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«WISB^I

NON-EUCLIDEAN
GEOMETRY

A CRITICAL AND
HISTORICAL STUDY OF ITS DEVELOPMENT

BY
ROBERTO BONOLA

Professor in the University of Pavia

AUTHORISED ENGLISH TRANSLATION WITH
ADDITIONAL APPENDICES

H. S. CARSLAW

Professor in the University of Sydney, N. S.W.
WITH AN INTRODUCTION

FEDERIGO ENRIQUES

Professor in the University of Bologna
-«m

CHICAGO

THE OPEN COURT PUBLISHING COMPANY

1912

THE OPEN COURT PUBLISHING COMPANY

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1912

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Printed by W. Drugulin, Leipzig, (Germany)

Introduction.

The translator of this little volume has done me -the
honour to ask me to write a few lines of introduction. And
I do this willingly, not only that I may render homage to the
memory of a friend^ prematurely torn from life and from
science, but also because I am convinced that the work of
Roberto Bonola deserves all the interest of the studious.
In it, in fact, the young mathematician will find not only
a clear exposition of the principles of a theory now classical,
but also a critical account of the developments which
led to the foundation of the theory in question.

It seems to me that this account, although concerned
with a particular field only, might well serve as a model
for a history of science, in respect of its accuracy and
its breadth of information, and, above all, the sound philo-
sophic spirit that permeates it. The various attempts of
successive writers are all duly rated according to their
relative importance, and are presented in such a way
as to bring out the continuity of the progress of science,
and the mode in which the human mind is led through
the tangle of partial error to a broader and broader view
of truth. This progress does not consist only in the ac-
quisition of fresh knowledge, the prominent place is taken
by the clearing up of ideas which it has involved; and it
is remarkable with what skill the author of this treatise has
elucidated the obscure concepts which have at particular
periods of time presented themselves to the eyes of the
investigator as obstacles, or causes of confusion. I will
cite as an example his lucid analysis of the idea of there

[V Introduction.

being in the case of Non-Euclidean Geometry, in contrast
to Euclidean Geometry, an absolute or natural measure of
geometrical magnitude.

The admirable simplicity of the author's treatment,
the elementary character of the constructions he employs,
the sense of harmony which dominates every part of this
little work, are in accordance, not only with the artistic
temperament and broad education of the author, but also
with the lasting devotion which he bestowed on the Theory
of Non-Euclidean Geometry from the very beginning of
his scientific career. May his devotion stimulate others to
pursue with ideals equally lofty the path of historical and
philosophical criticism of the principles of science! Such
efforts may be regarded as the most fitting introduction
to the study of the high problems of philosophy in general,
and subsequently of the theory of the understanding, in
the most genuine and profound signification of the term,
following the great tradition which was interrupted by the
romantic movement of the nineteenth century.

Bologna, October ist^ 191 1.

Federigo Enriques.

Translator's Preface.

Bonola's Non-Euclidean Geometry is an elementary
historical and critical study of the development of that subject.
Based upon his article in Enriques' collection of Monographs
on Questions of Elementary Geometry^, in its final form it still
retains its elementary character, and only in the last chapter
is a knowledge of more advanced mathematics required.

Recent changes in the teaching of Elementary Geometry
in England and America have made it more then ever ne-
cessary that those who are engaged in the training of the
teachers should be able to tell them something of the
growth of that science; of the hypothesis on which it
is built; more especially of that hypotheses on which rests
Euclid's theory of parallels; of the long discussion to which
that theory was subjected; and of the final discovery of the
logical possibility of the different Non-Euclidean Geometries.

These questions, and others associated with them, are
treated in an elementary way in the pages of this book.

In the English translation, which Professor Bonola
kindly permitted me to undertake, I have introduced some
changes made in the German translation.^ For permission
to do so I desire to express my sincere thanks to the firm of
B. G. Teubner and to Professor Liebmann. Considerable
new material has also been placed in my hands by Professor
Bonola, including a slightly altered discussion of part of

1 Enriques, F., Questioni riguardan/i la geometria elementare,
(Bologna, Zanichelli, 1900).

2 Wissenschaft und Hypothese, IV. Band : Die nichteuklidische
Geometrie. Historisch-kritische Darstellung ihrer Entwicklung. Von
R, Bonola, Deutsch v. H. Liebmann. (Teubner, Leipzig, 190SJ.

VI Translator's Preface.

Saccheri's work, an Appendix on the Independence of Pro-
jective Geometry from the Parallel Postulate, and some further
Non-Euclidean Parallel Constructions.

In dealing with Gauss's contribution to Non-Euclidean
Geometry I have made some changes in the original on the
authority of the most recent discoveries among Gauss's
papers. A reference to Thibaut's 'proof, and some addit-
ional footnotes have been inserted. Those for which I am
responsible have been placed within square brackets. I have
also added another Appendix, containing an elementary
proof of the impossibility of proving the Parallel Postulate,
based upon the properties of a system of circles orthogonal
to a fixed circle. This method offers fewer difficulties than
the others, and the discussion also establishes some of the
striking theorems of the hyperbolic Geometry.

It only remains for me to thank Professor Gibson of
Glasgow for some valuable suggestions, to acknowledge the
interest, which both the author and Professor Liebmann have
taken in the progress of the translation, and to express my
satisfaction that it finds a place in the same collection as
Hilbert's classical Grundlagen der Geometrie.

P. S. As the book is passing through the press I have
received the sad news of the death of Professor Bonola.
With him the Italian School of Mathematics has lost one Of
its most devoted workers on the Principles of Geometry.
Professor Enriques, his intimate friend, from whom I heard
of Bonola's death, has kindly consented to write a short
introduction to the present volume. I have to thank him,
and also Professor W. H. Young, in whose hands, to avoid
delay, I am leaving the matter of the translation of this
introduction and its passage through the press.

The University, Sydney, August 1 9 1 1 .

H. S. Carslaw.

Author's Preface.

The material now available on the origin and develop-
ment of Non-Euclidean Geometry, and the interest felt in
the critical and historical exposition of the principles of the
various sciences, have led me to expand the first part of my
article — Sulla teoria delle parallele e sulle geometrie iioji-
euclidee — which appeared sÌ5i years ago in the Questioni ri-
guardanti la geometria elemefiiare, collected and arranged
by Professor F. Enriques.

That article, which has been completely rewritten for the
German translation* of the work, was chiefly concerned with
the systematic part of the subject. This book is devoted, on
the other hand, to a fuller treatment of the history of parallels,
and to the historical development of the geometries of Lo-
fi atschewky-Boly ai and RiEMANN.

In Chapter I., which goes back to the work of Euclid
and the earliest commentators on the Fifth Postulate, I have
given the most important arguments, by means of which
the Greeks, the Arabs and the geometers of the Renaissance
attempted to place the theory of parallels on a firmer
foundation. In Chapter II., relying chiefly upon the work of
Saccheri, Lambert and Legendre, I have tried to throw
some light on the transition from the old to the new
ideas, which became prevalent in the beginning of the 19th
Century. In Chapters III. and IV., by the aid of the in-

I Enriques, F., Fragen der Elementargeometrie. I. Teil: Die
Grundlagen der Geometrie. Deutsch von H. Thieme. {1910.)
II. Teil: Die geometrischen Aufgaben, ihre Losung und Losbarkeit.
Deutsch von H. Fleischer. (1907.) Teubner, Leipzig.

YUj Author's Preface.

vestigations of GausS; Schweikart, TaurinuS; and the con-
structive work of Lobatschewsky and Bolyai, I have ex-
plained the principles of the first of the geometrical systems,
founded upon the denial of Euclid's Fifth Hypothesis. In
Chapter V., I have described synthetically the further deve-
lopment of Non-Euclidean Geometry, due to the work of
RiEMANN and Helmholtz on the structure of space, and
to Cayley's projective interpretation of the metrical proper-
ties of geometry.

In the whole of the book I have endeavoured to pre-
sent, the various arguments in their historical order. How-
ever when such an order would have made it impossible to
treat the subject simply, I have not hesitated to sacrifice it,
so that I might preserve the strictly elementary character of
the book.

Among the numerous postulates equivalent to Euclid's
Fifth Postulate, the most remarkable of which are brought
together at the end of Chapter IV., there is one of a statical
nature, whose experimental verification would furnish an
empirical foundation of the theory of parallels. In this we
have an important link between Geometry and Statics
(Genocchi); and as it was impossible to find a suitable place
for it in the preceding Chapters, the first of the two Notes'^
in the Appendix is devoted to it.

The second Note refers to a theory no less interesting.
The investigations of Gauss, Lobatschewsky and Bolyai on
the theory of parallels depend upon an extension of one of
the fundamental conceptions of classical geometry. But a
conception can generally be extended in various directions.
In this case, the ordinary idea of parallelism, founded on
the hypothesis of non-intersecting straight Unes, coplanar and

I In the English translation these Notes are called Appendix I.
and Appendix II.

Author's Preface. IX

equidistant, was extended by the above-mentioned geometers,
who gave up Euclid's Fifth Postulate (equidistance), and
later, by Clifford, who abandoned the hypothesis that the
lines should be m the same plane.

No elementary treatment of Clifford's parallels is avail-
able, as they have been studied first by the projective
method (Clifford-Klein) and later, by the aid of Different-
ial-Geometry (BiANCHi-FuBiNi). For this reason the second
Note is chiefly devoted to the exposition of their simplest
and neatest properties in an elementary and synthetical
manner. This Note concludes with a rapid sketch of Clif-
ford-Klein's problem, which is allied historically to the
parallelism of Clifford. In this problem an attempt is made
to characterize the geometrical structure of space, by assum-
ing as a foundation the smallest possible number of postul-
ates, consistent with the experimental data, and with the
principle of the homogeneity of space.

This is, briefly, the nature of the book. Before sub-
mitting the little work to the favourable judgment of its
readers, I wish most heartily to thank my respected teacher,
Professor Federigo Enriques, for the valuable advice with
which he has assisted me in the disposition of the material
and in the critical part of the work; Professor Corrado Segre,
for kindly placing at my disposal the manuscript of a course
of lectures on Non-Euclidean geometry, given by him, three
years ago, in the University of Turin; and my friend. Professor
Giovanni Vailati, for the valuable references which he has
given me on Greek geometry, and for his help in the cor-
rection of the proofs.

Finally my grateful thanks are due to my publisher
Cesare Zanichelli, who has so readily placed my book in
his collection of scientific works.

Pavia, March, 1906.

Roberto Bonola.

Table of Contents.

Chapter I. pages

The Attempts to prove Euclid's Parallel Postulate.

S I — 5. The Greek Geometers and the Parallel

Postulate I — 9

S 6. The Arabs and the Parallel Postulate .. .. 9 — 12
S 7 — 10. The Parallel Postulate during the Renais-
sance and the ly^^ Century 12 — 21

Chapter II.

The Forerunners of Non-Euclidean Geometry.

S II — 17. Gerolamo Saccheri (1667 — 1733) .. .. 22—44

S 18 — 22. Johann Heinrich Lambert (172S — 1777) 44—51
S 23 — 26. The French Geometers towards the End

of the l8th Century 51 — 55

S 27 — 28. Adrien Marie Legendre (1752 — 1833) .. 55— 60

S 29. Wolfgang Bolyai (1775 — 1856) 60—62

§ 30. Friedrich Ludwig Wachter (1792 — 1817) .. 62 — 63

§ 30 (bis) Bernhard Friedrich Thib/vut (1776—1832) 63

Chapter III.

The Founders of Non-Euclidean Geometry.

S 31—34- Karl Friedrich Gauss (1777— 1855) .. 64 — 75

S 35. Ferdinand Karl Schweikart (1780—1859) .. 75—77

S 36 — 38. Franz Adolf Taurinus (1794—1874) .. 77 — 83

Chapter IV.

The Founders of Non-Euclidean Geometry (Cont).
S 39—45- Nicolai Ivanovitsch Lobatschewsky

(1793—1856) , 84—96

S 46 — 55. Johann Bolyai (1S02 — 1860) 96—113

S 56—58. The Absolute Trigonometry I13— 118

§ 59. Hypotheses equivalent to Euclid's Postulate .. 118—121

§ 60 — 65. The Spread of Non-Euclidean Geometry 121 — 128

Chapter V.

The Later Development of Non-Euclidean Geometry.

S 66. Introduction 129

Table of Contents. XI

pages
Differential Geometry and Non-Euclidean Geometry.

S 67—69. Geometry upon a Surface .. 130 — 139

§ 70 — 76. Principles of Plane Geometry on the Ideas

of RiEMANN 139 — 150

§ 77. Principles of Riemann's Solid Geometry.. .. 151 — 152

g 78. The Work of Helmholtz and the Investigations

of Lie 152 — 154

Projective Geometry and Non-Euclidean Geometry.

S 79 "83. Subordination of Metrical Geometry to

Projective Geometry 154 — 164

S 84 — 91. Representation of the Geometry of Lobat-

SCHEWSKV-BOLYAI On the Euclidean Plane .. .. 164—175

S 92. Representation of Riemann's Elliptic Geometry

in Euclidean Space 175 — 176

S 93. Foundation of Geometry upon Descriptive Pro-
perties , 176 — 177

S 94. The Impossibility of proving Euclid's Postulate 177 — ^^o

Appendix I.

The Fundamental Principles of Statics and Euclid's

Postulate.

S I — 3. On the Principle of the Lever 181 — 184

S 4 — 8. On the Composition of Forces acting at

a Point 184 — 192

S 9 — 10. Non-Euclidean Statics 192 — 195

S II — 12. Deduction of Plane Trigonometry from

Statics 195 — 199

Appendix II.

Clifford's Parallels and Surface. Sketch of Clifford-
Klein's Problem.

S 1—4. Clifford's Parallels 2co— 206

S 5—8. Clifford's Surface 206—211

S 9 — 11. Sketch of Clifford-Klein's Problem .. 211 — 215
Appendix III.

The Non-Euclidean Parallel Construction and other
Allied Constructions.
S 1 — 3. The Non-Euclidean Parallel Construction .. 216—222
§ 4. Construction of the Common Perpendicular to

two non-intersecting Straight Lines 222-— 223

S 5. Construction of the Common Parallel to the

Straight Lines which bound an Angle 223—224

XII Table of Contents.

pages
S 6. Construction of the Straight Line which is per-
pendicular to one of the lines bounding an acute

Angle and Parallel to the other 224

S 7- The Absolute and the Parallel Construction .. 224 — 226
Appendix IV.

The Independence of Projective Geometry from Euclid's
Postulate.

S I. Statement of the Problem 227 — 228

§ 2. Improper Points and the Complete Projective

Plane 228—229

§ 3. The Complete Projective Line 229

S 4. Combination of Elements 229 — 231

§ 5. Improper Lines 231 — 233

S 6. Complete Projective Space 233

S 7. Indirect Proof of the Independence of Pro-
jective Geometry from the Fifth Postulate .. 233 — 234
S 8. Beltrami's Direct Proof of this Independence 234—236
S 9. Klein's Direct Proof of this Independence .. 236— -237
Appendix V.

The Impossibility of proving Euclid's Postulate.
An Elementary Demonstration of this Impossibility
founded upon the Properties of the System of
Circles orthogonal to a Fixed Circle.

§ I. Introduction 238

S 2 — 7. The System of Circles passing through a

Fixed Point 239 250

S 8 — 12. The System of Circles orthogonal to a

Fixed Circle - 250—264

Index of Authors .. 265

Chapter I.

The Attempts to prove Euclid's Parallel
Postulate

The Greek Geometers and the Parallel Postulate,

§ I. Euclid (circa 330 — 275, B. C.) calls two straight
lines parallel, when they are in the same plane and being
produced indefinitely in both directions, do not meet one
another in either direction (Def. XXIII.).^ He proves that
two straight lines are parallel, when they form with one of
their transversals equal interior alternate angles, or equal
corresponding angles^ or interior angles on the same side
which are supplementary. To prove the converse of these
propositions he makes use of the following Postulate (V.) :

If a straight Ime falling on t7V0 straight lines make the
ifiterior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, meet on that
side on which are the angles less than the two right angles.

The Euclidean Theory of Parallels is then completed
by the following theorems:

Straight lines which are parallel to the same straight
line are parallel to each other (Bk. I., Prop. 30).

I With regard to Euclid's text, references are made to the
critical edition of J. L. Heiberg (Leipzig, Teubner, 1883). [The
wording of this definition (XXIIF, and of Postulate V below, are
taken from Heath's translation of Heiberg's text. (Camb. Univ. Press,

1908Ì.]

2 I. The Attempts to prove Euclid's Parallel Postulate.

Through a given point one and only one straight line
can be drawn which will be parallel to a given straight line
(Bk. I. Prop. 31).

The straight lines joining the extremities of two equal
and parallel straight lines are equal and parallel (Bk. I.
Prop. 33).

From the last theorem it can be shown that two parallel
straight lines are equidistant from each other. Among the
most noteworthy consequences of the Euclidean theory are
the well-known theorem on the sum of the angles of a tri-
angle, and the properties of similar figures.

§ 2. Even the earliest commentators on Euclid's text
held that Postulate V. was not sufficiently evident to be
accepted without proof, and they attempted to deduce it as
a consequence of other propositions. To carry out their pur-
pose, they frequently substituted other definitions of parallels
for the Euclidean definition, given verbally in a negative
form. These alternative definitions do not appear in this
form, which was believed to be a defect.

Proclus (410 — 485) — in his Commefitary on the First
Book of Euclid^ — hands down to us valuable informa-
tion upon the first attempts made in this direction. He states,
for example, that Posidonius (i^' Century, B. C.) had pro-
posed to call two equidistant and coplanar straight lines par-
allels. However, this definition and the Euclidean one
correspond to two facts, which can appear separately, and

» Wher the text of Proclus is quoted, we refer to the edi-
tion of G. FriEDLEIN: Frodi Diadochi in primum Eudidis element-
orum librian commeyitarii, [Leipzig, Teubner, 1873). [Compare also
W. B. Frankland, The First Book of Eudid''s Elements with a
Commentary based prindpally upon that of Produs Diadochus, (Camb.
Univ. Press, 1905). Also Heath's Euclid, Vol. I., Introduction,
Chapter IV., to which most important work reference has been
made on p. l].

The Commentary of Proclus. •»

Proclus (p. 177), referring to a work by Geminus (1^' Cen-
tury, B. C), brings forward in this connection the examples
of the hyperbola and the conchoid, and their behaviour with
respect to their asymptotes, to show that there might be
parallel lines in the Euclidean sense, (that is, lines which
produced indefinitely do not meet), which would not be
parallel in the sense of Posidonius, (that is, equidistant).

Such a fact is regarded by Geminus, quoting still from
Proclus, as the most paradoxical [TrapaòoHÓTaTOv] in the
whole of Geometry.

Before we can bring Euclid's definition into line
with that of Posidonius, it is necessary to prove that if two
coplanar straight lines do not meet, they are equidistant; or,
that the locus of points, which are equidistant from a straight
line, is a straight line. And for the proof of this proposition
Euclid requires his Parallel Postulate.

However Proclus (p. 364) refuses to count it among
the postulates. In justification of his opinion he remarks
that its converse {The sum of hvo angles of a triangle is less
than two right angles), is one of the theorems proved by
Euclid. (Bk. I. Prop. 17);
and he thinks it impossible
that a theorem whose con-
verse can be proved, is not
itself capable of proof. Also
he utters a warning against
mistaken appeals to self-
evidence, and insists upon
the (hypothetical) possibi-
lity of straight lines which
are asymptotic (p. 191 — 2).

Ptolemy (2°^ Century, A. D.) — we quote again from
Proclus (p. 362 — 5)— attempted to settle the question by
means of the following curious piece of reasoning.

A I. The Attempts to prove Euclid's Parallel Postulate.

Let AB, CD, be two parallel straight lines and FG a
transversal (Fig. i).

Let a, P be the two interior angles to the left of FG,
and a, P' the two interior angles to the right.

Then a + P will be either greater than, equal to, or less
than a + p'.

It is assumed that if any one of these cases holds for
one pair of parallels (e. g. a + P ^ 2 right angles) this case
will also hold for every other pair.

Now FB, GD, are parallels; as are also FA and GC.

Since a + P ^ 2 right angles,

it follows that a' + P' ^ 2 right angles.

Thus a+P + a'+P'>>4 right angles,
which is obviously absurd.
Hence a + P cannot be greater than 2 right angles.

In the same way it can be shown that

a + P cannot be less than 2 right angles.

Therefore we must have

a + p = 2 right angles (Proclus, p. 365).
From this result Euclid's Postulate can be easily obtained.

§ 3. Proclus (p. 371), after a criticism of Ptolemy's
reasoning, attempts to reach the same goal by another path.
His demonstration rests upon the following proposition,
which he assumes as evident: — The distance between two
points upon two intersecting straight lines can be made as great
as 7V e please, by prolonging the two lines sufficiently}

From this he deduces the lemma : A straight line which
meets one of two parallels must also meet the other.

I For the truth of this proposition, which he assumes as self-
evident, Proclus relies upon the authority of Aristotle. Cf.
De Coelo I., 5. A rigorous demonstration of this very theorem
was given by Saccheri in the work quoted on p. 22.

Proclus (continued). C

His proof of this lemma is as follows:
Let AB, CD, be two parallels and £G a transversal,
cutting the former in ^ (Fig. 2).

— D

Fig. 2.

The distance of a variable point on the ray J^G from
the line AB increases without limit, when the distance of that
point from jF is increased indefinitely. But since the distance
between the two parallels is finite, the straight line EG must
necessarily meet CD.

Proclus, however, introduced the hypothesis that the
distance between two parallels remains finite; and from this
hypothesis Euclid's Parallel Postulate can be logically de-
duced.

§ 4. Further evidence of the discussion and research
among the Greeks regarding Euclid's Postulate is given by
the following paradoxical argument. Relying upon it, accord-
ing to Proclus, some held that it had been shown that two
straight lines, which are cut by a third, do not meet one
another, even when the sum of the interior angles on the
same side is less than two right angles.

Let -(4C be a transversal of the two straight lines AB,
CD and let E be the middle point of AC (Fig. 3).

On the side of ^C on which the sum of the two internal
angles is less than two right angles, take the segments AF
and CG upon AB and CD each equal to AE. The two
lines AB and CD cannot meet between the points AF and
CG, since in any triangle each side is less than the sum of
the other t\vo.

~ K

E
C

(

Ì L

6 I. The Attempts to prove Euclid's Parallel Postulate.

The points F and G are then joined, and the same
process is repeated, starting from the hne FG. The segments
FK and GL are now taken on AB and CD, each equal to
half of FG. The two lines AB, CD are not able to meet
between the points 7% K and G, L.

Since this operation
can be repeated indefini-
tely, it is inferred that the
two lines AB, CD will never
meet.

The fallacy in this ar-
gument is contained in the
use of infinity, since the
segments AF, FK could
tend to zero, while their
sum might remain finite. The author of this paradox has
made use of the principle by means of which Zeno (495 —
435 B. C.) maintained that it could be proved that Achilles
would never overtake the tortoise, though he were to travel
with double its velocity.

This is pointed out, under another form, by Proclus
(p. 369 — 70), where he says that this argument proves that
the point of intersection of the lines could not be reached
(to determine, ópiZieiv) by this process. It does not prove
that such a point does not exist.*

Proclus remarks further that 'since the sum of two
angles of a triangle is less than two right angles (Euclid Bk. I.
Prop. 17), there exist some lines, intersected by a third,
which meet on that side on which the sum of the interior

Fig- 3-

I [Suppose we start with a triangle ABC and bisect the base
BC in D. Then on BA take the segment BE equal to BD, and
on CA the segment CB' equal to CD, and join EF. Then repeat
this process indefinitely. The vertex A can never be reached by
this means, although it is at a finite distance.]

Proclus (continued). j

angles is less than two right angles. Thus if it is asserted
that for every difference between this sum and two right
angles the lines do not meet, it can be replied that for
greater differences the lines intersect.'

'But if there exists a point of section, for certam pairs
of lines, forming with a third interior angles on the same
side whose sum is less than two right angles, it remains to be
shown that this is the case for all the pairs of lines. Since
it might be urged that there could be a certain deficiency (from
two right angles) for which they (the lines) would not inter-
sect, while on the other hand all the other lines, for which the
deficiency was greater, would intersect.' (Proclus, p. 371.)

From the sequel it will appear that the question, which
Proclus here suggests, can be answered in the affirmative
only in the case when the segment AC of the transversal
remains unaltered, while the lines rotate about the points A
and C and cause the difference from two right angles to vary.

§ 5. Another very old proof of the Fifth Postulate,
reproduced in the Arabian Commentary of Al-Nirizi' (ptt
Century), has come down to us through the Latin translation
of Gherardo da Cremona* (12th Century), and is attributed
to Aganis.3

The part of this commentary relating to the definitions,
postulates and axioms, contains frequent references to the

1 Cf. R. O. Besthorn u. J. L. Heiberg, 'Codex Leidensis,'
399, I. Euclidis Elementa ex interpretatione Al-Hadsckdschadsch cum
commentariis Al-N^ariziif (Copenhagen, F. Hegel, 1893 — 97)-

2 Cf. M. CuRTZE, *Anariin in decern libros priores eleinentoriim
Euclidis Commentarii.' Ex interpretatione Gherardi Cremonensis in
Codice Cracoviensi 569 servata, (Leipzig, Teubner, 1899).

3 With regard to Aganis it is right to mention that he is
identified by Curtze and Heiberg with Geminus. On the other
hand P. Tannery does not accept this identification. Cf. Tannery,
*Z<f phibsophe Aganis est-il identique à Geminus?' Bibliotheca Math.
(3) Bd. II. p. 9— II [1901],

3 I. The Attempts to prove Euclid's Parallel Postulate.

the name of Sambelichius, easily identified with Simplicius,
the celebrated commentator on Aristotle, who lived in the
6^^ Century. It would thus appear that Simplicius had written
an Introduction to the First Book of Euclid, in which he ex-
pressed ideas similar to those of Geminus and Posidonius,
affirming that the Fifth Postulate is not self evident, and
bringing forward the demonstration of his friend Aganis.

This demonstration is founded upon the hypothesis that
equidistant straight lines exist. Aganis calls these parallels,
as had already been done by Posidonius. From this hypo-
thesis he deduces that the shortest distance between two
parallels is the common perpendicular to both the lines:
that two straight lines perpendicular to a third are parallel
to each other: that two parallels, cut by a third line, form
interior angles on the same side, which are supplementary,
and conversely.

These propositions can be proved so easily that it is
unnecessary for us to reproduce the reasoning of Aganis.
Having remarked that Propositions 30 and 33 of the First
Book of Euclid follow from them, we proceed to show how
Aganis constructs the point of intersection of two straight
lines which are not equidistant.

Let AB^ GD be two straight lines cut by the trans-
versal EZ^ and such that the sum of the interior angles AEZ^
EZD is less than two right angles (Fig. 4).

Without making our figure any less general we may sup-
pose that the angle AEZ is a right angle.

Upon ZD take an arbitrary point T.

From T draw TL perpendicular to ZE.

Bisect the segment EZ at P: then bisect the segment
PZ at M: and then bisect the segments MZ, etc. . . . until
one of the middle points P, M, . . . falls on the segment LZ.

Let this point, for example, be the point M.

Draw MN perpendicular to EZ, meeting ZD in N',

Equidistant Straight Lines. g

Finally from Z£> cut off the segment ZC, the same
multiple of ZiV as Z£ is of ZM.

In the case taken in the figure ZC = 4 ZiV.

The point C thus obiaified is the point of intersection of
the two straight ii?ies AB and GD.
C F

To prove this it would be necessary to show that the
equal segments ZN, JVS, . . ., which have been cut off one
after the other firom the line ZD, have equal projections on
Z£. We do not discuss this point, as we must return to it
later (p. 11). In any case the reasoning is suggested directly
by Aganis' figure.

The distinctive feature of the preceding construction is
to be noticed. It rests upon the (implicit) use of the so-called
Postulate of Archimedes, which is necessary for the deter-
mination of the segment J/Z, less than LZ and a submult-
iple of EZ.

The Arabs and the Parallel Postulate.

§ 6. The Arabs, succeeding the Greeks as leaders in
mathematical discovery, like them also investigated the Fifth
Postulate.

Some, however, accepted without hesitation the ideas
and demonstrations of their teachers. Among this number is
Al-Nirlzi (9th Century), whose commentary on the definitions,

IO I. The Attempts to prove Euclid's Parallel Postulate.

postulates and axioms ot the First Book is modelled on the
Introduction to the ''Elements^ of Simplicius, while his demon-
stration of the Fifth Euclideari Hypothesis is that of Aganis, to
which we have above referred.

Others brought their own personal contribution to the
argument. Nasìr-Eddìn [1201 — 1274], for example, although
in his proof of the Fifth Postulate he employs the criterion
used by Aganis, deserves to be mentioned for his original idea
of explicitly putting in the forefront the theorem on the sum
of the angles of a triangle, and for the exhaustive nature of
his reasoning.'

The essential part of his hypothesis is as follows: If two
straight lines ;- afid s are the 07ie perpendictilar and the other
oblique to the segment AB, the perpendiculars drawn frotn s
upon r are less than AB on the side on 7vhich s makes an acute
angle with AB, and greater on the side on which s makes an
obtuse angle with AB.

It follows immediately that \i AB and A'B' are two equal
perpendiculars to the line BB' from the same side, the line
AA' is itself perpendicular to both AB and A'B'. Further
we have AA' = BE' ; and therefore the figure AA'B'B is a
quadrilateral with its angles right angles and its opposite sides
equal, i. e., a rectangle.

From this result Nasìr-Eddìn easily deduced that the sum
of the angles of a triangle is equal to two right angles. For
the right-angled triangle the theorem is obvious, as it is half
of a rectangle; for any triangle we obtain it by breaking up
the triangle into two right-angled triangles.

With this introduction, we can now explain shortly how
the Arabian geometer proves the Euclidean Postulate [cf.
Aganis].

I Cf. : Eiiclidis elementorum libri XII studii N'assiredini, (Rome,
1594). This work, written in Arabic, was republished in 1657 and
1801. It has not been translated into any other language.

Nasìr-Eddìn's Proof.

o'c m' k' h' a

Fig- 5-

Let AB, CD be two rays, the one oblique and the other
perpendicular to the straight Hne AC (Fig. 5). From AB cut
oflf the part AH, and from ZTdraw the perpendicular HH'
to AC. If the point H' falls on C, or on the opposite side
of C from A^ the two rays AB and
CD must intersect. If, however, H'
falls between A and C, draw the line
AL perpendicular to AC and equal
to HH' . Then, from what we have
said above, HL = AH' . In AH^io-
<iuced take HK equal to AH. From
K draw KK' perpendicular to AC.
Since A'A" ^ HH', we can take
X'L' = i^'^, and join L'H The
quadrilaterals K'H'HL', H'ALHaxQ both rectangles. There-
fore the three points Z', ZT, Z are in one straight line. It fol-
lows that ^L'HK= <^AHL, and that the triangles AHL,
HL'K diXQ: equal. Thus L'H= HL, and from the properties
of rectangles, K'H' = H'A.

In HK produced, take KM equal to KH. From M
draw MM' perpendicular to AC. By reasoning similar to
•what has just been given, it follows that

M'K' = K'H = H'A.

This result obtained, we take a multiple of AH' greater
than AC [The Postulate of Archimedes]. For example, let
AO., equal to 4 AH' , be greater than AC. Then from AB
cuttoff AO = 4 AH, and draw the perpendicular from O
to AC.

This perpendicular will evidently be 00' . Then, in the
right-angled triangle AO' O, the line CD, which is perpendicu-
lar to the side AO' , cannot meet the other side 00' , and it
must therefore meet the hypotenuse OA.

By this means it has been proved that two straight lines
AB, CD, must intersect, when one is perpendicular to the

12 I- The Attempts to prove Euclid's Parallel Postulate.

transversal AC and the other obhque to it. In other words
the Euclidean Postulate has been proved for the case in which
one of the internal angles is a right angle.

Nasìr-Eddìn now makes use of the theorem on the sum
of the angles of a triangle, and by its means reduces the
general case to this particular one. We do not give his reas-
oning, as we shall have to describe what is equivalent to
it in a later article, [cf p. 37.]^

The Parallel Postulate during the Renaissance and
the i7^b Century.

§ 7. The first versions of the Elements made in the
12th and 13th Centuries on the Arabian texts, and the later
ones, made at the end of the 1 5th and the beginning of the
i6tl», based on the Greek texts, contain hardly any critical
notes on the Fifth Postulate. Such criticism appears after the
year 1550, chiefly under the influence of the Commentary of
Froclus.'^ To follow this more easily we give a short sketch
of the views taken by the most noteworthy commentators of
the 1 6th and 17 th centuries.

F. CoMM ANDINO [1509 — 1575] adds to the Euclidean
definition of parallels, without giving any justification for this

1 Nasìr-Eddìn's demonstration of the Fifth Postulate is given
in full by the English Geometer J. Wallis, in Vol. II. of his works
(cf. Note on p. 15), and by G. Castillon, in a paper published in
the Mém. de I'Acad. roy. de Sciences et Belles-Lettres of Berlin,
T. XVIII. p. 175—183, (1788— 17S9). In addition, several other
writers refer to it, among whom we would mention chiefly, G. S.
Klijgel, (cf. note, (3), p. 44), J. Hoffman, Kriiik der Parallelentheorie^
(Jena, 1807); V. Flauti, Niiova ditnostr azione del postulato quinto, (Na-
ples, 1818).

2 The Cotnmentary of Proclus was first printed at Basle (1533)
in the original text; and next at Padua (1560) in Barozzi's Latin
translation.

Italian Mathematicians of the Renaissance.

step, the idea of equidistance. With regard to the Fifth Postul-
ate he gives the views and the demonstration of Proclus/

C. S. Clavio [1537 — 1612], in his Latin translation of
Eudid's text^, reproduces and criticises the demonstration of
Proclus. Then he brings forward a new demonstration of the
Euclidean hypothesis, based on the theorem: The line equi-
distant from a straight line is a straight line; which he at-
tempts to justify by similar reasoning. His demonstration
has many points in common with that of Nasir-Eddin.

P. A. Cataldi [? — 1626] is the first modern mathema-
tician to pubhsh a work devoted exclusively to the theory of
parallels. 3 Cataldi starts from the conception of equidistant
and non-equidistant straight lines; but to prove the effective
existence of equidistant straight lines, he adopts the hypothesis
that straight lines which are not equidistant converge in one
direction and diverge in the other, [cf Nasìr-Eddìn.] ♦.

G. A. BoRELLi [1608 — 1679] takes the following Axiom
[XIV], and attempts to justify his assumption:

^If a straight line which remains always in the saine plane
as a second straight line, moves so that the o?ie end always touches
this line, and during the whole displacement the first remains
continually perpendicular to the second., then the other end, as it
moves, will describe a straight line.'

Then he shows that two straight lines which are perpen-
dicular to a third are equidistant, and he defines parallels as
equidistant straight lines.

The theory of parallels follows. 5

1 Elementonim libri XV, (Pesaro, 1572).

2 Euclidis elementorum libri XV, (Rome, 1574).

3 OpereUa delle linee rette equidistanti et non equidistanti, (Bologna,
1603).

4 Cataldi made some further additions to his argument in the
work, Aggiunta all' operetta delle linee rette equidistanti et noti equi-
distanti. (Bologna, 1604).

5 BORELLI: Euclides restitutus, (Pisa, 1658).

I A I- The Attempts to prove Ex:clid's Parallel Postulate.

§ 8. Giordano Vitale [1633 — 171 1] again returns to
the idea of equidistance put forward by Posidonius, and re-
cognizes, with Proclus, that it is necessary to exclude the pos-
sibiHty of the Euchdean parallels being asymptotic lines. To
this end he defines two equidistant straight lines as parallels,
and attempts to prove that the locus of the points equidistant
from one straight line is another straight line.^

His demonstration practically depends upon the follow-
ing lemma:

1/ two points, A, C itpo7i a curve, wJwse concavity is to-
7vards X, are joined by the straight line AC, and perpendiculars
are drawn from the infinite number of points of the arc AC
upon atiy straight line, theft these perpendiculars cannot be equal
to each other.

The words 'any straight line', in this enunciation, do not

refer to a straight line taken at random in the plane, but to

Q P a straight line constructed in

the following way (Fig. 6).
From the point B of the arc
AC draw BD perpendicular to

^^A D ~C^ the chordae. Then at ^ draw

^'s- ^- AG also perpendicular to AC.

Finally, having cut off equal segments AG and DF upon
these two perpendiculars, join the ends G and F. GF is the
straight line which Giordano considers in his demonstration,
a straight line with respect to which the arc AB is certainly
not an equidistant line.

But when the author wishes to prove that the locus of
points equidistant from a straight line is also a straight line^
he applies the preceding lemma to a figure in which the re-
lations existing between the arc ABC and the straight line

I Giordano Vitale: Euclide restiluio overo gli antichi elementi
geometrici ristaurati. e facilitati. Libri XV. (Rome, 1680).

Giordano Vitale's Proof.

GF do not hold. Thus the consequences which he deduces
from the existence of equidistant straight lines are not really
legitimate.

From this point of view Giordano's proof makes no ad-
vance upon those which preceded it. However it includes a
most remarkable theorem, containing an idea which will be
further developed in the articles which follow.

Let ABCD be a quadrilateral of which the angles A, B
are right angles and the sides AD, BC
equal (Fig. 7). Further, XqIHK be the per-
pendicular drawn from a point H, upon the
side DC, to the base AB of the quadri-
lateral. Giordano proves: (ij that the ang-
les D, C are equal; (ii) that, when the seg-
ment HK is equal to the segment AD, the
two angles D, C are right angles, and CD is equidistant
from AB.

By means of this theorem Giordano reduces the question
of equidistant straight lines to the proof of the existence of
one point H vc^QXi DC, whose distance from AB is equal to
the segments AD and BC. We regard this as one of the
most noteworthy results in the theory of parallels obtained
up to that date.^

§ g. J. Wallis [1616 — 1703] abandoned the idea of
equidistance, employed without success by the preceding
mathematicians, and gave a new demonstration of the Fifth
Postulate. He based his proof on the Axiom: To every figure
there exists a similar figure of arbitrary viagnitude. We now
describe shortly how Wallis proceeds:^

1 Cf. : BoNOLA: Uti teo?-e?na di Giordano Vitale da Bitonto sidle
rette equidistanti. Bollettino di Bibliografia e Storia delle Scienze
Mat. (1905).

2 Cf. : Wallis : De Postulato Quinto; et Definizione Quinta; Lib. 6.

1 6 I. The Attempts to prove Euclid's Parallel Postulate.

Let a, b be two straightlines intersected at A^ B by the
transversal c (Fig. 8). Let a, p be the interior angles on the

same side of c, such that a + p is
less than two right angles. Through
A draw the straight line b' so that
b and b' form with c equal corre-
sponding angles. It is clear that
b' will he in the angle adjacent to
a. Let the line b be now moved
continuously along the segment
AB^ so that the angle which it
makes with c remains always equal to p. Before it reaches
its final position b' it must necessarily intersect a. In this way
a triangle AB^C^ is determined, with the angles at A and B^
respectively equal to a and p.

But, by Wallis's hypothesis of the existence of similar
figures, upon AB, the side homologous to AB^, we must be
able to construct a triangle ABC^wcAzx to the triangle AB^ Ci.
This is equivalent to saying that the straight lines a, b must
meet in a point, namely, the third angular point of the triangle
ABC. Therefore, etc.

Wallis then seeks to justify the new position he has taken
up. He points out that Euclid, in postulating the existence
of a circle of given centre and given radius, [Post. III.], practi-
cally admits the principle of similarity for circles. But even
although intuition would support this view, the idea of form,
independent of the dimensions of the figure, constitutes a

Eiididis; disceptatio geometrica. Opera Math. t. II; p. 669 — 78 (Oxford,
1693). This work by Wallis contains two lectures given by him in
the University of Oxford; the first in 1651, the second in 1663. It
also contains the demonstration of Nasìr-Eddìn. The part containing
Wallis's proof was translated into German by Engel and StAckel in
their Theorie der ParaUellmien von Euclid bis auf Gauss, p. 21 — 36,
(Leipzig, Teubner, 1895). We shall quote this work in future as
Th. der P.

Wallis's Proof.

hypothesis, which is certainly not more evident than the Postu-
late of Euclid.

We remark, further, that Wallis could more simply have
assumed the existence of triangles with equal angles, or, as
we shall see below, of only two unequal triangles whose
angles are correspondingly equal.

[cf. p. 29 Note I.]

§ 10 . The critical work of the preceding geometers is
sufficient to show the historical development of our subject in
the i6tb and 17th Centuries, so that it would be superfluous
to speak of other able writers, such as, e. g., Oliver of
Bury [1604], Luca Valerio [1613], H. Savile [1621],
A. Tacquet [1654], A. Arnauld [1667].^ However, it seems
necessary to say a few words on the question of the position
which the different commentators on the '' Ele7nents' allot to
the Euclidean hypothesis in the system of geometry.

In the Latin edition of the "" Elements' [1482], based upon
the Arabian texts, by Campanus [13th Century], this hypothesis
finds a place among the postulates. The same may be said
of the Latin translation of the Greek version by B. Zamberti
[1505], of the editions of Luca Paciuolo [1509], of N. Tar-
taglia [1543], of F. Commanding [1572], and of G. A.Bor-
ELLi [1658].

On the other hand the first printed copy of the 'Ele-
ments' in Greek, [Basle, 1533], contains the hypothesis among
the axioms [Axiom XI]. In succession it is placed among the
Axioms by F. Candalla [1556], C. S. Clavio [1574], Gior-
dano Vitale [1680], and also by Gregory [1703], in his
well-known Latin version of Euclid's works.

To attempt to form a correct judgment upon these dis-

I For fuller information on this subject cf. Riccardi: Saggio
di una bibliografia euclidea. Mem. di Bologna, (5) T. I. p. 27 — 34,
(1890).

1 8 I- The Attempts to prove Euclid's Parallel Postulate.

crepancies^ due more to the manuscripts handed down from
the Greeks than to the aforesaid authors, it will be an advan-
tage to know what meaning the former gave to the words
'postulates' [aÌTniaara] and 'axioms' [dHid))uara].' First of all
we note that the word ^axioms' is used here to denote what
Euclid in his text calls '"commcni notions' [KOivai evvoiai].

Proclus gives three different ways of explaining the differ-
ence between the axioms and postulates.

The first method takes us back to the difference between
a problem and a theorem. A postulate differs from an axiom,
as a problem differs from a theorem, says Proclus. By this we
must understand that a postulate affirms the possibility of a
construction.

The second method consists in saying \\\2X a postulate is
a proposition with a geometrical meaning, while an axiom is a
propontio7i common both to geometry and to arithmetic.

Finally the third method of explaining the difference
between the two words, given by Proclus, is supported by the
authority of Aristotle [384 — 322 B. C.]. The words axiom
2.\\^ postulate à.0 not appear to be used by Aristotle exclusive-
1\- in the mathematical sense. An axiom is that which is true
in itself, that is, owing to the meaning of the words which it
contains; a postulate is that 7vhich, although it is not an axiom,
in the aforesaid sense, is admitted without demonstration.

Thus the word axiom, as is more evident from an ex-
ample due to Aristotle, \7i)hen equal things are subtracted from
equal things the remainders are equal\ is used in a sense which

I For the following, cf. Proclus, in the chapter entitled Pe-
ata et axiomata. In a Paper read at the Third Mathematical Congress
(Heidelberg, 1904) G. Vailati has called the attention of students
anew to the meaning of these words among the Greeks. Cf. : In-
torno al significato della distinzione tra gli assiotiii ed i postulati nella
geometria greca. Verh. des dritten Math. Kongresses, p. 575 — 5^'»
(Leipzig, Teubner, 1005).

Position of the Parallel Postulate. jg

corresponds, at any rate very closely, to that of the common
notions of Euclid, whilst the word postulate in Aristotle has
a different meaning from each of the two to which reference
has just been made.'

Hence according as one or other of these distinctions be-
tween the words is adopted, a particular proposition would be
placed among the postulates or among the axioms. If we
adopt the first, only the first three of the five postulates of
Euclid, according to Proclus, have a right to this name, since
only in these are we asked to carry out a construction [to
join two points, to produce a straight line, to describe a circle
v.'hose centre and radius are arbitrary]. On the other hand,
Postulate IV. [all right angles are equal], and Postulate V. ought
to be placed among the axioms.*

1 Cf. Aristotle: Analytica Posteriora. I, lo. § 8. We quote in
full this slightly obscure passage, where the philosopher speaks of
the postulate: 6aa fièv ouv beiKTÙ òvxa \a|updvei aÙTÒq \ì.t\ òeiEa^,
TaÙTO éàv nèv òokoOvto \a|Lipdvr] tuj |aav6dvovTi ÙTT0TÌ9eTai. Kaì
éariv oùx à-rrXuJq ÙTTÓGeoK; àWà irpò? éKeivov fióvov. 'Eàv bè f)
firibeiuià? évoùjriq òò^r\ii f) koì évavTia(; évouariq \a.\\.^6.yix\, tò auro
aÌTeìrm. Kaì toùtu; òiaq)épei OiróBean; koì airrìiuo, ?(Jti yàp
aitrina tò ÙTrevavTi'ov toO juavGdvovxoq Tf) bóEr).

2 It is right to remark that the Fifth Postulate can be enun-
ciated thus : The common point of two straight lines can be found, when
these two lines, cut by a transversal, form two interior angles on the
same side whose sum is less than two right angles. Thus it follows
that this postulate affirms, like the first three, the possibility of a
construction. However this character disappears altogether, if it
is enunciated, for example, thus : Through a point there passes only
one parallel to a straight line; or, thus : Two straight lines which are
parallel to a third km are parallel to each other. It would therefore

appear that the distinction noted above is purely formal. However
we must not let ourselves be deceived by appearances. The Fifth
Postulate, in whatever way it is enunciated, practically allows the
construction of the point of intersection of all the straight lines of
a pencil with a given straight line in the plane of the pencil, one
of these lines alone being excepted. It is true that there is a certain

20 I- The Attempts to prove Euclid's Parallel Postulate.

Again, if we accept the second or the third distinction,
the five Euclidean postulates should all be included among
the postulates.

In this way the origin of the divergence between the var-
ious manuscripts is easily explained. To give greater weight
to this explanation we might add the uncertainty which histor-
ians feel in attributing to Euclid the postulates, common no-
tions and definitions of the First Book. So tar as regards the
postulates, the gravest doubts are directed against the last
two. The presence of the first three is sufficiently in accord
with the whole plan of the work.' Admitting the hypothesis
that the Fourth and Fifth Postulates are not Euclid's, even if
it is against the authority of Geminus and Proclus, the ex-
treme rigour of the ''Elements'' would naturally lead the later
geometers to seek in the body of the work all those pro-
positions which are admitted without demonstration. Now
the one which concerns us is found stated very concisely in
the demonstration of Bk. I. Prop. 29. From this, the sub-
stance of the Fifth Postulate could then be taken, and added
to the postulates of construction, or to the axioms, according
to the views held by the transcriber of Euclid's work.

Further, its natural place would be, and this is Gregory's
view, after Prop. 27, of which it enunciates the converse.

Finally, we remark that, whatever be the manner of de-
ciding the verbal question here raised, the modern philo-
sophy of mathematics is inclined generally to suppress the

difference between this postulate and the three postulates of con-
struction. In the latter the data are completely independent. In
the former the data (the two straight lines cut by a transversal) are
subject to a condition. So that the Euclidean Hypothesis belongs
to a class intermediate between the postulates and axiom, rather
than to the one or the other.

I Cf. P. Tannery: Sur Pauthentuité des axiomes d'Euclide. Bull,
d. Sc. Math. (2), T. VIII. p. 162—175, (1884).

Postulates and Axioms. 21

distinction between postulate and axiom, which is adopted in
the second and third of the above methods. The generally
accepted view is to regard the fundamental propositions of
geometry as hypotheses resting upon an empirical basis,
while it is considered superfluous to place statements, which
are simple consequences of the given definitions, among the
propositions.

Chapter IL

The Forerunners of Non-Euclidean
Geometry.

Gerolamo Saccheri [1667 — 1733].

§ II. The greater part of the work of Gerolamo Sac-
cheri: EucHdes ab o/nni Jiaevo vindicatus : sive conatus gco-
meiricus quo stabiliuntur prima ipsa universae Geoinetriae
Principia, [Milan, 1733], is devoted to the proof of the Fifth
Postulate. The distinctive feature of Saccheri's geometrical
writings is to be found in his ^Logica de/i:o?!strativa' , [Turin,
1697J. It is simply a particular method of reasoning, already
used by Euclid [Bk. IX. Prop. 1 2J, according to which by
assuming as hypothesis that the proposition 7vhlch is to beproi ed
is false, one ts brought to the conclusion that it is true}

Adopting this idea, the author takes as data the first
twenty-six propositions of Euclid, and he assumes as a hypo-
thesis that the Fifth Postulate is false. Among the consequences
of this hypothesis he seeks for some proposition, which would
entitle him to affirm the truth of the postulate itself.

Before entering upon an exposition of Saccheri's work,
we note that Euclid assumes implicitly that the straight line
is infinite in the demonstration of Bk, I. 16 [the exterior angle
of a triangle is greater than either of the interior and opposite

' Cf. G. Vailati: Di iin^ o/era dimenlicata del P. Gerolamo Sac-
chtri. Rivista Filosofica (1903).

Saccheri's Quadrilateral. 23

angles], since his argument is practically based upon the
existence of a segment which is double a given segment.

We shall deal later with the possibihty of abandoning
this hypothesis. At present we note that Saccheri tacitly as-
sumes it, since in the course of his work he uses Xh^ proposition
of the exterior angle.

Finally, we note that he also employs the Postulate of
Archimedes^ and the hypothesis of the continuity of the straight
liae,^ to extend, to all the figures of a given type, certain pro-
positions admitted to be true only for a single figure of that
type.

§ 12. The fundamental figure of Saccheri is the two
right-ar.gled isosceles quadrilateral; that is, the quadrilateral of
which two opposite sides are equal to each other and perpen-
dicular to the base. The properties of such a figure are de-
duced from the following Lemma I. , which can easily be
proved :

If a quadrilateral ABCD has the consecutive angles A
a7id B right angles, and the sides AD and BC equal, then the
angle C is equal to the angle D [This is a special case of Sac-
cheri's Prop. I.]; but if the sides AD and BC are unequal, of
the two angles C, Z>, that one is greater which is adjacent to
the shorter side, and vice versa.

