Download ON THE THEORY OF

GEOMETRICAL RESEARCHES
ON THE THEORY OF
PARALLELS
NICHOLAS LOBACHEVSKI
TRANSLATED BY
DR.
GEORGE BRUCE HALSTED
TRANSLATOR'S PREFACE.
Lobachevski was the
first
man
ever to publish a non-Euclidean geom-
etry.
Of the immortal essay now
first
appearing in English Gauss said, "The
author has treated the matter with a master-hand and in the true geometer's spirit.
I think I
ought to
your attention to this book, whose
call
perusal can not fail to give you the most vivid pleasure."
Clifford says, "It is quite simple, merely Euclid without the vicious
way things come out of one another is quite lovely."
"What Vesalius was to Galen, what Copernicus was to Ptolemy,
was Lobachevski to Euclid."
assumption, but the
* * *
that
Says Sylvester, "In Quaternions the example has been given of Algebra released from the yoke of the commutative principle of multipli-
—
an emancipation somewhat akin to Lobachevski's of Geometry
from Euclid's noted empirical axiom."
cation
Cayley says, "It
Playfair's form of
is
it,
well
known that
Euclid's twelfth axiom, even in
has been considered as needing demonstration;
and that Lobachevski constructed a perfectly consistent theory, wherein this
axiom was assumed not
to hold good, or say a system of non-
Euclidean plane geometry. There
is
a
like
system of non-Euclidean solid
geometry."
GEORGE BRUCE HALSTED.
2407 San Marcos Street,
Austin, Texas.
May
1,
1891.
TRANSLATOR'S INTRODUCTION.
"Prove
hold fast that which
all things,
From nothing
onstrate everything.
is
good," does not
mean dem-
assumed, nothing can be proved.
"Geometry without axioms," was a book which went through several
and still has historical value. But now a volume with such a
title would, without opening it, be set down as simply the work of a
editions,
paradoxer.
The
set of
axioms far the most
influential in the intellectual history
was put together in Egypt; but really it owed nothing to
the Egyptian race, drew nothing from the boasted lore of Egypt's
of the world
priests.
The Papyrus of the Rhind, belonging
given to the world by the erudition of a
lohr,
and a German
knowledge of the
to the British
German
historian of mathematics, Cantor, gives us
state of
mathematics in ancient Egypt than
previously accessible to the
modern
world.
Its
firms with overwhelming force the position that
and
strict
Venus
same race whose bloom
more
all else
whole testimony con-
Geometry
as a science,
was created by the subtle
self-conscious deductive reasoning,
intellect of the
Museum, but
Egyptologist, Eisen-
in art
still
overawes us in the
of Milo, the Apollo Belvidere, the Laocoon.
In a geometry occur the most noted
set of axioms, the
geometry of
Euclid, a pure Greek, professor at the University of Alexandria.
Not only at
its
very birth did this typical product of the Greek genius
assume sway as ruler in the pure
sciences, not only does its first efflor-
escence carry us through the splendid days of Theon and Hypatia, but
unlike the
ages,
latter, fanatics
can not drown
it.
can not murder
with the new birth of culture.
finds
it
that dismal flood, the dark
An
its
each other.
it
is
Finally back again in
from
it rises
Then
and the new-born printing press confer honor on
its original
in queenly. Basel, then in stately Oxford.
Greek
native Egypt,
Anglo-Saxon, Adelard of Bath,
clothed in Arabic vestments in the land of the Alhambra.
clothed in Latin,
first
it;
Like the phoenix of
I-eipsic's
learned presses.
Greek,
The
it is
published
latest edition in
6
THEORY OP PARALLELS.
How the first translation
into our cut-and-thrust, survival-of-the-fittest
English was made from the Greek and Latin by Henricus Billingsly,
Lord Mayor
Magician,
of London,
may
and published with a preface by .John Dee the
own Princeton, where
own copy of the Arabic-
be studied in the Library of our
they have, by some strange chance, Billingsly's
Latin version of Campanus bound with the Editio Princeps in Greek
and enriched with his autograph emendations. Even to-day in the vast
system of examinations set by Cambridge, Oxford, and the British government, no proof will be accepted which infringes Euclid's order, a
sequence founded upon his set of axioms.
