\index{Mathematics!Determinism in}%
without being defeated, and if the relation
of the circumference to the diameter is an infinite
transcendent \emph{series}, being
\[
\tfrac{1}{4}\pi = 1 - \tfrac{1}{3} + \tfrac{1}{5} - \tfrac{1}{7} + \tfrac{1}{9} - \tfrac{1}{11} + \tfrac{1}{13} - \tfrac{1}{15} + \tfrac{1}{17} - \tfrac{1}{19} + \tfrac{1}{21} - \tfrac{1}{23} + \cdots
\]
we cannot expect to square the circle.

If we compute the series, $\pi$~becomes $3.14159265$\dots,
figures which seem as arbitrary as the most
whimsical fancy.

It does not seem less strange that $e = 2.71828$\Add{\dots};
and yet it is as little arbitrary as the equation $3 × 4 = 12$.

The definiteness of our mathematical constructions
and arithmetical computations is based upon
the inexorable law of determinism, and everything
is fixed by the mode of its construction.


\Section{One Space, but Various Systems of Space-Measurement}
\index{Space-measurement!Various systems of|indexff}%

Riemann has generalized the idea of space and
\index{Riemann}%
would thus justify us in speaking of ``spaces.'' The
\PageSep{105}
common notion of space, which agrees best with
that of Euclidean geometry, has been degraded
into a mere species of space, one possible instance
among many other possibilities. And its very legitimacy
has been doubted, for it has come to be looked
upon in some quarters as only a popular (not to
say vulgar and commonplace) notion, a mere working
hypothesis, infested with many arbitrary conditions
of which the ideal conception of absolute
space should be free. How much more interesting
and aristocratic are curved space, the dainty two-dimensional
space, and above all the four-dimensional
space with its magic powers!

The new space-conception seems bewildering.
Some of these new spaces are constructions that are
not concretely representable, but only abstractly
thinkable; yet they allow us to indulge in ingenious
dreams. Think only of two-dimensional creatures,
and how limited they are! They can have no conception
of a third dimension! Then think of four-dimensional
beings; how superior they must be to
us poor tridimensional bodies! As we can take a
figure situated within a circle through the third dimension
and put it down again outside the circle
without crossing the circumference, so four-dimensional
beings could take tridimensional things encased
in a tridimensional box from their hiding-place
and put them back on some other spot on the
outside. They could easily help themselves to all
the money in the \Typo{steal}{steel}-lined safes of our banks, and
they could perform the most difficult obstetrical
\PageSep{106}
feats without resorting to the dangerous Cæsarean
operation.

Curved space is not less interesting. Just as
\index{Curved space}%
light may pass through a medium that offers such
a resistance as will involve a continuous displacement
of the rays, so in curved space the lines of
greatest intensity would be subject to a continuous
modification. The beings of curved space may be
assumed to have no conception of truly straight
lines. They must deem it quite natural that if they
walk on in the straightest possible manner they will
finally but unfailingly come back to the same spot.
Their world-space is not as vague and mystical as
ours: it is not infinite, hazy at a distance, vague
and without end, but definite, well rounded off, and
perfect. Presumably their lives have the same advantages
moving in boundless circles, while our
progression in straight lines hangs between two
infinitudes---the past and the future!

All these considerations are very interesting because
they open new vistas to imaginative speculators
and inventors, and we cannot deny that the
generalization of our space-conception has proved
helpful by throwing new light upon geometrical
problems and widening the horizon of our mathematical
knowledge.

Nevertheless after a mature deliberation of Riemann's
\index{Riemann|indexff}%
proposition, I have come to the conclusion
that it leads us off in a wrong direction, and in
contrast to his conception of space as being one
instance among many possibilities, I would insist
\PageSep{107}
upon the uniqueness of space. Space is the possibility
of motion in all directions, and mathematical
space is the ideal construction of our scope of motion
in all directions.

The homogeneity of space is due to our abstraction
which omits all particularities, and its homaloidality
means only that straight lines are possible
not in the real world, but in mathematical thought,
and will serve us as standards of measurement.

Curved space, so called, is a more complicated
construction of space-measurement to which some
additional feature of a particular nature has been
admitted, and in which we waive the advantages of
even boundaries as means of measurement.

Space, the actual scope of motion, remains different
from all systems of space-measurement, be
they homaloidal or curved, and should not be subsumed
with them under one and the same category.

Riemann's several space-conceptions are not
spaces in the proper sense of the word, but systems
of space-measurement. It is true that space is a
tridimensional manifold, and a plane a two-dimensional
manifold, and we can think of other systems
of $n$-manifoldness; but for that reason all these different
manifolds do not become spaces. Man is a
mammal having two prehensiles (his hands); the
elephant is a mammal with one prehensile (his
trunk); tailless monkeys like the pavian have four;
and tailed monkeys have five prehensiles. Is there
any logic in extending the denomination man to all
these animals, and should we define the elephant
\PageSep{108}
as a man with one prehensile, the pavian as a man
with four prehensiles and tailed monkeys as men
with five prehensiles? Our zoologists would at once
protest and denounce it as an illogical misuse of
names.

Space is a manifold, but not every manifold is
\index{Space!a manifold}%
a space.

Of course every one has a right to define the
terms he uses, and obviously my protest simply rejects
Riemann's use of a word, but I claim that his
identification of ``space'' with ``manifold'' is the
source of inextricable confusion.

It is well known that all colors can be reduced
to three primary colors, yellow, red, and blue, and
thus we can determine any possible tint by three
co-ordinates, and color just as much as mathematical
space is a threefold, viz., a system in which
three co-ordinates are needed for the determination
of any thing. But because color is a threefold, no
one would assume that color is space.

Riemann's manifolds are systems of measurement,
and the system of three co-ordinates on
three intersecting planes is an \Foreign{a~priori} or purely
formal and ideal construction invented to calculate
space. We can invent other more complicated systems
of measurement, with curved lines and with
more than three or less than three co-ordinates.
We can even employ them for space-measurement,
although they are rather awkward and unserviceable;
but these systems of measurement are not
``spaces,'' and if they are called so, they are spaces
\PageSep{109}
by courtesy only. By a metaphorical extension we
allow the idea of system of space-measurement to
stand for space itself. It is a brilliant idea and quite
as ingenious as the invention of animal fables in
which our quadruped fellow-beings are endowed
with speech and treated as human beings. But such
poetical licences, in which facts are stretched and
the meaning of terms is slightly modified, is possible
only if instead of the old-fashioned straight rules of
logic we grant a slight curvature to our syllogisms.


\Section{Fictitious Spaces and the Apriority of all
Systems of Space-Measurement}
\index{Apriority of different degrees!of space-measurement|indexff}%
\index{Fictitious spaces|indexff}%
\index{Mathematical space}%
\index{Spaces, Fictitious|indexff}%
\index{Space!Mathematical}%
\index{Space-measurement!Apriority of|indexff}%

Mathematical space, so called, is strictly speaking
no space at all, but the mental construction of
a manifold, being a tridimensional system of space-measurement
invented for the determination of actual
space.

Neither can a manifold of two dimensions be
called a space. It is a mere boundary in space, it
is no reality, but a concept, a construction of pure
thought.

Further, the manifold of four dimensions is a
\index{Four dimensions}%
system of measurement applicable to any reality
for the determination of which four co-ordinates
are needed. It is applicable to real space if there
is connected with it in addition to the three planes
at right angles another condition of a constant nature,
such as gravity.