1 [The Postulate of Archimedes is stated by Hilbert thus: Let
Al be any point upon a straight line between the arbitrarily chosen
points A and B. Take the points A2, A^, . . . so that Ai lies
between A and A2, A2 between Ai and /i;„ etc.; moreover let the
segments AAi, A1A2, ^2^3, ... be all equal. Then among this
series of points, there always exists a ceitain point Ad, such that
B lies between A and Aa-\

2 This hypothesis is used by Saccheri in its intuitive form,
viz. : a segment, which passes continuously from the length a to
the length b, different from a, takes, during its variation, every
length intermediate between a and b.

II. The Forerunners of Non-Euclidean Geometry.

Let ABCD be a quadrilateral with two right angles A
and B, and two equal sides AD and BC (Fig. 9). On the
Euclidean hypothesis the angles Cand D are also right angles.
Thus, if we assume that they are able to be both obtuse, or
both acute, we implicitly deny the Fifth Postulate. Saccheri
discusses these three hypotheses regarding the angles C, D.
He named them:

The Hypothesis of the Right Angle

[<^ 6"= <^ Z> = I right angle] :
The Hypothesis of the Obtuse A/igle

[-^ C = <^ Z> > I right angle] :
The Hypothesis of the Acute Angle

[^ C= <^Z> < I right angle].
One of his first important results is the following:
Accordifig as the Hypothesis of the Eight Angle, of the
Obtuse Angle, or of the Acute Attgle is true i?i the two right-
angled isosceles quadrilateral, we must have AB = CD,
ABy- CD, or AB < CD, respectively. [Prop. 111.]

In fact, on the Hypothesis of the Eight Angle, by the
preceding Lemma, we have immediately
AB = CD.
On the Hypothesis of the Obtuse Angle, the perpendicular
00' at the middle point of the segment y^/>'
divides the fundamental quadrilateral into
two equal quadrilaterals, with right angles at
O and O'. Since the angle D ^ angle A,
then we must have AO ^ DO , by this
Lemma. Thus AB > CD.

On the Hypothesis of the Acute Angle these
^'^ 9- inequalities have their sense changed and

we have

AB < CD.
Using the reductio ad absurdum argument, we obtain
the converse of this theorem. [Prop. IV.]

•The Three Hypotheses. 25

If the Hypothesis of the Right Angle is true in only one
4ase, then it is true in every other case. [Prop. V.]

Suppose that in the two right-angled isosceles quadrilat-
eral AB CD the Hypothesis of the Eight Angle is verified.

In AD and BC (Fig. lo) take the points ZTand K equi-
distant from AB; join HK a.nd form the

quadrilateral ABKH. m

If HK is perpendicular to AH and
BK, the Hypothesis of the Right Angle is ^^j-
also verified in the new quadrilateral. H

If it is not, suppose that the angle
AHK is acute. Then the adjacent angle ^ ^

DHK is obtuse. Thus in the quadrilateral '^' ^ '

ABKH, from the Hypothesis of the Acute Angle, it follows
that AB <C HK: while in the quadrilateral HKCD, from the
Hypothesis of the Obtuse Angle, it follows that HK<^ CD.

But these two inequalities are contradictory, since by
4he Hypothesis of the Right Angle in the quadrilateral ABCD,
AB = CD.

Thus the angle AHK cannot be acute : and since by the
same reasoning we could prove that the angle AHK cannot
be obtuse, it follows that the Hypothesis of the Right Angle is
also true in the quadrilateral ABKH.

On AD and i)C produced, take the points M, iV equi-
distant from the base AB. Then the Hypothesis of the Right
Angle is also true for the quadrilateral AB JVM. In fact if
AM is a multiple of AD, the proposition is obvious, li AM
is not a multiple of AD, we take a multiple of AD greater
than AM \the Postulate of Archimedes^ and from AD and
BC produced cut off AF and BQ equal to this multiple.
Since, as we have just seen, the Hypothesis of the Right Angle
is true in the quadrilateral ABQF, the same hypothesis must
also hold in the quadrilateral ABNM.

Finally the said hypothesis must hold for a quadrilateral

26 II- The Forerunners of Non-Euclidean Geometry.

on any base, since, in Fig. lo, we can take as the base one
of the sides perpendicular to AB.

Note. This theorem of Saccheri is practically contained
in that of Giordano Vitale, stated on p. 15. In fact, refer-
ring to Fig. 7, the hypothesis

DA== HK^ CR
is equivalent to the other

<5C Z> = -^ H=- < C = I right angle.
Ikit from the former, there follows the equidistance of the
two straight lines DC, AB^; and thus the validity of the Hypo-
thesis of the Right Angle in all the two right-angled isosceles
quadrilaterals, whose altitude is equal to the line DA, is
established. The same hypothesis is also true in a quadri-
lateral of any height, since the line called at one time the
base may later be regarded as the height.

If the Hypothesis of the Obtuse Angle is true in only one
case, then it is true in every other case. [Prop. VI.]

Referring to the standard quadrilateral ^j9CZ> (Fig. 11),
n K 1 C suppose that the angles C and D are ob-
tuse. Upon AD and BC take the points
H and K equidistant from AB.

In the first place we note that the

segment HK cannot be perpendicular to

the two sides AD and BC, since in that

A Oj B case the Hypothesis of the Right Angle

Fig. II. would be verified in the quadrilateral

ABKH, and consequently in the fundamental quadrilateral.

Let us suppose that the angle AHK is acute. Then

I It is true that Giordano in his argument refers to the points
of the segment DC, which he shows are equidistant from the base
AB of the quadrilateral. However the same argument is applicable
to all the points which lie upon DC, or upon DC produced. Cf.
Bonola's Note referred to on p. 15.

Proof for one Quadrilateral Sufficient. 27

by the Hypothesis of the Acute Angle, HK^ AB. But as the
Hypothesis of the Obtuse Angle holds in AB DC, we have

AB^ CD.
Therefore HK^ AB > CD.

If we now move the straight line HK continuously, so that it
remains perpendicular to the median 00' of the fundamental
quadrilateral, the segment HK, contained between the oppo-
site sides AD, BC, which in its initial position is greater than
AB, will become less than AB in its final position DC. From
the postulate of continuity we may then conclude that,
between the initial position HK and the final position DC,
there must exist an intermediate position H' K' , for which
H'K' = AB.

Consequently in the quadrilateral ABK'H' the Hypo-
thesis of the Right Angle would hold [Prop. III.J, and therefore,
by the preceding theorem, the Hypothesis of the Obtuse Angle
could not be true in ABCD.

The argument is also valid if the segments ^j^, BK arc
greater than AD, since it is impossible that the angle AHK
could be acute. Thus the Hypothesis of the Obtuse Angle holds
in ABKH as well as in ABCD.

Having proved the theorem for a quadrilateral whose
sides are of any size, we proceed to prove it for one whose
base is of any size: for example the base BK [cf Fig. 12].

Since the angles K, H, are obtuse, the
perpendicular at K to KB will meet the
segment AH in the point M, making the
angle AMK obtuse [theorem of the ex-
terior angle].

Then in ABKM we have AB > KM,
by Lemma I. Cut off from AB the segment
-5iV equal to MK. Then we can construct
the two right angled isosceles quadrilateral BKMN, with the
angle MNB obtuse, since it is an exterior angle of the triangle

28 II- The Forerunners of Non-Euclidean Geometry.

ANAL It follows that the Hypothesis of the Obtuse A?igle
holds in the new quadrilateral.

Thus the theorem is completely demonstrated.

1/ the Hypothesis of the Acute Angle is true in only one
case, then it is true in every other case. [Prop. VII.]

This theorem can be easily proved by using the method
of reductio ad absurdum.

§ 13. From the theorems of the last article Saccher:
easily obtains the following important result with regard to
triangles :

According as the Hypothesis of the Right Angle, the Hy-
pothesis of the Obtuse Angle, or the Hypothesis of the Acute
Angle, is found to be true, the sum of the angles of a triangle
will be respectively equal to, greater than^ or less thafi two right
angles. [Prop. IX.]

Let ABC [Fig. 1 3] be a triangle of which ^ is a right
P angle. Complete the quadrilateral by draw-
ing AD perpendicular to AB and equal to
BC; and jon CD.

On the Hypothesis of the Bight Afigle,
the two triangles ABC and ADC are equal.
Therefore -^BAC^^^DCA.
It follows immediately that in the tri-
angle ABC,
^A-\- ^B + <^ C= 2 right angles.
On the Hypothesis of the Obtuse Angle,
sinc^AB^DC,
we have ^ACB^ <C DAC. '

I This inequality is proved by Saccheri in his Prop. VIII.,
and serves as Lemma to Prop. IX. It is, of course. Prop. 25 of
Euclid's First Book.

The Sum of the Angles of a Triangle. 2Q

Therefore, in this triangle we shall have

■^ A + ^J5 + -^ C^ 2 right angles.

On the Hypothesis of the Acute Angle,
since AB<^DC,
we have ^ACB<C ^£>AC,
and therefore, in the same triangle,

<CA-\- <^+ <:C<2 right angles.
The theorem just proved can be easily extended to the
case of any triangle, by breaking the figure up into two right
angled triangles. In Prop. XV. Saccheri proves the converse,
by a reductio ad absurdum.

The following theorem is a simple deduction from these
results :

If the sum of the angles of a triangle is equal to, greater
than, or less than two right at/gles iti only one triangle, this
sum will be respectively equal to, greater than, or less than t7vo
right angles in every other triangle.'^

This theorem, which Saccheri does not enunciate ex-
plicitly, Legendre discovered anew and published, for the
first and third hypotheses, about a century later.

§ 14. The preceding theorems on the two right-
angled isosceles quadrilaterals were proved by Saccheri, and

I Another of Saccheri's propositions, which does not concern
us directly, states that if the sum of the angles of only one quadri-
lateral is equal to, greater than, or less than four right angles, the
Hypothesis of the Right Angle, the Hypothesis of the Obtuse Angle, or
the Hypothesis of the Acute Angle zvould respectively be true. A note
of Saccheri's on the Postulate of Wallis (cf. % 9) makes use of
this proposition. He points out that Wallis needed only to assume
the existence of two triangles, whose angles were equal each to
each and sides unequal, to deduce the existence of a quadrilateral
in which the sum of the angles is equal to four right angles. From
this the validity of the Hypothesis of the Right Angle would follow,
and in its turn the Fifth Postulate.

■JO II- The Forerunners of Non-Euclidean Geometry.

later by other geometers^ with the help oi ^t Postulate of
Anhimedes and the principle of contifiuity [cf. Prop. V., VI].
However Dehn^ has shown that they are independent of
these hypotheses. This can also be proved in an elementary
way as follows.^

On the straight line r (Fig. 14) let two points B and D
be chosen, and equal perpendiculars BA and DC be drawn
to these lines. Let A and C be joined by the straight line s.
The figure so obtained, in which evidently <fiBAC= -^ Z>CA,
is fundamental in our argument and we shall refer to it con-
stantly.

Two points Jt, E' are now taken on j, of which the
first is situated between A and C, and the second not.

Further let the perpendiculars from E^ E' to the line
r meet it at E and E' .

The following theorems now hold:

\\{ EE^AB,\
I. I or L the angles BAC^ DCA areright angles.

I E'E' = AB J

\\i EE>AB,\
II. ' or ^, the angles ^^C, Z>6>y are obtuse.

I E'E'<iAB\

niEE<iAB,\
III. or i , the angles BAC, DCA are acute.

[ E'E'^AB]

We now prove Theorem I. [cf. Fig. 14.]

From the hypothesis EE = AB, the following equalities
are deduced:

1 Cf. Die Legendreschen Satze iiber die IVinkehiimme im Dreieck.
Math. Ann. Bd. 53, p. 405 — 439 (1900).

2 Cf. BoNOLA, / teoremi del Padre Gerolamo Sacrheri sulla
somma degli angoli di 111/ triangolo e le ricerche di M. Dehn, Rend.
Istituto Lombardo (2); Voi. \XX.VIII. (1905).

Postulate of Archimedes not needed.

<^ BAE = ^ FEA, and <C FEC = <r DCE.
These, together with the fundamental equality

^BAC=^DCA,
are sufficient to establish the equality of the two angles FEA
and FEC.

E A E C

— s

F' B

F D

Fig. 14.

Since these are adjacent angles, they are both right
angles, and consequently the angles BAC and DCA are
right angles.

The same argument is applicable in the hypothesis
E'F' = AB.

We proceed to Theorem 11 [cf Fig. 1 5].

Suppose, in the first place, EF > AB. From FE cut
off j^/= AB, and join / to A and C.

Then the following equalities hold:

^ BA/= <è: EIA and ^r DCJ ^ ^ FJC.
Further, by the theorem of the exterior angle [Bk. !. 16],
we have

■22 n. The Forerunners of Non-Euclidean Geometry.

^ FIA + <^ FIC^ <^ FEA + <: FEC = 2 right angles.
But

^BAC ^ <^E>CA-><3Z£AI+ ^DCI.
Therefore

^BAC ^ ^ DC A > <: FIA + <: i^/C> 2 right angles.
But, since < BAC^ <^ BCA,
it follows that -^BAC^ 1 right angle. . . . Q. E. D.

In the second place, suppose that E'F' <C AB. Then from

F'E' produced cut off F'F = BA, and join /' to C and A.

The following relations, as usual, hold:

^ i^'/'^ = -^ BAT', ^ FTC = <^ DCI';

^ /'^i5' > <^ rCE\ ^ F'rA<_ <ac i^'/'C.

Combining these results, we deduce, first of all, that

^ BAI'<^ <C ^C7'.
From this, if we subtract the terms of the inequality

■^i'ae':><:J'ce',

we obtain

< BAE'<i <:nCE' = ^ BAC.
But the two angles BAE' and BAC are adjacent. Thus we
have proved that <C BAC is obtuse. — Q. E. D.

Theorem III. can be proved in exactly the same way.
The converses of these theorems can now be easily
shown to be true by the reductio ad absurdum method. In
particular, if M and N are the middle points ot the two seg-
ments AC and BE>, we have the following results for the
segment MN which is perpendicular to both the hnes AC
and BD (Fig. 16).

If <r. BAC= r nCA = / right angle, then MN= AB.
If ^ BAC^- ^DCA > / right angle, then AIN^AB.
If ^ BAC== <^ nCA < / right angle, then MN< AB.
Further it is easy to see that

(i) If <f^ BAC = <^ DCA = / right angle,
then <^ FEM and -^ F'E'M are each i right angle.

Bonola's Proof.

(ii) If -^ BAC == <: nCA '> I right angle,
then <^ FEM and <f^ F' E' M are each obtuse.

(Hi) If < BAC = <^ DC A < I right angle,
then <^ FEM and -^ F'E'M are each acute.

A. E

F' B F

In fact, in Case (i), since the lines r and s are equi-
distant, the following equalities hold:
^NMA = ^FEM=^ <^ BAC= ^F'E'M=i right angle.

To prove Cases (ii) and (iii), it is sufficient to use the
reductio ad absurdum method, and to take account of the
results obtained above.

Now let P be a point on the line MN, not contained
between J/ and iV(Fig. 1 7). Let RP be the perpendicular to
MN and RK the perpendicular to BD. This last perpend-
icular will meet AC in a point H. On this understanding
the preceding theorems immediately establish the truth of
the following results:

If-^BAM^i right angle, then <^ KHM and ^ KRF
are each equal to i right angle.

If <^ BAAC> r right angle, then <^ KHM and ^ KRF
are each greater than i right angle.

If ^ BAM < / right angle, then <^ KHM and <^ KRF
are each less than i right angle.

These results are also true, as can easily be seen, if the
point F falls between M and N.

In conclusion, the last three theorems, which clearly

II. The Forerunners of Non-Euclidean Geometry.

coincide with Saccheri's theorems upon the two right-angled
isosceles quadrilateral, are equivalent to the following result,
proved without using Archimedes' Postulate: —

M C

K B

N D

If the truth of the Hypothesis of the Right Angle, of the
Obtuse Angle, or of the Acute Angle, respectively, is known in
only o?ie case, its truth is also kno2V7i in every other case.

If we wish now to pass from the theorems on quad-
rilaterals to the corresponding theorems on triangles, we need
only refer to Saccheri's demonstration [cf. p. 28], since this
part of his argument does not in any way depend upon
the postulate in question.

We have thus obtained the result which was to be
proved.

§ 15. To make our exposition of Saccheri's work
more concise, we take from Prop. XI. and XII. the contents
of the following Lemma II:

Let ABC be a tria?igle of ivhich C is a right angle: let
H be the tniddle point of AB, and K the foot of the perpen-
dicular fro?n H upon AC. Then we shall have

AK = KC, Oil the Hypothesis of the Right Angle;

AK <^ KC, on the Hypothesis of the Obtuse Angle;

AK ^ KC, on the Hypothesis of the Acute Angle.

On the Hypothesis of the Right Angle the result is
obvious.

The Projection of a Line.

On the Hypothesis of the Obtuse Angie, since the sum of
the angles of a quadrilateral is greater than four right angles,
it follows that ^AHK <,^HBC. Let HL be the perpendi-
cular from H to BC (Fig. i8). Then the result just obtained,
and the fact that the two triangles AHK, HBL have equal
hypotenuses, give rise to the following inequality : AK<^HL.
But the quadrilateral HKCL has three right angles and there-
fore the angle H is obtuse {Hypothesis of the Obtuse Angle],
It follows that

HL < KC,
and thus

AK<^KC.

The third part of this Lemma can be proved in the
same way.

It is easy to extend this Lemma as follows (Fig. 1 9) :

}

Fig. 19.

Lemma LLI. Lf oti the one arm of an angle A equal seg-
ments AA^, A-i_A^, A^A.^y . . . are taken, and AA^ , A^A^^
AjA^'. . . are their projections upon the other arm of the afigle,
then the following results are true:

AAi = A-iA^ = A^A^ = . . .
on the Hypothesis of the Right Angle;

aa,'<:a,'a,'<a;a, = <. . .

on the Hypothesis of the Obtuse Angle;

aa;>a,'a/>a;a:>...

on the Hypothesis of the Acute Angle.

To save space the simple demonstration is omitted.

II. The Forerunners of Non-Euclidean Geometry.

We can now proceed to the proof of Prop. XI. and XII.
of Saccheri's work, combining them in the following theorem:

On the Hypothesis of the Right Angle and on the Hypo-
thesis of the Obtuse Angle, a line perpendicular to a given
straight line and a lifie cutting it at an acute angle intersect
each other.

Fig. ao.

Let (Fig, 2 o) LP and AD be two straight lines of which
the one is perpendicular to AP, and the other is inclined to
AP at an acute angle DAP.

After cutting off in succession equal segments AD, DF^,
upon AD, draw the perpendiculars DB and P^M^ upon the
line AP.

From Lemma III. above, we have
PM^ > AB,
or AM^ ^ 2 AB,

on the two hypotheses.

Now cut off Pip2 equal to AP^, from AP^ produced,
and let M^ be the foot of the perpendicular from P2 upon AP.
Then we have

AM2 ^ 2 AMi,
and thus

AM2 > 2' AB.
This process can be repeated as often as we please.

In this way we would obtain a point Pu upon the line
AD such that its projection upon the line AP would deter-
mine a segment A.if„ satisfying the relation

Two Hypotheses give Postulate V.

AM" > 2" AB.

But if n is taken sufficiently great, [by the Postulate of
Archimedes'^^ we would have

2'' AB^AP,
and therefore

AMn > AP.
Therefore the point P lies upon the side AMn of the right-
angled triangle AM„ Fn- The perpendicular PL cannot
intersect the other side of this triangle; therefore it cuts the
hypotenuse.^ Q^. E. D.

It is now possible to prove the following theorem :

T?ie Fifth Postulate is true on the Hypothesis of the
Right Angle at id on the Hypothesis of the Obtuse Angle [Prop.
XIII.].

Let (Fig. 2i) AB, CD be two straight lines cut by the
line AC.

Let us suppose that

^ BAC + ^ ^CZ> < 2 right angles.

Then one of the angles
BAC, ACD, for example the
first, will be acute.

From C draw the perpen-
dicular CH upon AB. In the
triangle ACH, from the hypo-
theses which have been made, A
we shall have

<^A-\r <^C + <C-^>2 right angles.

1 The Postulate of Archimedes, of which use is here made,
includes implicitly the infinity of the straight line.

2 The method followed by Saccheri in proving this theorem
is practically the same as that of Nasìr-Eddìn. However Nasir-
Eddìn only deals with the Hypothesis of the Right Angle, as he had
formerly shown that the sum of the angles of a triangle is equal
to two right angles. It is right to remember that Saccheri was
familiar with and had criticised the work of the Arabian Geometer.

28 II. The Forerunners of Non-Euclidean Geometry.

But we have assumed that

<^ BAC + <^ ACD < 2 right angles.
These two results show that

<^ AHC > «9C BCD.
Thus the angle HCD must be acute, as ^ is a right angle.
It follows from Prop. XI., XII. that the lines AB and CD
intersect.^

This result allows Saccheri to conclude that the Hypo-
thesis of the Obtuse Angle is false [Prop. XIV.]. In fact, on
this hypothesis Euclid's Postulate holds [Prop. XIII.], and
consequently, the usual theorems which are deduced from
this postulate also hold. Thus the sum of the angles of the
fundamental quadrilateral is equal to four right angles, so
that the Hypothesis of the Eight Angle is true.^

§ i6. But Saccheri wishes to prove that the Fifth
Postulate is true in every case. He thus sets himself to
destroy the Hypothesis of the Acute Angle.

To begin with he shows that o?i this hypothesis, a straight
line being given, there can be drawn a perpendicular to it and
a line cutting it at an acute angle, which do not intersect each
other [Prop. XVII.].

To construct these lines, let ^-5C (Fig. 22) be a triangle
of which the angle C is a right angle. At B draw BD mak-
ing the angle ABD equal to the angle BAC. Then, on the

1 This proof is also found in the work of Nasìr-Eddìn, which
evidently inspifed the investigations of Saccheri.

2 It should be noted that in this demonstration SacCHERI
makes use of the special type of argument of which we spoke in
Sii. In fact, from the assumption that the Hypothesis of the Ob-
tuse Angle is true, we arrive at the conclusion that the Hypothesis
of the Right Angle is true. This is a characteristic form taken in
such cases by the ordinary reductio ad absurdum argument.

Saccheri and the Third Hypothesis.

Hypothesis of the Acute Atigle, the angle CBD is acute, and
of the two hnes CA, BD, which do not meet [Bk. I, 27],
one makes a right angle with BC.

In what follows we consider only the Hypothesis of the
Acute Aiigle.

Let (Fig. 23) a,b be two straight lines in the same plane
which do not meet.

Fig. 23

From the points A^, A^^ on a draw perpendiculars
A^Bt,, A.^B^ to b.

The angles A^, A, of the quadrilateral thus obtained
can be

(i) one right, and one acute:

(ii) both acute:

(iii) one acute and one obtuse.

In the first case, there exists already a common per-
pendicular to the two lines a, b.

In the second case, we can prove the existence of such
a common perpendicular by using the idea of continuity
[Saccheri, Prop. XXII.]. In fact, if the straight line A-, B^ is
moved continuously, while kept perpendicular to b, until it
reaches the position A^B^, the angle B^At_A2 starts as an
acute angle and increases until it becomes an obtuse angle.
There must be an intermediate position AB in which the
angle BAA^ is a right angle. Then AB is the common
perpendicular to the two lines a, b.

In the third case, the lines a, b do not have a common

II. The Forerunners of Non-Euclidean Geometry.

perpendicular, or, if such exists, it does not fall between B^
and B2.

Evidently there will be no such perpendicular if, for all
the points Ar situated upon a, and on the same side of A^,
the quadrilateral ^i^.^r^;- has always an obtuse angle at Ar.

With this hypothesis of the existence of two coplanar
straight lines which do not intersect, and have no common
perpendicular, Saccheri proves that such lines always ap-
proach nearer and nearer to each other [Prop. XXIII.], and that
their distance apart finally becomes smaller than any segment,
taken as small as we please [Prop. XXV.]. In other words,
if there are two coplanar straight Hues, which do not cut
each other, and have no common perpendicular, then these
lines must be asymptotic to each other."

To prove that such asymptotic lines effectively exist,
Saccheri proceeds as follows: — ^

Fig. 24.

Among the lines of the pencil through A, coplanar with
the line b, there exist lines which cut b, as, e. g., the line
AB perpendicular to b; and lines which have a common

1 With this result the question raised by the Greeks, as to
the possibility of asymptotic lines in the same plane, is answered
in the affirmative. Cf. p. 3.

2 The statement of Saccheri's argument upon the asymptotic
lines differs in this edition from that given in the Italian and
German editions. The changes introduced were suggested to me
by some remarks of Professor Carslaw.

The Existence of Asymptotic Lines ai

perpendicular with ò, as, e. g., the line AA' perpendicular
to A£ [cf. Fig. 24].

If AI' cuts Ò, every other line of the pencil, which
makes a smaller angle with AB than the acute angle BAjP,
also cuts è. On the other hand^ if the line A Q, different from
AA', has a common perpendicular with ò, every other line,
which makes with AB a larger acute angle than the angle
BAQ, has a common perpendicular with ò [cf § 39,
case (ii).]

Also it is clear that, if we take the lines of the pencil
through A, from the ray AB towards the ray AA', we shall
not find, among those which cut d, any line which is the last
line of that set. In other words, the angles BAB, which the
lines AB, cutting ^, make with AB, have an i/J'per limit, the
angle BAX, such that the line AX does not cut b.

Then Saccheri proves [Prop. XXX.] that, if we start with
AA and proceed in the pencil through A in the direction
opposite to that just taken, we shall not find any last line in
the set of lines which have a common perpendicular with b\
that is to say, the angles BA Q, where A Q has a common
perpendicular with b, have a lower limit, the angle BA V,
such that the line ^y does not cut b and has not a com-
mon perpendicular with b.

It follows that A Vis a. line asymptotic to b.

Further Saccheri proves that the two hnes AX and A V
coincide [Prop. XXXII.]. His argument depends upon the
consideration of points at infinity; and it is better to sub-
stitute for it another, founded on his Prop. XXI., viz., On the
Hypothesis of the Right Angle, and on that of the Acute Angle,
the distance of a point on one of the lines containing an angle
from the other bounding line increases indefinitely as this point
moves further and further along the line.

II. The Forerunners of Non-Euclidean Geometry.
The suggested argument is as follows:

Ar^= p

Fig. 25-

If AX [Fig. 25] does not coincide with A Y, we can take
a point P on AY, such that the perpendicular FF' from F
to AX satisfies the inequality

( 1 1 FF' > AB. [Prop. XXL]

On the other hand, if FQ is the perpendicular from F to b,
the property of asymptotic lines [Prop. XXIII] shows that
AB>FQ.

But F is on the opposite side of AX from I?.
Therefore PQ > FF.

Combining this inequality with the preceding, we find that

AB>PF.
which contradicts (i).

Hence AX coincides with A Y.

We may sum up the preceding results in the following
theorem : —

t B b

Fig. 26.

On the Hypothesis of the Acute Angle, there exist in the
pencil of lines through A two lities p and q, asymptotic to b,
one towards the right, and the other towards the left, which
divide the pencil into two parts. The first of these consists of
the lines which intersect b, and the second of those which have a
common perpendicular ivith it.^

I In Saccheri's work tliere will be found many other inter-
esting theorems before he reaches this result. Of these the

Saccheri's Conclusion.

§ 17. At this point Saccheri attempts to come to a
decision, trusting to intuition and to faith in the validity of
the Fifth Postulate rather than to logic. To prove that the
Hypothesis of the Acute Angle is absolutely false, because it is
repugnant to the 7iature of the straight line [Prop. XXXIIL] he
relies upon five LemmaS;, spread over sixteen pages. In sub-
stance, however, his argument amounts to the statement
that if the Hypothesis of the Acute Angle were true, the
lines p (Fig. 2 6) and b would have a comfnon perpendicular
at their conunon point at i?iftnity, which is contrary to the
nature of the straight lifie. The so-called demonstration of
Saccheri is thus founded upon the extension to irifnity of
certain properties which are valid for figures at a finite
distance.

However, Saccheri is not satisfied with his reasoning
and attempts to reach the wished-for proof by adopting
anew the old idea of equidistance. It is not worth while to
reproduce this second treatment as it does not contain any-
thing of greater value than the discussions of his prede-
cessors.

Stillj though it failed in its aim, Saccheri's work is of
great importance. In it the most determined eftort had been
made on behalf of the Fifth Postulate; and the fact that he
did not succeed in discovering any contradictions among
the consequences of the Hypothesis of the Acute Angle, could
not help suggesting the question, whether a consistent log-
ical geometrical system could not be built upon this hypo-

following is noteworthy: If two straight liiies continually approach
each other and their distance apart remains always greater than a
given segment, then the Hypothesis of the Acute Angle is impossible.
Thus it follows that, if we postulate the absence of asymptotic
straight lines, we must accept the truth of the Euclidean hypo-
thesis.

AA TI. The Forerunners of Non-Euclidean Geometry.

thesis, and the Euclidean Postulate be impossible of demon-
stration.^

Johann Heinrich Lambert [1728 — 1777]-
§ 18. It is difficult to say what influence Saccheri's
work exercised upon the geometers of the iS^li century.
However, it is probable that the Swiss mathematician
Lambert \vas familiar with it, ^ since in his Theorie der Par-
allellitiien [1766] he quotes a dissertation by G. S. Klugel
[1739 — i8i2]3, where the work of the Italian geometer
is carefully analysed. Lambert's Theorie der Fara/lellmien
was published after the author's death, being edited by
J. Bernoulli and C. F. Hindenburg. It is divided into
three parts. The first part is of a critical and philosophical
nature. It deals with the two-fold question arising out of the
Fifth Postulate: whether it can be proved with the aid of
the preceding propositions only, or whether the help of some
other hypothesis is required. The second part is devoted to

1 The publication of Saccheri's work attracted considerable
attention. Mention is made of it in two Histories of Mathematics:
that of J. C. Heilbronner (Leipzig, 1742) and that of Montucla
(Paris, 1758). Further it is carefully examined by G. S. Klugel
in his dissertation noted below (Note (3)). Nevertheless it was
soon forgotten. Not till 1889 did E. Beltrami direct the attention
of geometers to it again in his Note: Un precursore italiatio
di Legendre e di Lobatschewsky. Rend. Ace. Lincei (4), T. V. p. 441
— 448. Thereafter Saccheri's work was translated into English by
G. B. Halsted (Amer. Math. Monthly, Vol. I. 1S94 et seq.); into
German, by Engel and Stackel (77/. der P. 1895); into Italian,
by G. Boccardini (Milan, Hoepli, 1904).

2 Cf. SegrE: Congetture intorno alla influenza di Girolamo
Saccheri sulla forrjiazione della geometria ìion euclidea. Atti Acc.
Scienze di Torino, T. XXXVIIL (1903).

3 Conatiiufn praecipuorum theoriam parallelarum demonstrandi
recensio, guani publico examini submitteni A. G. Kaestner et auctor
respondens G. S. Kliigel, (Gòttingen, 1763).

Lambert's Three Hypotheses.

the discussion of different attempts in which the Euclidean
Postulate is reduced to very simple propositions, which
however, in their turn, require to be proved. The third, and
most important, part contains an investigation resembling
that of Saccheri, of which we now give a short summary/

§ 19. Lambert's fundamental figure is a quadrilateral
with three right angles, and three hypotheses are made as to
the nature of the fourth angle. The first is the Hypothesis
of the Right Angle; the second, the Hypothesis of the Obtuse
Angle; and the third, the Hypothesis of the Acute Angle. Also
in his treatment of these hypotheses the author does not
depart far from Saccheri's method.

^\vt first hypothesis leads easily to the Euclidean system.

In rejecting the second hypothesis, Lambert relies upon
a figure formed by two straight lines a, b, perpendicular to
a third line ^^ (Fig. 27). From points £, B^, B^y.-Bn,
taken in succession upon B Bj B, B„

the line b, the perpen-
diculars, BAy B-,A^, B^A^,
: . B„An are drawn to the
hne a. He proves, in the

first place, that these per- A A.^ Aj An

pendiculars continually ^'^- ^7-

diminish, starting from the perpendicular BA. Next, that
the difterence between each and the one which succeeds it
continually increases.

Therefore we have

BA—BnAn > n {BA—B^A^.

But, if n is taken sufficiently large, the second member

I Cf. Magazin fur reine und angewandte Math., 2. Stuck,
p. 137 — 164. 3. Stuck, p. 325 — 358, (1786). Lambert's work was
again published by Engel and Stackel {Th. der P.) p. 135 — 208,
preceded by historical notes on the author.

aF) li. The Forerunners of Non-Euclidean Geometry.

of this inequality becomes as great as we please {Postulate
of Archimedes] \ whilst the first member is always less than
£A. This contradiction allows Lambert to declare that the
second hypothesis is false.

In examining the third hypothesis, Lambert again avails
himself of the preceding figure. He proves that the perpen-
diculars £A, BxA^, . . B,iAn continually increase, and that
at the same time the difference between each and the one
which precedes it continually increases. As this result does
not lead to contradictions, like Saccheri he is compelled to
carry his argument further. Then he finds, that, on the third
hypothesis the sum of the angles of a triangle is less than
two right angles; and going a step further than Saccheri,
he discovers that the defect of a polygon, that is, the differ-
ence between 2 {n — 2) right angles and the sum of its angles,
is proportional to the area of the polygon. This result can
be obtained more easily by observing that both the area and
the defect of a polygon, which is the sum of several others,
are, respectively, the sum of the areas and of the defects of
the polygons of which it is composed.^

§ 20. Another remarkable discovery made by Lambert
has reference to the measurement of geometrical magnitudes.
It consists precisely in this, that, whilst in the ordinary geo-
metry only a relative meaning attaches to the choice of a

1 The Postulate of Archimedes is again used here in a form
which assumes the infinity of the straight line (cf. Saccheri, Note
P- 37)-

2 It is right to point out that in the Hypothesis af the Aade
Angle Saccheri had already met the defect here referred to, and
also noted implicitly that a quadrilateral, made up of several
others, has for its defect the sum of the defects of its parts (Prop.
XXV). However he did not draw any conclusion from this as to
the area being proportional to the defect.

Relative and Absolute Units.

particular unit in the measurement of lines, in the geometry
founded upon the third hypothesis^ we can attach to it an
absolute meaning.

First of all we must explain the distinction, which is
here introduced, between absolute and relative. In many
questions it happens that the elements, supposed given, can
be divided into two groups, so that those oi i\\Q first grotip
remain fixed, right through the argument, while those of the
second group may vary in a number of possible cases. When
this happens, the explicit reference to the data of the first
group is often omitted. All that depends upon the varying
data is considered relative; all that depends upon the fixed
data is absolute.

For example, in the theory of the Domain, of Ration-
ality, the data of the second group [the variable data] are
taken as certain simple irrationalities [constituting a base\.,
and "ùx^ first group consists simply of unity [i], which is
often passed over in silence as it is common to all domains.
In speaking of a number, we say that it is rational relatively
to a given base, if it belongs to the domain of rationality
defined by that base. We say that it is rational absolutely,
if it is proved to be rational with respect to the base i,
which is common to all domains.

Passing to Geometry, we observe that in every actual
problem, we generally take certain figures as given and
therefore the magnitudes of their parts. In addition to these
variable data [of the second group\ which can be chosen in
an arbitrary manner, there is always implicitly assumed the
presence of the fundamental figures, straight lines, planes,
pencils, etc. [fixed data or of the first group]. Thus, every
construction, every measurement, every property of any
figure ought to be held as relative, if it is essentially relative
to the variable data. It ought, on the other hand, to be
spoken of as absolute, if it is relative only to the fixed data

^8 II- The Forerunners of Non-Euclidean Geometry.

[the fundamental figures], or, if, being enunciated in terms
of the variable data, it only appears to depend upon them,
so that it remains fixed when these vary.

In this sense it is clear that in ordinary geometry the
measurement of lines has necessarily a relative meaning.
Indeed the existence of similar figures does not allow us in
any way to individualize the size of a line in terms of funda-
mental figures [straight line, pencil, etc.].

For an angle on the other hand, we can choose a method
of measurement which expresses one of its absolute pro-
perties. It is sufficient to take its ratio to the angle of a
complete revolution, that is, to the entire pencil, this being
one of the fundamental figures.

We return now to Lambert and his geometry corre-
sponding to the third hypothesis. He observed that with
every segment we can associate a definite angle, which can
easily be constructed. From this it follows that every seg-
ment is brought into correspondence with the fundamental
figure [the pencil]. Therefore, in the new [hypothetical]
geometry, we are entitled to ascribe an absolute meaning
also to the measurement of segments.

To show in the simplest way how to every segment we
can find a corresponding angle, and thus obtain an ab-
solute numerical measurement of lines, let us imagine an
equilateral triangle constructed upon every segment. We
are able to associate with every segment the angle of the
triangle corresponding to it and then the measure of this
angle. Thus there exists a one-one correspondence between
segments and the angles comprised between certain limits.

But the numerical representation of segments thus ob-
tained does not enjoy the distributive property which belongs
to lengths. On taking the sum of two segments, we do not
obtain the sum of the corresponding angles. However, a
function of the angle, possessing this property, can be ob-

The Absolute Unit of Length. aq

tained, and we can associate with the segment, not the said
angle, but this function of the angle. For every value of the
angle between certain limits, such a function gives an absolute
vieasure of segments. The absolute unit of length is that
segment for which this function takes the value i.

Now if a certain function of the angle is distributive in
the sense just indicated, the product of this function and an
arbitrary constant also possesses that property. It is there-
fore clear that we can always choose this constant so that
the absolute unit segment shall be that segment which corre-
sponds to any assigned angle: e. g., 45". The possibility of
constructing the absolute unit segment, given the angle, de-
pends upon the solution of the following problem :

To construct, on the Hypothesis of the Acute Angle, an
equilateral triangle with a given defect.

So far as regards the absolute m.easure of the areas of
polygons, we remark that it is given at once by the defect
of the polygons. We can also assign an absolute measure
for polyhedrons.

But with our intuition of space the absolute measure
of all these geometrical magnitudes seems to us impossible.
Hence if tue deny the existence of an absolute unit for segments,
we can, with Lambert, reject the third hypothesis.

§ 21. As Lambert realized the arbitrary nature of this
statement, let it not be supposed that he believed that he
had in this way proved the Fifth Postulate.

To obtain the wished-for proof, he proceeds with his
investigation of the consequences of the third hypothesis, but
he only succeeds in transforming his question into others
equally difficult to answer.

Other very interesting points are contained in the
Theorie der Parallellinien, for example, the close resemblance

co II. The Forerunners of Non-Euclidean Geometry.

to spherical geometry^ of the plane geometry which would
hold, if the second hypothesis were valid, and the remark that
spherical geometry is independent of the Parallel Postulate,
Further^ referring to the third hypothesis^ he made the follow-
ing acute and original observation: Froin this I should al-
most conchcde that the third hypothesis tvould occur in the case
of an imaginary sphere.

He was perhaps brought to this way of looking at the
question by the formula {A-\-B-\- C — it) r^, which expresses
the area of a spherical triangle. If in this we write for the
radius r, the imaginary radius K -i r we obtain

r^\yi—A—B—C\;
that is, the formula for the area of a plane triangle on
Lambert's third hypothesise

§ 22. Lambert thus left the question in suspense. In-
deed the fact that he did not publish his investigation allows
us to conjecture that he may have discovered another way
of regarding the subject.

Further, ,it should be remarked that, from the general
want of success of these attempts, the conviction began to
be formed in the second half of the a 8th Century that it
would be necessary to admit the Euclidean Postulate^ or
some other equivalent postulate, without proof.

In Germany, where the writings upon the question
followed closely upon each other, this conviction had al-
ready assumed a fairly definite form. We recognize it in
A, G. Kastner,^ a well-known student of the theory of
parallels, and in his pupil, G. S. Klugel, author of the

1 In fact, in Spherical Geometry the sum of the angles of a
quadrilateral is greater than four right angles, etc.

2 Cf. Engel u. Stackel; Th. der P. p. 146.

3 For some information about Kastner, cf. Engel u. StAckel;
Th. der P. p. 139 — 141.

Klùgel's Work. e i

valuable criticism of the most celebrated attempts to de-
monstrate the Fifth Postulate, referred to on p. 44 [note 3].
In this work Klugel finds each of the proposed proofs
insufficient and suggests the possibility of non-intersecting
straight lines being divergent YMoglich ware es freilick, da^
Gerade, die sich nihct schneiden, voiieinander abweiche?i\. He
adds that the apparent contradiction which this presents is
not the result of a rigorous proof, nor a consequence of the
definitions of straight lines and curves, but rather something
derived from experience and the judgment of our senses.
\Dafi so etwas widersinnig ist, wissen wir nicht infolge strenger
Sc/iiusse Oder vcrmoge deutlicher Begriffe V07i der geraden und
der kntmmen Linie, viebnehr durch die Erfahrung und durch
das Urteil unserer Augen].

The investigations of Saccheri and Lambert tend to
confirm Klugel's opinion, but they cannot be held to be
a proof of the impossibihty of demonstrating the Euclidean
hypothesis. Neither would a proof be reached if we proceed-
ed along the way opened by these two geometers, and de-
duced any number of other propositions, not contradicting
the fundamendal theorems of geometry.

Nevertheless that one should go forward on this path,
without Saccheri's presupposition that contradictions would
be found there, constitutes historically the decisive step in the
discovery that Euclìd's Postulate could not be proved, and
in the creation of the Non-Euclidean geometries.

But from the work of Saccheri and Lambert to that of
LoBATSCHEWSKY and B0LYAI, which is based upon the above
idea, more than half a century had still to pass !

The French Geometers tov,;'ards the End of the
i8th Century.
§ 23. The critical study of the theory of parallels,
which had already led to results of great interest in Italy and

(-2 II. The Forerunners of Non-Euclidean Geometry.

Germany, also made a remarkable advance in France to-
wards the end of the iSth Century and the beginning of
the 19th.

D'Alembert [1717 — 1783]; in one of his articles on
geometry, states that 'La definition et les propriétés de la
ligne droite, ainsi que des lignes parallèles sont l'écueil et
pour ainsi dire le scandale des elements de Geometrie.' ^
He holds that with a good definition of the straight line
both difficulties ought to be avoided. He proposes to define
a parallel to a given straight line as any other coplanar
straight line, which joins two points which are on the same
side of and equally distant from the given line. This definition
allows parallel lines to be constructed immediately. However
it would still be necessary to show that these parallels are
equidistant. This theorem was offered, almost as a challenge,
by D'Alembert to his contemporaries.

§ 24. De Morgan, in his Budget of Paradoxes^, relates
that Lagrange [1736 — 1813], towards the end of his life,
wrote a memoir on parallels. Having presented it to the
French Academy, he broke off" his reading of it with the ex-
clamation: 'II faut que j'y songe encore!' and he withdrew
the MSS.

Further Houel states that Lagrange, in conversation
with BiOT, affirmed the independence of Spherical Trigon-
ometry from Euclid's Postulate.-^ In confirmation of this
statement it should be added that Lagrange had made a spe-
cial study of Spherical Trigonometry,'^ and that he inspired.

1 Cf. D'Alembert: Melanges de Littcrature, d'Hisioire, et
de Philosophie, T. V. S II (l7S9)- Also: Encychfédie Méihodiqiie
Mathématique ; T. II. p. 519, Article: Parallèles (1785).

2 A. DE Morgan: A Budget of Paiadoxes,^.\'J2,. (London, 1872).

3 Cf. J. Houel: Essai critique sur les principes fondamenlaux
de la geometrie élèmentaire, p. 84, Note (Paris, G. VJLLARS, 1 883).

4 Cf. Miscellanea Taurinensia, T. II. p. 299—322 (1760 — 5i).

D'Alembert, Lagrange, and Laplace. £2

if he did not write, a memoir ''Sur les principes fondamentaux
de la Mecanique [1760 — i]^, in which Foncenex discussed
a question of independence, analogous to that above noted
for Spherical Trigonometry. In fact, Foncenex shows that
the analytical law of the Composition of Forces acting at a
point does not depend on the Fifth Postulate, nor upon any
other which is equivalent to it.^

§ 25. The principle of similarity, as a fundamental
notion, had been already employed by Wallis in 1663 [cf.
§ 9]. It reappears at the beginning of the 19th Century, sup-
ported by the authority of two famous geometers: L. N. M.
Carxot [1753 — 1823] and Laplace [1749 — 1827].

In a Note [p. 481] to his Geometrie de Position [1803]
Carnot affirms that the theory of parallels is allied to the
principle of similarity, the evidence for which is almost on
the same plane as that for equality, and that, if this idea is
once admitted, it is easy to establish the said theory rigorously.

Laplace [1824] observes that Newton's Law [the Law
of Gravitation], by its simplicity, by its generality and by the
confirmation which it finds in the phenomena of nature, must
be regarded as rigorous. He then points out that one of its
most remarkable properties is that, if the dimensions of all
the bodies of the universe, their distances from each other,
and their velocities, were to decrease proportionally, the
heavenly bodies would describe curves exactly similar to
those which they now describe, so that the universe, reduced
step by step to the smallest imaginable space, would always
present the same phenomena to its observers. These pheno-
mena, he continues, are independent of the dimensions of the
universe, so that the simphcity of the laws of nature only allows
the observer to recognise their ratios. Referring again to this

1 Cf. Lagrange: Oeiivres, T. VIL p. 331 — 2fil-

2 Cf. Chapter VL

Ca II. The Forerunners of Non-Euclidean Geometry.

astronomical conception of space, he adds in a Note: 'The
attempts of geometers to prove Euclid's Postulate on Parallels
have been up till now futile. However no one can doubt this
postulate and the theorems which Euclid deduced from it. Thus
the notion of space includes a special property, self-evident,
without which the properties of parallels cannot be rigorously
established. The idea of a bounded region, e. g., the circle,
contains nothing which depends on its absolute magnitude.
But if we imagine its radius to diminish, we are brought
without fail to the diminution in the same ratio of its circum-
ference and the sides of all the inscribed figures. This pro-
portionality appears to me a more natural postulate than
that of Euclid, and it is worthy of note that it is discovered
afresh in the results of the theory of universal gravitation.' ^

§ 26. Along with the preceding geometers, it is right
also to mention J. B. Fourier [1768 — 1830], for a discussion
on the straight line which he carried on with Monge.^ To
bring this discussion into line with" the investigations on
parallels, we need only go back to D'Alembert's idea that
the demonstration of the postulate can be connected with
the definition of the straight line [cf § 23].

Fourier, who regarded the distance between two points
as a prime notion^ proposed to define first the sphere; then
the plane, as the locus of points equidistant from two
given points;^ then the straight line, as the locus of the
points equidistant from three given points. This method

1 Cf. Laplace. Oeuvres, T. VI. Livre, V. Ch. V. p. 472.

2 Cf. Seances de P Ecole ftormale: De bats, T. I. p. 28 — ^^
(1795). This discussion was reprinted in Mathésis. T. IX. p. 139
-141 (1883)-

3 This definition of the plane was given by Leibnitz about
a century before. Cf. Opuscules et fragtnents incdiis, edited by
L. CouTURAT, p. 554 — 5. (Paris, Alcan, 1903).

Fourier and Lesfendre.

of presenting the problem of the foundations of geometry
agrees with the opinions adopted at a later date by other
geometers, who made a special study of the question of
parallels [W. Bolyai, N. Lobatschewsky, de Tilly]. In
this sense the discussion between Fourier and Monge finds
a place among the earliest documents which refer to NoJi-
Euclidea7i geometry}

Adrien Marie Legendre [1752 — 1833I.

§ 27. The preceding geometers confined themselves to
pointing out difficulties and to stating their opinions upon
the Postulate. Legendre, on the other hand, attempted to
transform it into a theorem. His "investigations, scattered
among the different editions of his Elements de Geometrie
[1794 — 1823], are brought together in his Reflexions sur
différentes manières de démontrer la théorie des paralleles ou
le [the'orème sur la somme des trois afigles du triangle. [Mém.
Ac. Se, Paris, T. XIII. 1833.]

In jthe most interesting of his attempts, Legendre, like
Saccheri, approaches the question from the side of the sum
of the angles of a triangle, which sum he wishes to prove
equal to two right angles.

With this end in view, at the commencement of his work
he succeeds in! rejecting Saccheri's Hypothesis of the Obtuse
Angle, since he estabhshes that the sum of the angles of any
triangle is either less than ^Hypothesis of the Acute Angle] or
equal to {Hypothesis of the Right Angle] two right angles.

We reproduce a neat and simple proof which he gives
of this theorem :

Let n equal segments ^1^2, -^2^3, • . . ^«^«+1 be taken

I To this we add that later memoirs and investigations
showed that Fourier's definition also fails to build up the Eucli-
dean theory of parallels, without the help of the Fifth Postulate,
or some other equivalent to it.

c5 II. The Forerunners of Non-Euclidean Geometry.

one after the other on a straight Hne [Fig. 28]. On the same
side of the Hne let n equal triangles be constructed, having
for their third angular points B^B^. . . .B,f The segments
Bj_B2i BiB^,... Bn—x Bni which join these vertices, are equal
and can be taken as the bases of n equal triangles, B^A2B2,

B, E^ B3 B^ B^^2^J^3'--- -^«-i

A,iB„. The figure

is completed by

adding the triangle

which is equal to
the others.

Let the angle ^i of the triangle A^B^Az be denoted by
P, and the angle A2 of the consecutive triangle by a.
Then p < a.

In fact, if P ^ a, by comparing the two triangles A^B^Az
and B1A2B2, which have two equal sides, we would deduce
A,A2>B,B2.
Further, since the broken line A^B^Bz . . . -^«+1 ^«-f i
is greater than the segment AiA^-^i ,

A^Bx + n. B^B^ + ^„+i ^«+1 > n. A^A^,
i. e., 2 Aj,Bi^n{AiA2 — B^B^).
But if n is taken sufficiently great, this inequality con-
tradicts the Postulate of Archimedes.