The American
ideal is success.
In twenty years the American maker
expects to be improved upon, superseded.
The Greek ideal was per.
The Greek Epic and Lyric poets, the Greek sculptors, remain
unmatched. The axioms of the Greek geometer remained unquestioned
fection.
for twenty centuries.
How
and where doubt came
to look toward them is of no ordinary
doubt was epoch-making in the history of mind.
Euclid's axioms was one differing from the others in pro-
interest, for this
Among
lixity,
whose place
fluctuates in the manuscripts,
in Euclid's first twenty-seven propositions.
and which
Moreover
it
is
is
not used
only then
brought in to prove the inverse of one of these already demonstrated.
All this suggested, at Europe's renaissance, not a doubt of the axiom,
but the possibility of getting along without it, of deducing it from the
other axioms and the twenty-seven propositions already proved.
Euclid
demonstrates things more axiomatic by far.
He proves what every dog
knows, that any two sides of a triangle are together greater than the
third.
Yet when he has
perfectly proved that lines
making with a
transversal equal alternate angles are parallel, in order to prove the inverse, that parallels cut
by a
transversal
make equal
alternate angles, he
brings in the un wieldly postulate or axiom:
" If a straight line meet two straight lines, so as to
terior angles
on the same side of
it
make
the two in-
taken together less than two right
angles, these straight lines, being continually produced, shall at length
meet on that side on which are the angles which are
less
than two right
angles."
Do you wonder
to
push
that succeeding geometers wished
this unwieldly thing
from the
set of
by demonstration
fundamental axioms.
7
translator's introduction.
Numerous and
desperate were the attempts to deduce
it
from reason-
ings about the nature of the straight line and plane angle.
In the
und Kunste; Von Ersch und Gru-
" Encyclopoedie der "Wissenschaften
ber;" Leipzig, 1838; under "Parallel," Sohncke says that in mathe-
matics there
and
nothing over which so
is
striven, as
over the theory of
much
tant," still given on
like
so far (up to his
all,
decision.
new
definition of parallels, as
lines are
everywhere equally
dis-
page 33 of Schuyler's Geometry, which that author,
of his unfortunate prototypes, then attempts to identify with
many
Euclid's definition
by pseudo-reasoning which
"For,
e. g.
he says
p.
where equally
distant;
and since they
postulate,
and
defeat by taking a
example the stupid one, "Parallel
for
and
parallels,
time), without reaching a definite result
Some acknowledged
has been spoken, written,
35:
if
tacitly
assumes Euclid's
not parallel, they are not everylie
in the
same plane; must
ap-
proach when produced one way or the other; and since straight lines
continue in the same direction, must continue to approach
farther,
and
if
sufficiently produced, must meet."
Euclid's assumption, diseased
This
is
if
produced
nothing but
and contaminated by the introduction of
the indefinite term "direction."
How much
who
better to have followed the third class of his predecessors
honestly assume a
which
intersect can not both
The German
list
new axiom
Of these the
not in essence.
article
differing
from Euclid's in form
best is that called Playfair's;
be
parallel to the
mentioned
is
same
"Two
if
lines
line."
followed by a carefully prepared
of ninety-two authors on the subject.
In English an account of
was given by Perronet Thompson, Cambridge, 1833, and
is brought up to date in the charming volume, "Euclid and his Modern
Rivals," by C. L. Dodgson, late Mathematical Lecturer of Christ Church,
like attempts
Oxford, the Lewis Carroll, author of Alice in Wonderland.
All this shows
forth of genius
how ready the world was for the extraordinary flaming-
from
different parts of the
overturn, explain, and
quence
all
philosophy,
remake not only
all
ken-lore.
world which was at once to
all this
As was
subject but as conse-
the case with the dis-
covery of the Conservation of .Energy, the independent irruptions
of genius,
whether in Russia, Hungary, Germany, or even in Canada
gave everywhere the same
At first
results.
these results were not fully understood even
by the brightest
THEORY OF PARALLELS.