At any rate, we must deny the applicability of a
\PageSep{110}
system of four dimensions to empty space void of
any such particularity. The idea of space being
four-dimensional is chimerical if the word space is
used in the common acceptance of the term as
juxtaposition or as the scope of motion. So long
as four quarters make one whole, and four right
angles make one entire circumference, and so long
as the contents of a sphere which covers the entire
scope of motion round its center equals $\frac{4}{3}\pi r^{3}$, there
is no sense in entertaining the idea that empty space
might be four-dimensional.

But the argument is made and sustained by
Helmholtz that as two-dimensional beings perceive
\index{Helmholtz!on two-dimensional beings|indexf}%
two dimensions only and are unable to think how
a third dimension is at all possible, so we tridimensional
beings cannot represent in thought the possibility
of a fourth dimension. Helmholtz, speaking
of beings of only two dimensions living on the surface
of a solid body, says:
\begin{Quote}
``If such beings worked out a geometry, they would of
course assign only two dimensions to their space. They
would ascertain that a point in moving describes a line, and
that a line in moving describes a surface. But they could
as little represent to themselves what further spatial construction
would be generated by a surface moving out of
itself, as we can represent what should be generated by a
solid moving out of the space we know. By the much-abused
expression `to represent' or `to be able to think how
something happens' I understand---and I do not see how
anything else can be understood by it without loss of all
meaning---the power of imagining the whole series of sensible
impressions that would be had in such a case. Now,
\PageSep{111}
as no sensible impression is known relating to such an unheard-of
event, as the movement to a fourth dimension
would be to us, or as a movement to our third dimension
would be to the inhabitants of a surface, such a `representation'
is as impossible as the `representation' of colors
would be to one born blind, if a description of them in general
terms could be given to him.

``Our surface-beings would also be able to draw shortest
lines in their superficial space. These would not necessarily
be straight lines in our sense, but what are technically called
\emph{geodetic lines} of the surface on which they live; lines such
as are described by a \emph{tense} thread laid along the surface,
and which can slide upon it freely.''~\dots

``Now, if beings of this kind lived on an infinite plane,
their geometry would be exactly the same as our planimetry.
They would affirm that only one straight line is possible
between two points; that through a third point lying without
this line only one line can be drawn parallel to it; that
the ends of a straight line never meet though it is produced
to infinity, and so on.''~\dots

``But intelligent beings of the kind supposed might also
live on the surface of a sphere. Their shortest or straightest
line between two points would then be an arc of the great
circle passing through them.''~\dots

``Of parallel lines the sphere-dwellers would know nothing.
They would maintain that any two straightest lines,
sufficiently produced, must finally cut not in one only but in
two points. The sum of the angles of a triangle would be
always greater than two right angles, increasing as the surface
of the triangle grew greater. They could thus have
no conception of geometrical similarity between greater and
smaller figures of the same kind, for with them a greater
triangle must have different angles from a smaller one.
Their space would be unlimited, but would be found to be
finite or at least represented as such.

``It is clear, then, that such beings must set up a very
\PageSep{112}
different system of geometrical axioms from that of the
inhabitants of a plane, or from ours with our space of three
dimensions, though the logical powers of all were the same.''
\end{Quote}

I deny what Helmholtz implicitly assumes that
sensible impressions enter into the fabric of our
concepts of purely formal relations. We have the
idea of a surface as a boundary between solids, but
surfaces do not exist in reality. All real objects
are solid, and our idea of surface is a mere fiction
of abstract reasoning. Two dimensional things are
unreal, we have never seen any, and yet we form
the notion of surfaces, and lines, and points, and
pure space, etc. There is no straight line in existence,
\index{Straight line}%
hence it can produce no sense-impression,
and yet we have the notion of a straight line. The
straight lines on paper are incorrect pictures of the
true straight lines which are purely ideal constructions.
Our \Foreign{a~priori} constructions are not a product
\index{Apriori@\textit{A priori}!constructions}%
of our sense-impressions, but are independent of
sense or anything sensed.

It is of course to be granted that in order to
have any conception, we must have first of all sensation,
and we can gain an idea of pure form only
by abstraction. But having gained a fund of abstract
notions, we can generalize them and modify
them; we can use them as a child uses its building
blocks, we can make constructions of pure thought
unrealizable in the concrete world of actuality. Some
of such constructions cannot be represented in concrete
form, but they are not for that reason unthinkable.
Even if we grant that two-dimensional
\PageSep{113}
beings were possible, we would have no reason to
assume that two-dimensional beings could not construct
a tridimensional space-conception.

Two-dimensional beings could not be possessed
of a material body, because their absolute flatness
substantially reduces their shape to nothingness.
But if they existed, they would be limited to movements
in two directions and thus must be expected
to be incredulous as to the possibility of jumping
out of their flat existence and returning into it
through a third dimension. Having never moved
in a third dimension, they could speak of it as the
blind might discuss colors; in their flat minds they
could have no true conception of its significance and
would be unable to clearly picture it in their imagination;
but for all their limitations, they could very
well develop the abstract idea of tridimensional
space and therefrom derive all particulars of its
laws and conditions and possibilities in a similar
way as we can acquire the notion of a space of four
dimensions.

Helmholtz continues:
\begin{Quote}
\index{Curved space!Helmholtz on}%
\index{Helmholtz!on curved space}%
\index{Space!Helmholtz on curved}%
``But let us proceed still farther.

``Let us think of reasoning beings existing on the surface
of an egg-shaped body. Shortest lines could be drawn between
\index{Egg-shaped body|indexf}%
three points of such a surface and a triangle constructed.
But if the attempt were made to construct congruent
triangles at different parts of the surface, it would
be found that two triangles, with three pairs of equal sides,
would not have their angles equal.''
\end{Quote}

If there were two-dimensional beings living on
\PageSep{114}
an egg-shell, they would most likely have to determine
the place of their habitat by experience just
as much as we tridimensional beings living on a
flattened sphere have to map out our world by measurements
made \Foreign{a~posteriori} and based upon \Foreign{a~priori}
systems of measurement.

If the several systems of space-measurement
were not \Foreign{a~priori} constructions, how could Helmholtz
who does not belong to the class of two-dimensional
beings tell us what their notions must be
like?

I claim that if there were surface beings on a
sphere or on an egg-shell, they would have the
same \Foreign{a~priori} notions as we have; they would be
able to construct straight lines, even though they
were constrained to move in curves only; they would
be able to define the nature of a space of three dimensions
and would probably locate in the third
dimension their gods and the abode of spirits. I
insist that not sense experience, but \Foreign{a~priori} considerations,
teach us the notions of straight lines.

The truth is that we tridimensional beings actually
do live on a sphere, and we cannot get away
from it. What is the highest flight of an æronaut
and the deepest descent into a mine if measured by
the radius of the earth? If we made an exact imitation
of our planet, a yard in diameter, it would
be like a polished ball, and the highest elevations
would be less than a grain of sand; they would not
be noticeable were it not for a difference in color
and material.
\PageSep{115}

When we become conscious of the nature of our
habitation, we do not construct \Foreign{a~priori} conceptions
accordingly, but feel limited to a narrow surface
and behold with wonder the infinitude of space beyond.
We can very well construct other \Foreign{a~priori}
notions which would be adapted to one, or two, or
four-dimensional worlds, or to spaces of positive
or of negative curvatures, for all these constructions
are ideal; they are mind-made and we select from
them the one that would best serve our purpose of
space-measurement.