Therefore A^A^ is not greater than B^Bz ,
and it follows that it is impossible that P i>> a.
Thus we have P < a.

From this it readily follows that the sum of the angles ot
the triangle A^B^A^ is less than or equal to two right angles.
This theorem is usually, but mistakenly, called Legendre's
First Theorem. We say mistakenly, because Saccheri had
already established this theorem almost a century earlier [cf
p. 38] when he proved that the Hypothesis of the Obtuse
Angle was false.

Lea:endre's First Proof.

The theorem usually called Legendre's Second Theorem
was also given by Saccheri, and in a more general form
[cf. p. 29]. It is as follows:

If the sum of the angles of a triangle is less than or
equal to two right afigles in only one triangle, it is respectively
less than or equal to two right angles in every other triangle.

We do not repeat the demonstration of this theorem, as
it does not differ materially from that of Saccheri.

We shall rather show how Legendre proves that the
sum of the three angles of a tria?igle is equal to two right
angles.

Suppose that in the triangle ABC [cf. Fig. 2 9]
^A ■\- <iB -\r <fi C<C 2 right angles.

A point D being taken on AB, the transversal DE is
drawn, making the angle ADE
equal to the angle B. In the quadri-
lateral DBCE the sum of the angles
is less than 4 right angles.

Therefore ^AED^^ACB.
The angle E of the triangle ADE
is then a perfectly definite [decreas-
ing] function of the side AD: or,
what amounts to the same thing, the
length of the side AD is fully determined when we know the
size (in right angles) of the angle E, and of the two fixed
angles A, B.

But this result Legendre holds to be absurd, since the
length of a line has not a meaning, unless one knows the unit
of length to which it is referred, and the nature of the question
does not indicate this unit in any way.

In this way the hypothesis

<^A-^ ^B + -^ C< 2 right angles
is rejected, and consequently we have

<^A + <f:B + ^C=2 right angles.

II. The Forerunners of Non-Euclidean Geometry,

Also from this equality the proof of Euclid's Postulate
follows easily.

Legendre's method is thus based upon Lambert's postu-
late, which denies the existence of an absolute 7init segment.

§ 28. In another demonstration Legendre makes use of
the hypothesis:

From any point whatever, taken within an angle, we can
always draw a straight line which 7vill cut the two arms of
the angled

He proceeds as follows:

Let ABC he a triangle, in which, if possible, the sum of
the angles is less than two right angles.

Let 2 right angles— <^ A—^£— <^ C= a [the defect].
Find the point A', symmetrical to A, with respect to the
side BC. [cf. Fig. 30.]

The defect of the new tri-
angle BCA' is also a. In virtue
of the hypothesis enunciated
above, draw through A' a
transversal meeting the arms
of the angle A in Bj^ and C^.
It can easily be shown that the
defect of the triangle AB^ C^ is
the sum of the defects of the
four triangles of which it is
composed, [cf. also Lambert p. 46.]

Thus this defect is greater than 2 a.
Starting now with the triangle ABiQ and repeating the
same construction, we get a new triangle whose defect is
greater than 4 a.

Fis

I J. F. Lorenz had already used this hypothesis for the same
purpose. Cf. GnaidnjS der reinoi unci angewandlen Mathcmatik,
(Helmstedt, 1791).

Lesjendre's Second Proof.

After n operations of this kind a triangle will have been
constructed whose defect is greater than 2" a.

But for n sufficiently great, this defect, 2" a, must be
greater than 2 right angles [Postulate of Archimedes], which
is absurd.

It follows that (X = o, and ■^A-^^B-^^C=2
right angles.

This demonstration is founded upon the Postulate of
Archimedes. We shall now show how we could avoid using
this postulate [cf. Fig. 31].

Let AB and HK be two straight lines, of which AB
makes an acute angle, and HK a right angle, with AH.

Fig. 31-

Draw the straight line AB' symmetrical to AB with re-
gard to AH. Through the point H there passes, in virtue of
Legendre's hypothesis, a line r which cuts the two arms of
the angle BAB' . If this line is different from HK^ then also
the line /, symmetrical to it with respect to AH, enjoys the
same property of intersecting the arms of the angle. It fol-
lows that the line HK also meets them.

Thus the line perpendicular to AH and a line making
an acute angle with AH always meet.

From this result the ordinary theory of parallels follows,
and <5C^ + <^^+ ^C= 2 right angles.

In other demonstrations Legendre adopts the methods
of analysis and also makes an erroneous use of infinity.

6o n. The Forerunners of Non-Euclidean Geometry.

By these very varied investigations Legendre believed
that he had finally removed the serious difficulties surrounding
the foundations of geometry. In substance, however, he
added nothing new to the material and to the results ob-
tained by his predecessors. His greatest merit lies in the
elegant and simple form which he was able to give to all his
writings. For this reason they gained a wide circle of readers
and helped greatly to increase the number of disciples of the
new ideas, which at that time were beginning to be formed.

Wolfgang Bolyai [1775 — 1856].

§ 29. In this article we come to the work of the Hungarian
geometer W. Bolvai. His interest in the theory of parallels
dates back to the time when he was a student at Gottingen
[1796 — 99], and is probably due to the advice of Kastner
and of his friend, the young Professor of Astronomy, K. F.
Seyffer [1762 — 1822].

In 1804 he sent Gauss, formerly one of his student
friends at Gottingen, a Theoria Parallelarum, which contained
an attempt at a proof of the existence of equidistant straight
lines.^ Gauss showed that this proof was fallacious. Bolvai
however, did not on this account give up his study of Axiom
XL, though he only succeeded in substituting for it others,
more or less evident. In this way he came to doubt the possib-
ility of a demonstration and to conceive the impossibility
of doing away with the Euclidean hypothesis. He asserted
that the results derived from the denial of Axiom XI
could not contradict the principles of geometry, since the
law of the intersection of two straight lines, in its usual

I The Theoria Parallelarum was written in Latin. A German
translation by Engel and StAckel appears in Math. Ann. Bd.
XLIX. p. 168—205 (1897).

\V. Bolyai's Postulate.

form, represents a new datum, independent of those which
precede it.^

Wolfgang brought together his writings on the principles
of mathematics in tlie work: Tentamen juventutem studiosam
in elementa Matheseos [1832 — 33]; and in particular his in-
vestigations on Axiom XI., while in each attempt he pointed
out the new hypothesis necessary to render the demon-
stration rigorous.

A remarkable postulate to which Wolfgang reduces
Euclid's is the following:

Four povits, not on a plane, always lie 2ip07i a sphere;
or, what amounts to the same thing: A circle can always be
dratvn through three points not on a straight lifie.^

The Euclidean Postulate can be deduced from this as
follows [cf. Fig. 32]:

Let AA, BB' be two straight lines, one of them being
perpendicular to AB., and the other inclined to it at an acute
angle.

If we take a point M on the seg-
ment AB between A and ^, and
the points M' M". symmetrical to M
with respect to the lines BB' and
AA , we obtain two points M' , M"
not in the same straight line with M.
These three points M, M\ M" lie
on the circumference of a circle. Also
the lines AA , BB' must intersect,
since they both pass through the cen- Fig. 32.

tre of this circle.

But from the fact that a line which is perpendicular to

1 Cf. StackeL: Die Enideckiing der nichteuklidischen Geometrie
diirch y. Bolyai, Math. u. Naturw. Ber. aus Ungarn, Bd. XVII. (1901).

2 Cf. W. BoLYAi: Kurzer Grundriss eines Vermchs etc., p. 46.
(Maros Vàsarhely, '85 r).

52 !!• The Forerunners of Non-Euclidean Geometry.

another straight line and a line which cuts it at an acute angle
intersect, it follows immediately that there can be only one
parallel.

Friedrich Ludwig Wachter [1792 — 1817].

§ 30. When it had been seen that the Euclidean Postulate
depends on the possibiHty of a circle being drawn through
any three points not on a straight line, the idea at once sug-
gested itself that the existence of such a circle should be
established as a preliminary to any investigation of parallels.

An attempt in this direction was made by F. L. Wachter.

Wachter, a student under Gauss in Gottingen [1809],
and Professor of Mathematics in the Gymnasium of Dantzig,
had made several attempts at the demonstration of the Postu-
late. He believed that he had been successful, first in a letter
to Gauss [Dec, 1816], and later, in a tract, printed at Dantzig
in 1817.'

In this pamphlet he seeks to establish that given any four
points in space, (not on a plane), a sphere will pass through
them. He makes use of the following postulate :

Any four points of space fully determifie a surface [the
surface of four poifits], and two of these surfaces intersect in a
single line^ completely determi?icd by three points.

There is no advantage in following the argument by
means of which Wachter seeks to prove that the surface of
four points is a sphere, since he fails to give a precise defini-
tion of that surface in his tract. His deductions have thus
only an intuitive character.

On the other hand a passage in his letter of 1816 de-
serves special notice. It was written after a conversation with
Gauss, when they had spoken of an Anti- Euclidean Geometry.
In this letter he speaks of the surface to which a sphere tends

I Demonstratio axiomatis geometrici in Euclideis undechni.

Wachter and Thibaut. 63

as its radius approaches infinity, a siirface on the Euclidean
hypothesis identical with a plane. He affirms that eveti in the
case of the Fifth Postulate being false, there would be a geo-
metry on this surface identical with that of the ordifiary plane.
This statement is of the greatest importance as it con-
tains one of the most remarkable results which hold in the
system of geometry^ corresponding to Saccheri's Hypo-
thesis of the Acute Angle [cf. Lobatschewsky, § 40].'

Bernhard Friedrich Thibaut [1775 — 1832].

§ 30 (bis). One other erroneous proof of the theorem that the
sum of the angles of a triangle is equal to two right angles should
be mentioned, since it has recently been revived in English textbooks,
and to some extent received official sanction. It depends upon
the idea of diredion, and assumes that translation and rotation are
independent operations. It is due to Thibaut [Gì-icndrij] der reincn
Mathetnatik, 2. Aufl., Gottingen, 1809). Gauss refers to this "proof"
in his correspondence with Schumacher, and shows that it involves
a proposition which not only needs proof, but is, in essence, the
very proposition to be proved. Thibaut argued as follows : 2 —

"Let ABC be any triangle whose sides are traversed in order
from A along AB, BC, CA. While going from ^ to i? we always
gaze in the direction ABb [AB being produced to b), but do not
turn round. On arriving at B we turn from the direction Bb by a
rotation through the angle bBC, until we gaze in the direction BCc.
Then we proceed in the direction BCc as far as C, where again
we turn from Cc to CAa through the angle cCA; and at last arriving
at A, we turn from the direction Aa to the first direction AB
through the external angle aAB, This done, we have made a
complete revolution, — just as if, standing at some point, we had
turned completely round; and the measure of this rotation is 2 ir.
Hence the external angles of the triangle add up to 2 ir, and the
internal angles A-\- B -\- C = -n. Q. E. D."

1 With regard to Wachter, cf. P. StAckel: Friedrich Ludwig
Wachter, ein Beitrag zur Geschichte der nichtetiklidischen Geometrie.
Math. Ann. Bd. LIV. p. 49—85. (1901). In this article are reprinted
Wachter's letters upon the subject and the tract of 1S17 referred
to above.

2 [For further discussion of this "proof" see W. B. Frank-
LÀNd's Theories of Parallelism, (Camb. Univ. Press, 19 lo), from which
this version is taken, and Heath's Euclid, Vol. I., p. 321.]

Chapter III.
The Founders of Non-Euclidean Geometry.

Carl Friederich Gauss [1777 — 1855].

§ 31. Twenty centuries of useless effort, and in particular
the last unsuccessful investigations on the Fifth Postulate, con-
vinced many of the geometers, who flourished about the be-
ginning of last century, that the final settlement of the theory
of parallels involved a problem whose solution was impossible.
The Gottingen school had officially declared the necessity
of admitting the Euclidean hypothesis. This view, expressed
by Klugel in his Conatuum [cf p. 44] was accepted and sup-
ported by his teacher, A. G. Kastner, then Professor in the
University of Gottingen.^

Nevertheless keen interest was always taken in the
subject; an interest which still continued to provide those
who sought for a proof of the postulate with fruitless labour,
and led finally to the discovery of new systems of geometry.
These, founded like ordinary geometry on intuition, extend
into a far wider field, freed from the principle embodied in
the Euclidean Postulate.

How difficult was this advance towards the new order
of ideas will be clear to any one who carries himself back to
that period, and remembers the trend of the Kantian Philo-
sophy, then predominant.

§ 32. Gauss was the first to have a clear view of a
geometry independent of the Fifth Postulate, but this re-

I Cf. Enc.el u. StAckel: Tit. der P. p. 139—142,

Gauss and W. Bolyai. gc

mained for quite fifty' years concealed in the mind of the
great geometer, and was only revealed after the works of
LoBATSCHEWSKY [1829 — 30] and J. Bolyai [1832] appeared.

The documents which allow an approximate reconstruct-
ion of the lines of research followed by Gauss in his work
on parallels, are his correspondence with W. Bolyai, Olbers,
Schumacher, Gerling, Taurinus and Bessel [1799 — 1844];
two short articles in the Goti, gelehrten Anzeigm{\2>i6^ 1822];
and some notes found among his papers, [1831].^

Comparing the various passages in Gauss's letters, we
can fix the year 1792 as the date at which he began his 'Med-
itations'.

The following portion of a letter to W. Bolyai [Dec. 1 7,
1799] proves that Gauss, Hke Saccheri and Lambert before
him, had attempted to prove the truth of Postulate V. by as-
suming it to be false.

'As for me, I have already made some progress in my
work. However the path I have chosen does not lead at
all to the goal which we seek, and which you assure me you
have reached.3 It seems rather to compel me to doubt the
truth of geometry itself.

'It is true that I have come upon much which by most
people would be held to constitute a proof: but in my eyes
it proves as good as nothing. For example, if one could
show that a rectilinear triangle is possible, whose area would
be greater than any given area, then I would be ready to
prove the whole of geometry absolutely rigorously.

'Most people would certainly let this stand as an Axiom;
but I, no! It would, indeed, be possible that the area might

1 [It would be more correct to say over thirty.]

2 Cf. Gauss, Werke, Bd. VIE. p. 157—268.

3 It is to be remembered that W. Bolyai was working at
this subject in Gottingen and thought he had overcome his diffi-
culties. Cf. 3 29.

^S III. The Founders of Non-Euclidean Geometry.

always remain below a certain limit, however far apart the
three angular points of the triangle were taken.'

In 1804, replying to W. Bolyai on his Theoria parall-
elamm, he expresses the hope that the obstacles by which
their investigations had been brought to a standstill would
finally leave a way of advance open.^

From all this, Stackel and Engel, who collected and
verified Gauss's correspondence on this subject, come to the
conclusion that the great geometer did not recognize the
existence of a logically sound Non-Euclidean geometry by
intuition or by a flash of genius : that, on the contrary, he
had spent upon this subject many laborious hours before he
had overcome the inherited prejudice against it.

Did Gauss, when he began his investigations, know the
writings of Saccheri and Lambert? What influence did they
exert upon his work? Segre, in his Congetture^ already re-
ferred to [p. 44 note 2], remarks that both Gauss and W.
Bolyai, while students at Gottingen, the former from 1795
— 98, the later from 1796 — 99, were interested in the theory
of parallels. It is therefore possible that, through Kastner
and Seyffer, who were both deeply versed in this subject
they had obtained knowledge both of the Euclides ab omni
naevo vindicatus and of the Theorie der Faralleiiinien. But
the dates of which we are certain, although they do not con-
tradict this view, fail to confirm it absolutely.

§ 33. To this first period of Gauss's work, after 1 8 1 3
there follows a second. Of it we obtain some knowledge
chiefly from a few letters, one written by Wachter to Gauss
[18 1 6]; others [sent |by Gauss to Gerling [i8i9],jTaurinus
[1824] and Schumacher [183 i]; and also from some notes
found among Gauss's papers.

I [It should be noticed that these efforts were still directed
towards proving the truth of Euclid's postulate.]

Gauss's "Meditations".

These documents show us that Gauss, in this second
period, had overcome his doubts, and proceeded with his de-
velopment of the fundamental theorems of a new geometry,
which he first czWs, Anti-Euclidean [cf.WACHTER's letter quoted
on p. 62]; then Astral Geometry [following Schweikart, cf.
p. 76]; dina ^nsWy, Non-Euclidean [cf letter to Schumacher].
Thus he became convinced that the Non-Euclidean Geometry
did not in itself involve any contradiction, though at first
sight some of its results had the appearance of paradoxes
[letter to Schumacher, July 12, 183 1].

However Gauss did not let any rumour of his opinions
get abroad, being certain that he would be misunderstood.
[He was afraid of the clamour of the Boeotiatis; letter to Bessel,
Jan. 27, 1829]. Only to a few trusted friends did he reveal
something of his work. When circumstances compel him to
write to Taurinus [1824] on the subject, he begs him to
keep silence as to the information which he imparted to him.

The notes found among Gauss's papers contain two
brief synopses of the new theory of parallels, and probably
belong to the projected exposition of the Non-Euclidean Geo
metry, with regard to which he wrote to Schumacher [on
May 17, 1 831]: *In the last few weeks I have begun to put
down a few of my own Meditations, which are already to
some extent nearly 40 years old. These I had never put in
writing, so that I have been compelled three or four times
to go over the whole matter afresh in my head. Also I wished
that it should not perish with me.'

§ 34. Gauss defines parallels as follows : ^
If the coplanar straight lines AM, BN, do not intersect
each ether, while, on the other hand, every straight line through

I [In this section upon Gauss's work on Parallels fuller use
has been made of the material in his Collected Works (Gauss,
Werke, Bd. VIII, p. 202—9)].

Fig. 33-

68 III. The Founders of Non-Euclidean Geometry.

A between AM and AB cuts BN, then AM is said to be paral-
lel to BN{^g. ii\

He supposes a straight
B !—-.__ line passing through A^ to

start from the position AB,
and then to rotate continu-
ously on the side towards
^^^ which BN is drawn, till it
reaches the position AC^ in
Cèjt BA produced. This line be-

gins by cutting j^iVand in the
end it does not cut it. Thus
there can be one and only
one position, separating the lines which intersect ^iVfrom
those which do not intersect it. This must be "ùxt first of the
lines, which do not cut BN: and thus from our definition it
is the parallel AM) since there can obviously be no last line
of the set of lines which intersect BN.

It will be seen in what way this definition differs from
Euclid's. If Euclid's Postulate is rejected, there could be dif-
ferent lines through A, on the side towards which BN is
drawn, which would not cut BN. These lines would all be
parallels to BN according to Euclid's Definition. In Gauss's
definition only the first of these is said to be parallel
\.oBN.

Proceeding with his argument Gauss now points out
that in his definition the starting points of the lines AM and
BN are assumed, though the lines are supposed to be pro-
duced indefinitely in the directions of AM and BN.

I. He proceeds to show that the parallelism of the line
AM to the line BN is independent of the points A a?id B, pro-
vided the sense in which the lines are to be produced indefinitely
remain the same.

It is obvious that we would obtain the same parallel AM

Gauss's Theory of Parallels.

if we kept A fixed and took instead of B another point B'
on the line BN, or on that Hne produced backwards.

It remains to prove that \i AMis parallel to BJV (or the
point A, it is also the parallel to BNiox any point upon AM,
or upon AM produced backwards.

Instead of ^ [Fig. 34] take another starting point A' upon
AM. Through A\ between
A'B and A'M, draw the line
A'F in any direction. B|<

Through Q, any point on
A'F, between A' and F, draw
the line AQ.

Then, from the definition, A ■
AQ must cut BN, so that it
is clear QF must also cut
BN.

Thus AA'M is the first of
the lines which do not cut BN, and A'M is parallel to BN.
Again take the point A' upon AM produced backwards
[Fig- 35]-

Fig- 34.

Fig. 35-

Draw through A', between A'B and A'M, the line A'F
in any direction.

Produce A'F backwards and upon it take any point Q.
Then, by the definition, QA must cut BN, for example,

III. The Founders of Non-Euclidean Geometry.

in R. Therefore AP lies within the closed figure AARB,
and must cut one of the four sides AA^ AR, RB, and BA.
Obviously this must be the third side RB, and therefore
AM is parallel to BN.

II. The Reciprocity of the Parallelism can also be estab-
lished.

In other words, if AM is parallel to BN, then BN is
also parallel to AM.

Gauss proves this result as follows:

From any point B upon BN draw BA perpendicular to
AM. Through B draw any line BN' between BA and BN.

At B, on the same side of AB as BN, make
<^ ABC^ V2 ^N'BN
There are two possible cases:

Case (i), when BC cuts AM [cf. Fig. 36].

Case (ii), when BC does not cut AM [cf. Fig. 37].

Fig. 36.

Case (i). Let BC cut AM in D. Take AE = AD, and
join BE. Make ^BDF^^BED.

Since AM is parallel to BN^ DF must cut BM^ for
example, in G.

From EM cut off EH equal to DG.

Then, in the triangles BEH s^nà BDG, it follows that

Gauss's Theory of Parallels (contd.).

JM.

Therefore «^ EBD = ^HBG.
But <^ EBD = ^N'BN.
Therefore BJV and BII coincide, and BN' must cut

But BN" is any line through B, between BA and BN.
Therefore BN is parallel to AM.

Fig. 37-

Case (ii). In this case let Z> be any arbitrary point upon
AM. Then with the same argument as above,
■^ EBB = <^ GBH,

But ^ ABD < < ^^C.

Therefore <^ ^^Z> < ^ iV^'^iV.

Therefore <^ GBH<.^N'BN.

Therefore BN' must cut AM.

But ^iV" is any line through B, between BA and BN.

Therefore BN is parallel to AM.

Thus in both cases we have proved that \iAM\5 parallel
to BN, then BN is parallel to AM. "■

The next theorem proved by Gauss in this synopsis is
as follows:

[I Gauss's second proof of this theorem is given in the German
translation. However it will be found that in it he assumes that BC
cuts AM, and to prove this the argument used above is necessary.]

72 III- The Founders of Non-Euclidean Geometry.

III. If the line (i) is parallel to the line (2) arid to the
line (3), then (2) and (3) are parallel to each other.

Case (i). Let the line (i) lie between (2) and (3) [cf.

Fig. 38].

Let A and B be two points on (2) and (3), and let AB
cut (i) in C.

Through A let an arbitrary line AD be drawn between

AB and (2). Then it must
^é: ^ cut (i), and on being pro-

duced must also cut (3).

Since this holds for every
line such as AD, (2) is
parallel to (3).

Case (ii). Let the line

(i) be outside both (2) and

(3), and let (2) He between

(i) and (3) [cf. Fig. 39].

If (2) is not parallel to (3), through any point chosen at

random upon (3), a line different from (3) can be drawn

which is parallel to (2).

This, by Case (i), is also par-
allel to (i), which is absurd.
This short Note on Parall-
els closes with the theorem
that if tivo lities AM and BN
are parallel, these lines produced
backwards cannot tneet.
From all this it is evident that the parallelism of Gauss
xtitzxis parallelism in a given sense. Indeed his definition of
parallels deals with a line drawn from A on a. definite side of
the transversal AB: e. g., the ray drawn to the right, so that
we might speak of AM as the parallel to BJV towards the right.
The parallel from A to BJV towards the left is not necessari-
ly AM. If it were, we would obtain the Euclidean hypothesis.

Fig. 38.

?• 39-

Corresponding Points.

The two lines, in the third theorem, which are each pa-
rallel to a third line, are thus both parallels in the same sense
(both left-hand, or both right-hand parallels).

In a second memorandum on parallels, Gauss goes over
the same ground, but adds the idea of Corresponding Points
on two parallels AA , BB' . Two points A, B are said to corre-
spond^ when AB makes equal internal angles with the parallels
en the same side [cf. Fig. 40].

Fig. 40.

Fig. 41.

With regard to these Corresponding Points he states the
following theorems:

(i) If A, B are two correspofiding points upon tivo paral-
lels, and M is the middle poitit of AB, the line MN, perpen-
dicular to AB, is parallel to the two given lines, and every
point on the same side of MN as A is nearer A than B.

(ii) If A, B are two corresponding points upon the
parallels {\) and {2), and A', B' two other correspo7iding points
on the same lifies, then AA = BB', and co?iversely.

(iii) If A, B, C are three points on the parallels (i), (2)
and (3), such that A and B, B and C, correspond, then A and
C also correspond.

>jA III. The Founders of Non-Euclidean Geometry.

The idea of Corresponding Points, when taken in con-
nection with three Hnes of a pencil (that is, three concurrent
lines [cf. Fig. 41] allows us to define the circle as the locus of
the points on the lines of a pencil which correspond to a given
point. But this locus can also be constructed when the lines
of the pencil are parallel. In the Euclidean case the locus
is a straight line : but putting aside the Euclidean hypothesis,
the locus in question is a line, having many properties in
common with the circle, but yet not itself a circle. Indeed if
any three points are taken upon it, a circle cannot be drawn
through them. This line can be regarded as the limiting case
of a circle, when its radius becomes infinite. In the Non-
Euclidean geometry of Lobatschewsky and Bolyai, this locus
plays a most important part, and we shall meet it there under
the name of the Horocycle.'

This work Gauss did not need to complete, for in 1832
he received from Wolfgang Bolyai a copy of the work of
his son Johann on Absolute Geometry.

From letters before and after the date at which he
interrupted his work, we know that Gauss had discovered in
his geometry an Absolute Unit of Length [cf. Lambert and
Legendre], and that a constant k appeared in his formulae,
by means of which all the problems of the Non-Euclidean
Geometry could be solved [letter to Taurinus, Nov. 8,
1824].

Speaking more fully of these matters in 1831 [letter to

I [Lobatschewsky ; Gremkreis, Courbe-Umite or Iloricycle. BOL-
YAI; Parazykl, L-lÌ7iie.

It is interesting to notice that Gauss, even at this date,
seems to have anticipated the importance of the Ilorocycle. The
definition of Corresponding Points and the statement of their
properties is evidently meant to form an introduction to the dis-
cussion of the properties of this curve, to which he seems to have
given the name Trope.']

The Perimeter of a Circle. 7C

Schumacher], he gave the length of the circumference of a
circle of radius r in the form

■nk\e^—e ^) .

With regard to k, he says that, if we wish to make the new
geometry agree wth the facts of experience, we must suppose
k infinitely great in comparison with all known measurements.
For >è ^ 00 , Gauss's expression takes the usual form
for the perimeter of a circle. ' The same remark holds for the
whole of Gauss's system of geometry. It contains Euclid's
system, as the limiting case, when /è = 00 . ^

Ferdinand Karl Schweikart [1780 — 1859].

§ 35. The investigations of the Professor of Jurispru-
dence, F. K. ScHWEiKART,3 date from the same period as
those of Gauss, but are independent of them. In 1807 he
published Die Theorie der Parallellinien nebst dem Vorschlage
ihrer Verbannung aus der Geometrie. Contrary to what one
might expect from its title, this work does not contain a
treatment of parallels independent of the Fifth Postulate,
but one based on the idea of the parallelogram.

But at a later date, Schweikart, having discovered a
new order of ideas, developed a geometry independent of
Euclid s hypothesis. When in Marburg in December, 1818,
he handed the following memorandum to his colleague Ger-
LiNG, asking him to communicate it to Gauss and obtain his
opinion upon it:

1 To show this we need only use the exponential series.

2 For other investigations by Gauss, cf. Note on p. 90.

3 He studied law at Marburg and from 1796 — 98 attended the
lectures on Mathematics given in that University by Professor J. K,
F. Hauff, the author of various memoirs on parallels, cf. Th. der
P. p. 243.

n^ III. The Founders of Non-Euclidean Geometry.

Memorandum.

'There are two kinds of geometry— a geometry in the
strict sense — the Eudidean; and an astral geometry [astra-
hsche Grofienlehre].

'Triangles in the latter have the property that the sum
of their three angles is not equal to two right angles.'

'This being assumed, we can prove rigorously:

a) That the sum of the three angles of a triangle is less
than two right angles;

b) that the sum becomes ever less, the greater the area
of the triangle;

c) that the altitude of an isosceles right-angled triangle
continually grows, as the sides increase, but it can
never become greater than a certain length, which
I call the Cofistant.

Squares have, therefore, the following form [Fig. 42].
'If this Constant were for us the Radius of the Earth,
(so that every line drawn in the
universe from one fixed Star
to another, distant 90° from the
first, would be a tangent to the
surface of the earth), it would be
infinitely great in comparison with
the spaces which occur in daily
life.

'The Euclidean geometry holds
only on the assumption that the
Constant is infinite. Only in this
case is it true that the three angles of every triangle are equal
to two right angles: and this can easily be proved, as soon
as we admit that the Constant is infinite.' ^

Schweikart's Astral Geometry and Gauss's Non-Euclid-

Fig. 42.

Schweikart's Work. nj

ean Geometry exactly correspond to the systems of Sac-
CHERi and Lambert for the Hypothesis of the Acute Angle.
Indeed the contents of the above memorandum can be ob-
tained directly from the theorems of Saccheri, stated in
Klùgel's Conatuum, and from Lambert's Theorem on the
area of a triangle. Also since Schweikart in his Theorie of
1807 mentions the works of the two latter authors, the direct
influence of Lambert, and, at least, the indirect influence of
Saccheri upon his investigations are established.^

In March, 1 8 1 9 Gauss replied to Gerling with regard
to the Astral Geometry. He compliments Schweikart, and
declares his agreement with all that the sheet of paper sent
to him contained. He adds that he had extended the Astral
Geometry so far that he could completely solve all its pro-
blems, if only Schweikart's Constant were given. In con-
clusion, he gives the upper limit for the area of a triangle
in the form J

[log hyp (I + \2)Y '
Schweikart did not publish his investigations.

Franz Adolf Taurinus [1794 — 1874].
§ 36. In addition to carrying on his own investigations
on parallels, Schweikart had persuaded [1820] his nephew
Taurinus to devote himself to the subject, calling his atten-

1 Cf. Gauss, Werke, Bd. VIII, p. iSo— 181.

2 Cf. Segre's Congetture, cited above on p. 44.

3 The constant which appears in this formula is Schweikart's
Constant C, not Gauss-'s constant /', in terms of which he expressed
the length of the circumference of a circle, (cf. p. 75). The two
constants are connected by the following equation:

log (1+1/2)-

78 in. The Founders of Non-Euclidean Geometry.

tion to the Astral Geometry, and to Gauss's favourable ver-
dict upon it.

Taurinus appears to have taken up the subject seriously
for the first time in 1824, but with views very different from
his uncle's. He was then convinced of the absolute truth of
the Fifth Postulate, and always remained so, and he cherish-
ed the hope of being able to prove it. FaiHng in his first at-
tempts, under the influence of Gauss and Schweikart, he
again began the study of the question. In 1825 he publish-
ed a Theorie der Parallellinien^ containing a treatment of the
subject on Non-Euclidean lines, the rejection oi the Hypothesis
of the Obtuse Angle, and some investigations resembling those
of Saccheri and Lambert on the Hypothesis of the Acute
Angle. He found in this way Schweikart's Constant, which
he called a Parameter. He thought an absolute unit of
length impossible, and concluded that all the systems, corre-
sponding to the infinite number of values of the parameter,
ought to hold simultaneously. But this, in its turn, led to con-
siderations incompatible with his conception of space, and
thus Taurinus was led to reject the Hypothesis of the Acute
Angle while recognising the logical compatibility of the propo-
sitions which followed from it.

In the next year Taurinus published his Geometriae Pri-
ma Elementa [Cologne, 1826], in which he gave an improved
version of his researches of 1825. This work concludes with
a most important appendix, in which the author shows how
a system of analytical geometry could be actually constructed
on the Hypothesis of the Acute Angle. ^

With this aim Tauriuus starts from the fundamental for-
mula of Spherical Trigonometry —

I For the final influence of Saccheri and Lambert upon Tau-
rinus, cf. SeGRE's Congetture, quoted above on p. 44.

The Work of Taurinus. yg

a b C.Ò.C .

COS -r = COS -7 COS -r + sm ^ sm -, cos A,

In it he transforms the real radius k into the imaginary radius

ik. Using the notation of the hyperboHc functions, we thus

have

(i) cosh -T = cosh — cosh — sinh — sinh -^ cos A.

This is the fundamental formula of the Logarithmic-
Spherical Geometry \logarithmisch-spharischen Geometrie'\ of
Taurinus.

It is easy to show that in this geometry the sum of the
angles of a triangle is less than 180°. For simplicity we take
the case of an equilateral triangle, putting a=b=c in (i).

Solving, for cos A, we obtain

cosh —
(i*) cos ^ =

cosh— + I

But sech T<C I-

Therefore cos ^ ]> ^/a-

Thus A is less than 60°, and the sum of the angles of
the triangle is less than 180°.

It is instructive to note, that, from (i*).
Lt. (cos A) = Vz.

a == o

So that in the Hmit when a becomes zero, A is equal to 60°.
Therefore, in the log. -spherical geotnetry, the sum of the angles
of a triangle tends to x8o° when the sides tend to zero.
We may also note that from (i*)

Lt. (cos A) = V2 ;

k «
so that in the limit when k is infinite, A is equal to 60°. There-
fore, when the constant k tends to infinity, the angles of the
equilateral triangle are each equal to 60°, as in the ordinary
geometry.

8o ni. The Founders of Non-Euclidean Geometry.

More generally, using the exponential forms for the hy-
perbolic functions, it will be seen that in the limit when k is
infinite (i) becomes

a^ = b"^ -^ c- — 2bc cos A,

the fundamental formula of Euclidean Plane Trigonometry.

§ 37. The second fundamental formula of Spherical
Trigonometry,

cos A = — cos B cos C + sin ^ sin C cos -y >

by simply interchanging the cosine with the hyperbolic cosine,

gives rise to the second fundamental formula of the log.-spher-

zVa/ geometry:

(2) COS A = — cos B cos C + sin B sin C cosh -r.

For A = o and C= 90°, we have

(3) cosh X = •' ^"
^■^ k sin B

The triangle corresponding to this formula has one angle
zero and the two sides containing it are infinite and parallel
[asymptotic]. [Fig. 43.] The angle B^ between the side which

Fig. 43-

is parallel and the side which is perpendicular to CA, is seen
from (3) to be a function of a. From this onward we can
call it the Angle of Parallelism for the distance a [cf. Lobat-
SCHEWSKY, p. 87].

For B = 45°, the segment BC^ which is given by (3), is
Schweikart's Constant [cf. p. 76]. Thus, denoting it by P,

The Angle of Parallelism. gl

cosh ^ = V2,
from which, solving for k, we have

k^ ^-_.

log (I + V2)

This relation connecting the two constants /' and ^ was
given by Taurinus. The constant k is the same as that em-
ployed by Gauss [cf. p. 75] in finding the length of the cir-
cumference of a circle.

§ 38. Taurinus deduced other important theorems in
the log.-spherical geometry by further transformations of the
formulae of Spherical Trigonometry, replacing the real radius
by an imaginary one.

For example, that the area of a triangle is proportional
to its defect [Lambert, p. 46] :

that the superior limit of that area is

„ , , -.,-.,, [Gauss, p. 77 ;

[log(l-{-^2)]2

that the length of the circumference of a circle of radius r is

2Tr/è sinh -. [Gauss, p. 75];
that the area of a circle of radius r is

2TtZ'^ (cosh -T- — i);

that the area of the surface of a sphere and its volume, are
respectively

■y

47T/&^ smh^ -,,

and 2TT/è3 (sinh , cosh y — — ).

We shall not devote more space to the different anaiyt-

32 III- The Founders of Non-Euclidean Geometry.

ical developments, since a fuller discussion would cast no
fresh light upon the method. However we note that the
results of Taurinus confirm the prophecy of Lambert on
the Third Hypothesis [cf. p. 50], since the formulae of the
log.-spherical geometry, interpreted analytically, give the fun-
damental relations between the elements of a triangle traced
upon a sphere of imaginary radius.^

To this we add that Taurinus in common with Lambert
recognized that Spherical Geometry corresponds exactly to
the system valid in the case of the Hypothesis of the Obtuse
Angle: further that the ordinary geometry forms a hnk be-
tween spherical geometry and the log.-spherical geometry.

Indeed, if the radius k passes continuously from the real
domain to the purely imaginary one, through infinity, we pro-
ceed from the spherical system to the log.- spherical system,
through the Euclidean.

Although Taurinus, as we have already remarked, ex-
cluded the possibility that a log.-spherical geometry could be
vahd on the plane, the theoretical interest, which it offers,
did not escape his notice. Calling the attention of geo-
meters to his formulae, he seemed to prophecy the existence

I At this stage it should be remarked that Lambert, simul-
taneously with his researches on parallels, was working at the tri-
gonometrical functions with an imaginary argument, whose connection
with Non-Euclidean Geometry was brought to light by Taukinus.
Perhaps Lambert recognised that the formulae of Spherical Trig-
onometry were still real, even when the real radius was changed
in a purely imaginary one. In this case his prophecy with regard
to the Hypothesis of the Acute Angle (cf. p. 50) would have a firm
foundation in his own work. However we have no authority for
the view that he had ever actually compared his investigations on
the trigonometrical functions with those on the theory of parallels.
Cf. P. StAckel: Bcmerkungen sit Lamberts Theorie der Parallellinien.
Biblioteca Math. p. 107 — lio. (1899).

Some Conclusions by Taurinus. 83

of some concrete case in which they would find an inter-
pretation. *

I The important service rendered by Schweikart and Tau-
rinus towards the discovery of the Non-Euclidean Geometry was
recognised and made known by Engel and Stackel. In their
Th. der P., they devote a whole chapter to those authors, and
quote the most important passages in Taurinus' writings, besides
some letters which passed between him, Gauss and Schweikart.
Cf. Stackel: Franz Adolf Taurinus, Abhandl. zur Geschichte der
Math., IX, p. 397 — 427 (1899).

Chapter IV.

The Founders of Non-Euclidean Geometry
(Contd.).

Nicolai Ivanovitsch Lobatschewsky [1793 — 1856],'

§ 39. Lobatschewsky studied mathematics at the Uni-
versity of Kasan under a German J. M. C. Bartels [1769 —
1836], who was a friend and fellow countryman of Gauss.
He took his degree in 18 13 and remained in the University,
first as Assistant, and then as Professor. In the latter position
he lectured upon mathematics in all its branches and also
upon physics and astronomy.

As early as 181 5 Lobatschewsky was working at paral-
lels, and in a copy of his notes for his lectures [1815 — 17]
several attempts at the proof of the Fifth Postulate, and
some investigations resembling those of Legendre have been
found.

However it was only after 1823 that he had thought of
the Imaginary Geometry. This may be inferred from the
manuscript for his book on Elementary Geometry, where he
says that we do not possess any proof of the Fifth Postulate,
but that such a proof may be possible-^

1 For historical and critical notes upon Lobatschewsky we
refer once and for all to F. Engel's book: N. I. Lobàtschefskij :
Zzaci geo7netrische Abhandlungen ans de?n Russischen ubersetzt tiitf
Anmerktoigen und mit einer Biographic dcs Verfassers, (Leipzig,
Teubner, 1899).

2 [This manuscript had been sent to St. Petersburg in 1823
to be published. However it was not printed, and it was dis-

Lobatschewsky's Works. ge

Between 1823 and 1825 Lobatschewsky had turned
his attention to a geometry independent of Euclid's hypothe-
sis. The first fruit of his new studies is the Exposition suc-
cincie des principes de la geometrie avec une demonstration ri-
goureuse dii théorcme des parallcles, read on 1 2 [24] Feb., 1826,
to the Physical Mathematical Section of the University of
Kasan. In this "Lecture", the manuscript of which has
not been discovered, Lobatschewsky explains the prin-
ciples of a geometry, more general than the ordinary geo-
metry, where two parallels to a given line can be drawn
through a point, and where the sum of the angles of a tri-
angle is less than two right angles [The Hypothesis of the Acute
Angle of Saccheri and Lambert].

Later, in 1829 — 30, he published a memoir On the Prin-
ciples of Geometry ^'^ containing the essential parts of the
preceding "Lecture", and further apphcations of the new
theory in analysis. In succession appeared the Imaginary
Geometry [1835],^ New Principles of Geometry, with a Com-

covered in the archives of the University of Kasan. in 1898. It
is clear from some other remarks in this work that he had made
further advance in the subject since 1815 — 17. He was now con-
vinced that all the first attempts at a proof of the Parallel Postulate
were unsuccessful, and that the assumption that the angles of a
triangle could depend only on the ratio of the sides and not upon
their absolute lengths was unjustifiable (cf. Engel, loc.cit. p. 369 — 70).]

1 Kasan Bulletin, (1829 — 1830). Geometrical Works of Lobat-
schewsky (Kasan 1883 — 18S6), Vol. I p. 1 — 67. German translation
by F. Engel p. i — 66 of the work referred to on the previous page.

Where the titles are given in English we refer to works pub-
lished in Russian. The Geometrical Works of Lobatschewsky contain
two parts; the first, the memoirs originally published in Russian;
the second, those published in French or German. It will be seen
below that of the works in Vol. i. several translations are now
to be had.

2 The Scientific Publications of the University of Kasan (1835).
Geometrical Works, Vol. I, p. 71 — 120. German translation by

86 IV. The Founders of Non-Euclidean Geometry (Contd.).

plett Theory of Parallels [1835 — 38]^ the Applications of the
Lnaginary Geometry to Some Integrals [1836]^, then the
Geometrie Imaginaire [183 7] 3, and in 1840, a small book
containing a summary of his work, Geometrische Unter'
suchungen zur Theorie der Farallellinien,'^ written in German
and intended by Lobatschewsky to call the attention of
mathemiaticans to his researches. Finally, in 1855, a year
before his death, when he was already blind, he dictated and
pubHshed in Russian and French a complete exposition of his
system of geometry under the title : Pangéométrie ou precis
de geometrie fondée sur une theorie generale et rigoureuse des
par alleles, s

§ 40. Non-Euclidean Geometry, just as it was conceived
by Gauss and Schweikart in 1816, and studied as an ab-

H. LlEBMANN, with Notes. Abhandlungen zur Geschichte der Mathe-
matik, Bd. XIX, p. 3—50 (Leipzig, Teubner, 1904).

1 Scientific Publications of the University of Kasan (1835 — 38).
Geom. Works. Vol. I: p. 219 — 486. German translation by F. Engel,
p. 67 — 235 of his work referred to on p. 84. English translation
of the Introduction by G. B. Halsted, (Austin, Texas, 1897).

2 Scientific Publications of the University of Kasan. (1836).
Geom. Works, Vol. I, p. 121 — 2l8. German translation by H, LlEB-
MANN; loc. cit: p. 51 — 130.

3 Crelle's Journal, Bd. XVII, p. 295—320. (1837). Geom.
Works, Vol. II, p. 581—613.

4 Berlin (1840). Geo7)i. Works, Vol. II, p. 553—578. French
translation by J. Houel in Mém. de Bourdeaux, T. IV. (1866), and
also va. Recherches géomèiriques sur la theorie des parallèles {?a.xis, Her-
mann, 1900). English translation by G. B. Halsted, (Austin,
Texas, 1891). Facsimile reprint (Berlin, Mayer and Muller, 1887).

5 Collection of Memoirs by Professors of the Royal University of
Kasan on the ^o*''- anniversary of its foundation. Vol. I, p. 279 — 340.
(1856). Also in Geom. Works, Vol. II, p. 617— 680. In Russian, in
Scientific Publications of the University of Kasan, (1855). Italian
translation, by G. Battaglini, in Giornale di Mat. T. V. p. 273—336,
(1867). German translation, by H. Liebmann, Ostwald's Klassiker
der exakten Wissenschaften, Nr, 130 (Leipzig, 1 902).

Lobatschewsky^s Theory of Parallels. 37

stract system by Taurinus in 1826, became in 1829 — 30
a recognized part of the general scientific inheritance.

To describe, as shortly as possible, the method followed
by LoBATSCHEWSKY in the construction of the Imaginary Geo-
metry or Pangeometry, let us glance at his G eovietrische Unter-
suchungeii zur Theorie der ParallellÌ7iien of 1840.

In this work Lobatschewsky states, first of all, a group
of theorems independent of the theory of parallels. Then he
considers a pencil with vertex
A, and a straight line BC^ in
the plane of the pencil, but
not belonging to it. Let AD
be the line of the pencil which
is perpendicular to BC^ and
AE that perpendicular to
AD. In the Euclidean system
this latter line is the only line which does not intersect BC.
In the geometry of Lobatschewsky there are other lines of the
pencil through A which do not intersect BC. The non-inter-
secting lines are separated from the intersectijig lines by the
two hues h, k (see Fig. 44), which in their turn do not meet
BC. [cf. Saccheri, p. 42.] These lines, which the author calls
parallels, have each a definite direction of parallelistn. The
line //, of the figure, is the parallel to the right: k, to the left.
The angle which the perpendicular AD makes with one of
the parallels is the ajtgle 0/ parallelism for the length AD.
Lobatschewsky uses the symbol TT {a) to denote the angle
of parallelism corresponding to the length a. In the ordinary
geometry, we have TT {a) = ()o° always. In the geometry of
Lobatschewsky, it is a definite function of a, tending to
90° as a tends to zero, and to zero as a increases without
limit.

From the definition of parallels the author then deduces
their principal properties:

2,S IV. The Founders of Non-Euclidean Geometry (Contd.).

That if AD is the parallel to £C for the point A, it is
the parallel to BC in that direction for every point on AD
[permanency];

That if AD is parallel to BC, then BC is parallel to .
AD [reciprocity] :

That if the lines (2) and (3) are parallel to (i), then (2)
and (3) are parallel to each other [transitivity] [cf. Gauss,
p. 72]; and that

If AD and BC are parallel, AD is asymptotic to BC.

Finally, the discussion of these questions is preceded by
the theorems on the sum of the angles of a triangle, the
same theorems as those already given by Legendre, and
still earlier by Saccheri. There can be little doubt that Lo-
BATSCHEWSKY was familiar with the work of Legendre.^

But the most important part of the Imaginary Geometry
is the construction of the formulae of trigonometry.

To obtain these, the author introduces two new figures:
the Horocycle [circle of infinite radius, cf. Gauss, p. 74], and
the Horosphere ^ [the sphere of infinite radius], which in the
ordinary geometry are the straight line and plane, respect-
ively. Now on the Horosphere, which is made up of 00 *
Horocycles, there exists a geometry analogous to the
ordinary geometry, in which Horocycles take the place of
straight lines. Thus Lobatschewsky obtains this first re-
markable result:

The Euclidean Geometry [cf. Wachter, p. 63], and., in
particular, the ordinary plane trigonometry, hold upon the Hor-
osphere.

1 Cf. LoBATSCHEWSKv's Criticism of I.egendre's attempt to
obtain a proof of Euclid's Postulate in his Nexu Pnnciples of Geometry
(Engel's translation, p. 68).

2 [Lobatschewsky uses the terms Grcnzkreis, Grenzkugel in
his German work: courbe-limite, horicycle, horisphere, su7-/ace-limite in
his French work.]

The Horocycle snd Horocyclic Surface. 30

This remarkable property and another relating to Co-
axal Horocydes [concentric circles with infinite radius] are
employed by Lobatschewsky in deducing the formulae of
the new Plane and Spherical Trigonometries \ The formulas
of spherical trigonometry in the new system are found to be
exactly the same as those of ordinary spherical trigonometry,
when the elements of the triangle are measured in right- angles.

§ 41. It is well to note the form in which Lobatschewsky
expresses these results. In the plane triangle ABC, let the
sides be denoted by a^ b, c, the angles by A, B, C; and let
T7 (a), TT (a), TT (c) be the angles of parallelism corresponding
to the sides a, b, c. Then Lobatschewsky's fundamental
formula is

, . , TT /7\ TT / \ fin T\ (l>) sin TT \c)

(4) cos A cos TT {b) cos TT {c) + ^^^ = 1.

^^^ ' sm IT [cij

It is easy to see that this formula and that of Taurinus
[(i), p. 79] can be transformed into each other.

To pass from that of Taurinus to that of Lobatschew-
sky, we make use of (3) of p. 80, observing that the angle B,
which appears in it, is TT {a).

For the converse step, it is sufficient to use one* of Lo-
batschewsky's results, namely :

TT (x) _ ^

(5) tan-^' = a. "^

This is the same as the equation (3) of Taurinus, under
another form.

The constant a which appears in (5) is indeterminate.
It represents the constant ratio of the arcs cut off two Coaxal

I It can be proved that the formulae of Non-Euclidean Plane
Trigonometry can be obtained without the •introduction of the
Ho)-ospke7e. The only result required is the relation between the
arcs cut off two Horocydes by two of their axes (cf. p. 90). Cf.
H. LlEBMANN, Elementare Ableitutrg der nichteuklidiscken Trigonometrie.
Ber. d. kòn. Sach. Ges. d. Wiss., Math. Phys. Klasse, (1907).

QO IV. The Founders of Non-Euclidean Geometry (Contd.).

Horocycles by a pair of axes, when the distance between

these arcs is the unit of length.

[Fig- 45-]

If we choose, with Lobatschew-
SKY, a convenient unit, we are able
to take a equal to e, the base of
Natural Logarithms. If we wish,
on the other hand, to bring Lo-
'^ '' ' batschewsky's results into accord

with the log.-spherical geometry of Taurinus, or the Non-Eu-
clidean geometry of Gauss, we take

Then (5) becomes x

.r U(x) ~~T

(5) tan^-— = ir

2 ,

which is the same as

(6) cosh 7- = -. — TT-— ,•

A sin 1 1 (x)

This result at once transforms Lobatschewsky's equa-
tion (4) into the equation (i) of Taurinus.

It follows that:

T/ie log.-spherical geometry of Taurinus is identical with
the imagiftary geometry \_pa?igeometry] of Lobatschewsky.

§ 42. We add the most remarkable of the results which
Lobatschewsky deduces from his formulae:

(a) In the case of triangles whose sides are very small
[infinitesimal] we can use the ordinary trigonometrical for-
mulae as the formulae ol Imaginary Trigonometry, infinitesi-
mals of a higher order being neglected \

I Conversely, the assumption that the Euclidean Geometry
holds for the infinitesimally small can be taken as the starting
point for the development of Non-Euclidean Geometry. It is one
of the most interesting discoveries from the recent examination of

Lobatschewsky's Trigonometry. gj

(b) If for a, b, c are substituted ia^ ib, ic, the formulae
of Imaginary Trigonometry are transformed into those of or-
dinary Spherical Trigonometry.^

(c) If we introduce a system of coordinates in two and
three dimensions similar to the ordinary Cartesian coordinates,
we can find the lengths of curves, the areas of surface^- and
the volumes of solids by the methods of analytical geometry.