8
Thirty years after the publication of the book he mentions,
intellects.
we
see the brilliant Clifford writing
April
2,
have
procured
:
-
from Trinity
ideas have
Lobachevski,
- -
des Parallels'
says
new
1870, "Several
"Etudes
come
to
College, Cambridge,
me
Geometriques
lately:
First I
sur
Theorie
la
a small tract of which Gauss, therein quoted,
L'auteur a traits la matiere en main de maitre et avec
le veritable
esprit geometrique.
Je crois devoir appeler votre attention sur ce livre,
dont la lecture ne peut manquer de vous causer le plus vif plaisir.'"
Then says
Clifford:
"It is quite simple, merely Euclid without the
vicious assumption, but the
way
the things come out of one another
is
quite lovely."
The
man
axiom doubted
more clearly than
first
sees
He had
vicious.
is
called a "vicious assumption," soon
Clifford that all are assumptions
no
and none
been reading the French translation by Houel, pub-
lished in 1866, of a little book of 61 pages published in 1840 in Berlin
under the
linien
title Geometrische Untersuchungen zur Theorie der Parallelby Nicolas Lobachevski (1793-1856), the first public expression
whose
of
February
Under
discoveries, however, dates back to
this
covery of a
A new
all
a discourse at Kasan on
12, 1826.
commonplace
new space
in
to hold our universe
kind of universal space
the space in which
we
and occupancy
;
the idea
is
suspected the dis-
and ourselves.
a hard one.
To name it,
move and
think the world and stars live and
have their being was ceded
description,
who would have
title
which
;
to Euclid as his
then the
new
by right of pre-emption,
space and
its
quick-following
fellows could be called Non-Euclidean.
Gauss in a
letter to
Schumacher, dated Nov.
28, 1846,
as far back as 1792 he had started on this path to a
Again he says:
mentions that
new
universe.
"La geometrie non-euclidienne ne renferme en
elle
rien de contradictoire, quoique, a premiere vue, beaucoup de ses resultats aieu-Tair de paradoxes.
regardees
comme
Ces contradictions apparents doivent etre
d'une illusion, due a l'habitude que nous avons
prise de bonne heure de considerer la geometrie euclidienne comme
l'effet
rigoureuse."
But here we
word the same imperfection of view as in
The perception has not yet come that though the non-
see in the last
Clifford's letter.
Euclidean geometry
is rigorous,
Euclid
is
not one whit less
so.
V
TRANSLATOR'S INTRODUCTION.
former friend of Gauss at Goettingen was the Hungarian Wolfgang
Bolyai. His principal work, published by subscription, has the follow-
A
ing title:
Tentamen Juventutem studiosam in elementa Matheseos purae,
ele-
mentaris ac sublimioris, methodo intuitiva, evidentiaque huic propria,
in-
Tomus Primus,
troducendi.
1832;
Secundus, 1833.
Maros-Va-
8vo.
sarhelyini.
In the
volume with special numbering, appeared the celebrated
first
Appendix of his son John Bolyai with the following
title
APPENDIX.
Scientiam spatii absolute veram exhibens
XI
Axiomatis
tem.
Euclidei (a priori hand
:
a veritate aut falsitate
unquam decidenda) independen-
Auctore Johanne Bolyai de eadem, Geometrarum in Exercitu
Caesareo Regio Austriaco Castrensium Capitaneo.
(26 pages of text).
This marvellous Appendix has been translated into French, Italian,
English and German.
In the
Wolfgang Bolyai's
of
title
last work, the only
one he com-
posed in German (88 pages of text, 1851), occurs the following:
"und da die Frage, ob zwey von der dritten geschnittene Oeraden,
wenn
die
summe
der inneren Winkel nicht=2R, sich schneiden oder
nicht? niemand auf der Erde ohne ein
beantworten wird; die
aufzustellen,
Axiom (wie Euclid das XI)
davon
unabhsngige
Geometrie
abzusondern; und eine auf die Ja-Antwort, andere auf das Nein so zu
bauen, dass die Formeln der letzen, auf ein
Wink auch
in der ersten
giiltig seyen."