The claim is made that if we were four-dimensional
beings, our present three-dimensional world
would appear to us as flat and shallow, as the plane
is to us in our present tridimensional predicament.
That statement is true, because it is conditioned by
an ``if.'' And what pretty romances have been
built upon it! I remind my reader only of the ingenious
story \Title{Flatland, Written by a Square}, and
\index{Flatland@\textit{Flatland}}%
portions of Wilhelm Busch's charming tale \Title{Edward's
\index{Edward's Dream@\textit{Edward's Dream}}%
\index{Busch, Wilhelm}%
\index{Open Court@\textit{Open Court}|indexnote}%
Dream};\footnote
  {\Title{The Open Court}, Vol.~VIII, p.~4266 et seq.}
but the worth of conditional truths
depends upon the assumption upon which they are
made contingent, and the argument is easy enough
that if things were different, they would not be what
they are. If I had wings, I could fly; if I had gills
I could live under water; if I were a magician I
could work miracles.
\PageSep{116}


\Section{Infinitude}
\index{Infinitude|indexff}%
\index{Infinite directions of space!Space is}%
\index{Infinite directions of space!Time is}%

The notion is rife at present that infinitude is
self-contradictory and impossible. But that notion
originates from the error that space is a thing, an
\index{Space!is infinite}%
objective and concrete reality, if not actually material,
yet consisting of some substance or essence.
It is true that infinite things cannot exist, for things
are always concrete and limited; but space is pure
potentiality of concrete existence. Pure space is
materially considered nothing. That this pure space
(this apparent nothing) possesses some very definite
positive qualities is a truth which at first sight
may seem strange, but on closer inspection is quite
natural and will be conceded by every one who comprehends
the paramount significance of the doctrine
of pure form.

Space being pure form of extension, it must be
infinite, and infinite means that however far we go,
in whatever direction we choose, we can go farther,
and will never reach an end. Time is just as infinite
\index{Time is infinite}%
as space. Our sun will set and the present day will
pass away, but time will not stop. We can go backward
to the beginning, and we must ask what was
before the beginning. Yet suppose we could fill the
blank with some hypothesis or another, mythological
or metaphysical, we would not come to an absolute
beginning. The same is true as to the end.
And if the universe broke to pieces, time would
continue, for even the duration in which the world
would lie in ruins would be measurable.
\PageSep{117}

Not only is space as a totality infinite, but in
\index{Infinite directions of space}%
\index{Infinite directions of space!division of line}%
\index{Space!Infinite directions of}%
every part of space we have infinite directions.
\index{Directions of space, Infinite}%

What does it mean that space has infinite directions?
If you lay down a direction by drawing a line
\index{Line created by construction!Infinite division of}%
from a given point, and continue to lay down other
directions, there is no way of exhausting your possibilities.
Light travels in all directions at once;
but ``all directions'' means that the whole extent of
the surroundings of a source of light is agitated,
and if we attempt to gather in the whole by picking
up every single direction of it, we stand before a
task that cannot be finished.

In the same way any line, though it be of definite
length, can suffer infinite division, and the fraction
$\frac{1}{3}$ is quite definite while the same amount if expressed
in decimals as $0.333$\dots, can never be completed.
Light actually travels in all directions,
which is a definite and concrete process, but if we
try to lay them down one by one we find that we can
as little exhaust their number as we can come to
an end in divisibility or as we can reach the boundary
of space, or as we can come to an ultimate
number in counting. In other words, reality is actual
and definite but our mode of measuring it or
reducing it to formulas admits of a more or less approximate
treatment only, being the function of an
infinite progress in some direction or other. There
is an objective \Foreign{raison d'être} for the conception of
the infinite, but our formulation of it is subjective,
and the puzzling feature of it originates from treating
the subjective feature as an objective fact.
\PageSep{118}

These considerations indicate that infinitude
does not appertain to the thing, but to our method
of viewing the thing. Things are always concrete
and definite, but the relational of things admits of
a progressive treatment. Space is not a thing, but
the relational feature of things. If we say that
space is infinite, we mean that a point may move incessantly
and will never reach the end where its
progress would be stopped.

There is a phrase current that the finite cannot
\index{Infinite directions of space!not mysterious}%
comprehend the infinite. Man is supposed to be
finite, and the infinite is identified with God or the
Unknowable, or anything that surpasses the comprehension
of the average intellect. The saying is
based upon the prejudicial conception of the infinite
as a realized actuality, while the infinite is not a
concrete thing, but a series, a process, an aspect,
or the plan of action that is carried on without stopping
and shall not, as a matter of principle, be cut
short. Accordingly, the infinite (though in its completeness
unactualizable) is neither mysterious nor
incomprehensible, and though mathematicians be
finite, they may very successfully employ the infinite
in their calculations.

I do not say that the idea of infinitude presents
no difficulties, but I do deny that it is a self-contradictory
notion and that if space must be conceived
to be infinite, mathematics will sink into mysticism.
\PageSep{119}


\Section{Geometry Remains A Priori}
\index{Geometry!\textit{a priori}|indexff}%

Those of our readers who have closely followed
our arguments will now understand how in one important
point we cannot accept Mr.\ B.~A.~W. Russell's
statement as to the main result of the metageometrical
inquisition. He says:
\begin{Quote}
\index{Russell, Bertrand A. W.!on non-Euclidean geometry}%
``There is thus a complete divorce between geometry and
the study of actual space. Geometry does not give us certain
knowledge as to what exists. That peculiar position
which geometry formerly appeared to occupy, as an \Foreign{a~priori}
science giving knowledge of something actual, now appears
to have been erroneous. It points out a whole series of possibilities,
each of which contains a whole system of connected
propositions; but it throws no more light upon the
nature of our space than arithmetic throws upon the population
\index{Population of Great Britain determined|indexf}%
of Great Britain. Thus the plan of attack suggested
by non-Euclidean geometry enables us to capture the last
stronghold of those who attempt, from logical or \Foreign{a~priori}
considerations, to deduce the nature of what exists. The
conclusion \emph{suggested} is, that no existential proposition can
be deduced from one which is not existential. But to prove
such a conclusion would demand a treatise upon all branches
\index{Apriori@\textit{A priori}!Geometry is|indexnote}%
of philosophy.''\footnote
  {In \Title{the new volumes of the Encyclopædia Britannica}, Vol.~XXVIII,
  of the complete work, \Foreign{s.~v.}\ Geometry, Non-Euclidean, p.~674.}
\end{Quote}

It is a matter of course that the single facts as to
the population of Great Britain must be supplied
by counting, and in the same way the measurements
of angles and actual distances must be taken by \Foreign{a~posteriori}
transactions; but having ascertained
some lines and angles, we can (assuming our data
to be correct) calculate other items with absolute
\PageSep{120}
exactness by purely \Foreign{a~priori} argument. There is
no need (as Mr.\ Russell puts it) ``from logical or
\Foreign{a~priori} considerations to deduce the nature of what
exists,''---which seems to mean, to determine special
features of concrete instances. No one ever assumed
that the nature of particular cases, the qualities
of material things, or sense-affecting properties,
could be determined by \Foreign{a~priori} considerations.
The real question is, whether or not the theorems
of space relations and, generally, purely formal conceptions,
such as are developed \Foreign{a~priori} in geometry
and kindred formal sciences, will hold good in
actual experience. In other words, can we assume
that form is an objective quality, which would imply
that the constitution of the actual world must be
the same as the constitution of our purely \Foreign{a~priori}
sciences? We answer this latter question in the
affirmative.