§ 43. How was LoBATSCHEWSKY led to investigate the
theory of parallels and to discover the Imaginary Geometry?

We have already remarked that Bartels, Lobatschew-
sky's teacher at Kasan, was a friend of Gauss [p. 84]. If we
now add that he and Gauss were at Brunswick together dur-
ing the two years which preceded his call to Kasan [1807],
and that later he kept up a correspondence with Gauss, the
hypothesis at once presents itself that they were not without
their influence upon Lobatschewsky's work.

We have also seen that before 1807 Gauss had attempted
to solve the problem of parallels, and that his efforts up till
that date had not borne other fruit than the hope of overcom-
ing the obstacles to which his researches had led him. Thus
anything that Bartels could have learned from Gauss before
1807 would be of a negative character. As regards Gauss's

Gauss's MSS. that the Princeps mathematicorum had already fol-
lowed this path. Cf. Gauss, Werke, Bd. VIII, p. 255—264.

Both the works of Flye St. Marie, [Thhrie analytlque sur la
thèorie des parallèies, (Paris, 1871)], and of KILLING [Die 7iichteuklid-
ischen Raiwiformen in analytischer Behandlung, (Leipzig, 1881)], are
founded upon this principle. In addition, the formulae of trigono-
metry have been obtained in a simple manner by the application
of the same principle, and the use of a few fundamental ideas, by
M. Simon. [Cf. M. Simon, Die Trigonometrie in der absoluten Geotnetrie,
Crelle's Journal, Bd. 109, p. 187 — 198 (1892)].

I This result justifies the method followed by Taurinus in
the construction of his log. -spherical geometry.

Q2 IV. The Founders of Non-Euclidean Geometry (Contd.).

later views, it appears quite certain that Bartels had no news
of them^ so that we can be sure that Lobatschewsky created
his geometry quite independently of any influence from Gauss.*
Other influences might be mentioned: e. g., besides Legendre,
the works of Saccheri and Lambert, which the Russian geo-
meter might have known, either directly or through Klugel
and MoNTUCLA. But we can come to no definite decision
upon this question^. In any case, the failure of the demon-
strations of his predecessors, or the uselessness of his own
earlier researches [1815 — 17], induced Lobatschewsky, as
formerly Gauss, to believe that the difficulties which had
to be overcome were due to other causes than those to
which until then they had been attributed. Lobatschewsky
expresses this .thought clearly in the Nau Principles of
Geometry of 1825, where he says:

'The fruitlessness of the attempts made, since Euclid's
time, for the space of 2000 years, aroused in me the suspicion
that the truth, which it was desired to prove, was not contained
in the data themselves; that to establish it the aid of experi-
ment would be needed, for example, of astronomical obser-
vations, as in the case of other laws of nature. When I had
finally convinced myself of the justice of my conjecture and
beheved that I had completely solved this difficult question,
"^I wrote, in 1826, a memoir on this subject {Exposition suc-
cincte des principes de la Géomctrie\.' ^

The words of Lobatschewsky afford evidence of a phil-
osophical conception of space, opposed to that of Kant,
which was then generally accepted. The Kantian doctrine
considered space as a subjective intuition, a necessary presup-
position of every experience. Lobatschewsky's doctrine was

1 Cf. the work of F. Engel, quoted on p. 84. Zweiter Teil;
Lobatschefskijs Leben unci Schriftett. Cap. VI, p. 373 — 383.

2 Cf. Segre's work, quoted on p. 44.

3 Cf. p. 67 of Engel's work named above.

The Pangeometry. q-ì

rather allied to sensualism and the current empiricism, and
compelled geometry to take its place again among the ex-
perimental sciences.^

§44. It now remains to describe the relation of Lobat-
scHE\vsK"S''s Paiigeo7netry to the debated question of the Eu-
clidean Postulate. This discussion, as we have seen, aimed
at constructing the Theory of Parallels with the help of the
first 28 propositions of Euclid.

So far as regards this problem, Lobatschewsky, having
defined parallelism, assigns to it the distinguishing features
of reciprocity and transitivity. The property of equidistance
then presents itself to Lobatschewsky in its true light. Far
from being indissolubly bound up with the first 28 proposit-
ions of Euclid, it contains an element entirely new.

The truth of this statement follows directly from the ex-
istence of the Pangeometry [a logical deductive science founded
upon the said 28 propositions and on the negation of the
Fifth Postulate], in which parallels are not equidistatit, but are
asymptotic. Further, we can be sure that the Pangeometry
is a science in which the results follow logically one from the
other, i. e., are free from internal contradictions. To prove
this we need only consider, with Lobatschewsky, the analyt-
ical form in which it can be expressed.

This point is put by Lobatschewsky toward the end of
his work in the following way:

'Now that we have shown, in what precedes, the way in
which the lengths of curves, and the surfaces and volumes of
solids can be calculated, we are able to assert that the Pan-
geometry is a complete system of geometry. A single glance

I Cf. The discourse on Lobatschewsky by A. Vasiliev,
(Kasan, 1893). German translation by Engel in Schlomilch's Zeit-
schrift, Bd. XI, p. 205 — 244 (1895). 'English translation by Halsted,
(Austin, Texas, 1 895).

IV. The Founders of Non-Euclidean Geometry (Contd.

at the equations which express the relations existing between
the sides and angles of plane triangles, is sufficient to show
that, setting out from them, Pangeometry becomes a branch of
analysis, including and extending the analytical methods of
ordinary geometry. We could begin the exposition of Pan-
geometry with these equations. We could then attempt to
substitute for these equations others which would express the
relations between the sides and angles of every plane triangle.
However, in this last case, it would be necessary to show
that these new equations were in accord with the fundamental
notions of geometry. The standard equations, having been
deduced from these fundamental notions, must necessarily be
in accord with them, and all the equations which we would
substitute for them, if they cannot be deduced from the equa-
tions, would lead to results contradicting these notions. Our
equations are, therefore, the foundation of the most general
geometry, since they do not depend on the assumption that
the sum of the angles of a plane triangle is equal to two right
angles.' '

§ 45. To obtain fuller knowledge of
the nature of the constant k contained im-
plicity in Lobatschewsky's formulae, and
exphcitly in those of Taurinus, we must
apply the new trigonometry to some actual
case. To this end Lobatschewsky used a
triangle ABC, in which the side BC {a) is
equal to the radius of the earth's orbit,
and ^ is a fixed star, whose direction is
perpendicular to BC (Fig. 46). Denote
hy 2 p the maximum parallax of the star
A. Then we have

1 Cf. the Italian translation of the Pangéomélrie, Giornale di
Mat,, T. V. p. 334; or p. 75 of the German translation referred to
on p. 86.

Astronomy and Lobatschewsky's Theory. ge

Therefore

I /it \ I — tan/

tan -T](a)> tan (-- -/j = .^^^^

a
But tan l- T\ (a) = e J [cf. p. 90],

Therefore .'^<i^'-"^.
^ I — tan/

But on the hypothesis / <C j we have

Also, tan 2/ = 2 tan/
I — tan2/
= 2 (tan/ + tan3/ + tan^/ + ...).

Therefore we have

-^ < tan 2/.

Take now, with Lobatschewsky, the parallax of Sirius
as i", 24.

From the value of tan 2 /, we have

— <C 0,000006012.

This result does not allow us to assign a value to k,
but it tells us that it is very great compared with the diam-
eter of the earth's orbit. We could repeat the calculation
for much smaller parallaxes, for example o",i, and we
would find k to be greater than a million times the diameter
of the earth's orbit.

Thus, if the EucUdean Geometry and the Fifth Postul-
ate are to hold in actual space, k must be infinitely great.
That is to say, there must be stars whose parallaxes are in-
definitely small.

However it is evident that we can never state whether
this is the case or not, since astronomical observations will

q5 IV. The Founders of Non-Euclidean Geometry (Contd.).

always be true only within certain limits, Yet^ knowing the
enormous size of k in comparison with measurable lengths,
we must, with Lobatschewsky, admit that the Euclidean
hypothesis is valid for all practical purposes.

We would reach the same conclusion if we regarded
the question from the standpoint of the sum of the angles of
a triangle. The results of astronomical observations show that
the defect of a triangle, whose sides approach the distance
of the earth from the sun, cannot be more than o",ooo3.
Let us now consider, instead of an astronomical triangle, one
drawn on the Earth's surface, the angles of which can be
directly measured. In consequence of the fundamental theorem
that the area of a triangle is proportional to its defect, the
possible defect would fall within the limits of experimental
error. Thus we can regard the defect as zero in experimental
work, and Euclid's Postulate will hold in the domain of ex-
perience.^

Johann Bolyai [1802 — 1860].

§ 46. J. Bolyai a Hungarian officer in the Austrian
army, and son of Wolfgang Bolyai, shares with Lobat-
schewsky the honour of the discovery of Non-Euclidean geo-
metry. From boyhood he showed a remarkable aptitude for
mathematics, in which his father himself instructed him. The
teaching of Wolfgang quickly drew Johann's attention to
Axiom XL To its demonstration he set himself, in spite of
the advice of his father, who sought to dissuade him from >
the attempt. In this way the theory of parallels formed the
favourite occupation of the young mathematician, during his .
course [1817 — 22] in the Royal College for Engineers at
Vienna.

I For the contents of this section, cf. Lobatschewsky, On
the Principles of Geometry, See p. 22 — 24 of Engel's work named
on p. 84. Also Engel's remarks on p. 24S — 252 of the same work.

Johann Boiyai's Earlier Work.

At this time Johann was an intimate friend of Carl
SzAsz [1798-185 3] and the seeds of some of the ideas, which
led BoLYAi to create the Absolute Science of Space, were sown
in the conversations of the two eager students.

It appears that to Szasz is due the distinct idea of con-
sidering the parallel through £ to the line AM as the limit-
ing position of a secant BC turning in a definite direction
about JB; that is, the idea of consid-
ering BC as parallel to AM, when
BC, in the language of Szasz, de-
taches itself ^springs away) from AM
(Fig. 47). To this parallel Bolyai
gave the name of asymptotic parallel
or asymptote, [cf Saccheri]. From
the conversations of the two friends
were also derived the conception of
the line equidistant from a straight line,
and the other most important idea of
the Paracycle {lÌ7ìiiting curve or horo-
ry*;/.? of Lobatschewsky). Further they
recognised that the proof of Axiom XI would be obtained
if it could be shown that the Paracycle is a straight line.

When Szasz left Vienna in the beginning of 1821 to
undertake the teaching of Law at the College of Nagy-Enyed
(Hungary), Johann remained to carry on his speculations
alone. Up till 1820 he was filled with the idea of finding
a proof of Axiom XI, following a path similar to that of
Saccheri and Lambert. Indeed his correspondence with
his father shows that he thought he had been successful in
his aim.

The recognition of the mistakes he had made was the
cause of Johann's decisive step towards his future discoveries,
since he realised 'that one must do no violence to nature,
nor model it in conformity to any blindly formed chimsera;

q3 IV. The Founders of Non-Euclidean Geometry (^Contd.)

that, on the other hand, one must reguard nature reasonably
and naturally, as one would the truth, and be contented only
with a representation of it which errs to the smallest possible
extent.'

Johann Bolyai, then, set himself to construct an abso-
hite theory of space, following the classical methods of the
Greeks: that is, keeping the deductive method^ but without
deciding a priori on the truth or error of the FifthPostulate.

§ 47. As early as 1823 Bolyai had grasped the real
nature of his problem. His later additions only concerned
the material and its formal expression. At that date he had
discovered the formula:

a
e ■^ = tan - - ,

connecting the angle of parallelism with the line to which it
corresponds [cf. Lobatschewsky, p. 89]. This equation is
the key to all Non- Euclidean Trigonometry. To illustrate the
discoveries which Johann made in this period, we quote the
following extract from a letter which he wrote from Temesvar
to his father, on Nov. 3, 1823: 'I have now resolved to pub-
lish a work on the theory of parallels, as soon as I shall have
put the material in order, and my circumstances allow it. I
have not yet completed this work, but the road which I have
followed has made it almost certain that the goal will be
attained, if that is at all possible: the goal is not yet reached,
but I have made such wonderful discoveries that I have been
almost overwhelmed by them, and it would be the cause of
continual regret if they were lost. When you will see them,
you too will recognize it. In the meantime I can say only
this : / have created a ne^v universe from nothing. All that I
have sent iyou till now is but a house of cards compared to
the tower. I am as fully persuaded that it will bring me
honour, as if I had already completed the discovery.'

J. Bolyai's Theory of Parallels. gg

Wolfgang expressed the wish at once to add his son's
theory to the Tentatnen since 'if you have really succeeded
in the question, it is right that no time be lost in making it
public, for two reasons: first, because ideas pass easily from
one to another, who can anticipate its publication; and se-
condly, there is some truth in this, that many things have an
epoch, in which they are found at the same time in several
places, just as the violets appear on every side in spring.
Also every scientific struggle is just a serious war, in which
I cannot say when peace will arrive. Thus we ought to
conquer when we are able, since the advantage is always to
the first comer.'

Little did Wolfgang Bolyai think that his presentiment
would correspond to an actual fact (that is, to the simulta-
neous discovery of Non-Euclidean Geometry by the work of
Gauss, Taurinus, and Lobatschewsky).

In 1825 Johann sent an abstract of his work, among
others, to his father and to J.Walter von Eckwehr [1789 —
1857], his old Professor at the Military School. Also in 1829
he sent his manuscript to his father. Wolfgang was not
completely satisfied with it, chiefly because he could not see
why an indeterminate constant should enter into Johann's
formulae. None the less father and son were agreed in
pubhshing the new theory of space as an appendix to the
first volume of the Tentamen: —

The title of Johann Bolyai's work is as follows.

Appendix scientiam spatii absolute veram exhibens: a
ventate aut falsitate Axiomatis XI. Euclidei, a priori haud
unquam decidenda, independentem : adjecta ad casum falsitaiis
quadratura circuii geometrica.^

I A reprint — Edition de Luxe — was issued by the Hungarian
Academy of Sciences, on the occasion of the first centenary of
the birth of the author (Budapest, 1902). See also the English

lOO IV. The Founders of Non-Euclidean Geometry (Contd.).

The Appendix was sent for the first time [June, 1831]
to Gauss, but did not reach its destination; and a second
time, in January, 1832. Seven weeks later (March 6, 1832),
Gauss replied to Wolfgang thus:

"If I commenced by saying that I am unable to praise
this work (by Johann), you would certainly be surprised
for a moment. But I cannot say otherwise. To praise it,
would be to praise myself. Indeed the whole contents of
the work, the path taken by your son, the results to which he
is led, coincide almost entirely with my meditations, which
have occupied my mind partly for the last thirty or thirty-
five years. So I remained quite stupefied. So far as my
own work is concerned, of which up till now 1 have put little
on paper, my intention was not to let it be published during
my lifetime. Indeed the majority of people have not clear
ideas upon the questions of which we are speaking, and I
have found very few people who could regard with any special
interest what I communicated to them on this subject. To
be able to take such an interest it is first of all necessary
to have devoted careful thought to the real nature of what is
wanted and upon this matter almost all are most uncertain.
On the other hand it was my idea to write down all this later
so that at least it should not perish with me. It is therefore a
pleasant surprise for me that I am spared this trouble, and I
am very glad that it is just the son of my old friend, who
takes the precedence of me in such a remarkable manner."

Wolfgang communicated this letter to his son, adding:
"Gauss's answer with regard to your work is very satis-

translation by Halsted, T/ie Science Absolute of Space, (Austin, Texas^
1896). An Italian translation by G. B. Battagmni appeared in the
Giornale di Mat., T. VI, p- 97 — 115 (1868). Also a French trans-
lation by HOUEL, in Mém. de la Soc des Se. de Bordeaux, T. V-
p. 189 — 248 (1867). Cf. also Frischauf, Absoluie Geometrie nach
Johann Bolyai, (Leipzig, Teubner, 1872).

Gauss's Praise of Eolyai's Work. lOI

factory and redounds to the honour of our country and of
our nation."

Altogether different was the effect Gauss's letter pro-
duced on Johann. He was both unable and unwilling to
convince himself that others, earlier than and independent of
him, had arrived at the No7i- Euclidean Geometry. Further he
suspected that his father had communicated his discoveries
to Gauss before sending him the Appendix and that the latter
wished to claim for himself the priority of the discovery.
And although later he had to let himself be convinced that
such a suspicion was unfounded, Johann always regarded
the "Prince of Geometers" with an unjustifiable aversion. *

§ 48. We now give a short description of the most
important results contained in Johann Bolyai's work:

a) The definition of parallels and their properties in-
dependent of the Euchdean postulate.

b) The circle and sphere of infinite radius. The geo-
metry on the sphere of infinite radius is identical with ordi-
nary plane geometry.

c) Spherical Trigonometry is independent of Euclid's
Postulate. Direct demonstration of the formulae.

d) Plane Trigonometry in Non-Euclidean Geometry.
Applications to the calculation of areas and volumes.

e) Problems which can be solved by elementary me-
thods. Squaring the circle, on the hypothesis that the Fifth
Postulate is false.

While LoBATSCHEWSKY has given the Imaginary Geo-
metry a fuller development especially on its analytical side,

I For the contents of this and the preceding article seeSxAcKEL,
Die Entdeckung der 7iichteuklidischeii Geometrie durch jfohatin Bolyai.
Math. u. Naturw. Berichte aus Ungarn. Bd. XVII, [1901].

Also StAckel u. Engel. Gauss, die beiden Bolyai und die
nichteuklidische Geometrie. Math. Ann. Bd. XLIX, p. 149 — 167 [1897].
Bull. So. Math. (2) T. XXI, pp. 206—228 [1897].

I02 IV. The Founders of Non-Euclidean Geometry (Contd.).

BoLYAi entered more fully into the question of the depen-
dence or independence of the theorems of geometry upon
Euclid's Postulate. Also while Lobatschewky chiefly sought
to construct a system of geometry on the negation of the
said postulate, Johann Bolyai brought to light the pro-
positions and constructions in ordinary geometry which are
independent of it. Such propositions, which he calls ab-
solutely true, pertain to the absolute science of space. We
could find the propositions of this science by comparing
EucUd's Geometry with that of Lobatschewsky. Whatever
they have in common, e. g. the formulae of Spherical Trigon-
ometry, pertains to the Absolute Geometry. Johann Bolyai,
however, does not follow this path. He shows directly, that
is independently of the Euchdean Postulate, that his propos-
itions are absolutely true.

§ 49. One of BoLYAi's absolute theorems, remarkable
for its simplicity and neatness, is the following:

The sines of the angles of a rectilinear triangle are to one
another as the circumferences of the circles whose radii are
equal to the opposite sides.

A ^

Fig. 48.

Let ABC be a triangle in which C is a right angle, and
BB' the perpendicular through B to the plane of the triangle.

Draw the parallels through A and C to BB' in the
same sense.

Then let the Horosphere be drawn through A (eventually
the plane) cutting the lines AA\ BB' and CC, respectively,
in the points A, M, and N.

Bolyai's Theorem. - 10^

If we denote by a\ b\ c the sides of the rectangular
triangle AAIN on the Horosphere, it follows from what has
been said above [cf. § 48 (b)] that

sin AMN = — .

But two arcs of Horocycles on the Horosphere are pro-
portional to the circumferences of the circles which have
these arcs for their (horocyclic) radii.

If we denote by circumf. x the circumference of the
circle whose (horocyclic) radius is x', we can write:

A T^jT^T circumf. b'

Sin AMN = -. J—,.

circumf. c

On the other hand, the circle traced on the Horosphere
with horocyclic radius of length x\ can be regarded as the
circumference of an odinary circle whose radius (rectilinear)
is half of the chord of the arc 2 x' of the Horocycle.

Denoting by O •^ the circumference of the circle whose
(rectilinear) radius is x, and observing that the angles ABC
and AMN are equal, the preceding equation taken from

sin ABC = -^.

From the property of the right angled triangle ABC
expressed by this equation, we can deduce Bolyai's theorem
enunciated above, just as from the Euclidean equation

sin ABC = —

we can deduce that the sines of the angles of a triangle are
proportional to the opposite sides. {Appendix § 25.]

Bolyai's Theorem may be put shortly thus:
(i) O^ '• O^ '• O^ = sin ^ : sin B : sin C.

If we wish to discuss the geometrical systems separately
we will have

(i) In the case of the Euclidean Hypothesis,

O-^ = 2 TTAT.

I04 ^^' ^^^ Founders of Non-Euclidean Geometry (Contd.).

(!')

2 11;^ sinh -r*

(i ") sinh — : sinh ^ : sinh -r == sin A : sin B : sin C.

Thus, substituting in (i), we have

a:b'.C'. == sin ^ : sin ^ : sin C.
(ii) In the case of the Non-Eudidean Hypothesis,

Then substituting in (i) we have

k ' T • " /•

This last relation may be called the Sine Theorem of the
Bolyai-Lobatschewsky Geometry.

From the formula (i) Bolyai deduces, in much the
same way as the usual relations are obtained from (i), the
proportionality of the sines of the angles and the opposite sides
in a spherical triangle. From this it follows that Spherical
Trigonometry is independent of the Euclidean Postulate
{Appendix 8 26].

This fact makes the importance of Bolyai's Theorem
still clearer.

§ 50. The following construction for a parallel through
the point Z> to the straight line ^iV belongs also to the Ab-
solute Geometry [Appendix % 34].

Draw the perpendiculars DB and AE to AN [Fig. 49].
fi D

Fig. 49.

Also the perpendicular DE to the line AE. The angle
EDB of the quadrilateral ABDE, in which three angles

Bolyai's Parallel Construction. IO5

are right angles, is a right angle or an acute angle, according
as ED is equal to or greater than AB.

With centre A describe a circle whose radius is equal
to ED.

It will intersect DB at a point (9, coincident with B or
situated between B and D.

The angle which the line AO juakes with DB is the
angle of parallelisin corresponding to the segmefit BD.^
[Appendix §27.]

Therefore a parallel to AN through D can be con-
structed by drawing the line DM so that <C BDM is
equal to <^ AOB.^

1 We give a sketch of Bolyai's proof of this theorem: The
circumferences of the circles with radii AB and ED, traced out
by the points B and D in their rotation about the line AE, can
be considered as belonging, the first to the plane through A per-
pendicular to the axis AE, the second to an Equidistant Surface
for this plane. The constant distance between the surface and
the plane is the segment BD^^d. The ratio between these two
circumferences is thus a function of d only. Using Bolyai's
Theorem, S 49> and applying it to the two rightangled triangles
ADE and ADB, this ratio can be expressed as

O AB : O ED = sin u : sin v.
From this it is clear that the ratio sin k : sin v does not vary if
the line AE changes its position, remaining always perpendicular
to AB, while d remains fixed. In particular, if the foot of AE
tends to infinity along AJV, it tends to TT {d) and z/ to a right angle.
Consequently,

QAB ■.QED = sm T\{d):\.

On the other hand in the right-angled triangle AOB, we have
the equation

QAB:QiAO = sin AOB : i.
This, with the preceding equation, is sufficient to establish the
equality of the angles TT {d) und AOB.

2 Cf. Appendix III to this volume.

I06 IV. The Founders of Non-Euclidean Geometry (Contd.).

§ 51. The most interesting of the Non-Euclidean con-
structions given by Bolvai is that for the squaring of the
circle. Without keeping strictly to Bolyai's method, we shall
explain the principal features of his construction.

But we first insert the converse of the construction of
§ 50, which is necessary for our purpose.

On the Non-Euclidea7i Hypothesis to draw the segment
which corresponds to a given {acute) angle of parallelism.

Assuming that the theorem, that the three perpendiculars
from the angular points of a triangle on the opposite sides
intersect eventually, is also true in the Geometry of Bolyai-
LoBATSCHEWSKY, on the line AB bounding the acute angle
BAA' take a point B, such that the parallel BB' to AA
through B makes an acute angle {ABB') with AB. [Fig. 50.]

Fig. 50.

The two rays AA , BB', and the line AB may be
regarded as the three sides of a triangle of which one angular
point is Coo ) common to the two parallels AA, BB'. Then
the perpendiculars from A, B, to the opposite sides, meet in
he point O inside the triangle, and the perpendicular from
Coo to AB also passes through O.

Thus, if the perpendicular OL is drawn from O to AB,
the segment AL will have been found which corresponds to
the angle of parallelism BAA .

Bolyai's Parallel Construction (Contd.).

107

As a particular case the angle BAA' could be 45°.
Then AL would be Schweikart's Constant [cf. p. 76].

We note that the problem which we have just solved
could be enunciated thus:

To draw a line which shall be parallel to one of the lines
bounding an acute angle and perpendicular to the other. "^

§ 52. We now show how the preceding result is used
to construct a square equal in area to the maximum triangle.

The area of a triangle being

k'{M—KA — ^B—^C),
the maximum triangle, i. e. that for which the three angular
points are at infinity, will have for area

A = k^ TT.

To find the angle oi of a square whose area is k'^n, we need
only remember (Lambert, p. 46) that the area of a polygon^
as well as of a triangle, is proportional to its defect. Thus
we have the equation

k^ 11 = k'^ (2 TT — 4 uj),
from which it follows that

UU = " IT

45^

Assuming this^ let us consider the
right-angled triangle 0AM (Fig, 51),
which is the eighth part of the required
square. Putting OM = <?, and ap-
plying the formula (2) of p, 80 we \/
obtain

/ 2
/45°

F<g. 51-

cosh -r =

or cosh -r =
k

cos 22" 30

sin 45"

sin 670 30'

sin 450

cated.

I Bolyai's solution [Appendix., § 35] is, however, more compii-

I08 IV. The Founders of Non-Euclidean Geometry (Contd.).

If we now draw, as in § 51, the two segments b' , c,
which correspond to the angles 67° 30' and 45°, and if we
remember that [cf. p. 90 (6)]

cosh -T- = ^-TfT-Tj
k smTT (;>:)'

the following relation must hold between a, b' and c,
cosh -J cosh -J = cosh -j.

Finally if we take b' as side, and / as hypotenuse of a right-
angled triangle, the other side of this triangle, by formula (i)
of p. 7 9, is determined by the equation

cosh -,~ cosh -,- = cosh -r-.

Then comparing these two questions, we obtain

a = a.
Constructing a in this way, we can immediately find the
square whose area is equal to that of the maximum triangle.

§ 53. To construct a circle whose area shall be equal
to that of this square, that is, to the area of the maximum
triangle, we must transform the expression for the area of
a circle of radius r

2 TT /C'^ ( cosh -, I ) ,

given on p. 81, by the introduction of the angle of parallelism
TT( — j, corresponding to half the radius.

Then we have' for the area of this circle

On the other hand if the two parallels AA and BB'
are drawn from the ends of the segment AB^ making equal
angles with AB^ we have

I Using the result tan

TT {x) ^ -- xlk

The Square of area it/t*.

109

<^ A AB == <: B'BA = n (^),

where AB == r [Fig. 52].

Now draw ^C, perpendicular to BB\ and ^Z> perpen-
dicular to AC; also put

<^ CAB = a, <^ Z)^^' = z.
Then we have

cot TT ( ~ j cot a + I

tan z = cot f TT r ' j — cc j

cot a — cot

Ki)

It is easy to eliminate a from this last result by means
of the trigonometrical formulae for the triangle ABC and so

obtain

2
tan z = — ^

.a„ n (^)

Substituting this in the expression found for the area of
the circle, we obtain for that area

IT k^ tan^ z.
This formula, proved in an- D
other way by Bolyai {Appendix
% 43], allows us to associate a
definite angle z with every circle.
If 3 were equal to 45", then we
would have

for the area of the correspond-
ing circle.

Fig. 52.

2 C0t2

I Indeed, in the rightangled triangle ABC, we have cot TT ( — )

= cosh _

TT ( "^ ) + I' we deduce, first, that

cosh — From this, since cosh -^ = 2 sinh2 L. 4- i

k k 2k

I IO IV. The Founders of Non-Euclidean Geometry (Contd.).

That is : the a7'ea of the circle, for which the angle z is
4j°, is equal to the area of the maximtim triangle, and thus
to that of the square of § 52.

If z = ^AAD (Fig. 51) is given, we can find r by
the following construction:

(i) Draw the line AC perpendicular to AD.

(ii) Draw BB' parallel to AA and perpendicular to
^C7(S5i).

(iii) Draw the bisector of the strip between AA and
BB\

[By the theorem on the concurrency of the bisectors of
the angles of a triangle with an infinite vertex.]

(iv) Draw the perpendicular AB to this bisector. The
segment AB bounded by AA and BB' is the required
radius r.

§ 54. The problem of constructing a polygon equal to
a circle of area tc k'^ tan^ z is, as Bolyai remarked, closely
allied with the numerical value of tan z. It is resolvable
for every integral value of tan- z, and for every fractional
value, provided that the denominator of the fraction, re-
duced to its lowest terms, is included in the form assigned by
Gauss for the construction of regular polygons [Appendix

§43]-

The possibility of constructing a square equal to a
circle leads Johann to the conclusion "habeturque aut Axi-
oma XI Euclidis verutn, aut quadratura circuii geometrica;

cot TT I — ) cot a = 2 cot2 TT ( — ) +1,

and next that
cot a — cot

TT(T)=(^+tan3n(;))cotn(^).

These equations allow the expression for tan z to be written down
in the required form.

The Quadrature of the Circle. Ill

etsi hucusque indecisum manserit, quodnam ex his duobus
revera locum habeat."

This dilemma seemed to him at that time [1831] im-
possible of solution, since he closed his work with these
words: "Superesset denique (ut res omni numero absolvatur),
impossibilitatem (absque suppositione aliqua) decidenda,
num X (the Euclidean system) aut aliquod (et quodam) S (the
Non-Euclidean system) sit, demonstrare : quod tamen occasi-
oni magis idoneae reservatur."

Johann, however, never published any demonstration
of this kind.

§ 55. After 183 1 BoLYAi continued his labours at his
geometry, and in particular at the following problems:

1. The connection between Spherical Trigonometry and
Non-Euclidean Trigonometry.

2. Can one prove rigorously that Euclid's Axiom is
not a consequence of what precedes it ?

3. The volume of a tetrahedron in Non-Euclidean geo-
metry.

As regards the first of these problems, beyond estab-
lishing the analytical relation connecting the two trigono-
metries [cf. LoBATSCHEWSKY, p. 90], BoLYAi recognized that
in the Non-Euclidean hypothesis there exist three classes of
Uniform Surfaces^ on which the Non-Euclidean trigono-
metry, the ordinary trigonometry, and spherical trigonometry
respectively hold. To the first class belong planes and hyper-
spheres [surfaces equidistant from a plane]; to the second,
the paraspheres [Lobatschewsky's Horospheres] ; to the
third, spheres. The paraspheres are the limiting case
when we pass from the hyperspherical surfaces to the
spherical. This passage is shown analytically by making a

I BoLYAl seems to indicate by this name the surfaces which
behave as planes, with respect to displacement upon themselves.

I [2 IV, The Founders of Non-Euclidean Geometry (Contd.).

certain parameter, which appears in the formulae, vary con-
tinuously from the real domain to the purely imaginary
through infinity [cf Taurinus, p. 82].

As to the second problem, that regarding the impos-
sibility of demonstrating Axiom XT, Bolyai neither succeeded
in solving it, nor in forming any definite opinion upon it.
For some time he believed that we could not, in any way,
decide which was true, the Euclidean hypothesis or the
Non-Euclidean. Like Lobatschewsky, he relied upon the
analytical possibility of the new trigonometry. Then we find
Johann returning again to the old ideas, and attempting a
new demonstration of Axiom XI. In this attempt he applies
the Non-Euclidean formulae to a system of five coplanar
points. There must necessarily be some relation between
the distance of these points. Owing to a mistake in his
calculations Johann did not find this relation, and for some
time he believed that he had proved, in this way, the false-
hood of the Non-Euclidean hypothesis and the absolute truth
of Axiom XI .^

However he discovered his mistake later, but he did
not carry out further investigations in this direction, as the
method, when applied to six or more points, would have in-
volved too comphcated calculations.

The third of the problems mentioned above, that re-
garding the tetrahedron, is of a purely geometrical nature.
BoLYAi's solutions have been recently discovered and pub-

I The title of the paper which contains Johann's demon-
stration is as follows: "Beweis des bis mm auf der Erde itnmer
nock zwei/elkafi gewesenen, weltberuhmten ujid, ah der gesamtnten
Raum- und Bewegungslehre zu Grunae dienend, auch in der That
allerh'òchst7uichtigsten 11. Eudid'schen Axioms von J. Bolyai von Bolya,
k. k.Geiiie-Stabs/iauptmann in Pension. Cf. StaCKEL's paper: Untcr-
suchungen aiis der Absoluten Geotnetrie aus yohann Bolyais N'achlafi.
Math. u. Naturw. Berichte aus Ungarn. Ed. XVIII, p. 2S0— 307 (1902).
We are indebted to this paper for this section S 55-

Bolyai^s Later Work. 1 1 5

lished by Stackel [cf. p. 112 note i]. Lobatschewsky
had been often occupied with the same problem from 1829',
and Gauss proposed it to Johann in his letter quoted on
p. 100.

Finally we add that J. Bolyai heard of Lobatschewsky's
Geometrisc/ie Untersuchimgen in 1848: that he made them
the object of critical study ^: and that he set himself to com-
pose an important work on the reform of the Principles of
Mathematics with the hope of prevailing over the Russian.
He had planned this work at the time of the publication of
the Appendix, but he never succeeded in bringing it to a
conclusion.''

The Absolute Trigonometry.

§ 56. Although the formulae of Non-Euclidean trigono-
metry contain the ordinary relations between the sides and
angles of a triangle as a limiting case [cf. p. 80], yet they do
not form a part of what Johann Bolyai called Absolute Geo-
metry. Indeed the formulse do not apply at once to the two
classes of geometry, and they were deduced on the suppos-
ition of the validity of the Hypothesis of the Acute Angle.
Equations directly applicable both to the Euclidean case and
to the Non-Euclidean case were met by us in § 49 and they
make up Bolyai's Theorem. They are tliree in number, only
two of them being independent. Thus they furnish a first
set of formulae of Absolute Trigonometry.

» Cf. p. 53 et seq., of the work quoted on p. 84. Also
Liebmann's translation, referred to in Note 2, p. 85.

2 Cf. P. Stackel und J. KurschA'k: Johann Bolyals Be-
nierkungen iiber JV. Lobaischefskijs Geofneh-iscke Untermchungen zur
Theorie der Parallellinicn, Math. u. Naturw. Berichte aus Ungarn,
Bd. XVIII, p. 250—279 (1902).

3 Cf. P. StAckel: Johann Bolyais Raumlehre, Math. u. Naturw.
Berichte aus Ungarn, Bd. XIX (1903).

jj_^ IV. The Founders of Non-Euclidean Geometry (Contd.).

Other formulae of Absolute Trigonometry were given in
1870 by the Belgian geometer, De Tilly, in ins Etudes de
Mécaniqiie Abstraite. ^

The formulae given by De Tilly refer to rectilinear tri-
angles, and were deduced by means of kinematical con-
siderationS; requiring only those properties of a bounded
region of a plane area, which are independent of the value
of the sum of the angles of a triangle.

In addition to the function 0-'*^> which we have already
met in Bolyai's formulae, -those of De Tilly contain another
function Ex defined in the following way:

Let r be a straight line and / the equidistant curve,
distant x from r. Since the arcs of / are proportional to their
projections on r, it is clear that the ratio between a (recti-
fied) arc of / and its projection does not depend on the
length of the arc, but only on its distance x from r. De
Tilly's function Ex is the function which expresses this ratio.

On this understanding, the Formulae of Absolute Trigon-
ometry for the right-angled triangle ABC 2^0. as follows:
(i) \C)a = Qjc sin A

[0'^ = O^sin^'

(2) fcos A = Ea. sin B
[cos B = Eb. sin A

(3) Ec = Ea. Eb.
The set (i) is equivalent to Bolyai's

Theorem for the Right- Angled Triangle.
All the formulae of Absolute Trigono-
metry could be derived by suitable com-
bination of these three sets. In particular, for the right-angled
triangle, we obtain the following equation: —

I Mémoires couronnés et autres Mémoires, Acad, royale de
Belgique. T. XXI (1870). Cf. also the work by the same author:
Essai sur les principes Jmtdamentaux dc la p-rométrie et de la Mccanique,
Mém. de la Soc. des Se. de Bordeaux. T. Ili (cah. I) (1878).

The Absolute Trigonometry. II c

O^a {Ea + Eb. Ec) + Q'-^- ^Eb + Ec. Ea)
= O'^ (^^ + Ea. Eb).
This can be regarded as equivalent to the Theorem of
Pythagoras in the Absolute Geometry.^

§ 57. Let us now see how we can deduce the results
of the Euclidean and Non-Euclidean geometries from the
equations of the preceding article.

Euclidean Case.

The Equidistant Curve (/) is a straight line [that is, Ex
= 1], and the perimeters of circles are proportional to
their radii.

Thus the equations (i) become
(i') {a = c sin A

\b = c sin B.

The equations (2) give
(2') cos A = sin B, cos B = sin A.

Therefore A A^ B = 90°.

Finally the equation (3) reduces to an identity.

The equations (i') and (2') include the whole of ordin-
ary trigonometry.

Non- Euclidean Case.

Combining the equations (i) and (2) we obtain

E^a—l E2b—\

If we now apply the first of equations (2) to a right-
angled triangle whose vertex A goes oft" to infinity, so that
the angle A tends to zero, we shall have

Lt cos A = Lt {Ea. sin B).
But Ea is independent of A; also the angle B, in the
limit, becomes the angle of parallelism corresponding to a,
i. e. n {a).

^ Cf. R. BoNOLV, La trigonometria assoluta secondo Giovann:
Bolyai. Rend. Istituto Lombardo (2). T. XXXVIII (1905).

Il6 IV. The Founders of Non-Euclidean Geometry (Contd.).

Therefore we have

sin n (a)
A similar result holds for Eb.

Substituting these in equation (5) we obtain

cot2 TT {a) cot2 TT {b) '

from which

cot IT {a) cot jr (^}

This result, with the expression for Ex, allows us at
once to obtain from the equations (i), (2), (3), the formulae
of the Trigonometry of Bolyai-Lobatschewsky:

fcot TT {a) = cot TT {c) sin A
^^ ' jcot n {b) = cot n {c) sin B,
, „^ fsin A = cos B sin TT {b)

I 2 ) -I

Ì sin j9 = cos A sin TT («;),
(3") sin TT {c) = sin TT {a) sin TT {b).

These relations bet\veen the elements of every right-
angled triangle were given in this form by Lobatschewsky/
If we wish to introduce direct functions of the sides, instead
of the angles of parallelism TT (a), TT {b) and TT (^), it is
sufficient to remember [p. 90] that

tan —^ = e "'*.

We can thus express the circular functions of TT {x) in
terms of the hyperbolic functions of x. In this way the pre-
ceding equations are replaced by the following relations:

(i"'_) sinh -7- = sinh -j sin A

k k

b
J

sinh -r = sinh -7- sin B,

I Cf. e. g., The Geometrische Untersuchungen of LOBATSCHEWSKY
referred to on p. 86.

Absolute Trigonometry and Spherical Trigonometry. 117

(2"') COS A = sin B cosh -r

cos B = sin A cosh ^r.

and

(•?'") cosh -,- = cosh -y- cos /i cosh -7-.

§ 58. The following remark upon Absolute Trigono-
metry is most important: // we regard the elements in its
formulae as elements of a spherical triangle, we obtain a system
of equations which hold also for Spherical Triangles.

This property of Absolute Trigonometry is due to the
fact, already noticed on p. 114, that it was obtained by the
aid of equations which hold only for a limited region of the
plane. Further these do not depend on the hypothesis of the
angles of a triangle, so that they are valid also on the sphere.

If it is desired to obtain the result directly, it is only
necessary to note the following facts: —

(i) In Spherical Trigonometry the circumferences of
circles are proportional to the sines of their (spherical) radii,
so that the first formula for right-angled spherical triangles

sin <? = sin ^ sin A
is transformed at once into the first of the equations (i),

(ii) A circle of (spherical) radius b can be [con-
sidered as a curve equidistant from the concentric great
circle, and the ratio Eb for these two circles is given by

(v - 0

sm J^
2

= cos b.

Thus the formulae for right-angled spherical triangles
cos ^ = sin ^ cos a,
cos c = cos a cos b.

jl8 IV. The Founders of Non-Euclidean Geometry (Contd.).

are transformed immediately into the equations (2) and (3)
by means of this result.

Thus the formulae of Absolute Trigonometry also hold
on the sphere.

Hypotheses equivalent to Euclid's Postulate.

§ 59. Before leaving the elementary part of the sub-
ject, it seems right to call the attention of the reader to the
position occupied in the general system of geometry by certain
propositions, which are in a certain sense hypotheses equivalent
to the Fifth Postulate.

That our argument may be properly understood, we
begin by explaining the meaning of this equivalence.

Two hypotheses are absolutely equivaletit when each of
them can be deduced from the other without the lielp of any
new hypothesis. In this sense the two following hypotheses
are absolutely equivalent:

a) Two straight lines parallel to a third are parallel to
each other;

b) Through a point outside a straight line one and only
one parallel to it can be drawn.

This kind of equivalence has not much interest, since
the two hypotheses are simply two different forms of the
same proposition. However we must consider in what way
the idea of equivalence can be generalised.

Let us suppose that a deductive science is founded
upon a certain set of hypotheses, which we will denote by
\A,B, C,...If\. 'LttM and ^be two new hypotheses such
that N can be deduced from the set [A, B, C ... IT, J/|,
and M from the set {a, B, C . . . H, N)

We indicate this by writing

{A,B, C ...H,M) .) .jV,

Absolute and Relative Equivalence. ng

and

{A,B, C .. .H,N) .). M.

We shall now extend the idea of equivalence and say that
the two hypotheses J/, N are equivalent relatively to the
fundamental set \A, B, C . . . HY

It has to be noted that the fundamental set {A, B, C
. . . Jl^f has an important place in this definition. Indeed it
might happen that by diminishing this fundamental set, leav-
ing aside, for example, the hypothesis A, the two deductions

{B, C,.,.JI,M} .). JV
and

{B,C,...If,Ar\.).M

could not hold simultaneously.

In this case the hypotheses M, N are not equivalent
with respect to the new fundamental set \B^ C . . . H^

After these explanations of a logical kind, let us see
what follows from the discussion in the preceding chapters
as to the equivalence between such hypotheses and the
Euclidean hypothesis.

We assume in the first place as fundamental set of
hypotheses that formed by the postulates of Association {A),
and of Distribution {B)^ which characterise in the ordinary
way the conceptions of the straight line and the plane: also
by the postulates of Congruence (C), and the Postulate of
Archimedes (Z>).

Relative to this fundamental set, which we shall denote
by \a, B, C, D\, the following hypotheses are mutually
equivalent, and equivalent also to that stated by Euclid in
his Fifth Postulate:

a) The internal angles, which two parallels make with a
transversal on the same side, are supplementary [Ptolemy].

b) Two parallel straight lines are equidistant.

c) If a straight line intersects one of two parallels, it
also intersects the other (Proclus);

120 IV. The Founders of Non-Euclidean Geometry (Contd.).

or,

Two straight lines, which are parallel to a third, are
parallel to each other;
or again.

Through a point outside a straight line there can be
drawn one and only one parallel to that line.

d) A triangle being given, another triangle can be con-
structed similar to the given one and of any size whatever.
[Wallis.]

e) Through three points, not lying on a straight line, a
sphere can always be drawn. [W. Bolyai.]

f) Through a point between the Hnes bounding an angle
a straight hne can always be drawn which will intersect these
two lines. [Lorenz.]

a) If the straight line r is perpendicular to the trans-
versal AB and the straight line s cuts it at an acute angle?
the perpendiculars from the points of s upon r are less than
AB^ on the side in which AB makes an acute angle with s.
[Nasìr-Eddìn.]

P) The locus of the points which are equidistant from
a straight line is a straight line.

f ) The sum of the angles of a triangle is equal to two
right angles. [Saccherl]

Now let us suppose that we diminish the fundamental
set of hypotheses, cutii?ig oiit the Archimedean Hypothesis.
Then the propositions (a), (b), (c), (d), (e) and (f) are
mutually equivalent, and also equivalent to the Fifth Postu-
late of Euclid, with respect to the fundamental set |^, ^, C].
With regard to the propositions (a), (P), (t), while they are
mutually equivalent with respect to the set \A, B, C| no one
of them is equivalent to the Euclidean Postulate. This result
brings out clearly the importance of the Postulate of Archi-
medes. It is given in the memoir of Dehn' [19°°] to which

I Cf. Note on p. 30.

Hypotheses Equivalent to Euclid's Postulate. i2I

reference has already been made. In that memoir it is sho^vn
that the hypothesis (f) on the sum of the angles of a triangle
is compatible not only with the ordinary elementary geo-
metry, but also with a new geometry— necessarily Non-Archi-
medean—where the Fifth Postulate does not hold, and in
which an infinite number of lines pass through a point and
do not intersect a given straight line. To this geometry the
author gave the name of Semi-Euclidean Geometry.

The Spread of Non-Euclidean Geometry.

§ 60. The works of Lobatschewsky and Bolyai did
not receive on their publication the welcome which so many
centuries of slow and continual preparation seemed to
promise. However this ought not to surprise us. The
history of scientific discovery teaches that every radical change
in its separate departments does not suddenly alter the con-
victions and the presuppositions upon which investigators
and teachers have for a considerable time based the present-
ation of their subjects.

In our case the acceptance of the Non-Euclidean Geo-
metry was delayed by special reasons, such as the difficulty
of mastering Lobatschewsky's works, written as they were in
Russian, the fact that the names of the two discoverers were
new to the scientific world, and the Kantian conception of
space which was then in the ascendant.

Lobatschewsky's French and German writings helped
to drive away the darkness in which the new theories were
hidden in the first years; more than all availed the constant
and indefatigable labors of certain geometers, whose names
are now associated with the spread and triumph of Non-
Euclidean Geometry. We would mention particularly: C. L.
Gerling [1788— 1864], R. Baltzer [1818— 1887] and Fr.
Schmidt [1827 — 1901], in Germany; J. Hoùel [1823 —

122 IV. The Founders of Non-Euclidean Geometry (Contd.).

1886], G. Battaglini [1826 — 1894], E. Beltrami [1835—
1900], and A. Forti, in France and Italy.

§ 61. From 181 6 Gerling kept up a correspondence
upon parallels with Gauss % and in 181 9 he sent him
Schweikart's memorandum on Astra lgeo?/ietrie [cf. p. 75].
Also he had heard from Gauss himself [1832], and in terms
which could not help exciting his natural curiosity, of a
kleine Schrift on Non-Euclidean Geometry written by a
young Austrian officer, son of W. Bolyai.* The bibliograph-
ical notes he received later from Gauss [1844] on the works
of Lobatschewsky andBoLYAi^ induced Gerling to procure
for himself the Geomdrischeii Uiitersuchtingen and the Appen-
dix, and thus to rescue them from the oblivion into which
they seemed plunged.

§ 62. The correspondence between Gauss and Schu-
macher, published between i860 and iSós,"^ the numerous
references to the works of LoBATSCHEWSKy and Bolyai, and
the attempts of Legendre to introduce even into the elemen-
tary text books a rigorous treatment of the theory of pa-
rallels, led Baltzer, in the second edition of his Elemmte der

1 Cf. Gauss, Werke, Bd. VIII, p. 167—169.

2 Cf. Gauss's letter to Gerling (Gauss, Werke, Bd. VIII,
p. 220). In this note Gauss says with reference to the contents
of the Appendix: "worin ich alle meine eigenen Ideen taid Resultate
wlederfiiìde mit g7-ofier Eleganz entwickelt." And of the author of
the work : „Ich halte diesm jiingC7i Geonietei' v. Bolyai fib- eni Genie
erster Grafie".

3 Cf. Gauss, IVerke, Bd. VIII, p. 234—238.

4 Briefwechsel ziuischen C, F. Gauss 7cnd H. C. Schuinacher,
Bd. II, p. 268—431 Bd. V, p. 246 (Altona, 1860—1863). As to
Gauss's opinions at this time, see also, Sartorius von Walters'
HaUSEN, Gatifi zutn Geddr/Unis, p. 8o— 8l (Leipzig, 1S56). Cf. GAUSS,
Werke, Bd. VIII, p. 267—268.

The Spread of Non-Euclidean Geometry. 1 23

Mathemafik {1^6'j), to substitute, for the Euclidean definition
of parallels one derived from the new conception of space.
Following LoBATSCHEWSKY he placed the equation A-\-B
+ C = 180°, which characterises the Euclidean triangle,
among the experimental results. To justify this innovation,
Baltzer did not fail to insert a brief reference to the possi-
bility of a more general geometry than the ordinary one,
founded on the hypothesis of two parallels. He also gave
suitable prominence to the names of its founders.^ At the
same time he called the attention of Houel, whose interest
in the question of elementary geometry was well known to
scientific men, ^ to the Non-Euclidean geometry, and re-
quested him to translate the Geometrischen Untersiichungen
and the Appendix into French.

§ 63. The French translation of this little book by
LoBATSCHEWSKY appeared in 1866 and was accompanied
by some extracts from the correspondence between Gauss
and Schumacher.^ That the views of Lobatschewsky,
Bolyai, and Gauss were thus brought together was extremely
fortunate, since the name of Gauss and his approval of the
discoveries of the two geometers, then obscure and unknown,

1 Cf. Baltzer, Elemente der Mathematik, Bd. 11 (5. Auflage)
p. 12 — 14 (Leipzig, 1878). Also T. 4, p. 5 — 7, of Cremona's trans-
lation of that work (Genoa, 1867).

2 In 1863 HoiJEL had published his wellknown Essai d'une
exposition 7-ationelIe des principes fondametitmcx de la Geometrie èli-
7!ientaire. Archiv d. Math. u. Physik, Bd. XL (1863).