The author mentions Lobachevski's Geometrische TJntersuchungen,
and compares it with the work of his son John Bolyai,
Berlin, 1840,
"au sujet duquel
il
'Quelques exemplaires de l'ouvrage publie
dit:
ont §t6 envoyes a cette epoque a Vienne, a Berlin, a Goettingue.
Goettingue
le
embrasse du
.
ici
De
geant mathematique, [Gauss] qui du sommet des hauteurs
meme
qu'il Stait ravi
In fact this
regard les astres et la profondeur des abimes, a ecrit
de voir execute
laisser apres lui
clid's
.
le travail qu'il avait
dans ses papiers.'
first
le
of the Non-Euclidean geometries accepts all of Eu-
axioms but the
tradictory, that the
commence pour
""
last,
sum
which
it flatly
denies and replaces by its con-
of the interior angles
made on
the
same
side of
10
THEORY OF PARALLELS.
a transversal by two straight lines
without the lines meeting.
A
may
be less than a straight angle
and elegant geometry
then follows, in which the sum of the angles of a triangle is always less
than a straight angle, and not every triangle has its vertices concyclic.
perfectly consistent
THEORY OF PARALLELS.
In geometry I find certain imperfections which I hold to be the
son
why
this science, apart
from
make no advance from that state in which it has come to us from
As
rea-
transition into analytics, can as yet
Euclid.
belonging to these imperfections, I consider the obscurity in the
fundamental concepts of the geometrical magnitudes and in the manner
and method of representing the measuring of these magnitudes, and
finally the momentous gap in the theory of parallels, to fill which all efforts of
For
mathematicians have been so far in vain.
endeavors have done nothing, since he
this theory Legendre's
was forced to leave the only rigid
way
to turn into a side path
refuge in auxiliary theorems which he
necessary axioms.
lished in the
My first essay on
Kasan Messenger
under the
title
11
parts in the " Ge.
1836, 1837, 1838,
Elements of Geometry, with a complete Theory
The extent
of Parallels."
In the hope of having
my work in separate
Kasan for the years
and published
lehrten Schriften der Unwersitcet
"New
the foundations of geometry I pub-
for the year 1829.
undertook hereupon a treatment of the whole
satisfied all requirements, I
of this science,
and take
illogically strove to exhibit as
of this
work perhaps hindered
my
country-
men from
following such a subject, which since Legendre had lost
interest.
Yet
not lose
its
am
I
of the opinion that the
Theory of
its
Parallels should
claim to the attention of geometers, and therefore I aim to
give here the substance of
my investigations,
contrary to the opinion of Legendre,
ample, the definition of a straight line
and without any
real influence
In order not to fatigue
my
remarking beforehand that
— for
— show themselves foreign here
all
other imperfections
on the theory of
ex-
parallels.
reader with the multitude of those theo-
rems whose proofs present no difficulties, I prefix here only those of
which a knowledge is necessary for what follows.
1.
A straight line
fits
upon
itself in all its positions.
that during the revolution of the surface containing
does not change
surface:
(i.
e.,
its
if
place
we
if it
it
By this I mean
the straight line
goes through two unmoving points in the
turn the surface containing
the line, the line does not move.)
[ii]
it
about two points of
12
THEORY OF PARALLELS.
2.
Two
3.
A
beyond
way
bounds, and in such
all
Two
4.
straight lines can not intersect in
two
points.
straight line sufficiently produced both
cuts a
ways must go out
bounded plain into two parts.
straight lines perpendicular to a third never intersect,
how
far soever they be produced.
A straight line always cuts another in going
5.
over to the other side:
(t.
has points on both sides of
from one side of
one straight line must cut another
e.,
it
if it
it.)