We cannot determine by \Foreign{a~priori} reasoning the
population of Great Britain. But we can \Foreign{a~posteriori}
count the inhabitants of several towns and
districts, and determine the total by addition. The
rules of addition, of division, and multiplication can
be relied upon for the calculation of objective facts.

Or to take a geometrical example. When we
measure the distance between two observatories
and also the angles at which at either end of the
line thus laid down the moon appears in a given
moment, we can calculate the moon's distance from
the earth; and this is possible only on the assumption
that the formal relations of objective space
\PageSep{121}
are the same as those of mathematical space. In
\index{Mathematical space!Apriority of}%
\index{Space!Apriority of mathematical}%
other words, that our \Foreign{a~priori} mathematical calculations
can be made to throw light upon the nature
of space,---the real objective space of the world in
which we live.

\Thoughtbreak

The result of our investigation is quite conservative.
It re-establishes the apriority of mathematical
space, yet in doing so it justifies the method of metaphysicians
in their constructions of the several non-Euclidean
systems. All geometrical systems, Euclidean
as well as non-Euclidean, are purely ideal
constructions. If we make one of them we then
and there for that purpose and for the time being,
exclude the other systems, but they are all, each
one on its own premises, equally true and the question
of preference between them is not one of truth
or untruth but of adequacy, of practicability, of usefulness.

The question is not: ``Is real space that of Euclid
\index{Question in geometry}%
or of Riemann, of Lobatchevsky or Bolyai?''\
for real space is simply the juxtapositions of things,
while our geometries are ideal schemes, mental constructions
of models for space measurement. The
real question is, ``Which system is the most convenient
to determine the juxtaposition of things?''

\Foreign{A~priori} considered, all geometries have equal
\index{Apriority of different degrees!of mathematical space}%
rights, but for all that Euclidean geometry, which
\index{Geometry!Question in}%
in the parallel theorem takes the bull by the horn,
will remain classical forever, for after all the non-Euclidean
\index{Euclidean geometry, classical}%
\PageSep{122}
systems cannot avoid developing the notion
of the straight line or other even boundaries.
\index{Even boundaries}%
\index{Straight line}%
Any geometry could, within its own premises, be
utilized for a determination of objective space; but
\index{Space!Sense-experience and|indexff}%
we will naturally give the preference to plane geometry,
not because it is truer, but because it is
simpler and will therefore be more serviceable.

How an ideal (and apparently purely subjective)
construction can give us any information of
the objective constitution of things, at least so far as
space-relations are concerned, seems mysterious but
the problem is solved if we bear in mind the objective
nature of the \Foreign{a~priori},---a topic which we have
\index{Apriori@\textit{A priori}!constructions verified by experience}%
elsewhere discussed.\footnote
  {See also the author's exposition of the problem of the \Foreign{a~priori}
\index{Carus, Paul!Kant's Prolegomena@\textit{Kant's Prolegomena}|indexnote}%
  in his edition of \Title{Kant's Prolegomena}, pp.~167--240.}


\Section{Sense-Experience and Space}
\index{Sense-experience and space|indexff}%

We have learned that sense-experience cannot
be used as a source from which we construct our
fundamental notions of geometry, yet sense-experience
justifies them.

Experience can verify \Foreign{a~priori} constructions as,
\eg, tridimensionality is verified in Newton's laws;
but experience can never refute them, nor can it
change them. We may apply any system if we only
remain consistent. It is quite indifferent whether
we count after the decimal, the binary or the duo-decimal
system. The result will be the same. If
experience does not tally with our calculations, we
have either made a mistake or made a wrong observation.
\PageSep{123}
For our \Foreign{a~priori} conceptions hold good
for any conditions, and their theory can be as little
wrong as reality can be inconsistent.

However, some of the most ingenious thinkers
and great mathematicians do not conceive of space
\index{Space!On the nature of|indexf}%
as mere potentiality of existence, which renders it
formal and purely \Foreign{a~priori}, but think of it as a
concrete reality, as though it were a big box, presumably
round, like an immeasurable sphere. If it
were such, space would be (as Riemann says)
\index{Riemann}%
boundless but not infinite, for we cannot find a
boundary on the surface of a sphere, and yet the
sphere has a finite surface that can be expressed in
definite numbers.

I should like to know what Riemann would call
that something which lies outside of his spherical
space. Would the name ``province of the extra-spatial''
perhaps be an appropriate term? I do not
know how we can rid ourselves of this enormous
portion of unutilized outside room. Strange though
it may seem, this space-conception of Riemann
counts among its advocates mathematicians of first
rank, among whom I will here mention only the
name of Sir Robert Ball.
\index{Ball, Sir Robert, on the nature of space|indexf}%

It will be interesting to hear a modern thinker
who is strongly affected by metageometrical studies,
on the nature of space. Mr.\ Charles S.~Peirce, an
\index{Peirce, Charles S., on the nature of space|indexf}%
uncommonly keen logician and an original thinker
of no mean repute, proposes the following three
alternatives. He says:
\PageSep{124}
\begin{Quote}
``First, space is, as Euclid teaches, both \emph{unlimited} and
\emph{immeasurable}, so that the infinitely distant parts of any
plane seen in perspective appear as a straight line, in which
case the sum of the three angles amounts to~$180°$; or,

``Second, space is \emph{immeasurable} but \emph{limited}, so that the
infinitely distant parts of any plane seen in perspective appear
as in a circle, beyond which all is blackness, and in this
case the sum of the three angles of a triangle is less than
$180°$ by an amount proportional to the area of the triangle;
or

``Third, space is \emph{unlimited} but \emph{finite} (like the surface of
a sphere), so that it has no infinitely distant parts; but a
finite journey along any straight line would bring one back
to his original position, and looking off with an unobstructed
view one would see the back of his own head enormously
magnified, in which case the sum of the three angles of a
triangle exceeds $180°$ by an amount proportional to the
area.

``Which of these three hypotheses is true we know not.
The largest triangles we can measure are such as have the
earth's orbit for base, and the distance of a fixed star for
altitude. The angular magnitude resulting from subtracting
the sum of the two angles at the base of such a triangle
from $180°$ is called the star's \emph{parallax}. The parallaxes of
only about forty stars have been measured as yet. Two of
them come out negative, that of Arided\Note{Deneb} ($\alpha$~\Typo{Cycni}{Cygni}), a star
of magnitude~$1\frac{1}{2}$, which is $-0.''082$, according to C.~A.~F.
Peters, and that of a star of magnitude~$7\frac{3}{4}$, known as
Piazzi III 422, which is $-0.''045$ according to R.~S. Ball.
But these negative parallaxes are undoubtedly to be attributed
to errors of observation; for the probable error of
such a determination is about $±0.''075$, and it would be
strange indeed if we were to be able to see, as it were, more
than half way round space, without being able to see stars
with larger negative parallaxes. Indeed, the very fact that
of all the parallaxes measured only two come out negative
\PageSep{125}
would be a strong argument that the smallest parallaxes
really amount to~$+0.''1$, were it not for the reflexion that
the publication of other negative parallaxes may have been
suppressed. I think we may feel confident that the parallax
of the furthest star lies somewhere between $-0.''05$ and
$+0.''15$, and within another century our grandchildren will
surely know whether the three angles of a triangle are
greater or less than~$180°$,---that they are \emph{exactly} that amount
is what nobody ever can be justified in concluding. It is
true that according to the axioms of geometry the sum of
the three sides of a triangle is precisely~$180°$; but these
axioms are now exploded, and geometers confess that they,
as geometers, know not the slightest reason for supposing
them to be precisely true. They are expressions of our inborn
conception of space, and as such are entitled to credit,
so far as their truth could have influenced the formation of
the mind. But that affords not the slightest reason for supposing
them exact.'' (\Title{The Monist}, Vol.~I, pp.~173--174.)
\index{Monist@\textit{Monist}}%
\end{Quote}