3 Ména, de la Soc. des Sci. de Bordeaux, T. IV, p. 88 — 120
(1S66). This short work was also published separately under the
title Etudes géométriques sur la théorie des parallèles par N. I. LoBAT-
SCHEWSKY, Conseiller d'État de l'Empire de Russie et Professeur
à rUniversité de Kasan: traduit de l'allemand par J. Houel, suivie
d'un Extrait de la correspondance de Gauss et de Schumacher, (Paris,
G. VU.LARS, 1866).

124 ^^' T^ti^ Founders of Non-Euclidean Geometry (Contd.).

helped to bring credit and consideration to the new doctrines
in the most efficacious and certain manner.

The French translation of the Appendix appeared in
1867.' It was preceded by a Notice sur la vie et les travaux
des deux viathématiciens hotigrois W. et J. Bolyai de Bolya,
written by the architect Fr. Schmidt at the invitation of
HoiJEL,^ and was supplemented by some remarks by W. Bol-
yai, taken from Vol. I of the Tentameli and from a short
analysis, also by Wolfgang, of the Principles of Arithmetic
and Geometry.3

In the same year [1867] Schmidt's discoveries regard-
ing the BoLYAis were published in the Archiv d. Math. u.
Phys. Also in the following year A. Forti, who had already
written a critical and historical memoir on Lobatschewsky,'*

1 Mém. de la Soc. des Se. de Bordeaux, T. V, p. 189 —
248. This short work was also published separately unter the
title: La Science absolute de l' espace, indèpendante de la vérité on

fausseti de l'Axiome XI d'Euclide {que l'on ne pourra jamais établiì- a
priori); suivie de la quadrature géometrique du cercle, dans le cas de
la fausseté de l'Axiome XI, par Jean Bolyai, Capitaine au Corps
du genie dans l'armée autrichienne; Précède d'iene notice sur la vie
it les travaux de W. et de J. Bolyai, par M. Fr. Schmidt, (Paris,
G. VlLLARS, 1868).

2 Cf. P. StAckel, Franz Schmidt, Jahresber. d. Deutschen
Math. Ver., Bd. XI, p. 141 — 146 (1902).

3 This little book of \V. BoLYAl's is usually referred to
shortly by the first words of the title Kicrzer Grtmdriss. It was pub-
lished at Maros-Visàrhely in 1851.

■* Intorno alla geometria itnmaginaria o non euclidiana. Consid-
erazioni storico-critiche. Rivista Bolognese di scienze, lettere, arti
e scuole, T. Il, p. 171 — 184 (1867). It was published separately
as a pamphlet of 16 pages (Bologna, Fava e Garagnani, 1867).
The same article, with some additions and the title, Studii geo-
metrici sulla teorica delle parallele di N. J. Lobatschewky, appeared
in the politicai journal La Provincia di Pisa, Anno III, Nr. 25, 27,

Hoùel and Schmidt. 1 25

made the name and the works of the two now celebrated
Hungarian geometers known to the Italians/

To the credit of Hoùel there should also be mentioned
his interest in the manuscripts of Johann Bolyai, then [1867]
preserved, in terms of Wolfgang's will, in the library of
the Reformed College of Maros-Vàsàrhely. By the help of
Prince B. Boncampagni [182 i — 1894], who in his turn in-
terested the Hungarian Minister of Education, Baron Eòtvòs,
he succeeded in having them placed [1869] in the Hungarian
Academy of Science at Budapest.^ In this way they became
more accessible and were the subject of painstaking and
careful research, first by Schmidt and recently by Stackel.

In addition Houel did not fail in his efforts, on every
available opportunity, to secure a lasting triumph for the Non-
Euclidean Geometry. If we simply mention his Essai cri-
tique sur les principes fondameìiteaux de la geometrie:'^ his ar-
ticle, Sur l' impossibilité de démontrer par tene construction
plane le postulatum d'Euclide; "* the Notices sur la vie et les
iravaux de N. J. Lobatschewsky; 5 and finally his translations
of various writings upon Non-Euclidean Geometry into French,^

29, 30 (1867); and part of it was reprinted under the original title
(Pisa, Nistri, 1867).

* Cf. Iniorito alia vita ed agli sa-itti di Wolfgang e Giovanni
Bolyai di Bolya, rnatemalici ungheresi. Boll, di Bibliografia e di
Storia delle Scienze Mat. e Fisiche. T. I, p. 277—299 (1869),
Many historical and bibliographical notes were added to this article
of Forti's by B. Boncompagni.

2 Cf. Stackel's article on Franz Schmidt referred to above.

3 I. Ed., G. ViLLARS, Paris, 1867; 2 Ed., 1883 (cf. Note 3
p. 52).

4 Giornale di Mat. T. VII p. 84— 89; Nouvelles Annales (2)
T. IX, p. 93-96.

5 Bull. des. Sc. Math. T. I, p. 66—71, 324—328, 384—388
(1870).

0 In addition to the translations mentioned in the text, Hoùel

126 IV. The Founders of Non-Euclidean Geometry (Contd.).

it will e understood how fervent an apostle this science had
found in the famous French mathematician.

Hoùel's labours must have urged J. Frischauf to per-
form the service for Germany which the former had rendered
to France. His book — Absolute Geometrie nach J. Bolyai —
(1872)" is simply a free translation of Johann's Appendix, to
which were added the opinions of W. Bolyai on the Found-
ations of Geometry. A new and revised edition of Frisch-
auf's work was brought out in 1876^. In that work reference
is made to the writings of Lobatschewsky and the memoirs
of other authors who about that time had taken up this study
from a more advanced point of view. This volume remained
for many years the only book in which these new doctrines
upon space were brought together and compared.

§ 64. With equal conviction and earnestness Giuseppe
Battaglini introduced the new geometrical speculations into
Italy and there spread them abroad. From 1867 the Gior-
nale di Matematica, of which he was both founder and editor,
became the recognized organ of Non-Euclidean Geometry.

Battaglini's first memoir — Sulla geometria immaginaria
di Lobatschewsky^— y^z.% written to establish directly the prin-
ciple which forms the foundation of the general theory of
parallels and the trigonometry of Lobatschewsky. It was

translated a paper by Battaglini (cf. note 3), two by Beltrami
(cf. note 2 p. 127 and p. 147); one, by Rif.mann (cf. note p. 138).
and one by Helmholtz (cf. note p. 152).

1 (xii ■\- 96 pages) (Teubner, Leipzig).

2 Eletnente der Ahsoluteii Geometrie, (vi -|- 142 pages) (Teubner,
Leipzig).

3 Giornale di Mat. T. V, p. 217 — 231 (1S67). Rend. Ace.
Science Fis. e Matem. Napoli, T. VI, p. 157 — 173 (1867). French
translation, by HoUEL, Nouvelles Annales (2) T. VII, p. 209—21,
2Ó5— 277 (i8óS).

Battaglini and Beltrami. 127

followed, a few pages later, by the Italian translation of the
Pangéométrie'^; and this, in its turn, in 1868, by the translation
of the Appendix.

At the same time, in the sixth volume of the Giornale di
Matematica, appeared E. Beltrami's famous paper, Saggio di
ititerpretazione della geometria non euclidea. ^ This threw an
unexpected light on the question then being debated regard-
ing the fundamental principles of geometry, and the concep-
tions of Gauss and Lobatschewsky.-^

Glancing through the subsequent volumes of the Giorn-
ale di Matematica we frequently come upon papers upon
Non-Euclidean Geometry. There are two by Beltrami [1872]
connected with the above— named Saggio; several by Batt-
aglini [1874 — 78] and by d'OviDio [1875 — 77]? which treat
some questions in the new geometry by the projective me-
thods discovered by Cayley; Houel's paper [1870] on the
impossibility of demonstrating Euclid's Postulate; and others
by Cassani [1873 — 81], Gunther [1876], De Zolt [1877],
Frattini [1878], Ricordi [1880], etc.

§ 65. The work of spreading abroad the knowledge of
the new geometry, begun and energetically carried forward
by the aforesaid geometers, received a powerful impulse from
another set of publications, which appeared about this time
[1868—72]. These regarded the problem of the foundations
of geometry in a more general and less elementary way than
that which had been adopted in the investigations of Gauss,

1 This was also published separately as a small book, entitled,
Pangeometria 0 sunto di geometria fondata sopra una teoria generate
e rigorosa delle parallele (Naples, 1867; 2a Ed. 1874).

2 It was translated into French by Houel in the Ann. Sc. de
l'École Normale Sup., T. VI, p. 251—288 (1869).

3 Cf. Commemorazione di E. Beltrami by L. CREMONA: Giornale
di Mat. T. XXXVIII, p. 362 (1900). Also the Nachruf by E.
Pascal, Math. Ann. Bd. LVII, p. 65—107 (1903).

128 IV. The Founders of Non-Euclidean Geometry (Contd.).

LoBATSCHEWSKY, and BoLYAi. In Chapter V. we shall shortly
describe these new methods and developments, which are asso-
ciated with the names of some of the most eminent mathe-
maticians and philosophers of the present time. Here it is
sufficient to remark that the old question of parallels, from
which all interest seemed to have been taken by the in-
vestigations of Legendre forty years earlier, once again and
under a completely new aspect attracted the attention of geo-
meters and philosophers, and became the centre of an
extremely wide field of labour. Some of these efforts were
simply directed toward rendering the works of the founders
of Non-Euclidean geometry more accessible to the general
mathematical public. Others were prompted by the hope of
extending the results, the content, and the meaning of the
new doctrines, and at the same time contributing to the pro-
gress of certain special branches of Higher Mathematics,^

I Cf. e. g., É. Picard, La Science Moderne et son état
actual, p. 75 (Paris, Flammarion, 1905).

Chapter V.

The Later Development of Non-Euclidean
Geometry.

§ 66. To describe the further progress of Non-Eudidean
Geometry in the direction of Differential Geometry and Pro-
jective Geometry, we must leave the field of Elementary Mathe-
matics and speak of some of the branches of Higher Mathe-
matics, such as the Differential Geometry of Manifolds, the
Theory of Continuous Transformation Groups^ Pure Projec-
tive Geometry (the system of Staudt) and the Metrical
Geometries which are subordinate to it. As it is not consistent
with the plan of this work to refer, even shortly, to these
more advanced questions, we shall confine ourselves to those
matters without which the reader could not understand the
motive spirit of the new researches, nor be led to that other
geometrical system, due to Riemann, which has been alto-
gether excluded from the previous investigations, as they
assume that the straight line is of infinite length.

This system is known by the name of its discoverer and
corresponds to the Hypothesis of the Obtuse A?igle of Sac-
CHERi and Lambert.^

^ The reader, who wishes a complete discussion of the sub-
ject of this chapter, should consult Klein's Vorlesungen uber die
iiickteuklidische Geometrie, (Gòttingen, 1903); and BlANCHl's Lezioni
sulla Geometria differenziale, 2 Ed. T. I, Cap. XI — XIV (Pisa, Spoerri,
1903). German translation by Lukat, i^t Ed. (Leipzig, 1899). Also
The Elements of Non-Eicclidean Geometry by T. L. CoOLlDGE which
has recently (1909) been published by the Oxford University Press.

I 20 V. The Later Development of Non-Euclidean Geometry.

Differential Geometry and Non-Euclidean Geometry.

The Geometry upon a Surface.

§ 67. What follows will be more easily understood if
we start with a few observations:

A surface being given, let us see how far we can establish
a geometry upon it analogous to that on the plane.

Through two points A and B on the surface there will
generally pass one definite line belonging to the surface,
namely, the shortest distance on the surface between the two
points. This line is called the geodesic joining the two points.
In the case of the sphere, the geodesic joining two points, not
the extremities of a diameter, is an arc of the great circle
through the two points.

Now if we wish to compare the geometry upon a surface
with the geometry on a plane, it seems natural to make the
geodesies, which measure the distances on the one surface,
correspond to the straight lines of the other. It is also natural
to consider two figures traced upon the surface as {geodetical-
ly) equal, when there is a point to point correspondence be-
tween them, such that the geodesic distances between corre-
sponding points are equal.

We obtain a representation of this conception of equality,
if we assume that the surface is made of z. flexible and itiex-,
tensible sheet. Then by a movement of the surface, which does
not remain rigid, but is bent as described above, those figures
upon it, which we have called equal, are to be superposed
the one upon the other.

Let us take, for example, a piece of a cylindrical surface.
By simple bending, without stretching, folding, or tearing, this
can be applied to a plane area. It is clear that in this case
two figures ought to be called equal on the surface, which
coincide with equal areas on the plane, though of course two
such figures are not in general equal in space.

Differential Geometry and Non-Euclidean Geometry. j^j

Returning now to any surface whatsoever, the system of
conventions, suggested above, leads to a geometry on the sur-
face, which we propose to consider ahvays for suitably bounded
regions {^Normal Regions]. Two surfaces which are applicable
the one to the other, by bending without stretching, will have
the same geometry. Thus, for example, upon any cylindrical
surface whatsoever, we will have a geometry similar to that on
any plane surface, and, in general, upon any developable surface.

The geometry on the sphere affords an example of a
geometry on a surface essentially different from that on the
plane, since it is impossible to apply a portion of the sphere
to the plane. However there is an important analogy be-
tween the geometry on the plane and the geometry on the
sphere. This analogy has its foundation in the fact that the
sphere can be freely moved upon itself, so that propositions
in every way analogous to the postulates of congruence on
the plane hold for equal figures on the sphere.

Let us try to generalize this example. In order that a
suitably bounded surface, by bending but without stretching,
can be moved upon itself in the same way as a plane, a cer-
tain number \K\ invariant with respect to this bending, must
have a constant value at all points of the surface. This number
was introduced by Gauss and called the Curvature.'^ [In
English books it is usually called Gauss's Curvature or tlie
Measure of Curvature.]

I Remembering that the curvature at any poir t of a plane
curve is the reciprocal of the radius of the osculating circle for
that point, we shall now show that the curvature at a point M of the
surface can be defined. Having drawn the normal n to the surface
at M, we consider the pencil of planes through n, and the corre-
sponding pencil of curves formed by their intersections with the
surface. In this pencil of (plane) curves, there are two, orthogonal
to each other, whose curvatures, as defined above, are maximum
and minimum. The product of their curvatures is Gauss's Curva-
ture of the Surface at M. This Curvature has one most marked

I 32 ^ • The Later Development of Non-Euclidean Geometry.

Surfaces of Constant Curvature can be actually con-
structed. The three cases

K^O, A'>6>, K<^0,
have to be distinguished.

For K^= 6>, we have the developable surfaces [applic-
able to the plane].

For K^ O, we have the surfaces applicable to a sphere
of radius i : "j/ A', and the sphere can be taken as a model
for these surfaces.

For K<^ O, we have the surfaces applicable to the
Pseudosphere, which can be taken as a model for the surfaces
of constant negative curvature.

Pseudosphere.

Fig. 54-

The Pseudosphere is a surface of revolution. The equat-
ion of its meridian curve (the tractrix ^) referred to the axis

characteristic. It is unchanged for every bending of the surface
which does not involve stretching. Thus, if two surfaces are
applicable to each other in the sense of the text, they ought to
have the same Gaussian Curvature at corresponding points [Gauss],

This result, the converse of which was proved by Minding
to hold for vSurfaces of Constant Curvature, shows that surfaces,
freely movable upon themselves, are characterised by constancy of
curvature.

^ The tractrix is the curve in which the distance from the

Surfaces of Constant Curvature.

133

of rotation z, and to a suitably chosen axis of ;c perpendicular
to z, is

kJ^y k^—x2

(i) z = k\og'^ I —Vk'-x%

where k is connected with the Curvature K by the equation

To the pseudosphere generated by (i) can be applied
any portion of the surface of constant curvature — ,-.

Surface of Constant Negative Curvature.^
Fig. 56.

point of contact of a tangent to the point where it cuts its
asymptote is constant.

I Fig. 56 is reproduced from a photograph ef a surface con-
structed by Beltrami. The actual model belongs to the collection
of models in the Mathematical Institute of the University of Pavia.

I 34 V. The Later Development of Non-Euclidean Geometry.

§ 68. There is an analogy between the geometry on a
surface of constant curvature and that of a portion of a plane,
both taken within suitable boundaries. We can make this
analogy clear by tratislatiug the fundamental definitions and
properties of the one into those of the other. This is indicat-
ed shortly by the positions which the corresponding terms
occupy in the following table:

(a) Surface. (a) Portion of the plane.

(b) Point. (b) Point.

(d) Arc of Geodesic. (d) Rectilinear Segment.

(e) Linear properties of the (e) Postulates of Order for
Geodesic. points on a Straight Line.

(f) A Geodesic is determined (f) A Straight Line is deter-
by two points. mined by two points.

(g) Fundamental properties (g) Postulates of Congruence
of the equality of Geode- for Rectilinear Segments
sic Arcs and Angles. and Angles.

(h) If two Geodesic triangles (h) If two Rectilinear triang-
have their two sides and les have their two sides

the contained angles e- and the contained angles

qual, then the remaining equal, then the remaining

sides and angles are equal. sides and angles are equal.

It follows that we can retain as common to the geome-
try of the said surfaces all those properties concerning bound-
ed regions on a plane, which in the Euclidean system are
independent of the Parallel Postulate, when no use is made
of the complete plaiic [e. g., of the infinity of the straight
line] in their demonstration.

We must now proceed to compare the propositions for
a bounded region of the plane, depending on the Euclidean
hypothesis, with those which correspond to them in the geo-
metry on the surface of constant curvature. We have, e. g.,
the proposition that the sum of the angles of a triangle is

Geometry on a Surface of Constant Curvature. i^c

equal to two right angles. The corresponding property does
not generally hold for the surface.

Indeed Gauss showed that upon a surface whose curva-
ture K is constant or varies from point to point, the surface
integral

over the whole surface of a geodesic triangle ABC, is eqtial
io the excess of its three angles over two right angles. '

i. e. \[ KdS =A-VB+ C— IT.

ABC

Let us apply this formula to the surfaces of constant
curvature, distinguishing the three possible cases —
Case 1. K=^0.

In this case we have

UxdS = O; that is ^ + ^ + C=tx.

ABC

Thus the sum of the angles of a geodesic triangle on sur-
faces of zero curvature is equal to two 7'ight angles.

Case II. ^=i> ^•

In this case we have

ABC ABC

But {^dS = area of the triangle ABC = A.

^^=A-\- B-\- C—-K.

From this equation it follows that
^ + ^ + C> TT,
and that L=k^ {A^ B ■\- C— tt).

1 Cf. BlANCHi's work referred to above; Chapter VI.

I ?5 V. The Later Development of Non-Euclidean Geometry.

That is:

a) The sum of the angles of a geodesic triangle on sur-
faces of constant positive curvature is greater than two right
afigles.

b) The area of a geodesic triangle is proportional to the
excess of the sum of its angles over two right angles.

X--T,<0

Case III.

In this case we have

ABC ABC

where we again denote the area of the triangle ABC hy A.
Then we have

From this it follows that

A-ir B + C<Tr,
and that A = /è^Tt — ^ - B— C).

That is:

a) The sum of the angles of a geodesic triangle on sur-
faces of constant negative curvature is less than two right angles.

b) The area of a geodesic triangle is proportional to the
difference between the sum of its angles and two right angles.

We bring these results together in the following table:

Surfaces of Constant Curvature.

"Value of the Curvature

Model
of the Surface

Character of the Triangle

K^O Plane

<^^+<^^+<^C=TT

^'T.

Sphere

^^ + ^^ + <^C>Tr

K^ —

Pseudosphere

^A-\-^B-^^C<rz

The Geodesic Triangle. 1 27

With the geometry of surfaces of zero curvature and of
surfaces of constant positive curvature we are already ac-
quainted, since they correspond to Euclidean plane geometry
and to spherical geometry.

The study of the surfaces of constant negative curvature
was begun by F. Minding [1806 — 1885] with the investiga-
tion of the surfaces of revolution to which they could be ap-
plied.* The following remark of Minding's, fully proved
by D. Codazzi [1824 — 1873], establishes the trigonometry
of such surfaces. In the formulae of spherical trigonometry let
the angles be kept fixed and the sides multiplied by i = Y--i-
Then we obtaifi the equations which are satisfied by the elements
of the geodesic triangles on the surf aces of cofistatit negative cur-
vature.^ These equations [the pseudospherical trigonometryl
evidently coincide with those found by Taurinus; in other
words, with the formulae of the geometry of Lobatschewsky-

BOLVAI.

§ 69. From the preceding paragraphs it will be seen that
the theorems regarding the sum of the angles of a triangle in
the geometry on surfaces of constant curvature, are related to
those of plane geometry as follows: —

For K= O they correspond to those which hold on the
plane in the case of the Hypothesis of the Right Angle.

For K'^ O they correspond to those which hold on the
plane in the case of the Hypothesis of the Obtuse Angle.

1 Wie sick entscheldeii lasst, ob zivei gegebene knimme Flachen
aufelnander abwickelbar sind oder tticht; nebst Bemerknngen iiber die
Fliichen von unveranderlichem Kriimtmtngsmasse. Crelle^s Journal,
Bd. XIX, p. 370-387 (1839).

2 Minding: Beitrage zur Theorie der kiirzesten Linien aiif krummen
Flachen. Crelle's Journal, Bd. XX, p. 323—327 (1^40). D. Codazzi:
Intorno alle superficie, le guali hanno costante il prodotto de' dice raggi
di curvatura. Ann. di Scienze Mat. e Fis. T. Vili, p. 346 — 355
O857).

I •^S V. The Later Development of Non-Euclidean Geometry.

For K<CO they correspond to those which hold on the
plane in the case of the Hypothesis of the Acute Angle.

The first of the results is evident a priori, since we are
concerned with developable surfaces.

The analogy between the geometry of the surfaces of con-
stant negative curvature, for example, and the geometry of
LoBATSCHEWSKY-BoLYAi, could be made still more evident by
arranging in tabular form the relations between the elements
of the geodesic triangles traced upon those surfaces, and the
formulse of Non-Euclidean Trigonometry. Such a comparison
was made by E. Beltrami in his Saggio di interpretazione della
geometria non-euclidea. '

In this way it will be seen that the geometry upon a sur-
face of constant positive or negative curvature can be con-
sidered as a concrete interpretation of the Non-Euclideati Geo-
inetry, obtained in a bounded plane area, with the aid of the
Hypothesis of the Obtuse Angle or that of the Acute Angle.

The possibility of interpreting the geometry of a two-
dimensional manifold by means of ordinary surfaces was ob-
served by B. RiEMANN [1826 — 1866] in 1854, the year in
which he wrote his celebrated memoir: Ober die Hypothesen
welche der Geometrie zugrunde liegen.^ The developments of

1 Giorn. di Mat., T. VI, p. 284—312 (1868). Opere Mat.,
T. I, p. 374 — 405 (Hoepli, Milan, 1902).

2 Riemanns iVerke, 1. Aufl. (1876), p. 254 — 312: 2. Aufl.
(1892), p. 272 — 287. It was read by RlEMANN to the Philosophical
Faculty at Gottingen as his Hahilitatioiisschrift, before an audience
not composed solely of mathematicans. For this reason it does
not contain analytical developments, and the conceptions intro-
duced are mostly of an intuitive character. Some analytical ex-
planations are to be found in the notes on the Memoir sent by RiE-
MANN as a solution of a problem proposed by the Paris Academy
[Rietnatiits IVerke, I, Aufl., p. 384 — 391). The philosophical basis
of the Habilitatioiisschriji is the study of the properties of things
from their behaviour as infinitesimals. Cf. Klein's discourse!

Beltrami and Riemann.

139

Non-Euclidean Geometry in the direction of Differential Ge-
ometry are directly due to this memoir.

Beltrami's interpretation appears as a particular case of
Riemann's. It shows clearly, from the properties of surfaces
of constant curvature, that the chain of deductions from the
three hypotheses regarding the sum of the angles of a triangle
must lead to logically consistent systems of geometry.

This conclusion, so far as regards the Hypothesis of the
Obtuse Angle, seems to contradict the theorems of Saccheri,
Lambert, and Legendre, which altogether exclude the possi-
bility of a geometry founded on that hypothesis. However
the contradiction is only apparent. It disappears if we remem-
ber that in the demonstration of these theorems, not only
the fundamental properties of the bounded plane are used, but
also those of the complete plane, e. g., the property that the
straight line is infinite.

Principles of Plane Geometry on the Ideas of
Riemann.

§ 70. The preceding observations lead us to the foun-
dation of a metrical geometry, which excludes Euclid's Postul-

Riemann and seme BedeiUimg in der Entwickelung der modertten
Mathematik. Jahresb. d. Deutschen Math. Ver., Bd. IV, p. 72 — 82
(1894), and the Italian translation by E.Pascal in Ann. di Mat., (2),
T. XXIII, p. 222. The Habilitationsschrift was first published in 1867
after the death of the author [Gott. Abh. XIII] under the editor-
ship of Dedekind. It was then translated into French by J. HoiJEL
[Ann. di Mat. (2). T. Ill (1870), Oeuvres de Riemann, (1876)]; into
English, by \V. K. Clifford [Nature, Vol. VIII, (1873)], and again
by G. B. Halsted [Tokyo sagaku butsurigaku kwai kiji, Vol. VII,
(1895); into Polish, by DiCKSTEiN (Comm. Acad. Litt. Cracov.
Vol. IX, 1877); into Russian, by D. Sintsoff [Mem. of the Phy-
sical Mathematical Society of the University of Kasan, (2), Vol. Ill,
App. (1893)].

I^O V. The Later Development of Non-Euclidean Geometry.

ate, and adopts a more general point of view than that for-
merly held :

(a) We assume that we start from a bounded plane area
{normal region), and not from the whole plane.

(Ò) We regard as postulates those elementary propositio7is,
which are revealed to us by the senses for the region originally
taken; the propositions relative to the straight line being determ-
ined by two points, to congruence, etc.

{c) We assume that the properties of the initial region can
be extended to the neighbourhood of any point on the plane \jve
do not say to the complete plane., viewed as a whole].

The geometry, built upon these foundations, will be the
most general plane geometry, consistent with the data which
rigorously express the result of our experience. These results
are, however, limited to an accessible region.

From the remarks in § 69, it is clear that the said geo-
metry will find a concrete interpretation in that of the sur-
faces of constant curvature.

This correspondence, however, exists only from the
point of view {differejitial) according to which only bounded
regions are compared. If, on the other hand, we place our-
selves at the {integral) point of view, according to which the
geometry of the whole plane and the geometry on the sur-
face are compared, the correspondence no longer exists. In-
deed, from this standpoint, we cannot even say that the same
geometry will hold on two surfaces with the same constant
curvature. For example, a circular cylinder has a constant
curvature, zero, and a portion of it can be applied to a region
of a plane, but the entire cylinder cannot be applied in this
way to the entire plane. The geometry of the complete cy-
linder thus differs from that of the complete Euclidean plane.
Upon the cylinder there are closed geodesies (its circular
sections), and, in general, two of its geodesies (helices) meet
in an infinite number of points, instead of in just two.

Riemann's New Geometry. 141

Similar differences will in general appear between a me-
trical Non-Euclidean geometry, founded on the postulates
enunciated above, and the geometry on a corresponding sur-
face of constant curvature.

When we attempt to consider the geometry on a surface
of constant curvature (e. g., on the sphere or pseudosphere)
as a whole, we see, in general, that the fundamental property
of a normal region that a geodesic is fully determined by two
points ceases to hold. This fact, however, is not a necessary
consequence of the hypotheses on which, in the sense above
explained, a general metrical Non-Euclidean geometry of the
plane is based. Indeed, when we examine whether a system
of plane geometry is logically possible, which will satisfy the
conditions (a), (b), and(c), and in which the postulates of con-
gruence and that a straight line is fully determined by two
points are valid on the complete plane, we obtain, in addition
to the ordinary Euclidean system, the two following systems
of geometry:

1. The system of Lobatsc/iewsky-Bolyai, already explain-
ed, in which two parallels to a straight line pass through a,
point.

2. A netv system (called Rietnann's system) which cor-
responds to Saccheri's Hypothesis of the Obtuse Angle, and
in which no parallel lines exist.

In the latter system the straight line is a closed line of
finite length. We thus avoid the contradiction to which we
would be led if we assumed that the straight line were open
(infinite). This hypothesis is required in proving Euclid's The-
orem of the Exterior Angle [I. 1 6] and some of Saccheri's
results.

§ 71. RiEMANN was the first to recognize the existence
of a system of geometry compatible with the Hypothesis of
the Obtuse Angle, since he was the first to substitute for the

142 V. The Later Development of Non-Euclidean Geometry.

hypothesis that the straight Hne is infinite^ the more general
one that it is unbounded. The difterence, which presents it-
self here, between infinite and imboimded is most important.
We quote in regard to this Riemann's own words : *

'In the extension of space construction to the infinitely
great, we must distinguish between unboundedness and iiifinite
extent; the former belongs to the extent relations; the latter to
the measure relations. That space is an unbounded three-fold
manifoldness is an assumption which is developed by every
conception of the outer world; according to which every in-
stant the region of real perception is completed and the pos-
sible positions of a sought object are constructed, and which
by these applications is for ever confirming itself. The un-
boundedness of space possesses in this way a greater empiri-
cal certainty than any external experience, but its infinite ex-
tent 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.'

Finally, the postulate which gives the straight line an in-
finite length, implicitly contained in the work of preceding
geometers, is to Riemann as fit a subject of discussion as that
of parallels. What Riemann holds as beyond discussion is
the iinboimdediiess of space. This property is compatible with
the hypothesis that the straight line is infinite (open), as well
as with the hypothesis that it is finite (closed).

The logical possibility of Riemann's system can be de-
duced from its concrete interpretation in the geometry of the
sheaf of tines. The properties of the sheaf of lines are trans-

I [This quotation is taken from Clifford's translation in
Nature, referred to above. (Teil III, S 2 of Riemann's Memoir.)].

The Geometry of the Sheaf.

143

lated readily into those of Riemann's plane, and vice versa,
with the aid of the following dictionary :

Sheaf

Plane

Line

Point

Plane [Pencil]

Straight line

Angle between two Lines

Segment

Dihedral Angle

Angle

Trihedron

Triangle

We now give, as an example, the 'translation' of some
of the best known propositions for the sheaf:

a) The sum of the three a) The sum of the three

dihedral angles of a trihedron
is greater than two right
dihedral angles.

b) All the planes which are
perpendicular to another
plane pass through a straight
line.

c) With every plane of
the sheaf let us associate the
straight line in which the
planes perpendicular to the
given plane all intersect. In
this way we obtain a corres-
pondence between planes and
straight lines which enjoys
the following property: The
straight lines corresponding
to the planes of a pencil
[Ebenenbiischel, set of planes
through one line, the axis of
the pencil] lie on a plane,

angles of a triangle is greater
than two right angles.

b) All the straight lines
perpendicular to another
straight line pass through a
point.

c) With every straight line
in the plane let us associate
the point in which the lines
perpendicular to the given
line intersect. In this way we
obtain a correspondence be-
tween lines and points, which
enjoys the following pro-
perty:

The points corresponding
to the lines of a pencil lie on
a straight line^ which in its turn
has for corresponding point
the vertex of the pencil.

144 ^" '^^^ Later Development of Non-Euclidean Geometry,

which in its turn has for cor- The correspondence thus
responding line the axis of defined is called absolute po-
the pencil. The correspond- larity of the plane,
enee thus defined is called
absolute [orthogonal] polarity
of the sheaf.

§ 72. A remarkable discovery with regard to the Hypo-
thesis of the Obtuse Angle was made recently by Dehn.

If we refer to the arguments of Saccheri [p. 37],
Lambert [p. 45], Legendre [p. 56], we see at once that
these authors, in their proof of the falsehood of the Hypo-
thesis of the Obtuse Angle, avail themselves, not only of the
hypothesis that the straight line is infinite, but also of the
Archimedea?i Hypothesis. Now we might ask ourselves if this
second hypothesis is required in the proof of this result. In
other words, we might ask ourselves if the two hypotheses,
one of which attributes to the straight line the character of
open lines, while the other attributes to the sum of the angles
of a triangle a value greater than two right angles, are com-
patible with each other, when the Postulate of Archimedes is
excluded. Dehn gave an answer to this question in his
memoir quoted above (p. 30), by the construction of a iVw/-
Archimedean geometry, in which the straight line is open,
and the sum of the angles of a triangle is greater than two
right angles. Thus the second of Saccheri's three hypotheses
is compatible with the hypothesis of the open straight line
in the sense of a Non- Archimedean system. This new
geometry was called by Dehn Noji-Legendrean Geometry [cf.
S 59, P- 121].

§ 73. We have seen above that the geometry of a
surface of constant curvature (positive or negative) does not
represent, in general, the whole of the Non-Euclidean geo-

Hubert's Theorem.

145

metry on the plane of Lobatschewky and of Riemann. The
question remains whether such a correspondence could not
be effected with the help of some particular surface of this
nature.

The answer to this question is as follows :
i) There does not exist any regular'^ analytic surface
on which the geometry of Lobatschrujsky-Bolyai is altogether
valid [Hilbert's Theorem].^

1 In other words, free from singularities.

2 Uber Flacheti von konstanter Gatissscher A'nimmung. Trans.
Amer. Math. Soc. Vol. II, p. 86 — 99 (1901); Grundlagen der Geo-
metrie, 2. Aufl. p. 162 — 175. (Leipzig, Teubner, 1903).

This question, which Hilbert's Theorem answers, was first
suggested to mathematicians by Beltrami's interpretation of the
LoBATSCHEWKY-BoLyAi Geometry. In 1870 Helmholtz— in his
lecture, Uber U}-spning und Bedeuhing der geometrischen Axiome,
(Vortrage und Reden, Bd. II. Brunswick, 1844)— had denied the
possibility of constructing a pseudospherical surface, extending
indefinitely in every direction. Also A. Gennocchi — in his Lettre
à M. Qiietelet sur diverses questions tiiathèmatiques, [Belgique Bull. (2).
T. XXXVI, p. 181— 198 (1873)], and more fully in his Memoir,
Sur Ulte mhnoire de D. Foncenex et sur les geometries non-euclidieunes,
[Torino Memorie (3), T. XXIX, p. 365—404 (1877)], showed the
insufficiency of some intuitive demonstrations, intended to prove
the concrete existence of a surface suitable for the representation
of the entire Non -Euclidean plane. Also he insisted upon the
probable existence of singular points — (as for example, those on
the line of regression of Fig. 54) — in every concrete model of a
surface of constant negative curvature.

So far as regards Hilbert's Theorem, we add that the
analytic character of the surface, assumed by the author, has been
shown to be unnecessary. Cf. the dissertation of G. Lutkemeyer :
Uber den analytischen Charakter der Integrale von partiellen Differ en-
tialgleichungen, (Gottingen, 1902). Also the Note by E. Holmgren:
Sur les surfaces à courbure constante negative, [Comptes Rendus, I Sem.,
p. 840—843 (1902)].

[In a recent paper Sur les surfaces à courbure constante negative,
(Bull. Soc. Math, de France, t. XXXVII p. 51—58, 1909) É. GouRSAT

IaQ V. The Later Development of Non-Euclidean Geometry.

2) A surface on which the geometry of the piatte of
Riema?in 7uould be altogether valid itiust be a closed surface.

The only regular analytic closed surface of constant posi-
tive curvature is the sphere [Liebmann's Theorem].^

But on the sphere, in normal regions of which Riemann's
geometry is valid, two lines always meet in two (opposite)
points.

We therefore conclude that:

In ordinary space there are no surfaces 7vhich satisfy in
their complete extent all the properties of the Non-Euclideati
planes.

§ 74. At this place it is right to observe that the sphere,
among all the surfaces whose curvature is constant and different
from zero, has a characteristic that brings it nearer to the
plane than all the others. Indeed the sphere can be moved
upon itself just as the plane, so that the properties of con-
gruence are valid not only for normal regions, but, as in the
plane, for the surface of the sphere taken as a whole.

This fact suggests to us a method of enunciating the
postulates of geometry, which does not exclude, a priori, the
possible existence of a plane with all the characteristics of
the sphere, including that of opposite points. We would

has discussed a problem slightly less general than that enunciated
by Hilbert, and has succeeded in proving — in a fairly simple
manner — the impossibility of constructing an analytical surface of
constant curvature, which has no singular points at a finite distance.]
I Eiiw nelle EigenschaJ't der Kiigel, Gott. Nachr. p. 44 — 54
(1899). This property is also proved by Hilbert on p. 172 — 175
of his Gnindlagen der Geometrie. We notice that the surfaces of
constant positive curvature are necessarily analytic. Cf. I.UTKE-
meyer's Dissertation referred to above (p. 163), and the memoir
by Holmgren : Ober eine K lasse voji partiellen Differcntialgleickuiigen
der ziveiten Ordntmg, Math. Ann. Bd. TVII, p. 407 — 420 (1903).

The Elliptic and Spherical Planes. jaj

need to assume that ilic following relations were true for
the plane:

i) The postulates (/>), (c-) [cf. § 70] in every normal
region.

2) The postulates of congruence in the whole of the
plane.

Thus we would have the geometrical systems of Euclid,
of LOBATSCHEWSKY-BoLVAi. and of RiEMANN {f/i£ elliptic type),
which we have met above, where two straight lines have
only one common point : and a second Riemann's system
Kthe spherical type), where two straight lines have always two
common points.

§ 75- We cannot be quiie certain what idea Riemann
had formed of his complete plane, whether he had thought
of it as the elliptic p'laiu, or the spherical plane, or had
recognized the possibility of both. This uncertainty is due
to the fact that in his memoir he deals with Differential
Geometry and devotes only a lew words to the complete
forms. Further, those who continued his labours in this direc-
tion, among them Beltrami, always considered Riemann's
geometry in connection with the sphere. They were thus led
to hold that on the complete Riemann's plane, as on the
sphere (owing to the existeurc of the opposite ends of a
diameter), the postulate that a straight line is determined by
two points had exceptions," and that the only form of the
plane compatible with the Hypothesis of the Obtuse Angle
would be the spherical plane.

Cf. for example, tlic sliort reference to the geometry of
space of constant positive curvature with which Beltrami concludes
his memoir: Teoria fondamt'iifale lit'^^li spazii di atrvatura costante,
Ann. di Mat. (2). T. 11, p. 354 — 355 (1868); or the French trans-
lation of this memoir by J. llouKi., Ann. So. d. I'Ecole Norm. Sup.
T. VI, p. 347-377.

10*

148 V. The Later Development of Non-Euclidean Geometry.

The fundamental characteristics of the elliptic plane
were given by A. Cayley [1821 — 1895] in 1859, but the
connection between these properties and Non-Euclidean
geometry was first pointed out by Klein in 187 i. To Klein
is also due the clear distinction between the two geometries
of RiEMANN, and the representation of the elliptic geometry
by the geometry of the sheaf [cf S 7i]-

To make the difference between the spherical and
elliptic geometries clearer, let us fix our attention on two
classes of surfaces presented to us in ordinary space: the
surface with two faces {two-sided) and the surface with one
face {one-sided).

Examples of two-sided surfaces are afforded by the
ordinary plane, the surfaces of the second order (conicoidal,
cylindrical, and spherical), and in general all the surfaces
enclosing solids. On these it is possible to distinguish two
faces.

An example of a one-sided surface is given by the
Leaf of MÒBIUS [MoBiussche Blatt], which can be easily
constructed as follows: Cut a rectangular strip AB CD. In-
stead of joining the opposite sides AB and CD and thus
obtaining a cylindrical surface, let these sides be joined
after one of them, e. g., CD, has been rotated through two
right angles about its middle point. Then what was the
upper face of the rectangle, in the neighbourhood of CD,
is now succeeded by the lower face of the original rectangle.
Thus on Mobius' Leaf the distinction between the tivo
faces becomes impossible.

If we wish to distinguish the one-sided surface from the
wo-sided by a characteristic, depending only on the intrinsic
properties of the surface, we may proceed thus: — We fix a
point on the surface, and a direction of rotation about it
Then we let the point describe a closed path upon the sur-
face, which does not leave the surface; for a two-sided sur-

A One-Sided Surface.

149

face the point returns to its initial position and the final
direction of rotation coincides with the initial one; for a one-
sided surface, [as can be easily verified on the Leaf of Mobius,
when the path coincides with the diametral line] there exist
closed paths for which the final direction of rotation is oppos-
ite to the initial direction.

Coming back to the two Riemann's
planes, we can now easily state in what
their essential difi"erence consists : the spher-
ical plaiie has the character of the two-sided
surface, and the elliptic plane that of the one-
sided surface.

The property of the elliptic plane here ^he Leaf of Mobius.
enunciated, as well as all its other propert- '^" ^^'

ies, finds a concrete interpretation in the sheaf of lines. In
fact, if one of the lines of the sheaf is turned about the vertex
through half a revolution, the two rotations which have this
line for axis are interchanged.

Another property of the eUiptic plane, allied to the
preceding, is this : The elliptic plane, unlike the Euclidean
plane and the other Non-Euclidean planes, is not divided by
its lines into two parts. We can state this property other-
wise: If two points A and A' are given upon the plane, and
an arbitrary straight hne, we can pass from A to A' by a
path which does not leave the plane and does not cut the
line.^ This fact is 'translated' by an obvious property of the
sheaf, which it would be superfluous to mention.

§ 76. The interpretation of the spherical plane by the
sheaf of rays (straight lines starting from the vertex) is ana-
logous to that given above for the elliptic plane. The trans-

I A surface which completely possesses the properties of the
elliptic plane was constructed by W. Boy. [Gott. Berichte, p. 20
—23 (1900); Math. Ann. Bd. LVII, p. 151 — 184 (1903)].

ICQ V. The Later Developmeru of Non-Euclidean Geometry.

lation of the properties of this plane into the properties of
the sheaf of rays is effected ])y the use of a 'dictionary'
similar to that of § 71, in which the word J>oÌ7ìf is found
opposite the word rav.

The comparison of the sheaf of rays with the sheaf of
lines affords a useful means of making clear the connections,
and revealing the differences, whic-h are to be found in the
two geometries of Rikmann.

We can consider two sheaves, with the same vertex, the
one of lines, the other of rays. Tt is clear that to every line
of the first correspond two ra)s of the second; that every
figure of the first is formed by two symmetrical figures of the
second; and that, with certain restrictions, the metrical pro-
perties of the two forms are the same. Thus if we agree to
regard the two opposite rays of tlie sheaf of rays as forming
one element only, the sheaf of rays and the sheaf of lines
are identical.

The same considerations :ipply to the two Riemann's
planes. To every point of the elliptic plane correspond
two distinct and opposite points of the spherical plane; to
two lines of the first, which pass through that point, corres-
pond two lines of the second, which have two points in
common; etc.

The elliptic plane, when compared with the spherical
plane, ought to be regarded as a tfoubh' plane.

With regard to the elliptic j)laue and the spherical
plane, it is right to remark th:it tlii- formulae of absolute tri-
gonometry, given in § 56, can he applied to them in every
suitably bounded region. This follows from the fact, al-
ready noted in S 58, that the formulae of absolute trigonom-
etry hold on the sphere, 'and the geometry of the sphere, so
far as regards normal regions, coincides with that of these
two planes.

Riemann's Solid Geometry. jci

Principles of Riemann's Solid Geometry.

§ 77. Returning now to solid geometry, we start from
the philosophical foundation that the postulates, although
we grant them, by hypothesis, an actual meaning, express
truths of experience, which can be verified only in a bounded
region. We also assume, that on the foundation of these postul-
ates points in space are represented by three coordinates.

On such an (analytical) representation, every line is
given by three equations in a single variable:

and we must now proceed to determine a function j, of
the parameter t, which shall express the length of an arc of
the curve.

On the strength of the distributive property, by which
the length of an arc is equal to the sum of the lengths of
the parts into which we imagine it to be divided, such a
function will be fully determined when we know the element
of distance (ds) between two infinitely near points, whose
coordinates are

jCi + dXi , x, + dx2 , X, + dxy

RiEMANN starts with very general hypotheses, which
are satisfied most simply by assuming that ds', the square
of the element of distance, is a quadratic expression in-
volving the differentials of the variables, which always re-
mains positive:

ds' == Zfly- dxi dxj ,
where the coefficients aij are functions oi x^, x^, Xy

Then, admitting the principle of superposition of figures,
it can be shown that the fimction a;j must be such that, with
the choice of a suitable system of coordinates,

ds'= ^ ^

I-U— (jri2+;<r22 4-jf32)
4

I e 2 V. The Later Development of Non-Euclidean Geometry.

In this formula the constant K is what Riemann, by an ex-
tension of Gauss's conception, calls the Curvature of Space.

According as K is greater than, equal to, or less than
zero, we have space of constant positive curvature, space
of zero curvature, or space of constant negative curvature.

AVe make another forward step when we assume that the
principle of superposition [the principle of movement] can be
extended to the whole of space, as also the postulate that a
straight line is always determined by two points. In this way
we obtain three forms of space; that is, three geometries
which are logically possible, consistent with the data from
which we set out.

The first of these geometries, corresponding to positive
curvature, is characterised by the fact that Riemann's system
is valid in every plane. For this reason space of positive
curvature will be unbounded and finite in all directions.
The second, corresponding to zero curvature, is the ordinary
Euclidean geometry. And the third, which corresponds to
negative curvature, gives rise in every plane to the geometry

of LOBATSCHEWSKV-BOLYAI.

The Work of Helmholtz and the Investigations
of Lie.

§ 78. In some of his philosophical and mathematical
writings,* Helmholtz [1821 — 1894] has also dealt with the

I Uber die Ihatsachlichen Gniiidlagcn der Geometrie, Heidelberg,
Verb. d. naturw.-med. Vereins, Bd. IV, p. 197 — 202 (1868); Bd. V,
p. 31 — 32 (1869). Wiss. Abhandlungen von H. Helmholtz, Bd. II,
p. 610—617 (Leipzig, 1883). French translation by J. HouEL in
Mém. de la Soc. des Se. Phys. et Nat. de Bordeaux, T. V, (1868),
and also, in book form, along with the Etudes Gèométriques of
LOBATSCHEWSKV and the Correspondance de Gauss et de Schumacher,
(^Paris, Hermann, 1895).

Uber dii'Thatsachen, die der Geometrie zum Gruude lie;^en. Cott.

Helmholtz and Lie. It2

question of the foundations of geometry. Instead of assum-
ing a priori the form

ds"^ = XiZ/;- dxi dxj,
as the expression for the element of distance, he showed
that this expression, in the form given to it by Riemann for
space of constant curvature, is the only one possible, when,
in addition to Riemann's hypotheses, we accept, from the
beginning, that of the mobility of figures, as it would be given
by the movement of Rigid Bodies.

The problem of Riemann-Helmholtz was carefully
examined by S. Lie [1842 — 1899]. He started from the
fundamental idea, recognized by Klein in Helmholtz's
work, that the congruence of two figures signifies that they are
able to be transfiormed the one into the other, by means of a
certain point transformation in space: and that the properties,
in virtue of which congruefice takes the logical character of
equality, depend upon the fact that displacements are given by
a group of transformations.^

In this way the problem of Riemann-Helmholtz was
reduced by Lie to the following form:

Nachr. Bd. XV, p. 193—221 (1868). Wiss. Abhandl., Bd. II, p. 618

—639-

The Axioms of Geometry. The Academy, Vol. I, p. 123 — l8i
(1870); Revue des cours scient., T. VII, p. 498—501 (1870).

Uber die Axiome der Geotjietrie. Populare wissenschaftliche Vor-
tràge. Heft 3, p. 21 — 54. (Brunswick, 1876). English translation;
Mind, Vol. I, p. 301 — 321. French translation; Revue scientifique
de la France et de l'Étranger (2). T. XII, p, 1 197— 1207 (1877)

Uber den Ur sprung , Sinn, ii7id Bedetiticng der geo?}tct)isc/ie?t
Salze, "Wiss. Abh. Bd. II, p. 640 — 660. English translation; Mind,
Vol. n, p. 212 — 224(1878).

I Cf. Klein : Vergleichende Betrachtungen iiber netiere geometrische
Forschungen, (Erlangen, 1872); reprinted in Math. Ann. Bd. XLIII,
p. 63 — 100 (1893). Italian translation by G. Fang, Ann. di Mat. (2),
T. XVII, p. 301-343 (1899)-

I e /I V. The Later Development of Non-Euclidean Geometry.

To determine all the continuous groups in space which,
in a bounded regio?t, have the property of displacements.

When these properties, which depend upon the free
mobiUty of Hne and surface elements through a point, are
put in a suitable form, there arise three types of groups,
which characterise the three geometries of Euclid, of

LOBATSCHEWSKY-BOLYAI and of RiEMANN. '

Projective Geometry and Non-Euclidean Geometry.

Subordination of Metrical Geometry to Projective

Geometry.

§ 79. In conclusion, there is an interesting connection
between Projective Geometry and the three geometrical
systems of Euclid, Lobatschewsky-Bolyai and Riemann.

To give an idea of this last method of treating the
question, we must remember that Projective Geometry, in
the system of G. C.Staudt [1798 — 1867], rests simply upon
graphical notions on the relations between points, lines
and planes. Every conception of congruence and movement
[and thus of measurement etc,,] is systematically banished.
For this reason Projective Geometry, excluding a certain
group of postulates, will contain a more restricted number of
general properties, which for plane figures are the [projective]
properties, remaining invariant by projection and section.

However, when we have laid the foundations of Pro-
jective Geometry in space, 7cie can introduce into this system

I Cf. Lie: Theorie der Transjormalionsgritppcu. Bd. Ill, p- 437
— 543 (Leipzig, 1893). In connection with the same subject, H.
Poincaré, in his memoir: Sur les hypotheses fondamctitanx de la
gioinitrie [Bull, de La Soc. Math, de France. T. XV, p. 203 — 2l6
('877)]» solved the problem of finding all the hypotheses, which
distinguish the fundamental group of plane Euclidean Geometry
from the other transformation groups.

Projective Geometry and Xon-Euclidean Geometry. jer

the metrical conceptions, as relations between its figures and
certain definite {metrical) entities.

Keeping to the case of the Euclidean plane, let us see
what graphical interpretation can be given to the fundamental
metrical conceptions of parallelism and of perpendicularity.

To this end we must specially consider the line at infi?i-
ity of the plane, and the absolute involution which the set of
orthogonal lines of a pencil determine upon it. The double
points of such an involution, conjugate imaginaries, are
called the circular poiftts (at infinity), since they are common
to all circles in the plane [Poncelet, 1822^].