Vertical angles, where the sides of one are productions of the
6.
sides of the other, are equal.
among themselves,
Two
7.
same
This holds of plane rectilineal angles
as also of plane surface angles
straight lines can not intersect,
if
(i. e.,
:
dihedral angles.)
a third cuts them at the
angle.
In
8.
inversely.
a
rectilineal triangle equal sides lie opposite equal angles,
In a
9.
angle.
rectilineal triangle,
a greater side
of the other sides, and the two angles adjacent to
10.
opposite a greater
lies
In a right-angled triangle the hypothenuse
is
greater than either
are acute.
it
Rectilineal triangles are congruent if they
and
have a side and two
angles equal, or two sides and the included angle equal, or two sides and
the angle opposite the greater equal, or three sides equal.
11.
A
straight
line
which stands
straight lines not in one plane with
lines
drawn through the common
at right angles
it is
upon two other
perpendicular to
all
straight
intersection point in the plane of those
two.
12.
The intersection
13.
A straight
of a sphere with a plane
pendicular planes, and in one,
14.
is
a
circle.
line at right angles to the intersection of
is
two
per-
perpendicular to the other.
In a spherical triangle equal sides He opposite equal angles, and
inversely.
15.
two
Spherical triangles are congruent (or symmetrical)
sides
if they have
and the included angle equal, or a side and the adjacent angles
equal.
From
proofs.
here follow the other theorems with their explanations and
13
THEORY OF PARALLELS.
All straight lines which in a plane go out from a point can,
16.
with reference to a given straight line in the same plane, be divided
into
two
classes
—
The boundary
From
line
BC
given
to the
the point
A (Fig.
In the right angle
lines
line
AD,
AE.
to
DC, as
for
upon the
which again
E AD either will all straight
which go out from the point
A meet
the
.
example AF, or some of them,
like the perpendicular
line
line.
1) let fall
the perpendicular
draw the perpendicular
not-cutting.
the one and the other class of those lines will
lines of
be called parallel
and
into cutting
AE,
will
not meet the
In the uncertainty whether the per-
DC.
AE
pendicular
the only line which does not
is
may be possible that
example AG,
Fig. 1.
which do not cut DC, how far soever they may be prolonged. In passing over from the cutting lines, as AF, to the not-cutting lines, as AG,
we must come upon a line AH, parallel to DC, a boundary line, upon
meet DC, we
there are
will
still
assume
other
one side of which
it
lines, for
all lines
AG are such
as
do not meet the
The angle
is
by
J] (p) for
AD =
we remark
which are made at the point
their prolongations
EE'
all others, if
intersect the line
<
DAK =
If IT (p)
tion
the perpen-
A
by
the perpendiculars
itself
AE and AD,
every straight line which goes
or at least
its
prolongation, lies in one
two right angles which are turned toward BC, so that except the
parallel
angle
AE' of
that in regard to the four right angles,
AE' and AD',
out from the point A, either
of the
will here
AE likewise be parallel to the prolongation DB of the line DC,
in addition to which
and
which we
p.
If IT (p) is a right angle, so will the prolongation
dicular
DC,
HAD between the parallel HA and the perpendicular AD
called the parallel angle (angle of parallelism),
designate
line
AF cuts the line DC.
while upon the other side every straight line
DB
make a
they are sufficiently produced both ways, must
BC.
-£ 7T,
AD, making
the same
AK,
parallel to the
prolongs
this
assumption
then upon the other side of
II (p) will
of the line
he
also
a line
DC, so that under
distinction of sides in parallelism.
we must
also
THEORY OF PARALLELS.
14
All remaining lines or their prolongations within the two right angles
turned toward
BC
pertain to those that intersect,
angle
HAK =
hand
to the non-intersecting
parallels
—
AD.
—
Upon
II
if
they
lie
within the
they pertain on the other
(p),
be parallel to
K'AH',
between the
parallels
and EE' the per-
the other side of the perpendicular
manner the prolongations
AK likewise
parallels;
=
E'AK — \k
pendicular to
angle
between the
AG, if they lie upon the other sides of the
and AK, in the opening of the two angles EAH
•£ x
AH
II (p),
in like
2 JJ (p)
BC
the remaining lines pertain,
;
EE'
will
AH' and AK' of the parallels AH and
to the intersecting,
but
if
in the angles
if
in the
K'AE, H'AE'
to the non-intersecting.