Now, let us for argument's sake assume that
\index{Measurement!of star parallaxes}%
\index{Star parallaxes, Measurements of}%
the measurements of star-parallaxes unequivocally
yield results which indicate that the sum of the
angles in cosmic triangles is either a trifle more
or a trifle less than~$180°$; would we have to conclude
that cosmic space is curved, or would we not have
to look for some concrete and special cause for the
aberration of the light? If the moon is eclipsed
while the sun still appears on the horizon, it proves
only that the refraction of the solar rays makes the
sun appear higher than it really stands, if its position
is determined by a straight line, but it does not
refute the straight line conception of geometry.
Measurements of star-parallaxes (if they could no
longer be accounted for by the personal equation
\PageSep{126}
of erroneous observation), may prove that ether can
slightly deflect the rays of light, but it will never
prove that the straight line of plane geometry is
really a \Typo{cirve}{curve}. We might as well say that the norms
of logic are refuted when we make faulty observations
or whenever we are confronted by contradictory
statements. No one feels called upon, on account
of the many lies that are told, to propose a
theory on the probable curvature of logic. Yet,
seriously speaking, in the province of pure being
the theory of a curved logic has the same right to
a respectful hearing as the curvature of space in the
province of the scope of pure motility.

Ideal constructions, like the systems of geometry,
\index{Infinite directions of space!Space is}%
\index{Space!curved}%
\index{Space!is infinite}%
logic, etc., cannot be refuted by facts. Our
observation of facts may call attention to the logical
mistakes we have made, but experience cannot
overthrow logic itself or the principles of thinking.
They bear their standard of correctness in
themselves which is based upon the same principle
of consistency that pervades any system of actual
or purely ideal operations.

But if space is not round, are we not driven to
the other horn of the dilemma that space is infinite?

Perhaps we are. What of it? I see nothing
amiss in the idea of infinite space.

By the by, if objective space were really curved,
would not its twist be dominated in all probability
by more than one determinant? Why should it be
a curvature in the plane which makes every straight
line a circle? Might not the plane in which our
\PageSep{127}
straightest line lies be also possessed of a twist so
\index{Line created by construction!Straightest}%
\index{Straightest line}%
as to give it the shape of a flat screw, which would
change every straightest line into a spiral? But
the spiral is as infinite as the straight line. Obviously,
curved space does not get rid of infinitude;
besides the infinitely small, which would not be
thereby eliminated, is not less troublesome than the
infinitely great.


\Section{The Teaching of Mathematics}
\index{Mathematics!Teaching of|indexff}%
\index{Teaching of mathematics|indexff}%

As has been pointed out before, Euclid avoided
\index{Axiom@``Axiom,''!Euclid avoided}%
\index{Euclid!avoided "axiom,"}%
the word axiom, and I believe with Grassmann, that
\index{Grassmann}%
its omission in the \Title{Elements} is not accidental but
the result of well-considered intention. The introduction
of the term among Euclid's successors is
due to a lack of clearness as to the nature of geometry
and the conditions through which its fundamental
notions originate.

It may be a flaw in the Euclidean \Title{Elements} that
the construction of the plane is presupposed, but it
does not invalidate the details of his glorious work
which will forever remain classical.

The invention of other geometries can only serve
to illustrate the truth that all geometries, the plane
geometry of Euclid included, are \Foreign{a~priori} constructions,
\index{Apriori@\textit{A priori}!constructions. Geometries are}%
\index{Geometries, \textit{a priori} constructions}%
and were not for obvious reasons Euclid's
plane geometry preferable, other systems might as
well be employed for the purpose of space-determination.
Neither homaloidality nor curvature belongs
to space; they belong to the several systems of
\PageSep{128}
manifolds that can be invented for the determination
of the juxtapositions of things, called space.

If I had to rearrange the preliminary expositions
of Euclid, I would state first the \emph{Common Notions}
\index{Common notions}%
\index{Definitions of Euclid}%
\index{Euclid!Expositions of, rearranged}%
which embody those general principles of Pure Reason
and are indispensable for geometry. Then I
would propose the \emph{Postulates} which set forth our
\index{Postulates}%
own activity (viz., the faculty of construction) and
the conditions under which we intend to carry out
our operations, viz., the obliteration of all particularity,
characterizable as ``anyness of motion.''
Thirdly, I would describe the instruments to be
employed: the ruler and the pair of compasses;
the former being the crease in a plane folded upon
itself, and the latter to be conceived as a straight
line (a stretched string) one end of which is stationary
while the other is movable. And finally I
would lay down the \emph{Definitions} as the most elementary
constructions which are to serve as tools and
objects for experiment in the further expositions
of geometry. There would be no mention of axioms,
nor would we have to regard anything as an assumption
or an hypothesis.

Professor Hilbert has methodically arranged
\index{Axiom@``Axiom,''!Hilbert's use of}%
\index{Hilbert's use of ``axiom''}%
the principles that underlie mathematics, and the
excellency of his work is universally recognized.\footnote
  {\Title{The Foundations of Geometry}, The Open Court Pub.\ Co.,
  Chicago, 1902.}
It is a pity, however, that he retains the term
``axiom,'' and we would suggest replacing it by
some other appropriate word. ``Axiom'' with Hilbert
\PageSep{129}
does not mean an obvious truth that does not
stand in need of proof, but a principle, or rule, viz.,
a formula describing certain general characteristic
conditions.

Mathematical space is an ideal construction, and
\index{Apriority of different degrees!of mathematical space}%
\index{Mathematical space!Apriority of}%
\index{Space!Apriority of mathematical}%
as such it is \Foreign{a~priori}. But its apriority is not as
rigid as is the apriority of logic. It presupposes
not only the rules of pure reason but also our own
activity (viz., pure motility) both being sufficient
to create any and all geometrical figures \Foreign{a~priori}.

Boundaries that are congruent with themselves
\index{Boundaries}%
being limits that are unique recommend themselves
as standards of measurement. Hence the significance
of the straight line, the plane, and the right
\index{Plane@Plane, a zero of curvature!Significance of}%
\index{Right angle!Significance of}%
\index{Straight line!Significance of}%
angle.

The theorem of parallels is only a side issue of
\index{Parallel lines in spherical space!theorem}%
the implications of the straight line.

The postulate that figures of the same relations
are congruent in whatever place they may be, and
also that figures can be drawn similar to any figure,
is due to our abstraction which creates the condition
of anyness.

The teaching of mathematics, now utterly neglected
in the public schools and not specially favored
in the high schools, should begin early, but Euclid's
method with his pedantic propositions and proofs
should be replaced by construction work. Let children
begin geometry by doing, not by reasoning.
The reasoning faculties are not yet sufficiently developed
in a child. Abstract reasoning is tedious,
but if it comes in as an incidental aid to construction,
\PageSep{130}
it will be welcome. Action is the main-spring
of life and the child will be interested so long as
there is something to achieve.\footnote
  {\Typo{Cp.}{Cf.}\ the author's article ``Anticipate the School'' (\Title{Open Court},
\index{Open Court@\textit{Open Court}|indexnote}%
  1899, p.~747).}

Lines must be divided, perpendiculars dropped,
parallel lines drawn, angles measured and transferred,
triangles constructed, unknown quantities
determined with the help of proportion, the nature
of the triangle studied and its internal relations
laid down and finally the right-angled triangle computed
by the rules of trigonometry, etc. All instruction
should consist in giving \emph{tasks to be performed,
not theorems to be proved}; and the pupil
should find out the theorems merely because he
needs them for his construction.