On this understanding, the parallelism of two lines is
expressed graphically by the property which they possess of
meetifjg in a point on the line at iiifinity : the perpendicularity
of two lines is expressed graphically by the property that
their points at infinity are conjugate in the absolute involution,
that is, form a harmonic range with the circular points.
[Chasles, 1850.^]

Other metrical properties, which can be expressed
graphicallyj are those relative to the size of angles, since
every equation

F{A,B, C...)= O,
between the angles A, B, C, . . ., can be replaced by
^/loga log_^ lo^._ \_

in which a, b, c . . . are the anharmonic ratios of the pencils
formed by the lines bounding the angles and the (imaginary)
ines Joining the angular points to the circular points. [La-

GUERRE, 1853.3]

1 Traiti des propriétés projectives des figures. 2. Ed., T.I. Nr. 94,
p. 48 (Paris, G, Villars, 1865).

2 Traile de Géoméùie supérieitre. 2. Ed., Nr. 660, p. 425 (Paris,
G. Villars, 1880).

3 Sur la thcorie des foyers. Nouv. Ann. T. XII, p. 57- Oeuvres
de Laguerre. T. II, p. 12—13 (Paris, G. Villars, 1902;.

IC.0 V. The Later Development of Xon-Euclidean Geometry.

More generally it can be shown that the congruence
of any two plane figures can be expressed by a graphical
relation between them, the line at infinity, and the absolute
involution.^ Also, since congruence is the foundation of all
metrical properties, it follows that the line at infinity and the
absolute involution allow all the properties of Euchdean
metrical geometry to be subordinated to Projective Geo-
metry. TÀUS the metrical properties appear in projective geometry,
not as graphical properties of the figures considered in them-
selves, but as graphical properties with regard to the funda-
mental metrical entities, made up of the line at infinity and the
absolute involution.

The complete set of fundamental metrical entities is
called the absolute of the plane (Cayley).

All that has been said with regard to the j)lane can
naturally be extended to space. The fundamental metrical
entities in space, which allow the metrical properties to be
subordinated to the graphical, are the plane at infinity and a
certain polarity {absolute polarity) on this plane. This polar-
ity is given by the polarity of the sheaf, in which every line
corresponds to a plane to which it is perpendicular [cf. § 7 1].
The fundamental conic of this polarity is imaginary, since
there are no real lines in the sheaf, which lie on the corre-
sponding perpendicular plane. It can easily be shown that
it contains all the pairs of circular points, which belong to
the different planes in space, and that it appears as the com-
mon section of all spheres. From this property the name
of circle at infinity is given to this fundamental metrical
entity in space.

» Cf., e. g. F. Enriques, Lezioni di Geomelria proielliva, 2a. Ed.
p. 177 — 188 (Bologna, Zanichelli, 1904). There is a German
translation of the first edition of this work by H. Fleischer
(Leipzig, 1903).

Cayley's Absolute. icj

§ 80. The two following questions naturally arise at
this stage:

(i) Can projective geometry be founded upon the Non-
Euclidean hypothesis Ì

(ii) If such a foundatiofi is possible, can the metrical
properties, as in the Euclidean case, he subordinated to the
projective?

To both these questions the reply is in the affirmative.
If Riemann's system is valid in space, the foundation of
projective geometry does not offer any difficulty, since those
graphical properties are immediately verified, which give rise
to the ordinary projective geometry, after the i^nproper entities
are introduced. If the system of Lobatschewsky-Bolyai is
valid in space, we can also again lay the foundation of the
projective geometry, by introducing, with suitable conventions,
improper or ideal points, lines a?id planes. This extension will
follow the same lines as were taken in the Euclidean case, in
completing space with the elements at infinity. It would be
sufficient, for this, to consider along with the proper sheaf
(the set of lines passing through a point), two improper
sheaves, one formed by all the lines which are parallel to a
given line in one direction, the other by all the lines perpen-
dicular to a given plane; also to introduce improper points,
to be regarded as the vertices of these sheaves.

Even if the improper points of a plane cannot in this
case, as in the Euclidean, be assigned to a straight line \the
lifie at infinity\ yet they form a complete region, separated
from the region of ordinary points {proper points) by a conic
[limiting conic, or conic at infinity]. This conic is the locus
of the improper points determined by the pencils of parallel
lines.

In space the improper points are separated from the
proper points by a non-ruled quadric [limiting qiiadric or

ic8 V. The Later Development of Non-Euclidean Geometry.

quadric at injinity], which is the locus of the improper points
determined by sets of parallel lines.

The validity of projective geometry having been estab-
hshed on the Non-Euclidean hypotheses [Klein ^], to obtain
the subordination of the metrical geometry to the projective
it is sufficient to consider, as in the Euclidean case, the
fundametital metrical entities {the absolute)^ and to interpret the
metrical properties of figures as graphical relations between
them and these entities. On the plane of Lobatschewsky-
BoLYAi the fundamental metrical entity is the limiting conic,
which separates the region of proper points from that of
improper points, on the plane of Riemann it is an imaginary
conic, defined by the absolute polarity of the plane [cf. p. 144].

In the one case as well as in the other, the metrical
properties of figures are all the graphical properties which
remain ufialtered in the projective transformatiotis^ leaving the
absolute fixed.

These projective transformations constitute the 00 ■J dis-
placements of the Non-Euchdean plane.

In the Euclidean case the said transformations, (which
leave the absolute unaltered), are the 00 ^ transformations of
similarity, among which, as a special case, are to be found
the 003 displacements.

In space the subordination of the metrical to the pro-

1 The question of the independence of Projective Geometry
from the theory of parallels is touched upon lightly by Klein in
his first memoir: Uber die sogenannte Nicht-Euklidische Geometrie,
Math. Ann. Bd. IV, p. 573 — 625 (1871). He gives a fuller treatment
of the question in Math. Ann. Bd. VI, p. 112 — 145 (1873). This
question is discussed at length in our Appendix IV p. 227.

2 By the term projective transformation is understood such a
transformation as causes a point to correspond to a point, a line to
a line, and a point and a line through it, to a point and a line
through it.

Metrical Properties as Graphical. I en

jective geometry is carried out by means of the limiting
quadric {the absolute of space). If this is real, we obtain the
geometry of Lobatschewsky-Bolyai; if it is imaginary, we
obtain Riemann's elliptic type.

The metrical properties of figures are therefore the graph-
ical properties of space in relation to its absolute; that is, the
graphical properties which remaiti unaltered in all the project-
ive transformatio7is wJiich leave the absolute of space fixed.

§ 8i. How will the ideas of distance and of angle be
expressed with reference to the absolute?

Take a system of homogeneous coordinates (ati, x^, x.^
on the projective plane. By their means the straight line is
represented by a linear equation, and the equation of the
absolute takes the form :

Qrj; = l-Uij Xi Xj = O.

Then the distance between two points X {x^, X2, x^),
V (y^ , ^2 , y^ is expressed, omitting a constant factor, by the
logarithm of the anharmonic ratio of the range consisting of
X, V, and the points M, JV, in which the line X Y meets the
absolute.

If we then put

Qjcy = ^aijxiyj,

and remember, from analytical geometry, that the anharm-
onic ratio of the four points X, Y, M, N is given by

Q.v+^

the expression for the distance D^cy will be : —
(i) Z>.j, = -J log -^ 17 -^ -- .

^^xy ^^xy ^^xx ^^yy

Introducing the inverse circular and hyperboUc functions,

l6o V- The Later Uevelopment of Non-Euclidean Geometry.

(2)

D^y == ik COS ~^ .

I D,y =/ècosh-^ ^ "'^ -

(3)

xD^y = ik Sin -^ ,- Jil- — -

The constant k, which appears in these formulae, is
connected with Riemann's Curvature K by the equation

Similar considerations lead to the projective interpret-
ation of the conception of angle. The atigle between hoo
lines is proportional to the logarithm of the anharmo7iic ratio
of the pencil which they fortn with the tajigents from their
point of intersection to the absolute.

If we wish the measure of the complete pencil to be

2 TT, as in the ordinary measurement, we must take the

fraction x : z i as the constant multiplier. Then to express

' analytically the angle between two lines u (ui, u^-, u^),

V (z/i, z^2j 2^3)} we put

Y«„ = Z bij Ui uj .

If bij is the CO factor of the element aij in the dis-
criminant of ^xxi the tangential equation of the absolute is
given by

and the angle between the two lines by the following
formulae: —

^nv + Ku/ 2 U/ uT"

(l) ^U,V^--j\Qg^

'nil ' uii ' vv

Formulie for the Angle.

i6i

(2)

-< ?/, 7' ^= COS ~^ — ,-

r I «j< I 21V

<^ 2^, e- = ^ cosh -^ ^"^L _

^ z/, Z' = sin -» ^"'^ ^^'^ ^""

(3')

z/, z/

sinh

' ■ UU ' VV

Similar expressions hold for the distance between two
points and the angle between two planes, in the geometry
of space. We need only suppose that

^xx = O, ¥„„ = O,

represent the equations (in point and tangential coordinates)
of the absolute of space, instead of the absolute of the plane.
According as Q^x = C> is the equation of a real quadric,
without generating lines, or of an imaginary quadric, the
formulae will refer to the geometry of Lobatschewky-Bolyai,
or that of RiEMANN.'

§ 82. The preceding formulae, concerning the angles
between two lines or planes, contain those of ordinary
geometry as a special case. Indeed if, for simplicity, we take
the case of the plane, and the system of orthogonal axes,
the tangential equation of the Euclidean absolute {f^e circular
points, § 79) is

The formula (2'j, when we insert

becomes

I For a full discussion of the subject of this and the pre-
ceding sections, see Clebsch-Lindemann, Vorlesungen ilber Geometrie,
Bd. II. Th. I, p. 461 — et seq. (Leipzig, 1891).

1 62 V. The Later Development of Non-Euclidean Geometry.
<fi U,V = COS ' ,

from which we have

COS (^^, v) =

But the direction cosines of the line ti (Ui, u^, u^) are

cos {U, X) = —-=--, COS (Z^_>')

so that this equation can be written

COS {u, V) = /j 4 -+- m^ 7)12.
the ordinary expression for the angle between the two lines
(/i ;//i) and (4 m^.

For the distance between two points the argument does
not proceed so simply, when the absolute degenerates into
the circular points. Indeed the points J/, N, where the line
XY intersects the absolute, coincide in the point at infinity on
this line, and the formula (i) gives in every case:

D^y = 4 log {M^N^XY) = A log I = o.

However, by a simple artifice we can obtain the
ordinary formula for the distance as the limiting case of
formula (3).

To do this more easily, let us suppose the equations
of the absolute (not degenerate), in point and line coor-
dinates, reduced to the form :

Q.i^. == ^Xi'^ + ^X^^ + X~^ = O,

Then^ putting

equation (3) of the preceding section gives

Euclid's Geometry as a Limiting Case. 1 53

D,y = ik sin-^ /eA.
Let e be infinitesimal. Omitting terms of a higher
order, we can substitute K € A for sin~' K e A in this formula
If we now choose k^ infinitely large, so that the product
ik Y^ remains finite and equal to unity for every value of e,
the said formula becomes

Let € now tend to the limit zero. The tangential
equation of the absolute becomes

«i^ + «2^ = O;
and the conic degenerates into two imaginary conjugate
points on the line u^ = o. The formula for the distance,
on putting

takes the form

which is the ordinary Euclidean formula. We have thus ob-
tained the required result.

We note that to obtain the special Euclidean case from
the general formula for the distance, we must let k^ tend to
infinity. Since Riemann's curvature is given by - — tj , this
affords a confirmation of the fact that Riemann's curvature
is zero in Euclidean space.

§ 83. The properties of plane figures with respect to a
conic, and those of space with respect to a quadric, together
constitute projective metrical geometry. This was first studied
by Cayley,'' apart from its connection with the Non-Euclid-

I Sixth Memoir upon Quantics. Phil. Trans. Vol. CXLIX, p. 6 1
-90 (1859). Also Collected Works, Vol. II, p. 561 — 592.

104 ^' ^^^ Later Development of Non-Euclidean Geometry.

ean geometries. These last relations were discovered and
explained some years later by F. Klein. ^

To Klein is also due a widely used nomenclature for
the projective metrical geometries. He gives the name hyper-
bolic geometry to Cayley's geometry, when the absolute is
real and not degenerate: elliptic geoinetry, to that in which
the absolute is imaginary and not degenerate: parabolic
geometry, to the limiting case of these two. Thus, in the
remaining articles, we can use this nomenclature to describe
the three geometrical systems of Lobatschewsky-Bolyai, of
RiEMANN (elliptic type), and of Euclid.

Representation of the Geometry of Lobatschewsky-
Bolyai on the Euclidean Plane.

§ 84. To the projective interpretation of the Non-
Euclidean measurements, of which we have just spoken, may
be added an interesting representation which can be given
of the Hyperbolic Geometry on the Euclidean plane. To ob-
tain it, we take on the plane a real, not degenerate, conic :
e. g. a circle. Then we make the following definitions, relative
to this circle :

Plane == region of points within the circle.

Point = point inside the circle.

Straight line = chord of the circle.

We can now easily verify that the postulate that a
straight line is determined by two points, and the postulates
regarding the properties of straight lines and angles, can be
expressed as relations, which are always valid, when the above
interpretations are given to these terms.

But in the further development of this geometry we add

I Cf. Uber die sogcnannte Nichi-Euklidische Geometrie. Math.
Aim. Bd. IV, p. 573-625 (1871).

Representation on the Euclidean Plane. 1 65

to these the postulates of congruence, contained in the
following principle of displacement.

If we are given two points A, A' on the plane, and the
straight lines a, a , respectively passing through them, there
are four methods of superposing the plane on itself, so that
A and a coincide respectively with A and a. More precisely:
ofie method of superposition is defined by taking as corre-
sponding to each other, one ray of a and one ray of a , one
section of the plane bounded by a and one section bounded
by a. Two of these displacements are dirt'ct co?igruèncès
and two converse congruences.

With the preceding interpretations of the entities, point,
Un; and plane, the principle here expressed is translated
into the following proposition:

If a conic {i\ g., a circle) is given in a platie, and two
internal points A, A' are taken, as also two chords a, a', re-
spectively passing through them, there are four projective trans-
formations of the plafie, which change into itself the space
within the conic, and which make A and a correspond respect-
ively to A' and a .

To fix one of them^ it is sufficient to make sure that a
given extremity of a corresponds to a given extremity of a ,
and that to one section of the plane bounded by a, cor-
responds a definite section of the plane bounded by ci . Of
these four transformations, two determine on the conic a
projective correspondence in the same sense, and two a pro-
jective correspondence i?i the opposite sense.

§ 85. We shall prove this proposition, taking for sim-
plicity two distinct conies T, t', in the same plane or other-
wise.

Let M, N be the extremities of the chord a [cf Fig. 5 8].
Also M\ N' those of a [cf Fig. 59].

1 66 V. The Later Development of Non-Euclidean Geometry.

Let F, P' be the poles of a^ a with respect to the two
conies.

On this understanding, the Hne PA intersects the
conic T in two real and distinct points i?, S: also the line
P' A intersects the conic t' in two real and distinct points

A projective transformation which changes t into t', the
line a into a, and the point A into A, will make the point P
correspond to P\ and the hne PA to the line P' A.

Fig. 59-

Thus this transformation determines a projective cor-
respondence between the points of the two conies, in which
the pair of points M', N' corresponds to the pair of points
M, N: and the pair of points R' , S' to R, S.

Vice versa, a projective transformation between the two
conies, which enjoys this property, is associated with a pro-
jective transformation of the two planes, such as is here de-
scribed.'

But if we consider the two conies t, t', we see that to

I For this proof, and the theorems of Projective Geometry
upon which it is founded, see Chapter X, p. 251 — 253 of the work
of Enriques referred to on p. 156.

Projective Transformations.

167

the points of the range MNRS on T may be made to cor-
respond the points of any one of the following ranges on t':
M'N'R'S'

n'm's'e:
m'n's'jr:

N'M'R'S'.
In this way we prove the existence of the four project-
ive transformations of which we have spoken in the propos-
ition just enunciated.

If we suppose that the two conies coincide, we do not
need to change the
preceding argument in
any way. We add, how- p
ever, that of the four
transformations only
one makes the segment
AM correspond to the
segment A'M\ if at the
same time the shaded
parts of the figure cor-
respond to each other.

Further the two transformations defined by the ranges

/ MNRS

\ M'lYR'S'

determine projections in the same sense, while the other two,
defined by the ranges :

f MNRS \ / MNRS \

\ MN'S'R' ) \ N'M'R'S' )
determine projections in the opposite setise.

\ / MNRS \
)' ), \ N'M'S'K )

§ 86. With these remarks, we now return to complete
the definitions of S 84, relative to a circle given on the
plane.

Flane = region of points within the circle.

l68 V. The Later Development of Non-Euclidean Geometry.

Point = point within the circle.

Straight Line = chord of the circle.

Displacements == projective transformations of the plane
which change the space within the circle into itself.

Semi-Revolutions = homographic transformations of the
circle.

Congn/ent Figures = figures which can be transformed
the one into the other by means of the projective trans-
formations named above.

The preceding arguments permit us to affirm at once
that all the propositions of elementary plane geometry, asso-
ciated with the concepts straight line, angle and congruence,
can be readily translated into proj)erties relative to the
system of points inside the circle, which we denote by {S).
In particular let us see what corresponds in {S) to two per-
pendicular lines in the ordinary plane.

To this end we note that if r, s are two perpendicular
lines, a semi-revolution of the plane about j will superpose
r upon itself, exchanging, however, the two rays in which it
is divided by s.

According to the above definitions, a semi-revolution in
{S) is a homographic transformation, which has for axis a
chord s of the circle and for centre the pole of the chord.
The lines which are unchanged in this transformation, in ad-
dition to s, are the lines passing through its centre. Thus
in the system (S) we must call two lines perpendicular, when
they are conjugate with respect to the fundamcjital circle.

We could easily verify in {S) all the propositions on
perpendicular lines. In particular, that if we draw the (imag-
inary) tangents to the fundamental circle from the common
point of two conjugate chords in (^), these tangents form
a harmonic pencil with the perpendicular lines [cf. p. 155].'

I This representation of the Non-Euclidean plane has been

The Distance between two Points. i5q

§ 87. Let US now see how the distance between two
points can be expressed in this conventional measurement,
which is being taken for the interior of the circle.

To this end we introduce a system of orthogonal coord-
inates {x, y), with origin at the centre of the circle.

The distance between two points A {x, j), B {x , y)
in the plane with which we are dealing cannot be represen-
ted by the usual formula

Y{x~xy\{y-y)\

since it is not invariant for the projective transformations
which we have called displacements. The distance must be a
function of the coordinates, invariant for the said transforma-
tions, which for points on the straight line possesses the dis-
tributive property given by the formula

dist. {AE) = dist. {AC) -f dist. {CB).

Now the anharmonic ratio of the four points A^ B, M,
N, where M, N are the extremities of the chord AB, is a
relation between the coordinates {x, y), {x\ y') of AB,
remaining invariant for all projective transformations which
leave the ~_ fundamental circle fixed. The most general ex-
pression, possessing this invariant property, will be an arbi-
trary function of this anharmonic ratio.

If we remember that the said function must be distrib-
utive in the sense above indicated, we must assume that,
except for a multiplier, it is equal to the logarithm of the
anharmonic ratio,

(ABM^T) = ^^: -^^,-

We shall thus have

distance (AB) = ^ log {ABMN).

employed by Grossmann in carrying out a number of the con-
structions of Non-Euclidean Geometry. Cf. Appendix, III, p. 225.

170 V- The Later Development of Non-Euclidean Geometry.

In a similar way we proceed to find the proper ex-
pression for the angle between two straight lines. In this case

we must notice that if we wish the right angle to be ex-
it
pressed by — , we must take as constant multiplier of the

logarithm the factor 1:22.

Then we shall have for the angle between a and b,

^^>^=^ 2/ ^^^ iabmn),

where m, n are the conjugate imaginary tangents from the
vertex of the angle to the circle, and {a b m n) is the an-
harmonic ratio of the four lines a, b, in and «, expressed
analytically by

sin ia ni) sin {a n)
sin [d m) ' sin (i n)

% 88. A glance at what was said above on the sub-
ordination of the metrical to the projective geometry (S 81)
will show clearly that the preceding formulge^ regarding the
distance and angle, agree with those which we would have in
the Non-Euclidean plane, if the absolute were a circle. This
would be sufficient to suggest that the geometry of the system
(6') gives a concrete representation of the geometry of
LoBATSCHEWSKY-BoLVAi. However, as we wish to discuss
this point more fully, let us see how the definition and pro-
perty of parallels are translated in \S).

Let r (z^i, U2, u^) and r {i\, V2, v^ be two difterent
chords of the fundamental circle.

Let the circle be referred to an orthogonal Cartesian
set of axes, with the centre for origin, and let us take the
radius as unit of length.

Then we have

x^ -Vy^ — 1=0,
u^-\-v^ — 1=0,
for the point and line equation of the circle.

The Angle between two Lines.

171

Making these equations homogeneous, we obtain

Xj,^ + x,' — x^' = O,

Ui'+U2' — //3^ = O.

The angle ^r, r between the two straight hnes r and
r can be calculated by means of the formula (3') of § 81,
if we put

^uu '-'■ u,' + u.

3 '

We thus obtain

sin <5C r, r

V (U1V3 ViUiY — (2^22^3 — "v-^ò^ — iu-^-i^ V-^U^^'

But the lines r, r are given by

Xt_H^-\- X-Jl^-V X.jLl.^ = O,
XtJ)-!^ + .^22^2 + -^3^3 ^= O ;

and they meet in the point,

x^ = u^v^, — U^^2 ,

X^ == U^^x—-U{U^,
X^ == U1V2 — 2/2 Z'l-

Thus the preceding expression for this angle takes
the form

/ \ • V ' ' \Xx Xx X^, )

(4) sm <^r,r= ^ — " .

l/(2<!.^ + «2^ — 2^3^) (Z'l^ + Z^a^ — t'3^)

From this it is evident that the necessary and sufficient
condition that the angle be zero is that the numerator of
this fraction should vanish.

Now if this numerator is zero, the point {x-^, x^, x.^, in
which the chords intersect, must lie on the circumference of
the fundamental circle, and vice versa (Fig. 61).

Therefore in our Ì7iterpretation of the geometrical pro-
positions by 77ieans of the system (S), we must call two chords
parallel, when they iueet in a point on the circumference of the

J 72 V. The Later Development of Non-Euclidean Geometry.

fundatnental circle, since the angle between those two chords
is zero.

Since there are two chords through any point within a
circle which join this point to the ends of any given chord,
the fundamental proposition of hyperbolic geometry will be
verified for the system {S).

§ 89. We proceed to find for the system {S) the
formula regarding the angle of parallelism. To do this we
first calculate the angle OMN, between the axis of_j' and
the line MN, joining a point M on the axis of^ to the ex-
tremity of the axis oi x (Fig. 62).

Fig. 61.

Fig. 62.

Denoting by a the ordinary distance of the two points
M and O, the homogeneous coordinates of the line MN and
the line OM axQ, respectively {a, i, — a), (i, O, o) and the
coordinates of their common point are (o, a, 1).

Then from (4) of the preceding article,
sin <^ OMN = \^i-a2.

On the other hand, the distance;, according to our con-
vention, between the two points O and M is given by (2) of
S 81 as

OM = h cosh -' -—- —

Thus

OM _ I

cosh ~ =

The Angle of Parallelism. 172

Comparing these two results, we have

, OM I

cosh —r- =

k sin <^ OMN'>

a relation which agrees with that given by TaurinuS; Lo-
BATSCHEWSKY and BoLYAi for the angle of parallehsm [cf.
p. 90].

§ 90. We proceed, finally, to see how the distance be-
tween two neighbouring points {the element of distance) is
expressed in the system (vS), so that we may be able to
compare this representation of the hyperbolic geometry with
that given by Beltrami [cf. g 69].

Let {x^y)^ (x + dx,y + dy) be two neighbouring points.
Their distance ds is calculated by means of (2) of § 81 if we
substitute :

Qxz = x'+y^— I,

Qj,y = (x + dxy + iy + dyY— i,

^xy = X {x + dx) -^-y (y + dy) — i .

Since the angle is small, we may substitute the sine for
the angle, and we have

_ ,2 (dx^ + dy»){l—x2 —y2) 4- [xdx +ydyy

(X2 +-y2 _ I) ((;, ^ dxY + fy + dyY - I))

Thus, omitting terms higher than the second order,
we have

^^. _ ^2 {dx^ 4- dy2) (I — x2 —y2) + ^xdx +ydyy

(l X2 —y2)2

-^ {l—X2—y2)2

Now we recall that Beltrami, in i868, interpreted the
geometry of Lobatschewsky-Bolvai by that on the surfaces
of constant negative curvature. The study of the geometry
on such surfaces depends upon the use of a system of coord-
inates on the surface, and the law according to which the
element of distance {ds) is measured. The choice of a suitable

I HA R. The Later Development of Non-Euclidean Geometry.

system (?/, v) enabled Beltrami to put the square of ds in
this form:

(I — v^') dit^ -f- zitvdudri -\-(\ — n^) dv^
k ,

(I «2 — 2/2)2

where the constant k^ is the reciprocal, with its sign changed,
of the curvature of the surface.'

In studying the properties of these surfaces and in mak-
ing a comparison between them and the metrical results of
the geometry of Lobatschewsky-Bolyai, Beltrami in his
classical memoir, quoted on p. 138, employed the following
artifice:

He represented the points of the surface on an aux-
iliary plane, such that the point {u, z') of the surface corre-
sponded to the point on the plane whose Cartesian coord-
inates (x,}>) were {u, v). The points on the surface were
then represented by points inside the circle

x^ +y^ — I = O;

the points at infinity on the surface by points on the cir-
cumference of the circle: its geodesies by chords: parallel
geodesies by chords meeting in a point on the circumference
of the said circle. Then the expression for {dsY took the
same form as that given in (5), which states the form to be
used for the element of distance in the system {S).

It follows that, by his representation of the surfaces of
constant negative curvature on a plane, Beltrami was
led to one of the projective metrical geometries of Cayley,
and precisely to the metrical geometry relative to a funda-
mental circle, given above in §§ 80, 81.

I Risoluzione del problema di riportare i punti di una superficie
sop>ra un piano in modo che le linee geodetiche vengano rappresentate
da linee rette. Ann. di Mat. T. VII, p. 185 — 204 (1866). Also
Opere Matematiche. T. I, p. 262 — 280 (Milan, 1902).

Beltrami's Geometry and Projective Geometry. jyc

§ gi. The representation of plane hyperbolical geo-
metry on theEudidean plane is capable of being extended to
the case of solid geometry. To represent the solid geometry
of LoBATSCHEWSKY-BoLYAi in Ordinary space we need only
adopt the following definitions for the latter:

Space = Region of points inside a sphere.

I^owt = Point inside the sphere.

Straight Line = Chord of the sphere.

Plane = Points of a plane of section which are inside
the sphere.

Displacements = Projective transformations of space,
which change the region of the points inside the
sphere into itself, etc.

With this 'Dictionary' the propositions of hyperbolic
solid geometry can be translated into corresponding proper-
ties of the Euclidean space, relative to the system of points
inside the sphere.'

Representation of Riemann's Elliptic Geometry in
Euclidean Space.

§ 92. So far as regards plane geometry, we have already
remarked [pp. 142 — 3] that the geometry of the ordinary
sheaf of lines gives a concrete interpretation of the elliptical
system of Riemann. Therefore, if we cut the sheaf by an
ordinary plane, completed by the line at infinity, we obtain
a representation on the Euclidean plane of the said Rie-
mann's plane.

I Beltrami considers the interpretation of Non-Euclidean Solid
Geometry, and, in general, of the geometries of manifolds of
higher order in space of constant curvature, in his memoir: Teoria
fondamentale degli spazii di curvatura costante. Ann. di Mat. (2),
T, II, p. 232—255 (1868). Opere Mat. T. I, p. 406—429 (Milan,
1902).

176 V. The Later Development of Non-Euclidean Geometry.

If we wish a representation of the elliptic space in the
Euclidean space, we need only assume in this a single-valued
polarity, to which corresponds an imaginary quadric, not
degenerate. We must then take, with respect to this quadric,
a system of definitions analogous to those indicated above
in the hyperbolic case. We do not pursue this point further,
as it offers no fresh difficulty.

However we remark that in this representation all the
points of the Euclidean space, including the points on the plane
at infinity, would have a one-one correspondence with the points
of Rietnann' s space.

Foundation of Geometry upon Descriptive
Properties.

§ 93. The principles explained in the preceding sections
lead to a new order of ideas in which the descriptive propert-
ies appear as the first foundations of geometry, instead of
congruence and displacement, of which Riemann and Helm-
HOLTZ availed themselves. We note that, if we do not wish
to introduce at the beginning any hypothesis on the inter-
section of coplanar straight lines, we must start from a
suitable system of postulates, valid in a boic7ided region of
space, and that we must complete the initial region later by
means oi improper points, lines and planes [cf. p. 157].^

When projective geometry has been developed, the
metrical properties can be introduced into space, by adding
to the initial postulates those referring to displacement or

I For such developments, cf. Klein, Ioc. cit. p. 158: Pasch,
Vorlesungen iiber neuere Geometrie, (Leipzig, l882)j SCHUR, Uber die
Einfichrting der sogenannten idea! en Elemenie in die projective Geometrie,
Math. Ann. Bd. XXXIX, p. 113 — 124 (1891): Bonola, Suila intro-
duzione degli elementi improprii in geometria proiettiva. Giornale di
Mat. T. XXXVIII, p. 105— 116 (1900).

Foundation of Geometry upon Descriptive Properties. 177

congruence. By so doing we find that a certain polarity of
space, allied to the metrical conceptions, becomes trans-
formed into itself by all displacements. Then it is shown
that the fundamental quadric of this polarity can only be:

a) A real, non-ruled quadric;

b) An imaginary quadric (with real equation);

c) A degetiérate quadric.

Thus the three geometrical systems, which Riemann and
Helmholtz reached from the conception of the element of
distance, are to be found also in this way.*

The Impossibility of proving Euclid's Postulate.

§ 94. Before we bring to a close this historical treat-
ment of our subject it seems advisable to say a few words
on the impossibility of demonstrating Euclid's Postulate.

The very fact that the innumerable attempts made to
obtain a proof did not lead to the wished-for result, would
suggest the thought that its demonstration is impossible. In-
deed our geometrical instinct seems to afford us evidence
that a proposition, seemingly so simple, if it is provable,
ought to be proved by an argument of equal simplicity. But
such considerations cannot be held to afford a proof of the
impossibility in question.

If we put Euclid's Postulate aside, following the devel-
opments of Gauss, Lobatschewsky and Bolyai, we can
construct a geometrical system in which no contradictions
are met. This seems to prove the logical possibility of the
Non-Euclidean hypothesis, and that Euclid's Postulate is
independent of. the first principles of geometry and therefore
cannot be demonstrated. However the fact that contradictions

I For the proof of this result see BONOLA, Determinazione
per via geometrica dei ire tipi de spazio; iperbolico, parabolico, ellittico.
Rend. Gire. Mat. Palermo, T. XV, p. 56—65 (1901).

J 73 V. The Later Development of Non-Euclidean Geometry.

have not been met is not sufficient to prove this; we must
be certain that, proceeding on the same Hnes, such con-
tradictions could never be met. This conviction can be
gained with absolute certainty from the consideration of the
formulae of Non-Euclidean geometry. If we take the system
of all the sets of three numbers (x, y, z), and agree to con-
sider each set as an analytical point, we can define the
distance between two such analytical points by the formulae
of the said Non-Euclidean Trigonometry. In this way we
construct an analytical system, which offers a conventional
interpretation of the Non-Euclidean geometry, and thus
demonstrates its logical possibility.

In this sense the formulae of the Non- Euclidean Trigon-
ometry of Lobatscheiusky-Bolyai give the proof of the independ-
ence of Euclid's Postulate from the first principles of geometry
(regarding the straight line, the plane and congruence).

We can seek a geometrical proof of the said independ-
ence, on the lines of the later developments of which we
have given an account. For this it is necessary to start from
the principle that the conceptions, derived from our intu-
ition, independently of the correspondence which they find
in the external world, are a priori logically possible; and that
thus the Euclidean geometry is logically possible and every
set of deductions founded upon it.

But the interpretation which the Non-Euclidean plane
hyperbolic geometry finds in the geometry on the surfaces
of constant negative curvature, offers, up to a certain point,
a first proof of the im.possibility of demonstrating the Eu-
clidean postulate. To put the matter in more exact terms:
by this means it is established that the said postulate cannot
be demonstrated on the foundation of the first principles of
geometry, held valid in a bounded region of the plane. In
fact, every contradiction, which would arise from the
other postulate, would be translated into a contradiction

Euclid's Postulate cannot be Proved.

1/9

in the geometry on the surfaces of constant negative curv-
ature.

However, since the comparison between the hyperbolic
plane and the surfaces of constant negative curvature, exists,
as we have seen, only for bounded regions^ we have not thus
excluded the possibility that the Euclidean postulate might
be proved for the complete plane.

To remove this uncertainty, it would be necessary to
refer to the abstract manifold of constant curvature, since no
concrete surface exists in ordinary space, in which the ^(?w-
//<?/<? hyperbolic geometry holds [cf. § 73].

But, even so, the impossibility of proving Euclid's Pos-
tulate would have been shown only for pla7ie geometry. There
would still remain the question of the possibility of proving
it by means of the considerations of solid geometry.

The foundation of geometry, on Riemann's principles,
whereby the ideas of the geometry on a surface are extended
to a tliree-dimensional region, gives the complete proof of the
impossibility of this demonstration. This proof depends on
the existence of a Non-Euclidean analytical system. Thus we
are brought to another analytical proof. The same remark
applies also to the investigations of Helmholtz and Lie,
though it might be argued that the latter also offer a geomet-
rical proof, from the existence of transformation groups of
the Euclidean space, similar to the groups of displacements of
the Non-Euclidean geometry. Of course, it must be under-
stood that we here consider geometry in its fullest sense.

But the proof of the impossibility of demonstrating Eu-
clid's Postulate^ which is based upon the projective measure-
ments of Cayley, is simpler and easier to follow geometrically.

This proof depends upon the representation of the
Non-Euclidean geometry by the conventional measurement
relative to a circle or to a sphere, an interpretation which we

I So ^- The Later Development of Non-Euclidean Geometry.

have developed at length in the case of the plane [§§ 84
—92].

Further the proof of the logical possibility of Riemann's
elliptic hypothesis can be just as easily derived from these
projective measurements. For the plane, the interpretation
which we have given of it as the geometry of the sheat
will be sufficient [§ 71]/

I Another neat and simple proof of the independence of the
Fifth Postulate is to be found in the representation of the Non-
Euclidean plane, employed by Klein and Poincaré. In this the
points of the Non-Euclidean plane appear as points of the upper
portion of the Euclidean plane, and the straight lines of the Non-
Euclidean plane as semicircles, perpendicular to the straight bound-
ary of this halfplane; etc. The Elliptic Geometry can be repres-
ented in a similar way; and the Hyperbolic and Elliptic Solid
Geometries can also be brought into correspondence with the
Euclidean Space. An account of these representations is to be
found in "Weber und Wellstein's Encyklopàdie der Elemetttar-
Mathematik, Bd. II S 9— n» P- 39 — 81 (Leipzig, I905) and in
Chapter II of the NUhi-Euklidische Geometrie by H. Liebmann
(Sammlung Schubert, 49, Leipzig, 1905).

In Appendix V of this volume a similar argument is given,
based upon the discussion in Weber-Wellstein's volume. Points
upon the Non-Euclidean plane are represented by pairs of points
inverse to a fixed circle on the Euclidean plane; and straight
lines upon the one, are circles orthogonal to the fixed circle on
the other.

Appendix I.

The Fundamental Principles of Statics and
Euclid's Postulate.

On the Principle of the Lever.

§ I. To demonstrate the Principle of the Lever, Archi-
medes [287 — 212] avails himself of several hypotheses, some
expressed and others imphed. Among the hypotheses
passed over in silence, in addition to that which we would
now call the hypothesis of increased constraint ', there is one
which definitely concerns the equilibrium of the lever, and
can be expressed as follows:

When a lever is suspended fro7n its middle point, it is in
equilibrium, if a weight 2 F is applied at one end, a?id at the
other another lever is hung by its ntiddle point, each of its ends
supportifig a weight P}

We shall not discuss the various criticisms upon Archi-
medes' use of this hypothesis, nor the different attempts made
to prove it.^ In this connection we shall refer only to the

1 This hypothesis can be enunciated as follows: If several bodies,
subjected to constraints, are in eqziilibritim under the action of given
forces, they will still be ifz equilibrium, if new constraints are added

to those already in existence. Ci., for example, J. Andrade, Legons
de Méca7iique Physique, p. 59 (Paris, 1 898).

2 Cf. Archimedis opera omnia: critical edition by J. L. HeiberG;
Bd. II, p. 142 et seq. (Leipzig, 1881).

3 Cf,, for example, E. Mach, Die Mechanik in ihrer Ent-

1 82 Appendix I. The Fundamental Principles of Statics etc.

arguments of Lagrange, since these will show, clearly and
simply, the important link between this hypothesis and the
Parallel Postulate.

§ 2. Let ABD be an isosceles triangle {AD =■ £D),
from whose angular points A and £ are suspended two
(cf Fig. 63) equal weights P, while a weight equal to 2P is
suspended from D.

This triangle will be in equilibrium
about the straight line MN, joining
the middle points of the equal sides,
since each of these sides may be
regarded as a lever from whose ex-
tremities equal weights are hung.

But the equilibrium of the figure
will also be secured, if the triangle
rests upon a line passing through
i^ the vertex Z) and the middle point
C of the side AB. Therefore, if E
is the common point of CD and MN,
the triangle will be in equilibrium, when suspended from E.

'Or', continues Lagrange, 'comme I'axe [MN] passe
par le milieu des deux cótés du triangle, il passera aussi
nécessairement par le milieu de la droite menée du sommet
du triangle au milieu [CJ de sa base; done le levier trans-
versal [CZ>] aura le point d'appui [E] dans le miheu et
devra, par consequent, étre charge également aux bouts
[C, D\. done la charge que supporte le point d'appui du
levier; qui fait la base du triangle, et qui est charge, à ses

ivickelung, (3. Aufl., Leipzig, 15^97); English translation by T. J. Mc-
CoRMACK (Open Court Publishing Co. Chicago, 1902). Also, for
the different hypotheses from which the proof of the principle of
the lever, can be obtained, see P. Duhem, Les origines de la stati-
qiie, (Paris, 1905), especially Appendix C, Sur les divers axiomes
d'ou se peut déduire la ihcorie du levier.

Statical Hypothesis equivalent to Postulate V. i8^

deux extrémités de poids égaux, sera égale au poids double
du sommet et, par consequent, égale à la somme des deux
poids.* ^

§ 3. Lagrange's argument contains implicitly some
hypotheses of a statical nature, regarding symmetry, addition
of constraints,^ etc.; and, in addition, it involves a geometrical
property of the Euclidean triangle. But if we wish to omit
the latter, a course which for certain reasons seems natural,
the preceding conclusions will be modified.

Indeed, though we may still assume that the triangle
ABD is in equilibrium about the point E^ where the lines
MN and CD intersect, we cannot assert that E is the middle
point of CD, as this would be equivalent to assuming
Euclid's Postulate. Consequently, we cannot assert that the
single weight 2 P, applied at C, can be substituted for the two
weights at A and B, since, if such a change could take place,
a lever would be in equilibrium, with equal weights at its ends,
about a point which cannot be its middle point.

Vice versa, if we assume, with Archimedes, that two
equal weights at the end can be replaced by a double
weight at the middle point of the lever, then we can easily
deduce that E is the middle point of CD, and from this it
will follow that ABD is a Euclidean triangle.

Hence we have established the equivalence of Euclid's
Fifth Postulate and the said hypothesis of Archimedes. Such
equivalence is, of course, relative to the system of hypotheses
which comprises, on the one hand, the above-named statical
hypotheses, and, on the other, the ordinary geometrical
hypotheses.

1 Oeuvres de Lagrange, T. XI, p. 4 — 5.

2 For an analysis of \!as. physical principles on which ordinary
statics is founded, cf. F. Enriques, Problemi della Scienza. Cap. V.
(Bologna, 1906). German translation, (Leipzig, 1910).

184 Appendix I. The Fundamental Principles of Statics etc.

With the modern notation, we can speak of forces,
of the composition of forces, oi resultants, m'ìXtz.à. oi weights,
levers, etc.

Then the hypothesis referred to takes the following
form:

The resultant of two equal forces in the same plane, applied
at right angles to the extremities of a straight line and towards
the same side of it, is a single force at the middle point of the
line, of double the intensity of the given forces.

From what we have said above, if this law for the com-
position of forces were true, it would follow that the ord-
inary theory of parallels holds in space.

On the Composition *of Forces Acting at a Point.

§ 4. The other fundamental principle of statics, the
law of the Parallelogram of Forces, from the usual geom-
etrical interpretation which it receives, is closely connected
with the Euclidean nature of space. However, if we examine
the essential part of this principle, namely, the analytical
expression for the resultant R of two equal forces P, acting
at a point, it is easy to show that it exists independently of any
hypothesis on parallels.

This can be made clear by deducing the formula
R = a/' cos a,
where 2 a is the angle formed by the two concurrent forces
from the following principles:

i) Two or more forces, acting at the same point, have
a definite resultant.

2) The resultant of two equal and opposite forces
is zero.

3) The resultant of two or more forces, acting at a
point, along the same straight line, is a force through the
same point, equal to the sum of tlie given forces, and along
the same line.

Composition of Concurrent Forces.

I8:

4) The resultant of two equal forces, acting at the same
point, is directed along the line bisecting the angle between
the two forces.

5) The magnitude of the resultant is a continuous funct-
ion of the magnitude of the components.

Let us see briefly how we establish our theorem. The
value i? of the resultant of two forces of equal magnitude /*,
enclosing the angle 2 a, is a function of P and a only.

Thus we can Avrite

i?= 2/(P,a).

A first application of the principles named above shows
that R is proportional to P, and this result is independent
of any hypothesis on parallels [cf note i, p. 195]. Thus the
preceding equation can be written more simply as

R == 2P/{0.).

We now proceed to find the form of/" (a).

§ 5. Let us calculate /(a) for some particular value
of the angle.
(I) Let a = 45°-

At the point O at which act 1 p Q

the two forces Ft,, P2, of equal
magnitude P, let us imagine two
equal and opposite forces applied,
perpendicular to R and of magni-

tude — (cf. Fig. 64).

At the same time let us imag-
ine R decomposed into two others,
directed along R and of magni-
tude

We can then regard each force F as the resultant of
two forces at right angles, of magnitude — .

J 86 Appendix I. The Fundamental Principles of Statics etc.

We thus have

Z' = 2 . ^ ./(45°).

On the other hand^ R being the resultant of i^i and Pa,
we have

R== 2 /y(45°)-
From these two equations we obtain

/(45°) = \ V^'
(II) Again let a = 60°.

In this case apply a.t O a. force R' equal and opposite
to R (cf. Fig. 65). The system of the two forces R and of
R' is in equilibrium.

Thus by symmetry, R' = P.
Therefore, R = F.
But, on the other hand,

i?= 2 /y(6o").

Therefore/ (60") = y.

(Ill) Again let a = 36°.

At O let the five forces P^, P^-.Pc^, of magnitude P^ be

Special Cases. 1 87

applied, such that each of them forms with the next an angle
of 72° (cf. Fig. 66).

This system is in equilibrium.

For the resultant R of P2 and P^, we have

R= 2/y(36°).
For the resultant i?' of /'i and P^ , we have
R' = 2Pf{U').

On the other hand, R has the same direction as Pc^ ;
that is, a direction opposite to that of R.

Therefore 2 /yCsó") = 2 i'/(72°) + P.

(i) Therefore 2/(36°) = 2/(72°) + i.

If, instead, we take the resultants of P^ and P^ , and of
P^ and P^, we obtain two forces of magnitude 2 P/ (36°),
containing an angle of 144°.

Taking the resultant of these two, we obtain a new
force R" of magnitude

4 ^7(36°)/ (7 2°).
Now R", by the symmetry of the figure, has the same
line of action as P^ , but acts in the opposite direction.
Thus, since equihbrium must exist,

i^=4/'/(36°)/(72°).
(2) Therefore i = 4/ (36°)/ (72°).

From the two equations (i) and (2) we obtain

/(36o)_ltV:5_/(;.o)^-f^^,
4 4

on solving for/ (36°) and/ (7 2°).

§ 6. By arguments similar to those used in the pre-
ceding section we could deduce other values for / (a).
However, if we restrict ourselves only to those just found,

1 88 Appendix I. The Fundamental Principles of Statics etc.

and compare them with the corresponding values of cos a,
we obtain the following table:

cos 0° = 1 /(0°) = I

cos 36°

cos 45^

I + Vs'

cos 60° = —

/(36°) =

/(45°) =
/(6o°) =

COS 72° =

-I + /5

/(72°)

+ Vi

/(90-) = o.

This table suggests the
identity of the two functions
y(a) and cos a. For fuller
p confirmation of this fact, we
determine the functional
equation which _/ (a) satis-
fy 2 fies (cf. Fig. 67).

To this end let us con-
sider four forces F^, P2,
F.^, P^ of magnitude P,
acting at one point, forming
with each other the following angles

-^ p,p, = <: p^^p, = 2 p

^P,P,=- 2(a-P)
-^ P,P, == 2 (a + P).
We shall determine the resultant P of these four forces
in two different ways.

Taking Pt_ with P^ , and P^ with F^ we obtain two forces
i?i and i?j, of magnitude

The General Case. i8q

inclined at an angle 2 p. Taking the resultant of Rt_ and R2,
we have a force li, such that

i? = 4-/y(a)/(P).
On the other hand, taking /'i with /'_,, and F^ with F^^,
we obtain two resultants, both along the direction of R, and
of magnitudes

2Ff{a + ^\2Ff{a-^),
respectively.

These two forces have a resultant equal to their sum,
and thus

F = 2^/(a + p) + 2i'/(a— p).

Comparing the two values of i?, we find that
(i) 2/(a)/(P) =/(a + p) +/(a-P)

is the functional equation required.

If we now remember that

cos (a + P) + cos (a — P) = 2 cos a cos P,

and take account of the identity between f (a) and cos a in
the preceding table for certain values of a, and the hy-
pothesis that f (a) is continuous, without further argument
we can write

/ (a) = cos a.
It follows that

F = 2 F cos a.

The validity of this formula of the Euclidean space is
thus also established for the Non-Euclidean spaces.

§ 7. The law of composition of two equal concurrent
forces leads to the solution of the general problem of the
resultant, since we can assign, without any further hypothesis,
the components of a force F along two rectangular axes
through its point of application O.

J go Appendix I. The Fundamental Principles of Statics etc.

Let the two perpendicular lines be taken as the axes
of X and y, and let i? make the angles a, P with them

Through O draw the line
which makes an angle a with
Ox and an angle P with Oy.
Imagine two equal and oppos-
ite forces Pi and Pz to act
along this line at O, their mag-
nitude being — . Also imagine
the force 7? replaced by the
two equal forces P, of magni-
tude — , actmg m the same
direction as P.
Then the system P^, P^, P, Pha.s R for resultant. But

Pi and P, taken together, have a resultant
X = P cos a

along Ox: and P2 and P, taken together, have a resultant

Y= Rcoi p
along Oy.

These two forces are the components of P along the
two perpendicular lines. As to their magnitudes, they are
identical with what we would obtain in the ordinary theory
founded upon the principle of the Parallelogram of Forces.
However, the lines OX and O V, which represent the com-
ponents upon the axes, are not necessmily the projections of R,
as in the Euclidean case. Indeed we can easily see that, if
these lines were the orthogonal projections of R upon the
axes, the Euclidean Hypothesis would hold in the plane.

§ 8. The functional method applied in S 6 to the
composition of two equal forces acting at a point, is derived
from D. DE FoNCENEx [1734 — 1799]- r>y a method ana-

Rectangular Components of a Fece. Iqi

logous to that which led us to the equation for / (a) (= y),
FoNCENEX arrived at the differential equation'

P + ^y=^ o.

From this, on integrating and taking account of the initial
conditions of the problem, he obtained the known expression
for/ (a).

However the application of the principles of the In-
finitesimal Calculus, requires the continuity and differentiabil-
ity of/ (a), conditions, which, as Foncenex remarks, involve
the (physical) nature of the problem. But as he wishes to
go 'jusqu'aux difficultes les moins fondees', he avails himself
of the Calculus of Finite Differences, and of a Difference
Equation, which allows him to obtain / (a) for all values of
a which are commensurable with it. The case a incom-
mensurable is treated 'par une méthode famiUère aux Géo-
mètres et frequente surtout le écrits des Anciens'; that is, by
the Method of Exhaustion.^

All Foncenex' argument, and therefore that given in

1 We could obtain this equation from (l) p. 189 as follows:
Put p = a'a and suppose that /(a) can be expanded by Taylor's
Series for every value of a.

Then we have

2/(a) (/ (o) + '/a /' (o) + 'l^ f" (o) . . .^

= 2/(a) + 2 ^-/" {«) + ..

Equating the coefficients of do^ and putting y = /(a) and k'i

= — /" (o), we have

d2y

— il 4- ^2^ = o.

da'

2 Cf. Foncenex : Si/r les prindpes /ondameittatix de la Mecan-
ique. Misc. Taurinensia. T. II, p. 305 — 315 (1760 — 1761). His
argument is repeated and explained by A. GENOCCm in his paper:
Sur un Mémoire de Daviet de Foncenex et sur les geometries non-
euclidiennes. Torino, Memorie (2), T. XXIX, p. 366 — 371 (1877).

IQ2 Appendix I. The Fundamental Principles of Statics etc.

§ 6, is independent of Euclid's Postulate. However, it
should be remarked that Foncenex' aim was not to make
the law of composition of concurrent forces independent of
the theory of parallels, but rather to prove the law itself.
Probably he held, as other geometers [D. Bernouilli,
D'Alembert], that it was a truth independent of any ex-
perimental foundation.