In accordance with
this,
be only intersecting or
we must
for the assumption 77 (p)
parallel;
but
if
we assume
=
•£ tz.
the lines can
that /7(p)
<£
tt,
then
allow two parallels, one on the one and one on the other side;
in addition
we must
distinguish the remaining lines into non-intersect-
ing and intersecting.
For both assumptions it serves as the mark of parallelism that the
becomes intersecting for the smallest deviation toward the side
line
where
cuts
lies
the parallel, so that
if
AH
DC, how small soever the angle
is parallel to
HAF may be.
DC, every
line
AF
THEORY OF PARALLELS.
A straight
17.
line
15
maintains the characteristic of parallelism at all
its
points.
Given
2) parallel to
CD, to which
AC
latter
is
perpendic
We will consider two points taken at random on the line AB and
ular.
its
AB (Fig.
production beyond the perpendicular.
Let the point
E lie
on that side of the perpendicular on which
AB
is
looked upon as parallel to CD.
Let
that
from the point
fall
it falls
E
Connect the points
into
struction,
line
2),
let
draw EP
so
CD somewhere in G. Thus we get a triangle
EF goes; now since this latter, from the con.
can not cut AC, and can not cut
(Theorem
Now
which the
EK on CD and
BEK.
A and F by a straight line, whose production then
(by Theorem 16) must cut
ACG,
a perpendicular
within the angle
therefore
it
must meet
CD
AG or EK a second
somewhere
E' be a point on the production of
at
AB and
time
H (Theorem
3).
E'K' perpendic-
CD; draw the line E'F' making so
AE'F' that it cuts AC somewhere in F' making the
with AB, draw also from A the line AF, whose production
ular to the production of the line
small an angle
same angle
will cut
CD in G (Theorem
Thus we get a
line
;
E'F'; since
triangle
now
16.)
AGC,
this line
can not cut AG, since the angle
must
it
meet
into
CD somewhere in
and however
little
they
they always cut CD, to which
AC
a second time, and also
BAG = BE'G', (Theorem
7),
therefore
G'.
Therefore from whatever Doints
out,
which goes the production of the
can not cut
may
E and E'
the lines
EF and
E'F' go
AB,
yet will
diverge from the line
AB is parallel.
16
THEORY OF PARALLELS.
Two
18.
if
always mutually parallel.
lines are
AC
Let
be a perpendicular
we draw from C
CE making any
ECD with CD,
from
on CD,
to
AB
which
is
parallel
the line
acute angle
and
let
fall
A the perpendicular AF
upon CE, we obtain a
angled triangle
right-
ACF, in which
AC, being the hypothenuse,
is
AF
greater than the side
(Theorem
9.)
AG =
Make
AF, and
EFAB
FlG 3
slide
AF
-
-
AB and FE wU
BAK = FAC, con
sequently AK must cut the line DC somewhere in K (Theorem 1 6), thus
forming a triangle AKC, on one side of which the perpendicular GH
the figure
take the position
AK and
intersects the line
tance
AL of
until
AK
in
coincides with
GH, such
L
AG, when
that the angle
(Theorem
3),
and thus determines the
the intersection point of the lines
AB
and
CE
on the
dis-
line
AB from the point A.
CE will always intersect AB, how small soever
ECD, consequently CD is parallel to AB (Theorem 16.)
In a rectilineal triangle the sum of the three angles can not he greater
Hence
may be
follows that
it
the angle
19.
than two right angles.
Suppose in the triangle
equal to
n
-f- a;
ABC
(Fig. 4) the
sum
of the three angles
is
then choose in case
of the inequality of the sides the
smallest
from
A
BC, halve
through
it
D
in D,
draw
the line
AD
and make the prolongation of
DE, equal
point
E
to
AD, then
to the point
it,
join the
C by
the
Fig.