In the triangle as well as in the circle we should
accustom ourselves to using the same names for the
\index{Names, Same, for parts of figures}%
same parts.\footnote
  {Such was the method of my teacher, Prof.\ Hermann Grassmann.
  \index{Grassmann|indexnote}%
}

Every triangle is~$ABC$. The angle at~$A$ is always~$\alpha$,
at $B$~$\beta$, at $C$~$\gamma$. The side opposite~$A$ is~$a$,
opposite $B$~$b$, opposite $C$~$c$. Altitudes (heights) are
$h_{a}$, $h_{b}$,~$h_{c}$. The lines that from $A$,~$B$, and~$C$ pass
through the center of gravity to the middle of the
opposite sides I propose to call gravitals and would
designate them $g_{a}$, $g_{b}$,~$g_{c}$. The perpendiculars erected
upon the middle of the sides meeting in the center of
the circumscribed circle are $p_{a}$, $p_{b}$,~$p_{c}$. The lines
that divide the angles $\alpha$, $\beta$,~$\gamma$ and meet in the center
\PageSep{131}
of the inscribed circle I propose to call ``dichotoms''\footnote
  {From \Grk{diq'otomos}. I purposely avoid the term \emph{bisector} and also
  the term \emph{median}, the former because its natural abbreviation~$b$ is
  already appropriated to the side opposite to the point~$B$, and the
  latter because it has \Typo{beeen}{been} used to denote sometimes the gravitals
  and sometimes the dichotoms. It is thus reserved for general use
  in the sense of any middle lines.}
and would designate them as $d_{a}$, $d_{b}$,~$d_{c}$. The radius
of the circumscribed circle is~$r$, of the inscribed
circle~$\rho$, and the radii of the three ascribed circles
are $\rho_{a}$, $\rho_{b}$,~$\rho_{c}$. The point where the three heights
meet is~$H$; where the three gravitals meet,~$G$; where
the three dichotoms meet,~$O$.\footnote
  {The capital of the Greek~$\rho$ is objectionable, because it cannot be
  distinguished from the Roman~$P$.}
The stability of
designation is very desirable and perhaps indispensable
for a clear comprehension of these important
interrelated parts.
\PageSep{132}


\Appendix{Epilogue}

\First{While} matter is eternal and energy is indestructible,
forms change; yet there is a
feature in the changing of forms of matter and
energy that does not change. It is the norm that
determines the nature of all formations, commonly
\index{Nature@\textit{Nature}!Laws of}%
called law or uniformity.

The term ``norm'' is preferable to the usual
word ``law'' because the unchanging uniformities
\index{Uniformities}%
of the domain of natural existence that are formulated
by naturalists into the so-called ``laws of nature,''
\index{Laws of nature}%
have little analogy with ordinances properly
denoted by the term ``law.'' The ``laws of nature''
are not acts of legislation; they are no ukases of a
Czar-God, nor are they any decrees of Fate or of
any other anthropomorphic supremacy that sways
the universe. They are simply the results of a
given situation, the inevitable consequents of some
event that takes place under definite conditions.
They are due to the consistency that prevails in
existence.

There is no compulsion, no tyranny of external
oppression. They obtain by the internal necessity
of causation. What has been done produces its
proper effect, good or evil, intended or not intended,
\PageSep{133}
pursuant to a necessity which is not dynamical
and from without, but logical and from
within, yet, for all that, none the less inevitable.
The basis of every so-called ``law of nature'' is
the norm of formal relations, and if we call it a law
of form, we must bear in mind that the term ``law''
\index{Form}%
is used in the sense of uniformity.

Form (or rather our comprehension of the formal
and of all that it implies) is the condition that
dominates our thinking and constitutes the norm
of all sciences. From the same source we derive
the principle of consistency which underlies our
ideas of sameness, uniformity, rule, etc. This norm
is not a concrete fact of existence but the universal
feature that permeates both the anyness of our
mathematical constructions and the anyness of objective
conditions. Its application produces in the
realm of mind the \Foreign{a~priori}, and in the domain of
facts the uniformities of events which our scientists
reduce to formulas, called laws of nature. On a
superficial inspection it is pure nothingness, but in
fact it is universality, eternality, and omnipresence;
and it is the factor objectively of the world order
and subjectively of science, the latter being man's
capability of reducing the innumerable sense-impressions
of experience to a methodical system of
knowledge.

Faust, seeking the ideal of beauty, is advised to
\index{Faust}%
search for it in the domain of the eternal types of
existence, which is the omnipresent Nowhere, the
ever-enduring Never. Mephistopheles calls it the
\PageSep{134}
Naught. The norm of being, the foundation of
natural law, the principle of thinking, is non-existent
to Mephistopheles, but in that nothing (viz.,
the absence of any concrete materiality, implying a
general anyness) from which we weave the fabric
of the purely formal sciences is the realm in which
Faust finds ``the mothers'' in whom Goethe personifies
the Platonic ideas. When Mephistopheles calls
it ``the nothing,'' Faust replies:
\begin{Quote}
``In deinem Nichts hoff' ich das All zu finden.'' \\
{}['Tis in thy Naught I hope to find the All.]
\end{Quote}

And here we find it proper to notice the analogy
which mathematics bears to religion. In the history
\index{Mathematics!Analogy of, to religion}%
\index{Religion, Analogy of mathematics to}%
of mathematics we have first the rigid presentation
of mathematical truth discovered (as it were)
by instinct, by a prophetic divination, for practical
purposes, in the shape of a dogma as based upon
axioms, which is followed by a period of unrest,
being the search for a philosophical basis, which
finally leads to a higher standpoint from which,
though it acknowledges the relativity of the primitive
dogmatism, consists in a recognition of the
eternal verities on which are based all our thinking,
and being, and yearning.

The ``Naught'' of Mephistopheles may be empty,
but it is the rock of ages, it is the divinity of existence,
and we might well replace ``All'' by ``God,''
thus intensifying the meaning of Faust's reply, and
say:
\begin{Quote}
``\,'Tis in thy naught I hope to find my God.''
\end{Quote}
\PageSep{135}

The norm of Pure Reason, the factor that shapes
the world, the eternal Logos, is omnipresent and
eternal. It is God. The laws of nature have not
been fashioned by a creator, they are part and parcel
of the creator himself.

Plutarch quotes Plato as saying that God is
\index{Plato}%
\index{Plutarch}%
always geometrizing.\footnote
  {Plutarchus \Title{Convivia}, VIII, 2: \Grk{p~ws Pl'atwn >'elege t`on Je`on >ae`i gewmetre~in}.
\index{Ziwet, Professor|indexnote}%
  Having hunted in vain for the famous passage, I am indebted
  for the reference to Professor Ziwet of Ann Arbor, Mich.}
In other words, the purely
formal theorems of mathematics and logic are the
thoughts of God. Our thoughts are fleeting, but
God's thoughts are eternal and omnipresent verities.
They are intrinsically necessary, universal,
immutable, and the standard of truth and right.