Non-Euclidean Statics.

§ 9. Having thus shown that the analytical law for
the composition of concurrent forces does not depend on
Euclid's Fifth Postulate, we proceed to deduce the law accord-
ing to which forces perpendicular to a line will be composed.

Let A, A be the points of application of two lorces
Pi, P2 of equal magnitude P (cf Fig. 69).

Let C be the middle point of AA, and B a point on
the perpendicular BC to AA.

Joining AB and AB, and putting

<^ BAC = a, <^ ABC = p,
it is clear that the force P^ can be regarded as a component
of a force T-s,, acting at A and along BA.
The magnitude of this force is given by

P
sin a

r= -. —

Equal Forces perpendicular to a Line. ig^

The other component Q^, at right angles to P^, is
given by

Q = T'cos a = /'cot a.

Repeating this process with the force F2 , we obtain the
following system of coplanar forces :
(i) System F^, F^-

(2) System/',, P,, Q,, Q,.

(3) System 7;, T^.

If we assume that we can move the point of application
of a force along its line of action, it is clear that the first two
systems are equivalent, and because (2) is equivalent to (3),
we can substitute for the two forces jPi , P2, the two forces
7; and 7;.

The latter, being moved along their lines of action to B,
can be composed into one force

P = 2rcosp = 2/'^-
^ sin a

This, in its turn, can be moved to C, its direction per-
pendicular to A A remaining unchanged.

This result, which is obviously independent of Euclid's
Postulate, can be applied to the three systems of geometry:

Euclid^s Geometry.

In the triangle ABC we have

cos P = sin a.
Therefore

R= 2 P.

Geometry of Lobatschewsky-Bolyai.
In the triangle ABC, if we denote the side AA by 2 b,
we have

cos p ^ ^ / \

-. == cosh -r (p. II 7).

sin a k ^^ '^

Thus

i? ==• 2 jP cosh ,

1 94 Appendix I. The Fundamental Principles of Statics etc.

Riemann's Geometry.

In the same triansjle we have

Therefore

cos 6 h

-. = COS -r- •

sin a /C

R = 2 P COS —

Conclusioti.

It is only in EucUdean space that the resultant of two
equal forces, perpendicular to the same line, is equal to the
sum of the two given forces. In the Non-Euclidean spaces
the resultant depends, in the manner indicated above, on
the distance between the points at which the two forces are
applied.^

§ IO. The case of two unequal forces P^ Q, per-
pendicular to the same straight line, is treated in a similar
manner.

In the Euclidean Geometry we obtain the known results;
R^ P -V (2,
R _ P _ Q
p-\- 1 q P
In the Geometry of Lobatschewsky-Bolyai the problem
of the resultant leads to the following equations:

R = P cosh y + <2 cosh y,

R _ P Q

sinh T sinh -r sinh -7-

k K K

Then, by the usual substitution of the circular functions
for the hyperboHc, we obtain the corresponding result for
Riemann's Geometry:

I For a fuller treatment of Non-Euclidean Statics, the reader
is referred to the following authors: J. M. de Tilly, Etudes de
Mécafiique abstraiie, Mém. couronnés et autres mém., T. XXI (1870).
J. Andrade, La Statique et les Géo??iétries de Lobatscheivsky , d'Euclide,
et de Riemann. Appendix (II) of the work quoted on p. 181.

Unequal Forces. igc

7? == Z' COS y + (2 COS -|-,
R P Q

■ p-\- 9 • <] ■ P

sm — T— sin -T- sm -r

k k k

In these formulce /, q, denote the distances of the
points of application of P and Q from that of R.

These results can be summed up in a single formula,
valid for Absolute Geometry;

R = F.EP+ Q. Eq,
R _ -P __ Q

07/ +7) ~0(^)~Ò(?)'

To obtain these results directly, it is sufficient to use the
formulas of Absolute Trigonometry, instead of the Euclidean
or Non-Euclidean, in the argument of which a sketch has
just been given.

Deduction of Plane Trigonometry from Statics.

§ II. Let us see, in conclusion, how it is possible to
treat the converse (\\XQsi\on: given the law of composition of
forces, to deduce the fundamental equations of trigonometry.

To this end we note that the magnitude of the resultant
R of two equal forces' F, perpendicular to a line A A' of
length 2 b, will in general be a function of P and b.

Denoting this function by

cp {P, b),

we have

or more simply^

if = cp (P, b),
R = P(?{b).

I The proportionality of R and P follows from the laiu of
association on which the composition of forces depends. In fact,
let us imagine each of the forces P, acting at A and A', to be

Iq6 Appendix I. The Fundamental Principles of Statics etc.

On the other hand in § 9 (p. 193), we were brought to
the following expression for J^:

sm a

Eliminating i? and J^, between these, we have

/7\ cos p
op (Ò) = - -*- •
^ ^ ^ sin a

Thus if the analytical expression for (p (/;) is known,
this formula will supply a relation between the sides and
angles of a right-angled triangle.

To determine qp (fi), it is necessary to establish the
corresponding functional equation.

With this view, let us apply perpendicularly to the line
AA', the four equal forces J^j, F^, P^,, P^,, in such a way that
the points of application of jP^ and F^, F^ and jP,, are
distant 2 {a-\-b) and 2 {b — a), respectively (cf. Fig. 70).

We can determine the resultant R of these four forces
in two different ways:

(i) Taking F,_ with F2, and F^^ with F^, we obtain two
forces Ri, R^ of magnitude:

F^{ay,

replaced by n equal forces, applied at A and A'. Combining
these, we would have for R the expression

y? = « cp (^, b\.
Comparing this result with the equation given in the text, we have

Similarly we have

cp (kP, b)-^k(^{P, b),
for every rational value of /c; and the formula may be extendeii
to irrational values.

Then putting P= i and k = P v^t obtain
9 [P, Ò) = P(^ (6). Q. E. D.

Deduction of Trigonometry from Statics.

197

and taking R^, R2 together, we obtain
R = F(^ {a) qp {ù).

(ii) Taking F^ with F^ , we obtain a force of magnitude :
F(p{è + a),
and taking F2 with F^, we obtain another of magnitude:
F(p(^ — a).
Taking these two together we have, finally,
R = F(i>(ò + a) + Fcp(ù—a).

r-a-

b~a

R-

Fig. 70.

From the two expressions for R we obtain the functional
equation which qp (^) satisfies, namely,

(2) cp(ò) (p{a) = cp(l> + a) + cp (i> — a).

This equation, if we put cp {ò) = 2 fib), is identical
with that met in § 6 (p. 189), in treating the composition of
concurrent forces.

The method followed in finding (2) is due to D'Alem-
bert.^ However, if we suppose a and b equal to each other,
and if we note that qp io) = 2, the equation reduces to

(3) [9(^)]' = qp (2:r) + 2.

This last equation was obtained previously by Foncenex,
in connection with the equilibrium of the lever.^

1 Opuscules mathématiqiies, T. VI, p. 371 (1779).

2 Cf. p. 319—322 of the work by FOxNCENEX, referred to

above.

Iq8 Appendix I. The Fundamental Principles of Statics etc.

§ 12. The statical problem of the composition of
forces is thus reduced to the integration of a functional
equation.

FoNCENEX, who was the first to treat it in this way^,
thought that the only solution of (3), was cp (x) = const. If
this were so, the constant would be 2, as is easily verified.

Later Laplace and D'Alembert integrated (3), obtaining

cp (x) = e <^ + e ^ .

where <: is a constant, or any function which takes the same
value when x is changed to 2 x/

The solution of Laplace and D'Alembert, applied to
the statical problem of the preceding section, leads to the
case in which c- is a function of x. Further, since we cannot
admit values of c such a.sa+i ù, where a, i> are both different
from zero, we have three possible cases, according as c is
real, a pure imaginary, or infinite.^ Corresponding to these

1 We have stated above (p. 53), when speaking of FoNCENEX'
memoir, that, if it v?as not the vv'ork of Lagrange, it was certainly
inspired by him. This opinion, accepted by Genocchi and other
geometers, dates from Delambre. The distinguished biographer
of Lagrange puts the matter in the following words: "// (Za-
gi-aiigé) fournissait à Fonceiiex la parile analyllque de ses mémoires en
ltd laissajtl le soin de développer les raisonnements sur lesqueh portaiettl
ses formules. En effet, on remarque drja dans ces mémoires (of
Foncenex) cede marche purement analitique, qui depuis a fait le
caractère des grandes productions de Lagrange. II avail trouvè tt?ie
nouvelle théorie dii levier". Notices sur la voie et les ouvrages de M.
le Comic Lagrange. Mém. Inst, de France, classe Math, et Physique,
T. Xm, p. XXXV (1 8 1 2).

2 Cf. D'Alembert: Sur les principes de la Mécaniqtce : Mém. de
l'Ac. des Sciences de Paris (1769). — Laplace: Recherches sur
l'intrgraiion des equations diffirentiellcs : Mém. Ac. sciences de Paris
(savants étrangers) T. VII (1733). Oeuvres de Laplace, T. Vili,
p. 106 — 7.

3 We can obtain this result directly by integrating the equa-

The Three Geometries,

199

three cases, we have three possible laws for the composition
of forces, and consequently three distinct types of equations
connecting the sides and angles of a triangle. These results
are brought together in the following table, where k denotes
a real positive number.

Value of c

Form of q) (^)

Trigonometri-
cal equations

Nature of
plane

c = k

X X

ek'^-e T_2cosh^
k

b cos p

cosh-; . — „

X' sin a

hyperbolic

c = ik

i X ix

,k -\-e k = 2 cos —
' k

b cos p

elliptic

c = 00

X X

e^+e «' = 2

cos 6

1 =-. — -
sm a

parabolic

Conclusion: The law for the composition of forces per-
pendicular to a straight line, leads, in a certain sense, to the
relations which hold between the sides and angles of a
triangle, and thus to the geometrical properties of the plane
and of space.

This fact was completely established by A. Genocchi
[181 7 — 1889] in two most important papers', to which the
reader is referred for full historical and bibliographical
notes upon this question.

tion (2), or, what amounts to the same thing, equation (l) of
S 6. Cf., for this, the elementary method employed by Cauchy
for finding the function satisfying (i). Oeuvres de Cauchy , (sér. 2).

T. ni, p. 106— 113.

I One of them is the Memoir referred to on p. 19 1. The
other, which dates from 1869, is entitled: Dei primi principii della
meccanica e della geometria in relazione al postulato d'Euclide. Annali
della Società italiana delle Scienze (3). T. II, p. 153 — 189.

Appendix II.

Clifford's Parallels and Surface.
Sketch of Clifford-Klein's Problem.

Clifford's Parallels.

§ I. Euclid's Parallels are straight lines possessing the
following properties:

a) They are coplanar.

b) They have no common points.

c) They are equidistant.

If we give up the condition (c) and adopt the views of
Gauss, Lobatschewsky and Bolyai, we obtain a first ex-
tension of the notion of parallelism. But the parallels which
correspond to it have very few properties in common with
the ordinary parallels. This is due to the fact that the most
beautiful properties we meet in studying the latter depend
principally on the condition (c). For this reason we are led
to seek such an extension of the notion of parallelism, that,
so far as possible, the new parallels shall still possess the
characteristics, which, in Euclidean geometry, depend on
their equidistance. Thus, following W. K. Clifford [1845 —
1879], we give up the property of coplanariiy, in the definition
of parallels, and retain the other two. The new definition of
parallels will be as follows:

Two straight lines, iti the same or in different planes, are
called parallel, when the points of the one are equidistant from
the points of the other.

Clifford's Parallels. 201

§ 2. Two cases, then, present tlieip.selves, according as
these parallels lie, or do not lie, in the same plane.

The case in which the equidistant straight lines are
coplanar is quickly exhausted, since the discussion in the
earher part of this book [§ 8] allows us to state that the
corresponding space is the ordinary Euclidean. We shall,
therefore, suppose that the two
equidistant straight lines r and s T

are not in the same plane, and

that the perpendiculars drawn

from r to J are equal. Obvi- s ,

A R

ously these lines will also be per-

, , Fig. 71.

pendicular to r. Let AA , BB

be two such perpendiculars (Fig. 71). The skew quad-
rilateral ABB' A , which is thus obtained, has its four angles
and two opposite sides equal. It is easy to see that the
other two opposite sides AB, AB' are equal, and that the
interior alternate angles, which each diagonal — e. g. AB' —
makes with the two parallels, are equal. This follows from
the congruence of the two right-angled triangles AAB' and
ABB'.

If now we examine the solid angle at A, from a theorem
valid in all the three geometrical. systems, we can write

<C AAB' -f <^ B'AB > -^ AAB = i right angle.

This inequality, taken along with the fact that the angles
AB' A and B' AB are equal, can be written thus:

<^ AAB' 4- <^ AB' A > i right angle.

Stated in this way, we see that the sum of the acute
angles in the right-angled triangle AA B' is greater than a
right angle. Thus in the said triangle the Hypothesis of the
Obtuse Angle is verified, and consequently parallels ?iot iti the
same plane can exist only in the space of Riemann.

202

Appendix II. Clifford's Parallels and Surface.

§ 3. Now to prove that in the elliptic space of Riemann
there actually do exist pairs of straight lines, not in the same
plane and equidistant, let us consider an arbitrary straight
line r and the infinite number of planes perpendicular to it.
These planes all pass through another line r, the polar
of r in the absolute polarity of the elliptic space. Any line
whatever, joining a point of r with a point of/, is perpend-
icular both to r and to /, and has a constant length, equal
to half the length of a straight line. From this it follows
that r, r are two equidistant straight lines^ not in the same
plane.

But two such equidistants represent a very particular
case, since all the points of r have the same distance not
only from /, but from all the points of r.

Fig. 72.

To establish the existence of straight lines in which the
last peculiarity does not exist, we consider again two lines
r and /, one of which is the polar of the other (Fig. 72).
Upon these let the equal segments AB^ AB' be taken, each
less than half the length of a straight line. Joining A with
A^ and B with B' , we obtain two straight lines ^, b, not
polar the one to the other, and both perpendicular to the
lines r, r .

It can easily be proved that a, b are equidistant. To
show this, take a segment AH upon AA; then on the

The Polars as Parallels. 203

supplementary line ^ to AHA, take the segment ^i^ equal to
AH. If the poinfs H and M are joined respectively with
£^ and B, we obtain two right-angled triangles A£H, ABM,
which, in consequence of our construction, are congruent.

We thus have the equality

HB' = B3f.

Now if H and B are joined, and the two triangles
HBB' and HBM zx^ compared, we see immediately that they
are equal. They have the side HB common, the sides HB'
and MB equal, by the preceding result, and finally BB' and
HM are also equal, each being half of a straight line.

This means, in other words, that the various points of
the straight line a are equidistant from the line b. Now since
the argument can be repeated, starting from the line b and
dropping the perpendiculars to a, we conclude that the line
HK^ in addition to being perpendicular to b, is also perpend-
icular to a.

We remark, further, that from the equality of the
various segments AB, HK, A B\ . . . the equality of the re-
spective supplementary segments is deduced, so that the two
lines a, b, can be regarded as equidistant the one from the
other, in two different ways. If then it happened that the
line AB were equal to its supplement, we would have the ex
ceptional case, which we noted previously, where a, b are
the polars of each other, and consequently all the points of
a are equidistant from the different points of b.

§ 4. The non-planar parallels of elliptic space were
discovered by Clifford in 1873.^ Their most remarkable
properties are as follows:

1 The two different segments, determined by two points on
a straight line, are called supplementary.

2 Preliminary Sketch of Biquaternions. Proc. Lond. Math. Soc.
Vol. IV. p. 381— 395(1873). Clifford's Mathematical Papers, p. 181—200.

204

Appendix II. Clifford's Parallels and Surface.

fi) If a siraigJit line meets two parallels, it makes with
the»! equal eorrespo?iding angles, equal interior alternate
angles, etc.

(ii) If in a skew quadrilateral the opposite sides are
equal and the adjacent angles supplemcjitary, then the opposite
sides are parallel.

Such a quadrilateral can therefore be called a ske:a
parallelogram .

The first of these two theorems can be immediately
verified; the second can be proved by a similar argument
to that employed in § 3.

(iii) If two straight lines are equal and parallel, ajid
their extremities are suitably joined, we obtain a skezv paral-
lelogram.

This result, which can be looked upon, in a certain

sense, as the converse of (ii), can also be readily established.

(iv) Through a?iy point (AI) in space, which does not

lie on the polar of a straight line (r), two parallels can be

drawn to that line.

Indeed, let the perpendicular MN be drawn from M
to r, and let N' be the point in which the polar of MN

meets r (Fig. 73). From
this polar cut off the two
segments N' M' , N'AI",
equal to NM, and join the
points M', M" to M. The
two lines /, r", thus ob-
tained, are the required par-
allels.

If M lay on the polar of r, then MN would be
equal to half the straight line; the two points M' , M"
would coincide: and the two parallels /, r" would also
coincide.

Fig- 73-

Properties of Clifford's Parallels.

205

The angle between the t.vo parallels /, r" can be
measured by the segment MM", which the two arms of the
angle intercept on the polar of its vertex. In this way we
can say that half of the angle between r and r", that is,
the angle 0/ parallelism, is equal to the distance of parallelism.

To distinguish the two parallels /, r", let us consider a
helicoidal movement of space, with MN for axis, in which
the pencil of planes perpendicular to MJV, and the axis J/' J/ '
of that pencil, obviously remain fixed. Such a movement
can be considered as the resultant of a translation along MJV,
accompanied by a rotation about the same axis: or by two
translations, one along MN, the other along M'M". If the
two translations are of equal amount, we obtain a space
vector.

Vectors can be right-handed or left-handed. Thus, referr-
ing to the two parallels /, r", it is clear that one of them
will be superposed upon r by a right-handed vector of
magnitude AfJV, while the other will be superposed on r by
a left-handed vector of the same magnitude. Of the two
lines r, r", one could be called the right-handed parallel
and the other the left-handed parallel to r.

(v) Two right-handed {or left-handed) parallels to a
straight line are I'ight-handed {or left-handed) parallels to
each other.

Let b, c be two right-hand-
ed parallels to a. From the
two points A, A of a, distant
from each other half the length
of a straight Hne, draw the
perpendiculars AB, AB' on b,
and the perpendiculars AC,
AC on c (cf. Fig. 74).

The lines AB', AC are the polars of AB and AC.

Therefore ^ BAC = <^B'AC.

/^/

Fig. 74-

206 Appendix II. Clifford's Parallels and Surface.

Further^ by the properties of parallels

AB = AB\ AC^AC.

Therefore the triangles ABC, A JS C are equal

Thus it follows that

BC = B'C.

Again, since

BB' = AA = CC\
the skew quadrilateral BBC' C has its opposite sides equal.

But to establish the parallelism of b, c, we must also
prove that the adjacent angles of the said quadrilateral are
supplementary (cf ii). For this we compare the two solid
angles B {AB' C) and B' (AB"C'). In these the following
relations hold:

^ABB' = -^ AB'B" = I right angle
^ ABC = <^ AB'C.

Further, the two dihedral angles, which have BA and
B'A' for their edges, are each equal to a right angle, dimin-
ished (or increased) by the dihedral angle whose normal
section is the angle ABB'.

Therefore the said two solid angles are equal. From
this the equality of the two angles B' BC, B'B'C follows.
Hence we can prove that the angles B, B' of the quadri-
lateral BB' C C are supplementary, and then (on drawing
the diagonals of the quadrilateral, etc.) that the angle B is
supplementary to C, and C supplementary to C, etc.

Thus b and c are parallel. From the figure it is clear
that the parallelism between b and c is right-handed, if that
is the nature of the parallelism between the said lines and
tlie line a.

Clifford's Surface.

§ 5. From the preceding argument it follows that all
the lifies which meet three right-handed parallels are left-handed
parallels to each other.

Clifford's Surface.

207

Indeed, if ABC is a transversal cutting the three lines
a, b, c, and if three equal segments AA\ BB\ CC are taken
on these lines in the same direction," the points A'B'C lie
on a line parallel to ABC. The psjallelism between ABC
and A'B'C is thus left-handed.

From this we deduce that three parallels a, b, c, define
a ruled surface of the second order (Clifford's Surface).
On this surface the lines cutting a, b, c form one system of
generators {g^: the second system of generators {gd) is
formed by the infinite number of lines, which, like a, ^, c,
meet {gs).

Clifford's Surface possesses the following charact-
eristic properties:

a) Two generators of the same system are parallel to
each other.

b) Two generators of opposite systems cut each other at a
constant atigle.

§ 6. We proceed to show that Clifford's Surface has
t7vo distinct axes of rcvolutiofi.

To prove this, from
any point M draw the
parallels d (right-hand-
ed), s (left-handed), to a
line r, and denote by Ò
the distance MN of
each parallel from r
(cf. Fig. 75).

Keeping d fixed, let
s rotate about r, and let /, /', /" ,
positions which s takes in this rotation

Fig. 75-

. be the successive

I It is clear that if a direction is fixed for one line, it is
then fixed for every line parallel to the first.

208 Appendix II. Clifford's Parallels and Surface.

It is clear that s, s', s" . . . are all left-handed parallels
to r and that all intersect the line d.

Thus s in its rotation about r generates a Clifford's
Surface.

Vice versa, if d and j- are two generators of a Clifford's
Surface, which pass through a point M of the surface, and 2 Ò
the angle between them, we can raise the perpendicular
to the plane sd at M and upon it cut off the lines
AIL = MiV = Ò.

Let Z> and ^ be the points where the polar of ZiV meets
the lines d and s, respectively, and let i^be the middle point
ofZ'^= 2Ò.

Then the lines HL and HIV are parallel, both to s
and d.

Of the two lines HZ and HIV choose that which is
a right-handed parallel to d and a left-handed parallel to s,
say the line HIV.

Then the given Clifford's Surface can be generated by
the revolution of s or d about HIV.

In this way it is proved that every Clifford's Surface
possesses one axis of rotation and that every point on the
surface is equidistant from it.

The existence of another axis of rotation follows im-
mediately, if we remember that all the points of space, equi-
distant from HN., are also equidistant from the line which is
the polar of HN.

This line will, therefore, be the second axis of rotation
of the Clifford's Surface.

§ 7. The equidistance of the points of Clifford's
Surface from each axis of rotation leads to another most
remarkable property of the surfaces. In fact, every plane
passing through an axis r intersects it in a line equidistant
from the a.xis. The points of this line, being also equally
distant from the point {O) in which the plane of section meets

The Axes of Clifford's Surface.

209

the other axis of the surface, lie on a circle, whose centre (O)
is the pole of /■ with respect to the said line. Therefore the
meridians and the parallels of the surface are circles.

The surface can thus be generated by making a circle
rotate about the polar of its cetitre, or by making a circle move
so that its centre describes a straight line, while its plane is
maintained constantly perpendicular to it (Bianchi).'

This last method of generating the surface, common
also to the Euclidean cylinder, brings out the analogy be-
tween Clifford's Surface and the ordinary circular cyhnder
This analogy could be carried further, by considering the
properties of the hehcoidal paths of the points of the surface,
when the space is submitted to a screwing motion about
either of the axes of the surface.

§ 8. Finally, we shall show that the geometry on Clif-
ford's Surface, understood in the sense explained in §§ 67,
68, is identical with Euclidean geometry.

To prove this, let us determine the law according to
which the element of distance between two points on the
surface is measured.

Let u, V, be respectively a parallel and a meridian
through a point O on the surface, and M any arbitrary point
upon it.

Let the meridian and parallel
through M cut off the arcs OP, OQ
from u and v. The lengths u, ?> of
these arcs will be the coordinates of Q
Jlf. The analogy between the system
of coordinates here adopted and the
Cartesian orthogonal system is evident 0
(cf. Fig. 76). Fig. 75.

I Sulla siipeificie a curvatiaa nulla in geometria ellittica. Ann.
di Mat. (2) XXIV, p. 107 (1896). Also Lezioni di Geometria Differ-
enziale. 2a Ed., Voi. I, p. 454 (Pisa, 1902).

2 IO Appendix II. Clifford's Parallels and Surface.

Let M' be a point whose distance from M is infini-
tesimal. If {u, v) are the coordinates of J/, we can take
{u + du, V + dv) for those of M' .

Now consider the infinitesimal triangle MM' N., whose
third vertex N is the point in which the parallel through AI
intersects the meridian through M' . It is clear that the angle
MNM' is a right angle, and that the sides MN, NM' are
equal to du^ dv.

On the other hand, this triangle can be regarded as
rectilinear (as it lies on the tangent plane at M). So that,
from the properties of infinitesimal plane triangles, its hypo-
tenuse and its sides, by the Theorem of Pythagoras, are con-
nected by the relation

ds^ = du^ -^ dv^.
But this expression for ds* is characteristic of ordinary
geometry, so that we can immediately deduce that the pro-
perties of the Euclidean plane hold i?i every normal region on
a Clifford's Surface.

An important application of this result leads to the
evaluation of the area of this surface. Indeed, if we break
it up into such congruent infinitesimal parallelograms by
means of its generators, the area of one of these will be
given by the ordinary expression

dx dy sin 9,
where dx, dy are the lengths of the sides and 0 is the con-
stant angle between them (the angle between two generators).
The area of the surface is therefore

E dx dy sin 0 = sin 9 2 dx • 2 dy.

But both the sums 2 dx, 2 dy represent the length / of
a straight line.

Therefore the area A of Clifford's Surface takes the
very simple form.

The Area of Clifford's Surface. 211

A = /^ sin e,
which is identical v/ith the expression for the area of a
EucHdean parallelogram (Clifford).'

Sketch of Clifford-Klein's Problem.

§ 9. Clifford's ideas, explained in the preceding
sections, led Klein to a new statement of the fundamental
problem of geometry.

In giving a short sketch of Klein's views, let us refer
to the results of § 68 regarding the possibility of interpret-
ing plane geometry by that on the surfaces of constant
curvature. The contrast between the properties of the Eu-
chdean and Non-Euclidean planes and those of the said
surfaces was there restricted to suitably bounded regions.
In extending the comparison to the unbounded regions, we
are met, in general, by differences; in some cases due to
the presence of singular points on the surfaces (e. g., vertex
of a cone); in others, to the different connectivities of the
surfaces.

Leaving aside the singular points, let us take the cir-
cular cylinder as an example of a surface of constant curv-
ature, everywhere regular, but possessed of a connectivity
different from that of the Euclidean plane.

The difference between the geometry of the plane and
that of the cylinder, both understood in the complete sense,
has been already noticed on p. 140, where it was observed
that the postulate of congruence between two arbitrary
straight lines ceases to be true on the cylinder. Nevertheless
there are numerous properties common to the two geometries,

I Preliminary Sketch, cf. p. 203 above. The properties of
this surface were referred to only very briefly by Clifford in 1873.
They are developed more fully by Klein in his memoir: Zur nichl-
euklidischen Geometrie, Math. Ann. Bd. XXXVII, p. 544—572 (1890).

14*

212 Appendix II. Clifford's Parallels and Surface.

which have their origin in the double characteristic, that
both the plane and the cylinder have the same curvature,
and that they are both regular.

These properties can be summarized thus:

i) The geometry of a?iy normal region of the cylinder
is identical with that of any normal region of the plane.

2) The geometry of any normal region whatsoever of
the cylinder, fixed with respect to an arbitrary point upon it,
is identical with the geometry of any normal region what-
soever of the plane.

The importance of the comparison between the ge-
ometry of the plane and that of a surface, founded on the
properties (i) and (2), arises from the following consid-
erations :

A geometry of the plane, based upon experimental
criteria, depends on two distinct groups of hypotheses. The
first group expresses the validity of certain facts, directly
observed in a region accessible to experiment {postulates of
the normal region); the second group extends to inaccessible
regions some properties of the initial region {postulates of
extension).

The postulates of extension could demand, e. g., that
the properties of the accessible region should be valid in the
entire plane. We would then be brought to the two forms,
the parabolic and the hyperbolic plane. If, on the other hand,
the said postulates demanded the extension of these pro-
perties, with the exception of that which attributes to the
straight line the character of an open line, we ought to take
account of the elliptic plane as well as the two planes mentioned.

But the preceding discussion on the regular surfaces of
constant curvature suggests a more general method of enun-
ciating the postulates of extension. We might, indeed, simply
demand that the properties of the initial region should hold
in the neighbourhood of every point of the plane. In this

Clifford-Klein's Problem. 21 3

case, the class of possible forms of planes receives con-
siderable additions. We could, e. g., conceive a form with
zero curvature, of double connectivity, and able to be com-
pletely represented on the cyhnder of Euclidean space.

The object of Clifford-Klein' s problem is the determination
of all the two dimensional manifolds of constant curvature,
which are everyiohere regular.

§ 10. Is it possible to realise, with suitable regular
surfaces of constant curvature, in the Euclidean space, all
the for7tis of Clifford-Klein ?

The answer is in the negative, as the following example
clearly shows. The only regular developable surface of the
Euclidean space, whose geometry is not identical with that
of the plane, is the cylinder with closed cross-section. On
the other hand, Clifford's Surface in the elliptic space is a
regular surface of zero curvature, which is essentially different
from the plane and cylinder.

However with suitable conventions we can represent
Clifford's Surface even in ordinary space.

Let us return again to the cylinder. If we wish to un-
fold the cylinder, we must first render it simply connected
by a cut along a generator {g); then, by bending without
stretching, it can be spread out on the plane, covering a
strip between two parallels igxigz)-

There is a one-one correspondence between the points
of the cylinder and those of the strip. The only exception is
afforded by the points of the generator (^), to each of which
correspond two points, situated the one on^i, the other on
g2. However, if it is agreed to regard these two points as
idefitical, that is, as a single point, then the correspondence
becomes one-one without exception, and the geometry of the
strip is completely identical with thai of the cylinder.

214 Appendix II. Clifford's Parallels and Surface.

A representation analogous to the above can also be
adopted for Clifford's Surface. First the surface is made
simply connected by two cuts along the intersecting gener-
ators {g, g). In this way a skew parallelogram is obtained
in the elliptic space. Its sides have each the length of a
straight line, and its angles G and 9' [O + 0'= 2 right angles]
are the angles between g and g.

This being done, we take a rhombus in the Eu-
clidean plane, whose sides are the length of the straight line
in the elliptic plane, and whose angles are 0, 6'. On this
rhombus Clifford's Surface can be represented congruenti}'
(developed). The correspondence between the points of the
surface and those of the rhombus is a one-one correspond-
ence, with the exception of the points of^ and^', to each
of which correspond two points, situated on the opposite
sides of the rhombus. However, if we agree to regard these
points as identical, two by two, then the correspondence
becomes one-one without exception, and the geometry of
the rhombus is completely identical 7oith that of Clifford's
Surface.'^

§ II. These representations of the cylinder and of
Clifford's Surface show us how, for the case of zero curva-
ture, the investigation of Clifford-Klein's forms can be
reduced to the determination of suitable Euclidean polygons,
eventually degenerating into strips, whose sides are two by
two transformable, one into the other, by suitable movements
of the plane, their angles being together equal to four right-
angles (Klein).* Then it is only necessary to regard the
points of these sides as identical, two by two, to have a
representation of the required forms on the ordinary plane.

I Cf. Clifford loc. cit. Also Klei.n's memoir referred to
on p. 2X1.

* Cf. the memoir just named.

Clifford-Klein's Problem. 215

It is possible to present, in a similar way, the investi-
gation of Clifford-Klein's forms for positive or negative
values of the curvature, and the extension of this problem
to space.'

I A systematic treatment of Clifford-Klein's problem is to
be found in Killing's Eiiifilhrung in die Gnindlagen der Geometrie.
Bd. I, p. 271 — 349 (Paderborn, 1893).

Appendix III.

The Non-Euclidean Parallel Construction
and other Allied Constructions.

§ I. The Non-Euclidean Parallel Construction depends
upon the correspondence between the right-angled triangle
and the quadrilateral with three right angles. Indeed, when
this correspondence is known, a number of different con-
structions are immediately at our disposal.*

To express this correspondence we introduce the
following notation:

In the right-angled triangle, as usual, a, b are the sides:
c is the hypotenuse: X is the angle opposite a and fi
that opposite b. Further the angles of parallelism for a, b
are denoted by a and p: and the lines which have X, ]x. for
angles of parallelism are denoted by /, tn. Also two lines,
for which the corresponding angles of parallelism are com-
plementary, are distinguished by accents, e. g.:

n {d) = I - n(^), n(/') = ^ - ^ (^^-

Then with this notation: To every right-angled triangle
{a, b, c, X, \x) there corresponds a quadrilateral with three
right-angles^ whose fourth angle (acute) is P, a7id whose sides
are c, m\ a, /, taken in order from the corner at which the
angle is p.

The converse of this theorem is also true.

I Cf. p. 256 of Engel's work referred to on p.

Correspondence between Quadrilateral and Triangle. 217

The following is one of the constructions, which can be
derived from this theorem, for drawing the parallel through
A to the line BC (cf. Fig. 77).

Let AB be the perpendicular from A to BC. At A draw
the line perpendicular to AB, and from any point C in BC
draw the perpendicular CD ^ 3

to this line.

With centre A and rad-
ius BC (equal to c) describe
a circle cutting CD in E.

Now we have

^ EAD = M,
and therefore

•^ BAE = -^ — ^ = n (;//).

But the sides of the quadrilateral are c, m', a, /, taken in
order from C.

Therefore A£ is parallel to BC.

If a proof of this construction is required without using
the trigonometrical forms, one might attempt to show direct-
ly that the line AE produced, (simply owing to the equality
of BC and A£), does not cut BC produced, and that the
two have not a common perpendicular. If this were the
case, they would be parallel. Such a proof has not yet been
found.

Again, we might prove the truth of the construction
using the theorem, that in a prism of triangular section the
sum of the three dihedral angles is equal to two right angles':
so that for a prism with n angles the sum is (2 n—4) right
angles. This proof is given in § 2 below.

. ^ Cf. LoBATSCHEWSKY (Engel's translation) p. 172.

2l8 Appendix III. The Non-Euclidean Parallel Construction.

Finally, the correspondence stated in the above theorem
— only part of which is required for the Parallel Construction
of Fig. 78 ■ — can be verified without the use of the geo-
metry of the Non-Euclidean space. This proof is given in S 3-

§ 2. Direct proof of the Parallel Construction by fneans
of a Prism.

Fig. 78.

Let ABCD be a plane quadrilateral in which the angles
at Z>, Ay B are right angles. Let the angle at C be denoted
by p, AD by a, DC by /, CB by c, and BA by m.

At A draw the perpendicular ^Q to the plane of the
quadrilateral. Through B, C, and Z? draw ^Q, CQ and Z>S2
parallel to A^.

Also through A draw AQ parallel to BC, cutting CD
in E {ED = b^, and let the plane through A^ and AE
cut CZPQ in EQ.. From the definition, we have

^EAD

n (?//)

Further the plane ^lAB is at right angles to a, and the
plane Q.DA at right angles to /, since ^A and AB are per-
pendicular to a, while QZ> and a; are perpendicular to /.

Direct Proof of the Parallel Construction. 2I9

Also <^ AB9. = <^ OAB = -^ — ^

In the prism Q {ABCD) the faces which meet in Q^,
^.B, QD are perpendicular. Also the four dihedral angles
make up four right angles. It follows that the faces of the
prism C (DBQ), which meet along CQ, are perpendicular.
Also it is clear that in £ (DQA) the faces which meet in £A
are perpendicular, while the dihedral angle for the edge CD
is the same as for £D (thus equal to a).

We shall now prove the equality of the other dihedral
angles in these prisms C {DBQ.) and E (DQA) — those con-
tained by the faces which meet in CB and AE.

In the first prism this angle is equal to the angle be-
tween the planes ABCD and CBQ. It is thus equal to

|U, i. e. it is equal to <^ ABQ..

In the second prism, the angle between the planes
meeting in EQ belongs also to the prism Q {ADE). In this
the angle at Q.D is a right-angle, and that at QA is equal

to H- Thus the third angle is equal to |li.

Therefore the prisms C {DB9.) and E (DQJ) are
congruent

Therefore ^ BCQ = ^ QEA,
and the lines which have these angles of parallelism are
also equal.

Thus c = BC and ^i = AE

are equal, which was to be proved.
Further it follows that

^ DEA = <^ DCQ;
i. e.the angle Xj, opposite the side a of the triangle, is given by

X^ = TT (/) = X.
Finally ^ DCB == ^ DEQ;

i.e. P = 17 (d,), or Ù, = a.

220 Appendix III. The Non-Euclidean Parallel Construction.

Thus the correspondence between the triangle and the
quadrilateral is proved.^

§ 3. Proof of the Correspondence by Plane Geometry.

In the right-angled triangle ABC produce the hypo-
tenuse AB to D, where the perpendicular at D is parallel to
C^(cf. Fig. 79).

Fig- 79-

Then with the above notation

BD = m.

Draw through A the parallel to Z>0 and CBQ.

Then

^ CAQ = p = n {b),

and it is also equal to

X + <C DA(ò = \ + TT (^ + w).

We thus obtain the first of the six following equations.^

The third and fifth can be obtained in the same way. The

second, fourth, and sixth, come each from the preceding, if

we interchange the two sides a and b^ and, correspondingly

the angles X and )li.

1 Bonola: 1st. Lombardo, Rend. (2). T. XXXVII, p. 255 —
258 (1904). The theorem had already been proved by pure
geometrical methods by F. Engel: Bull, de la Soc. Phys. Math.
de Kasan (2). T. VI (1896); and Bericht d. Kon. Sachs. Ges. d.
Wiss., Math.-Phys. Klasse, Bd. L, p. 181—187 (Leipzig, 1898).

2 Cf. LoBATSCHEWSKY (Engel's translation), p. 15 — 16, and
LlEBMANN, Math. Ann. Bd. LXI, p. 1S5, (1905).

Second Proof of the Parallel Construction. 221

The table for this case is as follows:
\ + TT (^ + w) = p, ^ + U (c + /) = a:

\ + p = TT (f — w), ^ + a = TT (^ — /);

T](è+/)+ Uim — a)^^jx, U (m + a) + U (I— à) =^-rx.

Similar equations can also be obtained for the quad-
rilateral with three right angles. Some of the sides have to
be produced^ and the perpendiculars drawn^ which are
parallel to certain other sides, etc.

If we denote the acute angle of the quadrilateral by p,,
and the sides, counting from it, by c^, m/, a^, and /i, we ob-
tain the following table:

K + Tl {c; + m,) = p, , T. + n (/, -h a,') = P,;

K + px = n (c, —m,), Tx + Pi = n (/, — a^');

The second, fourth, and sixth formulae come from inter-
changing Ci and ;«i', with /i and Ui , as in the right-angled
triangle.

Let us now imagine a right-angled triangle constructed
with the hypotenuse c and the adjacent angle \x\ and let the
remaining elements be denoted by a, b, X as above.

In the same way, let a quadrilateral with three right-
angles be constructed, in which c is next the acute angle, m
follows c, the remaining elements being a^, /, , and p,.

Then a comparison of the first and third formulae for
the triangle, with the first and third for the quadrilateral,
shows that

Pi = Pj ^i = ^•

The fifth formula of both tables then gives

Ui = a.
Hence the theorem is proved.

222 Appendix III. The Non-Euclidean Parallel Construction.

From the two tables it also follows that to a right-
angled triangle with the elements

a, b, c, X, \x,
there corresponds a second triangle with the elements

a, = a, b^<== I , c^ = m, \i = — P, \x^ = ^ ,

a result which is of considerable importance in further con-
structions. But we shall not enter into fuller details.

The possibility of the Non-Euclidean Parallel Construc-
tion, with the aid of the ruler and compass, allows us to
draw, with the same instruments, the common perpendicular
to two lines which are not parallel and do not meet each
other (the non-intersecting lines); the common parallel to the
two Hues which bound an angle; and the line which is per-
pendicular to one of the bounding lines of an acute angle
and parallel to the other. We shall now describe, in a few
words, how these constructions can be carried out, following
the lines laid down by Hilbert.^

§ 4. Construction of the common perpendicular to two
non-intersecting straight lines.

Fig. 80.

Let a = Ai_A^ b = Bj,B, be two non-intersecting lines;
that is, lines which do not meet each other, and are not
parallel (cf. Fig. 80).

I Neue Begiiindung der Bolyai- Lobatschefskyschen Geometrie.
Math. Ann. Bd. 57, p. 137 — 150 (1903). Hilbert's Gruyidlagen der
Geometrie, 2. Aufl., p. ro7 at seq.

Some Allied Constructions. 223

Let AiB,, AB be the perpendiculars drawn from the
points Al , A upon a to the Hne b, constructed as in ordinary
geometry.

If the segments A^Bi, AB, are equal, the perpendicular
to b from the middle point of the segment B^B is also per-
pendicular to a; so that, in this case, the construction of
the common perpendicular is already effected.

If, on the other hand, the two segments AiB^, AB are
unequal, let us suppose, e. g., that A^Bi is greater than AB.

Then cut off from A^Bi the segment A'Bj, equal to AB;
and through the point A', in the part of the plane in which
the segment AB lies, let the ray A'M' be drawn, such that
the angle B^A'iM' is equal to the angle which the line a
makes with AB (cf. Fig. 80).

The ray A'M' must cut the line a in a point M' (cf.
Hilbert, loc. cit.). From M' drop the perpendicular M' P'
to b^ and from the line a, in the direction A-^A^ cut off the
segment ^^ equal to AM'.

If the perpendicular MP is now drawn to b, we have a
quadrilateral ABPM which is congruent with the quad-
rilateral A'B^P'M'.

It follows that MF is equal to M'F'.

It remains only to draw the perpendicular to b from
the middle point of P'F to obtain the common perpendicular
to the two lines a and b.

§ 5. Construction of the common parallel to two straight
lines which bound any angle.

Let a = AO, and b = BO, be the two lines which con-
tain the angle A OB (cf. Fig, 81). From a and b cut off the
equal segments OA and OB; and draw through A the ray
b' parallel to the line b, and through B the ray a' parallel to
the line a.

224 Appendix III. The Non-Euclidean Parallel Construction.

Let «I and ^i be the bisectors of the angles contained
by the lines ab\ and db.

The two lines a^b^ are non-intersecting lines, and their
common perpendicular yii^i, the construction for which was
given in the preceding paragraph, is the common parallel to
the lines which bound the angle AOB.

\B'

A, B,

Fig. 8i.

Reference should be made to Hilbert's memoir, quot-
ed above, for the proof of this construction.

§ 6. Construction of the straight line 7vhich is perpendi-
cular to one of the lines bounding ati acute angle and parallel
to the other.

Let a = AO and b = BO, be

the two lines which contain the acute

angle ^C>j9; and let the ray b' = B' O

be drawn, the image of the line b in

0 a (cf. Fig. 82).

Then, using the preceding con-
struction^ let the line BiBj,' be drawn
parallel to the two lines which con-
tain the angle BOB .

This line, from the symmetry of
the figure with respect to a, is perpendicular to OA.

It follows that BiB\ is parallel to one of the lines which
contain the angle AOB and perpendicular to the other.

^.? 7. The constructions given above depend upon
metrical considerations. However it is also possible to make
use of the fact that to the metrical definitions of perpend-

Fig. Sz.

Projective Constructions. 225

icularity and parallelism a projective meaning can be given
(§ 79), and that projective geometry is independent of the
parallel postulate (§ 80).

Working on these lines, what will be the construction
for the parallels through a point A to a. given line?

Let the points /'i, 1*2, P^ ^^^ P^i ^2', P^ be given
on g so that the points P^ , P^ •, P^, are all on the same
side of Pi, Pi, P^, and

p,p,' = p,p; = p,p;.

Join AP-i, ÀP2, APt^ and denote these Hnes by s^, s^,
and Sy Similarly let AP^', AP^', AP.' be denoted by Si',
$2 and J3'. Then the three pairs of rays through A^ determ-
ine a projective transformation of the pencil is) into itself,
the double elements of which are obviously the two parallels
which we require. These double elements can be constructed
by the methods of projective geometry.^

The absolute is then determined by five points: i. e., by
five pairs of parallels; and so all further problems of metrical
geometry are reduced to those of projective geometry.

If we represent (cf. § 84) the Lobatschewsky-Bolyai
Geometry (e. g., for the Euclidean plane) so that the image
of the absolute is a given conic (not reaching infinity), then
it has been shown by Grossmann^ that most of the problems
for the Non-Euchdean plane can be very beautifully and
easily solved by this 'translation'. However we must not
forget that this simplicity disappears, if we would pass from
the 'translation' back to the 'original text'.

1 Cf. for example, Enriques, Geometria proiettiva, (referred to
on p. 156) S 73-

2 Gross.mann, Die fiiiidamentalen Konstriiklioneti der nicht-
eiiklidiscken Geometrie, Programm der Thurgauischen Kantonschule,
(Frauenfeld, 1904).

226 Appendix III. The Non-Euclidean Parallel Construction.

In the Non-Euclidean plane the absolute is inaccessible,
and its points are only given by the intersection of pencils
of parallels. The points Outside of the absolute, while they
are accessible in the 'translation', cannot be reached in the
'text' itself. In this case they are pencils Of straight lines,
which do not meet in a point, but go through the (ideal)
pole of a certain line with respect to the absolute.

If, then, we would actually carry out the constructions,
difficulties will often arise, such as those we meet in the
translation of a foreign language, when we must often sub-
stitute for a single adjective a phrase of some length.

Appendix IV.

The Independence of Projective Geometry
from Euclid's Postulate.

§ I. Statement of the Frobietn. In the following pages
we shall examine more carefully a question to which only
passing reference was made in the text (cf. § 80), namely, the
validity ofProjective Geometry in Non-Euclidean Space, since
this question is closely related to the demonstration of the
independence of that geometry from the Fifth Postulate.

In elliptic space (cf § 80) we may assume that the
usual projective properties of figures are true, since the
postulates of projective geometry are fully verified. Indeed
the absence of parallels, or, what amounts to the same thing,
the fact that two coplanar lines always intersect, makes the
foundation of projectivity in elliptic space simpler than in Eu-
clidean space, which, as is well known, must be first com-
pleted by the points at infinity.

However in hyperbolic space the matter is more com-
plicated. Here it is not sufficient to account for the absence
of the point common to two parallel lines, an exception
which destroys the validity of the projective postulate: — two
coplanar lines have a coinmon point. We must also remove
the Other exception — the existence of coplanar lines which
do not cut each other, and are not parallel {the non-inter-
secting lines). The method, which we shall employ, is the
same as that used in dealing with the Euclidean case. We
introduce fictitious points^ regarded as belonging to two co-
planar lines which do not meet.

IS*

228 ApP- I^' ^^^ Indcpend. of Proj. Geo. from Euclid's Post.

In the following paragraphs, keeping for simplicity to
two dimensions only, we show how these fictitious points
can be introduced on the hyperbolic plane, and how they
enable us to establish the postulates of projective geometry
without exception. Naturally no distinction is now made be-
tween \kit proper poitits, that is, the Ordinary points, and the
fictitious points, thus introduced.

§ 2. Improper Points and the Complete Projective Plane.
We start with the pencil of lines, that is, the aggregate of
the lines of a plane passing through a point. We note that
through any point of the plane, which is not the vertex of
the pencil^ there passes one, and only one, line of the pencil.

On the hyperbolic plane, in addition to the pencil, there
exist two other systems of lines which enjoy this property,
namely; —

(i) the set of parallels to a line iti one direction',

(ii) the set of perpendiculars to a line.

If we extend the meaning of the term, pencil of lines,
we shall be able to include under it the two systems of lines
above mentioned. In that case it is clear that t7vo arbi-
trary lines of a plane will determine a pencil, to 7a hie h they
belong.

If the two lines are concurrent , the pencil is formed by
the set of lines passing through their common point; if they
are parallel, by the set of parallels to both, in the same
direction; finally, if they are nofi-ifitersectifig, by all the lines
which are orthogonal to their common perpendicular. In
the first type of pencil (Ù^e proper pencil), there exists a point
common to all its lines, the vertex of the pencil; in the two
other types (the improper pencils), this point is lacking. IVe
shall now introduce, by convention, a fictitious entity, called an
improper point, and regard it as pertainitig to all the lines of
the pencil. With this convention, every pencil has a vertex,

The Complete Line and Plane. 22Q

which will be a proper point, or an improper point, accord-
ing to the different cases. The hyperbolic plane, regarded
as the aggregate of all its points, proper and improper, will
be called the complete projective plane.

§ 3. The Complete Projective Line. The improper
points are of two kinds. They may be the vertices of pen-
cils of parallels, or the vertices of pencils of non-intersecting
lines. The points of the first species are obtained in the
same way, and have the same use, as the points at infinity
common to two Euclidean parallels. For this reason we shall
call them points at infinity on the hyperbolic plane, when it
is necessary to distinguish them from the others. The points
of the second species will be called ideal points.

It will be noticed that, while every line has only one
point at infinity on the Euclidean plane, it has tivo points at
infinity on the hyperbolic plane, there being two distinct
directions of parallelism for each line. Also that, while the
line on the Euclidean plane, with its point at infinity, is
closed, the hyperbolic line, regarded as the aggregate of
its proper points, and of its two points at infinity, is open.
The hyperbolic line is closed by associating with it all the
ideal points, which are common to it and to all the lines on
the plane which do not intersect it.

From this point of view we regard the line as made
up of two segments., whose common extremities are the two
points at infinity of the line. Of these segments, one contains,
in addition to its ends, all the proper points of the line; the
other all its improper points. The line, regarded as the
aggregate of its points, proper and improper, will be called
the complete projective line.

§ 4. Combination of Elements. We assume for the
concrete representation of a point of the complete projective
plane: —

2'ZO App. IV. The Independ. of Proj. Geo. from Euclid's. Post.

(i) its physical image, if it is a proper point;

(ii) a line which passes through it, and the relative
direction of the line, if it is a point at infinity;

(iii) the common perpendicular to all the lines passing
through it, if it is an ideal point.