4.
ADB and CDE, the angle
BAD = DEC (Theorems 6 and 10); whence follows
that also in the triangle ACE the sum of the three angles must be equal
straight line
EC.
ABD = DCE.
In the congruent triangles
and
BAC (Theorem 9) of the triangle
ABC in passing over into the new triangle ACE has been cut up into
the two parts EAC and AEC.
Continuing this process, continually
to
jt -\-
a! but also the smallest angle
17
THEORY OF PARALLELS.
we must
halving the side opposite the smallest angle,
finally attain to
a
which the sum of the three angles is k -f- a, but wherein are
two angles, each of which in absolute magnitude is less than %a; since
now, however, the third angle can not be greater than it, so must a be
triangle in
either null or negative.
If in any
20.
rectilineal triangle the
is this also the case
two right angles, so
ABC
If in the rectilineal triangle
=
jT,
and
then must at
Let
0, be acute.
the third angle
least
B
two of
fall
its
sum of
(Fig. 5) the
angles,
triangle.
sum of
B
side
AC
This will cut the
tri-
A*
Fig
angle into two right-angled triangles, in each
of which the
sum
of the three angles must also be
be greater than
either
it,
the three angles
A
from the vertex of
upon the opposite
the perpendicular p.
the three angles is equal to
for every other
and in
it,
since
it
caD not in
their combination not less than
So we obtain a right-angled triangle with the perpendicular
and
and from
q,
whose adjacent
By
this a quadrilateral
sides
p and q are
whose opposite
np and
q,
and
we can make another with
sides
ABCD
AB = np, AD = mq, DC = np, BC = mq, where
q
p
/s
a
q
q
P
/
q
/
P
/
v
p
/
&L—±
L
Fig.
m and
p
and
with sides at right angles
a quadrilateral
finally
sides
sides are equal
at right angles (Fig. 6.)
repetition of this quadrilateral
to each other, such that
it.
1
1
6.
n are any whole numbers. Such a quadrilateral is divided by
DB into two congruent right-angled triangles, BAD and
the diagonal
BCD, in each of which the sum of the three angles = -.
The numbers n and m can be taken sufficiently great
angled triangle
ABC
(Fig. V)
whose perpendicular
sides
for the right-
AB = np, BC
= mq, to enclose within itself another given (right-angled) triangle BDE
as soon as the right-angles
fit
each other.
18
THEORY OP PARALLELS.
Drawing the line DC, we obtain right-angled triangles of which every
two have a side in common.
The triangle ABC is formed by the union of the two triangles ACD
and DCB, in neither of which can the sum of the angles be greater than
successive
arj
consequently
compound
it
triangle
must be equal to
may
be equal to
order that the
n, in
sum
in the
n.
BDC consists of the two triangles DEC
and DBE, consequently must in DBE the sum of the three angles be
equal to n, and in general this must be true for every triangle, since
La the same way the triangle
each can be cut into two right-angled
From this it follows that only two
is the sum of the three angles in all
this sum is in all less than jr.
21»
fall
straight line
A (Fig.
Fig
8)
;
or
choose.
upon the given
line
BC
the
8,
AB take upon BC at random
AD; make DE = AD, and draw AE.
perpendicular
tt,
straight line that shall
an angle as small as we
from the given point
Either
rectilineal triangles equal to
From a given point we can always draw a
make with a given
Let
triangles.
hypotheses are allowable:
the point D, draw the line
THEORY OF PARALLELS.
In the right-angled triangle ABD
in the isosceles triangle
rems 8 and
AEB,
any given
If two perpendiculars
22.
sum of
other, then the
ADB = a;
then must
ADE the angle AED be either £a or less (Theo-
Continuing thus
20).
as is less than
the angle
let
19
we
finally attain to
such an angle,
angle.
to the
same
the three angles in
a
straight line are parallel to each
rectilineal triangle is equal to
two
right angles.
Let the
lines
pendicular to
AF to
A
be
parallel to
each other and per-
any distances
AE
the lines
E
the points
which are taken on the
at
9)
AC.