Matter is eternal and energy is indestructible,
but there is nothing divine in either matter or energy.
That which constitutes the divinity of the
world is the eternal principle of the laws of existence.
That is the creator of the cosmos, the
norm of truth, and the standard of right and wrong.
If incarnated in living beings, it produces mind,
and it continues to be the source of inspiration for
aspiring mankind, a refuge of the struggling and
storm-tossed sailors on the ocean of life, and the
holy of holies of the religious devotee and worshiper.

The norms of logic and of mathematics are
uncreate and uncreatable, they are irrefragable and
immutable, and no power on earth or in heaven can
change them. We can imagine that the world was
made by a great world builder, but we cannot think
\PageSep{136}
that logic or arithmetic or geometry was ever fashioned
by either man, or ghost, or god. Here is the
\index{God, Conception of}%
rock on which the old-fashioned theology and all
mythological God-conceptions must founder. If
God were a being like man, if he had created the
world as an artificer makes a tool, or a potter shapes
a vessel, we would have to confess that he is a limited
being. He might be infinitely greater and more
powerful than man, but he would, as much as man,
be subject to the same eternal laws, and he would,
as much as human inventors and manufacturers,
have to mind the multiplication tables, the theorems
of mathematics, and the rules of logic.

Happily this conception of the deity may fairly
well be regarded as antiquated. We know now that
God is not a big individual, like his creatures, but
that he is God, creator, law, and ultimate norm of
everything. He is not personal but superpersonal.
The qualities that characterize God are omnipresence,
eternality, intrinsic necessity, etc., and surely
wherever we face eternal verities it is a sign that
we are in the presence of God,---not of a mythological
God, but the God of the cosmic order, the God
of mathematics and of science, the God of the human
soul and its aspirations, the God of will guided by
ideals, the God of ethics and of duty. So long as we
can trace law in nature, as there is a norm of truth
and untruth, and a standard of right and wrong,
we need not turn atheists, even though the traditional
conception of God is not free from crudities
and mythological adornments. It will be by far
\PageSep{137}
preferable to purify our conception of God and replace
the traditional notion which during the unscientific
age of human development served man as
a useful surrogate, by a new conception of God,
that should be higher, and nobler, and better, because
truer.
\PageSep{138}
%[** Blank page]
\PageSep{139}

\PrintIndex
\iffalse
INDEX.


Absolute@Absolute, The#absolute 25

Anschauung@\textit{Anschauung}#Anschauung 82, 97

Anyness|indexff#anyness 46

Anyness 76
  Space founded on 60

Apollonius 31

Aposteriori@\textit{A posteriori}#posteriori 43, 60

Apriori@\textit{A priori}#priori 38, 64
  and the purely formal|indexff#purely 40
  Apparent arbitrariness of the|indexff#arbitrariness 96
  constructions 112
  constructions. Geometries are#geometries 127
  constructions verified by experience#experience 122
  Geometry is|indexnote#geometry 119
  is ideal#ideal 44
  Source of the#source 51
  The logical 54
  The purely 55
  The rigidly 54, 55

Apriority of different degrees|indexff#apriority 49

Apriority of different degrees
  of mathematical space#mathematical 121, 129
  of space-measurement|indexff#space-measurement 109
  Problem of#apriority 36

Archimedes 31

As if 79

Astral geometry 15

Atomic fiction 81

Ausdehnungslehre@\textit{Ausdehnungslehre}, Grassmann's#Ausdehnungslehre 28, 30

Ausdehnungslehre@\textit{Ausdehnungslehre}, Grassmann's|indexnote#Ausdehnungslehre 29

Axiom@``Axiom,''
  Euclid avoided 1, 127
  Hilbert's use of 128

Axioms|indexff 1

Axioms
  not Common Notions 4

Ball, Sir Robert, on the nature of space|indexf#Ball 123

Bernoulli 9

Bessel, Letter of Gauss to|indexff#Bessel 12

Billingsley, Sir H.#Billingsley 82

Bolyai@Bolyai, János|indexff#Bolyai 22

Bolyai@Bolyai, János#Bolyai 98
  translated 27

Boundary concepts, Utility of 74

Boundaries 78, 129
  produced by halving, Even 85, 86

Bridges of Königsberg|indexf#Königsberg 102

Busch, Wilhelm 115

Carus, Paul
  Fundamental Problems@\textit{Fundamental Problems}|indexnote#Carus 39
  Kant's Prolegomena@\textit{Kant's Prolegomena}|indexnote#Carus 39, 122
  Primer of Philosophy@\textit{Primer of Philosophy}|indexnote 40

Causation
  apriori@\textit{a priori}#priori 53
  transformation@{and transformation}#transformation 54
  Kant on#Kant 40

Cayley 25

Chessboard, Problem of#chess 101

Circle
  Squaring of the 104
  the simplest curve 75

Classification 79

Clifford 16, 32, 60
  Plane constructed by 69

Common notions 2, 4, 128

Comte 38

Concreteness, Purely formal, absence of#concreteness 60

Continuum@Continuum|indexff 78

Curved space 106
  Helmholtz on 113

Definitions of Euclid 1, 128

Delboeuf@Delb{\oe}uf, B. J.#oe 27

De Morgan, Augustus 10

Determinism in mathematics 104

Dimension, Definition of#dimension 85

Dimensions, Space of four|indexff#four 90

Directions of space, Infinite#directions 117

Discrete units|indexff#discrete 78

Dual number 89

Edward's Dream@\textit{Edward's Dream}#Edward 115

Egg-shaped body|indexf#egg 113

Elliptic geometry 25

Empiricism, Transcendentalism and|indexff#empiricism 38

Engel, Friedrich 26

Euclid|indexff#Euclid 31

Euclid 1-4
  avoided "axiom," 1, 127
  Expositions of, rearranged 128
  Halsted on|indexf#Halstead 31

Euclidean geometry, classical 31, 121

Even boundaries 122
  as standards of measurement|indexff#measurement 69
  as standards of measurement#measurement 85, 86
  produced by halving#halving 85, 86

Experience, Physiological space originates through#experience 65

Faust 133

Fictitious spaces|indexff#fictitious 109
%\PageSep{140}

Flatland@\textit{Flatland}#Flatland 115

Form 60, 133
  and reason 48

Four dimensional@Four-dimensional space and tridimensional beings#tridimensional 93

Four dimensions 109
  Space of|indexff#space 90

Fourth dimension 25
  illustrated by mirrors|indexff#mirrors 93

Gauss|indexff#Gauss 11

Gauss
  his letter to Bessel|indexff#Bessel 12
  his letter to Taurinus|indexf#Taurinus 6, 13

Geometrical construction, Definiteness of|indexff#construction 99

Geometry
 \textit{a priori}|indexff#geometry 119
  Astral 15
  Elliptic 25
  Question in 72, 73, 121

Geometries, \textit{a priori} constructions#constructions 127

God, Conception of#God 136

Grassmann|indexff 27

Grassmann 127

Grassmann|indexnote 130

Halsted, George Bruce|indexnote#Halstead 4, 20, 28

Halsted, George Bruce#Halstead 23, 26, 27, 101
  on Euclid|indexf#Euclid 31

Helmholtz 26, 83
  on curved space#curved 113
  on two-dimensional beings|indexf#two-dimensional 110

Hilbert's use of ``axiom''#Hilbert 128

Homaloidal 18, 74

Homogeneity of space|indexff#homogeneity 66

Hypatia 31

Ideal@``Ideal'' and ``subjective,'' Kant's identification of|indexf#Kant 44
  not synonyms#synonyms 64