We shall denote a proper point by an ordinary capital
letter; an improper point by a Greek capital; and to this
we shall add, for an ideal point, the letter which will
stand for the representative line of that point. Thus a point
at infinity will be denoted, e. g., by Q, while the ideal point,
through which all lines perpendicular to the line o pass, will
be denoted by Qo-

On this understanding, if we make no distinction be-
tween proper points and improper points, not only can we
affirm the unconditional validity of the projective postulate:
two arbitrary lines have a common point: but we can also
construct this point, understanding by this construction the
process of obtaining its concrete representation. In fact, if the
lines meet, in the ordinary sense of the term, or are parallel,
the point can be at once obtained. If they are non-inter-
secting, it is sufficient to draw their common perpendicular,
according to the rule obtained in Appendix III S 4-

On the other hand, we are not able to say that the
second postulate of projective gtovaeXry—tivo points determine
a line — and the corresponding constructions, are valid un-
conditionally. In fact no line passes through the ideal point
Qo and through the point at infinity Q on the line <?, since
there is no line whicli is at tlie same time parallel and per-
pendicular to a line o.

Before indicating how we can remove this and other
exceptions to the principle that a line can be determined by
a pair of points, we shall enumerate all the cases in which
two points fix a line, and the corresponding constructions: —

a) Two proper poiiits. The line is constructed as usual.

Combination of Elements.

231

b) A proper point [0] ajid a point at infinity [Q]. The
line OQ is constructed by drawing the parallel through 0 to
the line which contains Q, in the direction corresponding
to Q. (Appendix III).

(c) A proper point [0] and an ideal point [PJ. The line
Or^ is constructed by dropping the perpendicular from 0 to
the line c.

(d) Two points at infinity [Q, Q']. The line QQ' is the
common parallel to the two lines bounding an angle, the
construction for which is given in Appendix III § 5.

(e) A71 ideal point [fj and a point at infinity [Q], not
lying 071 the representative lifie c of the ideal point. Tlie line
QP^ is the line which is parallel to the direction given by Q
and perpendicular to c. The construction is given in Append-
ix HI § 6.

(f) Two ideal points [f^ , f^'], whose representative lines
c, c do not intersect. The line VX è, is constructed by drawing
the common perpendicular to c and / (Appendix III § 4).

The pairs of points which do not determine a line are
as follows: —

(i) an ideal point and a point at infinity, lying on the
representative line of the ideal point;

(ii) two ideal points, whose representative lines are
parallel, or meet in a proper point.

§ 5. Itnproper Lines. To remove the exceptions men-
tioned above in (i) and (ii), new entities must be introduced.
These we shall call improper lines, to distinguish them firom
the ordinary or proper lines.

These improper hnes are of two types:—

(i) If Q is a point at infinity, every line of the* pencil Q
is the representative entity of an ideal point. The locus of
these ideal points, together with the point Q, is an im-
proper line of the first type, or line at infinity. It will be
denoted by iw.

2^2 App. IV. The Independ. of Proj. Geo. from Euclid's Post.

(ii) If ^ is a proper point, every line passing through A
is the representative entity of an ideal point. The locus of
these ideal points is an improper line of the second type, Or
ù/ea/ line. It will be denoted by a^. The proper point A
can be taken as representative of the ideal Hne <1a.

These definitions of the terms line at infinity and ideal
li?ie allow us to state that two points, which do not belong
to a proper line, determine either a line at infinity, or an
ideal line. Hence, dropping the distinction between proper
and improper elements, the projective postulate — two points
determine a line — is universally true.

We must now show that, with the addition of the im-
proper lines, any two lines have a common point. The
various cases in which the two lines are proper have been
already discussed (§ 4). There remain to be examined the
cases in which at least one of the lines is improper.

(i) Let r be a proper hne and uj an improper line,
passing through the point Q at infinity. The point uur is the
ideal point, which has the line passing through Q and per-
pendicular to r for representative line.

(ii) Let r be a proper line and a^ an ideal line. The
point ro.A is the ideal point, which has the line passing
through A and perpendicular to r for its representative line.

(iii) Let UJ and uj' be two lines at infinity, to which
belong the points Q and Q' respectively. The point ujuj' is
the ideal point, whose representative line is the line joining
the points Q and Q'.

(iv) Let a,^, ^B be two ideal lines. The point o.a'^b is
the ideal point, whose representative line is the line joining
A and B,

(v) Let UJ and a^ be a line at infinity and an ideal hne.
The point uja^ is the ideal point, whose representative line
is the line joining ^ to Q.

Thus we have demonstrated that the two fundamental

Use of Improper Elements. 2^3

postulates of projective plane geometry hold on the hyper-
bolic plane.

§ 6. Complete Projective Space a fid the Validity of Pro-
jective Geometry in the Hyperbolic Space. We can introduce
improper points, lines and planes, into the Hyperbolic Space
by the same method which has been followed in the preced-
ing paragraphs. We can then extend the fundamental pro-
positions of projective geometry to the complete projective
space. Thereafter, following the lines laid down by Staudt,
all the important projective properties of figures can be de-
monstrated. Thus the validity of projective geometry in the
LoBATSCHEWSKY-BoLYAi Space is established.

§ 7. Indepefidence of Projective Geometry from the Fifth
Postulate. Let us suppose that in a connected argument,

founded on the group of postulates A, B ^ H, the only

hypotheses which can be used are /j, /i , /„. Also that

from the fundamental postulates and any one whatever of
the Is, a certain proposition M can be derived. Then we
may say that M is independent of the I's.

It is precisely in this way that the independence of pro-
jective geometry from the Fifth Postulate is proved, since
we have shown that it can be built up, starting from the
group of postulates common to the three systems of geo-
metry, and then adding to them any one of the hypotheses
on parallels.

The demonstration of the independence of il/ from any
one of the I's, founded on the deduction (cf § 59)

{4^,.. H,Ir) D 7lf^,= i,2,...«)

may be called indirect, reserving the term direct demonstration
for that which shows that it is possible to obtain AI without
introducing any of the I's at all. Such a possibility, from the
theoretical point of view, is to be expected, since the

2 •34 "''^PP- ■^^* ^^^ Independ. of Proj. Geo. from Euclid's Post.

preceding relations show that neither any single /, nor any
group of them, is necessary to obtain M. If we wish to give
a demonstration of the type

[A, B,...Il}^ M,
in which the /'s do not appear at all, v/e may meet difficult-
ies not always easily overcome, difficulties depending on the
nature of the question, and on the methods we may adopt
to solve it. So far as regards the independence of projective
geometry from the Fifth Postulate, we possess two interesting
types of direct proofs, founded on two different orders of
ideas. One employs the method of analysis: the other that
of synthesis. We shall now briefly describe the views on
which they are founded.

§ 8. Beltra7ni's Direct Demonstration of the Independ-
ence of Projective Geometry from the Fifth Postulate. The
demonstration implicitly contained in Beltrami's ''Saggid of
1868 must be placed first in chronological order. Referring
to the ^Saggio\ let us suppose that between the points of a
surface F, {or of a suitably litnited region of the surface), a?id
the points of an ordinary plane area, there can be established
a one-one correspondence, such that the geodesies of the former
are represented by the straight lines of the latter. Tlien, to the
projective properties of plane figures, which express the
collinearity of certain points, the concurrence of certain
lines, etc., correspond similar properties of the correspond-
ing figures on the surface, which are deduced from the first,
by simply changing the words platie and line into surface
and geodesic. If all this is possible, we should naturally say
that the projective properties of the corresponding plane
area are valid on the surface F; or, more simply, that the
ordinary projectivity of the plane holds upon the surface.
We shall now put this result in an analytical form.

Let u and v be the (curvilinear) coordinates of a point

Beltrami's Direct Demonstration.

235

on F^ and x and y those of the representative point on the
plane. The correspondence between the points {u^ v) and
{x, y) will be expressed analytically by putting
u ^ f {x, y)\

z' = cp (^, y)\
where y^ and qp are suitable functions.

To the equation

ip {u, v) ^ 0
of a geodesic on F, let us now apply the transformation (i).

We must obtain a linear equation in x, y, since, by our
hypothesis, the geodesies of F are represented by straight
lines on the plane.

But the equations (i) can also be interpreted as formulae
giving a transformatiofi of coordinates on F. We can there-
fore conclude that: — If, by a suitable choice of a system of
curvilinear coordinates on the surface F, the geodesies of that
surface can be represented by linear equations, the ordinary
projective geometry is valid on the surface.

Now Beltrami has shown in his ^Saggio' that on surfaces
of constant curvature it is always possible to choose a system
of coordinates (u, v), for which the general integral of the
differential equation of the geodesies takes the form
ax + by + c =^ 0.

Hence, from what has been said above, it follows that: —

Plane projective geometry is valid on the surfaces of con-
stant curvature with respect to their geodesies.

But, according to the value of the curvature, the geo-
metry of these surfaces coincides with that of the Euclidean
plane, or of the Non-Euclidean planes.

It follows that: —

The method of Beltrami, applied to a plane on which are
valid the metrical concepts co?n?non to the three geometries, leads

2 ■36 ApP' IV. The Independ. of Proj. Geo. from Euclid's Post.

to the foundation of pla?ie projective geometry without the
assumption of any hypothesis on parallels.

This result and the argument we have employed in ob-
taining it are easily extended to space. Beltrami's memoir
referring to this is the Teoria fondamentale degli spazii di
curvatura costa?ite, quoted in the note to § 75.

§ 9. Klein's Direct Demonstration of the Independence
of Projective Geometry from the Fifth Fostulate. The method
indicated above is not the only one which will serve our
purpose. In fact, we might be asked if we could not construct
projective geometry independently of any metrical consider-
ation; that is, starting from the notions of point, line, plane,
and from the axioms of connection and order, and the prin-
ciple of continuity.' In 187 1 Klein was convinced of the
possibility of such a foundation, from the consideration of
the method followed by Staudt in the construction of his
geometrical system. There remained one difficulty, relative
to the improper points. Staudt, following Poncelet, makes
them to depend on the ordinary theory of parallels. To
escape the various exceptions to the statement that two
coplanar lines have a common point, due to the omission of
the Euclidean hypothesis, YiLEm proposed to construct projective
geometry in a limited {and convex) region of space, such, e. g.,
as that of the points inside a tetrahedron. With reference to
such a region, for the end he has in view, every point on
the faces of, or external to, the tetrahedron must be con-
sidered as non-existent. Also we must give the name of line
and plane only to the portions 0/ the line and plane belonging
to the region considered. Then the graphical postulates of
connection, order, etc., which are supposed true in the whole

I For this nomenclature for the Axioms, cf. Tuwnsend's
translation of Hilbert's Fowtdatioiis of Geometiy, p. I (Open Court
Publishing Co. 1902).

Klein's Direct Demonstration. 23/

of space, are verified in the interior of the tetrahedron. Thus
to construct projective geometry in this region, it is neces-
sary, with suitable conventions, that the propositions on the
concurrence of hues, etc. should hold without exception.
These are not always true, when the word point means
simply point inside the tetrahedron.

Klein showed briefly, while various later writers dis-
cussed the question more fully, how the space inside the
tetrahedron can be completed by fictitious entities, called
ideal points, lines and planes, so that when no distinction
is made between the proper entities (inside the tetrahedron)
and the ideal entities, the graphical properties of space, on
which all projective geometry is constructed, are completely
verified.

From this there readily follows the independence of
projective geometry from Euclid's Fifth Postulate.

Appendix V.

The Impossibility of Proving Euclid's
Parallel Postulate.'

An Elementary Demonstration of this Impossibility founded
upon the Properties of the System of Circles orthogonal to a
Fixed Circle.

§ I. In the concluding article (§ 94) various arguments
are mentioned, any one of which establishes the independence
of Euclid's Parallel Postulate from the other assumptions on
which Euclidean Geometry is based. One of these has been
discussed in greater detail in Appendix IV. In the articles
which follow there will be found another and a more ele-
mentary proof that the Bolyai-Lobatschewsky system of
Non-Euclidean Geometry cannot lead to any contradictory
results, and that it is therefore impossible to prove Euclid's
Postulate or any of its equivalents. This proof depends, for
solid geometry, upon the properties of the system of spheres all
orthogonal to a fixed sphere, while for plane geometry the
system of circles all orthogonal to a fixed circle is taken.
In the course of the discussion many of the results of Hyper-
bolic Geometry are deduced from the properties of this
system of circles.

I This Appendix, added to the English translation, is based
upon Wellstein's work, referred to on p. I So, and the following
paper by Carslaw; ^The Bolyai-Lobatschewsky Non-EiicUdeaii Geo-
metry: an Elementary Interpretation of this Geometry and some Results
which follow from this Interpretation, Proc. Edin. Math. See. Vol.
XXVIII, p. 95 (1 910).

Cf. also : J. WellstEIN, Zusammeiihang zwischen zwei euklid-
tscheft Bilderit der nichtei/klidischcn Geometric. Archiv der Math. u.
Physik (3). XVII, p. 19s (1910).

Ideal Lines.

239

The System of Circles passing through a fixed Point.

§ 2. We shall examine first of all the representation of
ordinary Euclidean Geometry by the geometry of the system
of spheres all passing through a fixed point. In plane geo-
metry this reduces to the system of circles through a fixed
point, and we shall begin with that case.

Since the system of circles through a point O is the
inverse of the system of straight lines lying in the plane, to
every circle there corresponds a straight line, and the circles
intersect at the same angle as the corresponding hnes. The
properties of the set of circles could be established from the
knowledge of the geometry of the straight lines, and every
proposition concerning points and straight lines in the one
geometry could at once be interpreted as a proposition con-
cerning points and circles in the other.

There is another way in which the geometry of these
circles can be established independently. We shall first de-
scribe this method, and weshall then see that from this inter-
pretation of the Euclidean Geometry we can easily pass to a
corresponding representation of the Non-Euclidean Geometry.

§ 3. Ideal Lines.

It will be convenient to speak ot the plane of the
straight lines and the plane of the circles, as two separate
planes. We have seen that to every straight line in the plane
of the straight lines, there corresponds a circle in the plane
of the circles. We shall call these circles Ideal Lines. The
Ideal Points will be the same as ordinary points, except that
the point O will be excluded from the domain of the Ideal
Points.

On this understanding we can say that Any two different
Ideal Points, A, B, determine the Ideal Line A£; just as, in
Euclidean Geometry, any two different points A, B deter-
mine the straight line AB.

240 Appendix V. Impossibility of proving Euclid's Postulate.

As the angle between the circles in the one plane is
equal to the angle between the corresponding straight lines
in the other, we define the angle between tivo Ideal Lines as
the angle between the corresponding straight lines. Thus we
can speak of Ideal Lines being perpendicular to each other,
or cutting at any angle.

§ 4. Ideal Parallel Lines.

Let BC (cf. Fig. 83) be any straight line and A a point
not lying upon it.

Let AM be the perpendicular to BC, and AM^ , AM^,
AAf,, . . . different positions of the line AM, as it revolves
from the perpendicular position through two right angles.

The lines begin by cutting BC on the one side of Af,
and there is one line separating the lines which intersect
BC on the one side, from those which intersect it on the
other. This line is the parallel through A to BC.

In the corresponding figure for the Ideal Lines (cf.
Fig. 84), we have the Ideal Line through A perpendicular to
the Ideal Line BC; and the circle which passes through A,
and touches the circle OBC at O, separates the circles
through A, which cut BC on the one side of 31, from those
which cut it on the other.

Ideal Parallels.

241

We are thus led to define Parallel Ideal Lines as follows:

The Ideal Line through a?iy point parallel to a given

Ideal Line is the circle of the system which touches at O the

circle coinciding with the given line and also passes through the

given point.

Thus any two circles of the system which touch each
other at O will be Ideal Parallel Lines. Two Ideal Lines,
which are each parallel to a third Ideal Line, are parallel to
each other, etc.

§ 5. Ideal Leiigths.

Since Euclid's Parallel Postulate is equivalent to the
assumption that one, and only one, straight line can be
drawn through a point parallel to another straight line, and
since this postulate is obviously satisfied by the Ideal Line,

242 Appendix V. Impossibility of proving Euclid's Postulate.

in the geometry of these Hnes, Euclid's Theory of Parallels
will be true.

But such a geometry will require a measurement of
length. We must now define what is meant by the Ideal
Lmgth of an Ideal Segment. In other words we must define
the Ideal Distance between two points. It is clear that if the
two geometries are to be identical two Ideal Segments must
be regarded as of equal length, when the corresponding
rectilinear segments are equal. We thus define the Ideal
Length of an Ideal Segment as the length of the rectilinear
segmmt to which it corresponds.

It will be seen that the Ideal Distance between two
points y^, B is such that, if C is any other point on the
segment,

'distance' AB = 'distance' AC ^ 'distance' CB.

The other requisite for 'distance' is that it is unaltered
by displacement, and when we come to define Ideal Dis-
placement we shall have to make sure that this condition is
also satisfied.

It is clear that on this understanding the Ideal Length
of an Ideal Line is infinite. If we take 'equal* steps along
the Ideal Line BC from the foot of the perpendicular (cf.

Fig. 84) the actual lengths of the arcs MMi , M^M^, etc ,

the Ideal Lengths of which are equal, become gradually
smaller and smaller, as we proceed along the line towards O.
It will take an infinite number of such steps to reach O, just
as it will take an infinite number of steps along BC from AI
(cf Fig. 83) to reach the point at which BC is met by the
parallel through A. We have already seen that the domain
of Ideal Points contains aU the points of the plane except
O. This was required so that the Ideal Line might always
be determined by two different points. It is also needed for
the idea of 'between-ness'. On the straight line AB we. can
say that C lies between line A and B if, as we proceed along

Ideal Lengths.

243

AB from A to B^ we pass through C. On the Ideal Line AB
(cf. Fig. 85) the points G and C2 would both lie between
A and B, unless the point O were excluded. In other words
this convention must be made so that the Axioms of Order ^
may appear in the geometry of the Ideal Points and Lines.

Fig 85.

On this understanding, and still speaking of plane geo-
metry, we can say that two Ideal Lines are parallel when they
do not meet, however far they are produced.

To obtain an expression for the Ideal Length of an
Ideal Segment we may take the radius of inversion — k — to
be unity.

Consider the segment AB and the rectihnear segment
aP to which it corresponds. Then we have (Fig. 86)
^P _ Op _ op. OB _ _^2
AB ~ OA ^ OA. OB ~ ÒA~.0B'

I See Note on p. 236.

16*

244 Appendix V. Impossibility of proving Euclid's Postulate.

Hence we define the Ideal Length of the segf>tent AB as
AB

OA. OB
We shall now show that the Ideal length of an Ideal
Segment is unaltered by inversivi with regard to any circle of
the system.

Fig. So.

Let OD be any circle of the system and let C be its
centre (Fig. 87).

Then inversion changes an Ideal Line into an Ideal
Line.

Let the Ideal Segment AB invert into the Ideal Segment
A'B'. These two Ideal Lines intersect at the point D, where
the circle of inversion^meets AB.
Then

the Ideal Length of AD AD 1 A'D

the Ideal Length of ^'^ ~ OA. ODj OA' . OD
__ AD OA
~' 'ad ' OA'

But from the triangles CAD, CAD and OAC, OA'C,
we find

Ideal Displacements.

245

CA
CD

CA
CO

AO
A^O'

Thus the Ideal Length oi AD = the Ideal Length oiA'D.
Similarly we find BD and B'D have the same Ideal Length,
and therefore AB and A'B' have the same Ideal Length.

Fig. 87.

§ 6. Ideal Displacements.

The length of a segment must be unaltered by dis-
placement. This leads us to consider the definition of Ideal
Displacement. Any displacement may be produced by re-
peated applications of reflection; that is, by taking the image
of the figure in a line (or in a plane, in the case of solid
geometry). For example, to translate the segment AB (cf.
Fig. 88) into another position on the same straight line, we

246 Appendix V. Impossibility of proving Euclid's Postulate.

may reflect the figure, first about a line perpendicular to and
bisecting BB' , and then another reflection about the middle
point of AB would bring the ends into their former positions
relative to each other. Also to move the segment AB into

— I
A'

Fig.

the position AB' (cf. Fig. 89) we can first take the image ot
AB in the line bisecting the angle between AB and AB\
and then translate the segment along AB' to its final
position.

We proceed to show
that inversion about any
circle of the system is
equivalent to reflection of
the Ideal Points and Lines
in the Ideal Line which
coincides with the circle
of Ì7iversion.

Let C (Fig. 90) be
the centre of any circle
of the system, and let A
be the inverse of any
point A with regard to
this circle. Then the
circleO AA' is orthogonal
to the circle of inversion.
In other words, such inversion changes any point A into a
point A on the Ideal Line perpendicular to the circle of in-
version. Also the Ideal Line AA is 'bisected' by that circle
at M, since the Ideal Segment AM inverts into the segment
AM, and Ideal Lengths are unaltered by such inversion.

Again let AB be any Ideal Segment, and by inversion

Fig. 89.

Ideal Reflection.

247

with regard to any circle of the system let it take up the
position AS (Fig. 8 7). We have seen that the Ideal Length
of the segment is unaltered: and it is clear that the two
segments, when produced, meet on the circle of inversion,
and make equal angles with it. Also the Ideal Lines A A

Fig. 90.

and BB' are perpendicular to, and 'bisected' by, the Ideal
Line with which the circle of inversion coincides.

Such an inversion is, therefore, the same as reflection,
and translation will occur as a special case of the above,
when the circle of inversion is orthogonal to the given
Ideal Line.

We thus define Ideal Reflection m an Ideal Line as in-
version with this line as the circle of inversion.

It is unnecessary to say more about Ideal Displace-
fne?its than that they will be the result of Ideal Reflection.

With these definitions it is now possible to 'translate'
every proposition in the ordinary plane geometry into a

248 Appendix V. Impossibility of proving Euclid's Postulate.

corresponding proposition in this Ideal Geometry. We have
only to use the words Ideal Points, Lines, Parallels, etc.,
instead of the ordinary points, lines, parallels, etc. The
argument employed in proving a theorem, or the con-
struction used in solving a problem, will be applicable,
word for word, in the one geometry as well as in the other,
for the elements involved satisfy the same laws. This is the
'dictionary' method so frequently adopted in the previous
pages of this book.

§ 7. Extension to Solid ^ Geometry. The System of
Spheres passing through a fixed point.

These methods may be extended to solid geometry. In
this case the inversion of the system of points, lines, and
planes gives rise to the system of points, circles intersecting
in the centre of inversion, and spheres also intersecting in
that point. The geometry of this system of spheres could be
derived from that of the system of points^ lines and planes,
by interpreting each proposition in terms of the inverse
figures. For our purpose it is better to regard it as derived
from the former by the invention of the terms: Ideal
Point, Ideal Line, Ideal Plane, Ideal Length and Ideal Dis-
placement.

The Ideal Point is the same as the ordinary point, but
the point O is excluded from the domain of Ideal Points.

The Ideal line through two Ideal Points is the circle of
the system which passes through these two points.

The Ideal Flafie through three Ideal Points, not on an
Ideal Line, is the sphere of the system which passes through
these three points.

Thus the plane geometry, discussed in the preceding
articles, is a special case of this plane geometry.

Ideal Parallel Lines are defined as before. The line
through A parallel to ^C is the circle of the system, lying

Extension to Solid Geometry. 249

on the sphere through O, Ay B, and C, which touches the
circle given by the Ideal Line .BC at O and passes through A.

It is clear that an Ideal Line is determined by two
points, as a straight line is determined by two points. An
Ideal Plane is determined by three points, not on an Ideal
Line, as an ordinary plane is determined by three points,
not on a straight line. If two points of an Ideal Line lie on
an Ideal Plane, all the points of the line do so : just as if two
points of a straight line lie on a plane, all its points do so.
The intersection of two Ideal Planes is an Ideal Line; just as
the intersection of two ordinary planes is a straight line.

The measurement of angles in the two spaces is the same.

For the measurement of length we adopt the same de-
finition of Ideal Length as in the case of two dimensions.
The Ideal Length of an Ideal Segment is the length of the
rectilinear segment to which it corresponds. To these defi-
nitions it only remains to add that of Ideal Displacement.
As in the two dimensional case, this is reached by means of
Ideal Reflection : and it can easily be shown that if the system
of Ideal Poitits, Lines and Planes is inverted with regard to
one of its spheres, the result is equivalent to a reflection of the
system in this Ideal Platie.

This Ideal Geometry is identical with the ordinary
Euclidean Geometry. Its elements satisfy the same laws:
every proposition vaUd in the one is also valid in the other:
and from the results of Euclidean Geometry those of the
Ideal Geometry can be inferred.

In the articles that follow we shall establish an Ideal
Geometry whose elements satisfy the axioms upon which the
Non-Euclidean Geometry of Bolyai-Lobatschewsky is based.
The points, lines and planes of this geometry will be figures
of the Euclidean Geometry, and from the known properties
of these figures, we could state what the corresponding the-
orems of this Non-Euclidean Geometry would be. Also from

2 co Appendix V. Impossibility of proving Euclid's Postulate.

some of its constructions, the Non-Euclidean constructions
could be obtained. This process would be the converse of
that referred to in dealing with the Ideal Geometry of the
preceding articles; since, in that case, we obtained the the-
orems of the Ideal Geometry from the corresponding Eu-
clidean theorems.

The Geometry of the System of Circles Orthogonal
to a Fixed Circle.

§ 8. Ideal Poiiiis, Ideal lines and Ideal Parallels.

In the Ideal Geometry discussed in the previous articles,
the Ideal Point was the same as the ordinary point, and the
Ideal Lines and Planes had so far the characteristics of
straight lines and planes that they were lines and surfaces
respectively. Geometries can be constructed in which the
Ideal Points, Lines and Planes are quite rem.oved from
ordinary points, lines, and planes: so that the Ideal Points
no longer have the characteristic of having no parts: and
the Ideal Lines no longer boast only length, etc. What is
required in each geometry is that the entities concerned
satisfy the axioms which form the foundations of geometry.
If they satisfy the axioms of Euclidean Geometry, the argu-
ments, which lead to the theorems of that geometry, will
give corresponding theorems in the Ideal Geometry: and if
they satisfy the axioms of any of the Non-Euclidean Geom-
etries, the arguments^ which lead to theorems in that Non-
Euclidean Geometry, will lead equally to theorems in the
corresponding Ideal Geometry.

We proceed to discuss the geometry of the system of
circles orthogonal to a fixed circle.

Let the fundamental circle be of radius k and centre O.

Let A, A" be any two inverse points, A being inside
the circle. Every such pair of points {A, A'), is an Ideal
Point {A) of the Ideal Geometry with which we shall 71010 deal.

Circles orthogonal to a fixed Circle.

251

If two such pairs of points are given — that is, two Ideal
Points (A, B), (Fig. 92) — these determine a circle which is
orthogonal to the fundamental circle. Every such circle is
a?i Ideal Line of this Ideal Geometry.

Fig. 91.

Hence any two different Ideal Points determine an Ideal
Line. In the case of the system of circles passing through a
fixed point O, this point O was excluded from the domain
of the Ideal Points. In this system of circles all orthogonal
to the fundamental circle, the coincident pairs of points lying
on the circumference of that circle are excluded from the
domain of the Ideal Points.

We define the angle between two Ideal Lines as the angle
between the circles which coincide with these lines.

We have now to consider in what way it will be proper
to define Parallel Ideal Lines.

Let AàI be the Ideal Line through A, perpendicular to
the Ideal Line BC; in other words, the circle of the system
passing through A', A'\ and orthogonal to the circle through
£', B", C and C" (cf Fig. 92).

2^2 Appendix V. Impossibilty of proving Euclid's Postulate.

Imagine AM to rotate about A so that those Ideal
Lines through A cut the Ideal Line BC at a gradually
decreasing angle. The circles through A which touch the given

Fig. 92.

circle £C at the points [/, V, where it meets the fundamental
circle, are Ideal Lines of the system. They separate the
lines of the pencil of Ideal Lines through A, which cut the
Ideal Line -BC, from those which do not cut that line. All
the lines in the angle q), shaded in the figure, do not cut
the line £C; all those in the angle ^), not shaded, do cut
this line. This property is exactly what is assumed in the
Parallel Postulate upon which the Non-Euclidean Geometry
of BoLYAi-LoBATSCHEWSKY is based. We therefore are led to
define Parallel Ideal Lines in this Plane Ideal Geometry as
follows:

TAe Ideal Lines through an Ideal Point parallel to a
given Ideal Line are the two circles of the syston passing

Some Theorems in this Geometry. 253

through the given pointy which touch the circle with 7vhich the
given line coincides at the poi?tts where it meets the fundam-
ental circle.

Thus we have in this Ideal Geometry two parallels
through a point to a given line: a right-handed parallel, and
a left-handed parallel: and these separate the lines of the
pencil which intersect the given line from those which do
not intersect it.

Some Theorems of this Non-Euclidean Geometry.

§ 9. At this stage we can say that any of the theorems
of the BoLYAi-LoBATSCHEwsKY Non-EucHdean Geometry, in-
volving angle properties only, will hold in this Ideal Geo-
metry and vice versa. Those involving lengths we cannot yet
discuss, as we have not yet defined Ideal Lengths. For
example, it is obvious that there are triangles in which all
the angles are zero (cf. Fig. 93). The sides of such triangles
are parallel in pairs. Thus the sum of the angles of an Ideal
Triangle is certainly not always equal to two right angles.
We can prove that this sum is always less than two right
angles by a simple application of inversion, as follows:

Let Ci, C2, C3 be three circles of the system, forming
an Ideal Triangle. Invert these circles from the point of
intersection / of C^ and C2 , which hes inside the fundament-
al circle. Then d and C2 become two straight lines d'
and C2' through /. Also the fundamental circle C inverts
into a circle C cutting Ci and C2 at right-angles, so that
its centre is /. Again, the circle C. inverts into a circle C3',
cutting C at right-angles. Hence its centre lies outside C.
We thus obtain a 'triangle', in which the sum of the angles
is less than two right-angles, and since these angles are equal
to the angles of the Ideal Triangle, this result holds also for
the Ideal Triangle.

2^4 Appendix V. Impossibility of proviug Euclid's Postulate.

Finally, it can be shown that there is always one, and
only one, circle of the system cutting two non-intersecting
circles of the system at right-angles. In other words, two

Fig. 93-

non-intersecting Ideal Lines have a common perpendicular.
All these results must be true in the Hyperbolic Geometry.

§ IO. Ideal Lengths and Ideal Displacements.

Before we can proceed to the discussion of the metrical
properties of this geometry, we must define the Ideal Length
of an Ideal Segment. It is clear that this must be such that
it will be unaltered, if we take the points A\ B", as defining
the segment AB, instead of the points A\ B'. It must make
the complete line infinite in length. It must satisfy the distri-
butive law 'distance' AB = 'distance' AC -\- 'distance' CB,

Ideal Lengths.

255

if C is any other point on the segment AB^ and it must
also remain \in.di\i&xQàhy Ideal Displacement.

We defaie the Ideal length of a?iy segment AB as

'V_ I B'^\
77/ WI/J

where U, V are the points where the Ideal Line AB meets the

fundamental circle (cf. Fig. 91).

lot

\a'i

Fig. 94.

This expression obviously involves the Anharmonic
Ratio of the points UABV. It will be seen that this de-
finition satisfies the first three of the conditions named above.
It remains for us to examine what must represent dis-
placement in this Ideal Geometry.

Let us consider what is the effect of inversion with
regard to a circle of the system upon the system of Ideal
Points and Lines.

Let A A" be any Ideal Point A (cf. Fig. 94). Let the

2CS Appendix V. Impossibility of proving Euclid's Postulate.

circle of inversion meet the fundamental circle in C, and let
D be its centre. Let A', A" invert into B', B" . Since the
circle A A' C touches the circle of inversion at C, its inverse
also touches that circle at C. But a circle passes through
A ^ A", B' and B'\ and the radical axes of the three circles

AA'C, B'B"C, AA'B'B"
are concurrent.

Hence B' B" passes through 6>, and OB' . OB" = 0C\

Therefore inversion with regard to any circle of the
system changes an Ideal Point into an Ideal Point.

But it is clear that the circle AA'B'B" is orthogonal to
the fundamental circle, and also to the circle of inversion.

Thus the Ideal Line joining the Ideal Point A and the
Ideal Point B, into which it is changed by this inversion, is
perpendicular to the Ideal Line coincidiiig with the circle of
itwersion.

We shall now prove that it is 'bisected' by that Ideal
Line.

Let the circle through AB meet the circle of inversion
at M, and the fundamental circle in U and V. It is clear
that U and V are inverse points with regard to the circle of
inversion [cf. Fig. 95].

Then we have:

B'V _ CV
'AU~'CA">

A'V CV

^^^ B'U ~ CB' '
Thus

A'V B'V CV2 CV2

A'U B'U CA'.CB' CM 2

Therefore

A'V I M'V M'V I B'V

/M'V\ 2

A'U M'U M'U\ B'U

Ideal Reflection.

257

Hence the Ideal Length of AM is equal to the Ideal
Length of MB.

Thus we have the following result:

Inversion with regard to a circle of the system changes
any Ideal Point A ifito an Ideal Point B, such that the Ideal
Line AB is perpendicular to, and ''bisected' by, the Ideal LÌ7ie
coinciding with the circle of inversion.

Fig. 95

In other words, inversion with regard to such a circle
causes any Ideal Point A to take the position of its image in
the corresponding Ideal Line.

We proceed to examine what effect such inversion has
upon an Ideal Line.

Since a circle^ orthogonal to the fundamental circle,

2 e 8 Appendix V. Impossibility of proving Euclid's Postulate.

inverts into a circle also orthogonal to the fundamental circle,
any Ideal Line AB inverts into another Ideal Line ab, pass-
ing through the point M, where AB meets the circle of in-
version (cf. Fig. 96). Also the points U, V invert into the

Fig. 96.

points ti and v on the fundamental circle; and the lines AB
and ab are equally inclined to the circle of inversion.

It is easy to show that the Ideal Lengths of AM and
BM are equal to those of aM a.nd ^ J/ respectively, and it
follows that the Ideal Length of the segment AB is unaltered
by this inversion. Also we have seen that Aa and Bb are
perpendicular to, and 'bisected' by, the Ideal Line coinciding
vnth this circle.

// follows from these results that inversion with regard
to any circle of the system has the same effect upon an Ideal
Segment as reflection in the corresponding Ideal Line.

We are thus agaifi able to defi7ie Ideal Reflection in any
Ideal Line as the inversion of the system of Ideal Points and

Ideal Displacement.

259

Lines 7mt/i regard to the circle which ^ciacides with this
Ideal Line.

It is unnecessary to define Ideal Displacements., as any
displacement can be obtained by a series of reflections and
any Ideal Displacement by a series of Ideal Reflections.

We notice that the definition of the Ideal Length of
any Segment fixes the Ideal Unit of Length. We may take
this on one of the diameters of the fundamental circle, since
these lines are also Ideal Lines of the system. Let it be the
segment OP (Fig. 97).

Fig. 97-

Then

must have

/0V\ PV\

l°g \ou\pu)

i. e.

1 PU

log py = I.

Therefore

PF ~ ^'

and the point P divides the diameter in the ratio e: 1.

The Unit Segment is thus fixed for any position in the

17="

200 Appendix V. Impossibility of proving Euclid's Postulate.

domain of the Ideal Points, since the segment OP can be
'moved' so that one of its ends coincides with any given
Ideal Point.

A different expression for the Ideal Length

would simply mean an alteration in the unit, and taking
logarithms to any other base than <? would have the same
effect.

§ ir. Some further Theorems in this Non-Euclidean
Geometry.

We are now in a position to establish some further
theorems of the Hyperbolic Geometry using the metrical
properties of this Ideal Geometry.

In the first place we can state that Similar Triangles
are impossible in this geometry.

We also see that Parallel Ideal Lines are asymptotic;
that is, these lines continually approach each other and the
distance between them tends to zero.

Further, it is obvious that as the point A moves away
along the perpendicular MA to the line BC (cf. Fig. 92), the
angle of parallelism dimmishes from — to zero m the limit.

Again, we can prove from the Ideal Geometry that the
Angle of Parallelism TT (/), corresponding to a segment /, is
given by

tan n (/) _ -P
2

Consider an Ideal Line and the Ideal Parallel to it
through a point A.

Let AM (Fig. 98) be the perpendicular to the given line
MU^ and A U the parallel.

Let the figure be inverted from the point J/", the radius of
inversion being the tangent from M" to the fundamental circle.

Further Theorems in this Geometry.

261

Then we obtain a new figure (cf. Fig. 99) in which the
corresponding Ideal Lengths are the same, since the circle
of inversion is a circle of the system. The lines AM and
MU become straight lines through the centre of the fund-

amental circle, which is the inverse of the point M'.
Also the circle A U becomes the circle a'u^ touching the
radius mu at ?/, and cutting via at an angle TT (/). These
radii, mu, nib, are also Ideal Lines of the system.

The Ideal Length of the Segment AM is taken as p.

Then

(A'B \M'B\

^=^^^^ [ax lire)

. /a'b I vi'b \

But ac = k — k tan ( )

(i-"f).

and db = /& + /& tan

202 Appendix V. Impossibility of proving Euclid's Postulate.

where k is the radius of the fundamental circle.

TT {p^
Thus p = log cot — ^ ;

and e

-p

tan

n(/)

Fig. 99.

Finally, in this geometry there will be three kinds of
circles. There will be the circle^ with its centre at a finite
distance; the Limiting Curve or Horocycle, with its centre at
infinity, (at a point where two parallels meet) ; and the Equi-
distant Curve, with its centre at the imaginary point of inter-
section of two lines with a common perpendicular.

The first of these curves would be traced out in the
Ideal Geometry by one end of an Ideal Segment, when it is
reflected in the lines passing through the other end; that is,
by the rotation of this Ideal Segment about that end. The
second occurs when the Ideal Segment is reflected in the
successive lines of the pencil of Ideal Lines all parallel to it
in the same direction; and the third, when the reflection

Application to Euclid's Parallel Postulate. 263

takes place in the system of Ideal Lines which all have a
perpendicular with this segment. That these correspond to
the common Circle^ the Horocycle and the Equidistant Curve
of the Hyperbolic Geometry is easily proved.

§ 12. The Impossibility of Proving Euclid! s Parallel
Postulate.

We could obtain other results of the Hyperbolic Geo-
metry, and find some of its constructions, by further examin-
ation of the properties of this set of circles; but this is not
our object. Our argument was directed to proving, by reas-
oning involving only elementary geometry, that it is impossible
for any inconsistency or contradiction to arise in this Non-
Euclidean Geometry. If such contradiction entered into this
Plane Geometry, it would also occur in the interpretation of
the result in the Ideal Geometry. Thus the contradiction
would also be found in the Euclidean Geometry. We can,
therefore, state that it is impossible that any logical incon-
sistency could be traced in the Plane Hyperbolic Geometry. It
could still be argued that such contradiction might be found
in the Solid Hyperbolic Geometry. An answer to this ob-
jection is at once forthcoming. The geometry of the system
of circles, all orthogonal to a fixed circle, can be at once
extended into a three dimensional system. The Ideal Points
are taken as th£ pairs of points inverse to a fixed sphere,
excluding the points on the surface of the sphere from their
domain. The Ideal Lines are the circles tlurough two Ideal
Points. The Ideal Planes are the spheres through three Ideal
Points, not lying on an Ideal Line. The ordinary plane enters
as a particular case of these Ideal Planes, and so the Plane
Geometry just discussed is a special case of a plane geo-
metry on this system. With suitable definitions of Ideal
Lengths, Ideal Parallels and Ideal Displacements, we have
a Solid Geometry exactly analogous to the Hyperbolic Solid

204 Appendix V. Impossibility of proving Euclid's Postulate.

Geometry. It follows that no logical inconsistency can exist
in the Hyperbolic Solid Geometry, since if there were such
a contradiction, it would also be found in the interpretation
of the result in this Ideal Geometry; and therefore it would
enter into the Euclidean Geometry.

By this result our argument is complete. However far
the HyperboUc Geometry were developed, no contradictory
results could be obtained. This system is thus logically
possible; and the axioms upon which it is founded are not
contradictory. Hence it is impossible to prove Euclid's
Parallel Postulate, since its proof would involve the denial
of the Parallel Postulate of Bolvai-Lobatschewsky.

Index of Authors.

[The Jtnmbers refer to pages.]

Aganis, (6th Century^ 8— ii.

Al-Nirizi, (9th Century). 7, 9.

Andrade, J. 181, 194.

Archimedes, (287—212). 9, Tl
23, 25, 30, 34, 37, 46, 56, 59
119 — 121, 144, 181, 183.

Aristotle, (384—322). 4, 8, 18, 19

Arnauld, (1612— 1694). 17.

Baltzer,R. (1818 — 1887). 121—3

Barozzi, F. (l6'h Century). 12.

Battels, J. M. C. (1769—1836)
84, 91—2.

Battaglini, G. (1826—1894). 86
100, 122, 126 — 7.

Beltrami, E. (1835—1900). 44
122, 126—7, ^33. 13S— 9. 145
147, T[73— 5. 234—6.

Bernoulli, D. (1700—1782). 192.

Bernoulli, J. (1744—1807). 44.

Bessel, F. W. (1784-1846). 65
-67.

Besthorn, R. O. 7.

Bianchi, L. 129, 135, 209.

Biot, J. B. (1774—1862). 52.

Boccardini, G. 44.

Bolyai, J. (1802 — 1860). 51, 61,
65, 74, 96—107, 109-116-
121—6, 128, 137, 141, 145, 147,
152, 154, 157— 8, 161, 164,
170, 173—5. 177—8, 193—4,
200, 222, 225, 233, 238, 249,
252—3, 264.

Bolyai, W. (1775 — 1856). 55» 60

— 1, 65 — 6, 74, 96. 98—101,

120, 125 — 6.
Boncampagni, B. (1821 —1894).

125.
Bonola, R. (1875—1611) 15, 26,

30, 115, 176—7, 220.
Borelli, G. A. (1608—1679). 11,

13, 17-
Boy, W. 149.

Campanus, G. (13'!^ Century). 17.
Candalla, F. (1502—1594). 17.
Carnet, L. N.M. (1753—1823). 53-
Carslaw, H. S. 40, 238.
Cassani, P. (1832 — 1905). 127.
Castillon, G, (1708 — 1791). 12.
Cataldi, P. A. (1548?— 1626). 13.
Cauchy, A. L. (1789—1857). 199.
Cayley, A. (1821 — 1895). 127,

148, 156, 163—4, 174, 179.
Chasles, M. (1793—1880). 155.
Clavio, C. (1537—1612). 13, 17.
Clebsch, A. (1833—1872). 161.
Clifford.W.K. (1845—1879). 139,

142, 200—215.
Codazzi, D. (1824—1873). 137.
Commandino, F. (1509—1575).

12, 17.
Coolidge, J. L. 129.
Couturat, L. 54.
Cremona, L. (1830—1903). 123,

127.

266

Index of Authors.

Curtze, M. (1837—1903). 7.
D'Alembert, J. le R. (1717—

1783)- 52' 54, 192, 197—8.
Dedekind, J.W.R. (1831-1899).

139-
Dehn, M. 30, 120, 144.
Delambre, J. B. J. (1749 — 1822).

198.
De Morgan, A. (1806—1870). 52.
Dickstein, S. 139.
Duhem, P. 182.
Eckwehr, J. W. v. (1789-1857).

99-

Engel, F. 16, 44—5, 50, 60, 64,
66, 83—6, 88, 92—3, 96, 101,
216 — 7, 220.

Enriques, F. 156, 166, 183, 225.

Eòtvòs, 125.

Euclid (circa 330—275). 1 — 8, 10,
12 — 14, 16—20, 22, 38, 51 — 2,
54—5, 61—2, 68, 75, 82, 85,
92, 95, loi— 2, 104, 110, 112,
118 — 120, 127, 139, 141, 147,
152, 154—5, ^57, 164, 176—
i8i, 183, 191 — 5, 199—201,
227, 237—9, 241, 267.

Fano, G. 153.

Flauti, V. (1782— 1863) 12.

Fleischer, H. 156.

Flye St. Marie, 91.

Foncenex, D. de, (1743 — 1799).
53, 146, 190—2, 197—8.

Forti, A. (1818 — ). 122, 124—5.

Fourier,J.B. (1768— 1830). 54 — 5.

Frattini, G. 127.

Frankland, W. B. 2, 63.

Friedlein, G. 2.

Frischauf, J. 100, 126.

Gauss, C. F. (1777—1855). 16,
60—68, 70—78, 83—4, 86, 88,
90 — 2, 99 — 101, 110, 113, 122

—3, 127, 131, 135, 152, 177,

200.
Geminus, (ist. Century, B. C). 3,

7, 20.
Genocchi, A. (1817 — 18S9). 145,

191, 198—9.
Gerling, Ch. L. (1788—1864).

65, 66, 76 — 7, 121 — 2.
Gherardo daCremona, (12th Cen-
tury). 7.
Giordano Vitale, (1633 — 1711). I4

— 15, 17, 26.
Goursat, E. 145.
I Gregory, D. (1661 — 1710) 17, 20.
I Grossman, M. 169, 225.

Giinther, S. 127.
i Halsted, G. B. 44, 86, 93, 100, 139.
Hauff, J. K. F. (1766—1846). 75.
Heath, T. L. 1, 2, 63.
Heiberg, J. L. 1, 2, 7, 181.
Heilbronner, J. C. (1706 — 1745).

44.
Helmholtz, H. v. (1821—1894).

126, 145, 152—3, 176—7, 179.
Hilbert, D. 23, 145 — 6, 222 — 4,

236.
Hindenburg, K. F. (1741 — 1808).

45-
Hoffman, J. (1777—1866). 12.
Holmgren, E. A. 145 — 6.
Hoiiel, J. (1823—1866). 52, 86,

100, 121, 123—7, 139,^47, 152-
Kant, I. (1724—1804). 64, 92,121.
Kastner, A. G. (1719—1800). 50,

60, 64, 66.
Killing, W. 91, 215.
Klein, K. F. 129, 138, 148, 153,

158, 164, 17Ó, 180, 200, 211,

213—5, 236—7.
Kliigel, G. S. (1739— 1812). 12,

44, 51, 64, 77, 92.

Index of Authors.

267

Kiirschàk, J. 113.

Lagrange, J. L. (1736— 1813). 53

—4, 182—3, 198.
Laguerre, E. N. (1S34— 1866).

155-
Lambert, J. H. (1728—1777). 44
—51, 58, 65—6, 74, 77-8,
81—2, 92, 97, 107, 129, 139,

144-
Laplace, P. S. (1749 — 1827), 53

-54, 198-
Legendre, A. M. (1752 — 1833Ì,

29. 44, 55—59. 74> §4, 88, 92,
122, 128, 139, 144-
Leibnitz, G. W. F. (1646—1716).

54-

Lie, S. (1842 — 1899). 152—4, 179.

Liebmann, H. 86, 89, 113, 145,
180, 220.

Lindemann, F. 161.

Lobatschewsky, N. J. (1793 —
1856). 44, 5L 55> 63, 65, 74
80, 84 — 99, 101—3, 104—6;
111—3, 116, 121 — 8, 137, 141
145, H7, 152, 154, 157—8
161, 164, 170, 173—5, 177—8
193 — 4, 217, 220, 222, 225
238, 249, 252—3, 264.

Lorenz, J. F. (1738—1807). 58,
120.

Lukat, M. 129.

Liitkemeyer, G. 145 — 6.

Mc Cormack, T. J. 182.

Mach, E. 181.

Minding, F. (1806—1885). 132,

^37-
Mobius, A. F. (1790—1868). 148

— 9-

Monge, G. (1746—1818). 54—5.
Montucla, J. E. (1725 — 1799)
44. 92.

Nasìr-Eddìn, (1201 — 1274). 10,
12—3, 16, 37—8, 120.

Newton, L (1642—1727). 53.

01bers,H.W.M.(i758— 1840). 65.

Oliver, (ist Half of the 17* Cent-
ury). 17.
j Ovidio, (d') E. 127.

Paciolo, Luca (circa 1445 — 1514)-

17-

Pascal, E. 127, 139.

Pasch, M. 176.

Picard, C. É. 128.

Poincaré, H. 154, 180.

Poncelet, J. V. (1788—1867). 155,
236.

Posidonius, (ist Century B.C.I 2,
3, 8, 14.

Proclus, (410— 485^ 2—7, 12—3,
18—20, 119.

Ptolemy, (87—165). 3—4, 119.

Riccardi, P. (182S— 1898). 17.

Ricordi, E. 127.

Riemann, B. (1826—1866). 126,
129, 138—9, 141—3,145 — 154,
157—8, 160—1, 163—4, 175
— 7, 179—180, 194, 201—2.

Saccheri, G. (1667—1733). 4,
22—4, 26, 28—30, 34, 36—46,
51, 55-7, 65—6, 78, 85, 87
—8, 97, 120, 129, 139, 141,

144-
Sartorius v. Waltershausen, W.

(1809—1876). 122.
Saville, H. (1549—1622). 17.
Schmidt, F. (1826 — 1901). 121,

124—5.
Schumacher, H. K. (1780—1850).

65-7, 75, 122—3, 152.
Schur, F. H. 176.
Schweikart, F. K. (1780—1859).

67, 75 -78, 80, 83,86, 107, 122.

268

Index of Authors.

Segre, C. 44, 66, 77— S, 92.

Seyffer, K. F. (1762— 1S22). 60, 66.

Simon, H. 91.

Simplicius, (6'h Century). 8, 10.

Sintsoff, D. 139.

Stackel, P. 16, 44—5, 50, 60 — 1,

63, 66, 82 — 3, lOT, 112—3,

124-5.
Staudt.G. C.v. (1798—1867). 129,

154, 233, 236.
Szasz, C. (1798—1853). 97.
Tannery, P. (1S43— 1904). 7, 20.
Tacquet, A. (1612 — 1660). 17.
Tartaglia, N. (1500—1557). 17.
Taurinus, F. A. (1794—1874).

65—6, 74, 77-9' Si— 3, 87,

89-91, 94, 99, 112, 137, 173.

Thibaut, B. F. (1775— 1S32). 63.

Tilly (de), J. M. 55, 114, 194.

Townsend, E. J. 236.

Vailati, G. 18, 22.

Valerio Luca (? 1522 — 1618). 17.

Vasiliev, A. 93,

Wachter, F. L. (1792—1817). 62

—3, 66, 88.
Wallis, J. (1616 — 1703). 12, 15

—7, 29, 53' 120.
Weber, H. 180.
Wellstein, J. 180, 23S.
Zamberti, B. (ist Half of the

l6th Century). 17.
Zeno, (495— 435V 6.
Zolt, A. (de) 127.

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