Infinite directions of space 117
  division of line#division 117
  not mysterious#mysterious 118
  Space is#space 116, 126
  Time is#time 116

Infinitude|indexff#infinitude 116

Kant 35, 40, 61, 84
  and the \textit{a priori}#priori 36, 38
  his identification of ``ideal'' and ``subjective,''|indexf#subjective 44
  his term \textit{Anschauung}#Anschauung 32, 97
  his use of ``transcendental,''#transcendental 41

Kant@\textit{Kant's Prolegomena}|indexnote#Prolegomena 39

Keyser, Cassius Jackson#Keyser 77

Kinematoscope 80

Klein, Felix#Klein 25

Konigsberg@Königsberg, Seven bridges of|indexf#Königsberg 102

Lagrange|indexf#Lagrange 10

Lambert, Johann Heinrich|indexf#Lambert 9

Laws of nature 132

Legendre 11

Line created by construction#line 83
  independent of position#position 62
  Infinite division of#division 117
  Shortest 84
  Straightest 75, 127

Littre@Littré#Littré 38

Lobatchevsky|indexff#Lobatchevsky 20

Lobatchevsky 10, 75, 98
  translated 27

Lobatchevsky's \textit{Theory of Parallels} 101

Logic is static#logic 53

Mach, Ernst 27, 65

Mathematical space|indexff#space 63

Mathematical space 67, 109
  a priori@\textit{a priori}#priori 65
  Apriority of#apriority 121, 129

Mathematics
  Analogy of, to religion#religion 134
  Determinism in#determinism 104
  Reality of#reality 77
  Teaching of|indexff#teaching 127

Measurement
  Even@Even boundaries as standards of|indexff#boundaries 69
  Even@Even boundaries as standards of#boundaries 85, 86
  of star parallaxes#parallax 125
  Standards for#standards 74

Mental activity, First rule of#activity 79

Metageometry|indexff 5

Metageometry
  History of#history 26
  Mathematics and|indexff#mathematics 82

Mill, John Stuart#Mill 38

Mind
  develops through uniformities#uniform 52
  Origin of#origin 51

Mirrors, Fourth dimension illustrated by|indexff#mirror 93

Monist@\textit{Monist}|indexnote#Monist 4

Monist@\textit{Monist}#Monist 27, 125

Names, Same, for parts of figures#names 130

Nasir Eddin 7

Nature@\textit{Nature}#Nature 16
  a continuum#continuum 78
  Laws of#law 132

Newcomb, Simon#Newcomb 25

Open Court@\textit{Open Court}|indexnote 20, 115, 130

Order in life and arithmetic#order 80

Pangeometry 22

Pappus 31

Parallel lines in spherical space#parallel 84
  theorem|indexnote 4
  theorem|indexf 25
  theorem 98, 129

Parallels, Axiom of#parallels 3

Path of highest intensity a straight line#path 58

Peirce, Charles S., on the nature of space|indexf#Peirce 123

Physiological space|indexff#physiological 63

Physiological space#physiological 67
  originates through experience#experience 65

Plane@Plane, a zero of curvature#plane 82
  constructed by Clifford#Clifford 69
  created by construction#construction 83
  Nature of#nature 73
  Significance of#significance 129

Plato 135

Plutarch 135

Poincare@Poincaré, H.#Poincar 27

Point congruent with itself#congruent 71
%\PageSep{141}

Population of Great Britain determined|indexf#population 119

Position 62

Postulates 2, 128

Potentiality 63

Proclus 4, 31

Pseudo-spheres 83

Pure form 63
  space, Uniqueness of|indexff#uniqueness 61

Purely \textit{a priori}, The 55
  formal, absence of concreteness 60

Question in geometry 72, 73, 121

Ray a final boundary 58

Reason, Form and#reason 48
  Nature of#nature 76

Rectangular pentagon 98

Religion, Analogy of mathematics to#religion 134

Riemann|indexff 15, 106

Riemann 37, 62, 96, 104, 123

Right angle
  created by construction#construction 83
  Nature of#nature 73
  Significance of#significance 129

Russell, Bertrand A. W.#Russel 26
  on non-Euclidean geometry 119

Saccheri, Girolamo|indexf#Saccheri 8

Schlegel, Victor#Schlegel 30

Schoute, P. H.|indexnote#Schoute 30

Schumaker, Letter of Gauss to#Schumaker 11

Schweikart 15

Sense-experience and space|indexff#sense-experience 122

Shortest line 84

Space
  a manifold 108
  a spread of motion|indexff#spread 56
  Apriority of mathematical#apriority 121, 129
  curved 126
  founded on ``anyness,''#anyness 60
  Helmholtz on curved#Helmholtz 113
  Homogeneity of|indexff#homogeneity 66
  homaloidal 74
  Infinite directions of 117
  Interference of|indexff#interference 101
  is infinite#infinite 116, 126
  Mathematical|indexff#mathematical 63
  Mathematical 67, 109
  Mathematical and actual#mathematical 62
  of four dimensions|indexff#four 90
  On the nature of|indexf#nature 123
  Physiological|indexff#Physiological 63
  Physiological 67
  Sense-experience and|indexff#sense-experience 122
  the juxtaposition of things#juxtaposition 67, 87
  the possibility of motion#motion 59
  the potentiality of measuring#measure 61
  Uniqueness of pure|indexff#uniqueness 61

Space-conception
  how far \textit{a priori?}|indexf 59
  product of pure activity 55

Space-measurement
  Apriority of|indexff#apriority 109
  Various systems of|indexff#systems 104

Spaces, Fictitious|indexff 109

Squaring of the circle 104

Stackel, Paul#Stackel 26

Standards of measurement 74
  Even@Even boundaries as|indexff 69
  Even@Even boundaries as 85, 86

Star parallaxes, Measurements of 125

Straight line 69, 71, 112, 122
  a path of highest intensity#intensity 59
  created by construction#construction 83
  does not exist#exist 72
  indispensable|indexff#indispensable 72
  Nature of#nature 73
  One kind of#kind 75
  Significance of#significance 129
  possible 74

Straightest line 75, 127

Subjective and ideal
  Kant's identification of|indexf#Kant 44
  not synonyms#synonym 64

Superreal, The|indexff 76

Taurinus, Letter of Gauss to|indexf#Taurinus 6, 13

Teaching of mathematics|indexff 127

Tentamen@\textit{Tentamen}#Tentamen 23

Theon 31, 32

Theory of Parallels@\textit{Theory of Parallels}, Lobatchevsky's#parallels 101

Thought-forms, systems of reference#thought-forms 61

Three, The number|indexf#three 88

Time is infinite#time 116

Transcendental@`Transcendental,'' Kant's use of#transcendental 41

Transcendentalism@Transcendentalism and Empiricism#Transcendentalism 35

Transcendentalism@Transcendentalism and Empiricism|indexff#Transcendentalism 38

Transformation, Causation and#causation 54

Tridimensional beings
  Four-dimensional space and#four 93
  space, Two-dimensional beings and#two 91

Tridimensionality|indexff 84

Trinity, Doctrine of the#Trinity 89

Two-dimensional beings and tridimensional space 91

Uniformities 132
  Mind develops through 52

Units
  Discrete|indexff 78
  Positing of 80

Wallis, John|indexf#Wallis 7

Why@Why? 100

Zamberti 32

Ziwet, Professor|indexnote 135
%% End of index text
\fi
\PageSep{142}
\iffalse
%% [** TN: Raw OCR of catalog text follows]

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