Draw from
and
AB and CD (Fig.
and
line
FC > EC
F,
CD
from
the point C.
Suppose in the right-angled tri-
ACE the sum of the
angle AEF equal to
—
angle
tz
—
ft,
where a and
Further,
ft
we
it
equal to
in triangle
lessen the angle b, consequently the
and
tz
every line
p,
also /7(p)
<£
The
first
a,
in the
tri-
ACF equal tz —
a
BAF = a, AFC = b, so is a + ft = a — b; now
AF away from the perpendicular AC we can make
AF and the parallel AB as small as we choose; so
=
at the
or for
same time
two angles a and
sum
of the three angles
also the parallel angle
all triangles this
ft
and ft=0.
It follows that in all rectilineal triangles the
either
—
can not be negative.
can have no other magnitude than a
is
tz
line
the angle a between
can
then must
is
the angle
let
by revolving the
also
ft,
three angles
sum
is
<
tz
and
U (p) = \
at the
tz
for
same time
tz.
assumption serves as foundation for
the ordinary geometry
and
plane trigonometry.
The second assumption can likewise be admitted without leading to
any contradiction in the results, and founds a new geometric science,
to which I have given the name Imaginary Geometry, and which I intend here to expound as far as the development of the equations be-
tween the sides and angles of the
rectilineal
and spherical
triangle.
For every given angle a there is a line p such that II (p)
Let AB and AC (Fig. 10) be two straight lines which at the
23.
section point
A make the acute angle a;
take at random on
=
a.
inter.
AB a point
20
THEORY OF PARALLELS.
B'; from this point drop
AA';
erect at
A" the
B'A'
AC; make A" A"
at right angles to
=
perpendicular A^B"; and so continue until a per-
Fig. 10
pendicular
CD
is
which no longer
attained,
intersects
AB.
This must
AA'B' the sum of all three
triangle AB' A" it equals -k — 2a,
of necessity happen, for if in the triangle
angles
m
is
equal to
triangle
it finally
—
7:
AA"B"
a,
less
then in the
—
than ^
2a (Theorem
20),
and so
forth, until
becomes negative and thereby shows the impossibility of con-
structing the triangle.
The perpendicular
point
CD may
A all others cut AB;
be the very one nearer than which to the
at least in the passing
cut to -those not cutting such a perpendicular
Draw now from
acute angle
H of
the point
HFG, on
the line
FH
F
that side
let fall
the line
where
upon
AC
longation consequently must cut
triangle
AKB,
into
AB
line
AB somewhere
FH
in
enters,
M.
and
Since the
GFH is arbitrary and can be taken as small as we wish, therefore
and
AF=
One easily sees that with the
p
HK, whose prosomewhere in B, and so makes a
which the prolongation of the
FG is parallel to AB
for
FH, which makes with FG the
the point A. From any point
lies
the perpendicular
therefore must meet the hypothenuse
angle
over from those that
FG must exist.
=
0, it
approaches the value
a decreases, while
Since
we
p.
it
(Theorems 16 and
18.)
lessening of
fat;
p the angle a increases, while,
with the growth of p the angle
continually approaches zero for
p =00
are wholly at liberty to choose what angle
we
.
will
under
THEORY OP PARALLELS.
stand by the symbol Tl (p)
number, so we
when
the line p
is
21
expressed by a negative
assume
will
/7(p)+/7(-p)=;r,
an equation which
shall
=
hold for
and for p
0.
The farther parallel
24.
ative,
lelism, the
AB
If to the line
A
line,
CABD
eral
values of
lines are
p, positive as
well as neg-
prolonged on the side of their paral-
more they approach one another.
(Fig. 11)
and their end-points C and
a straight
all
AC = BD
two perpendiculars
are erected
D joined by
then will the quadrilat-
have two right angles at
and B, but two acute angles at C
D (Theorem 22) which are equal
and
to
one another, as
we can
easily see
by thinking the quadrilateral super-
imposed upon
itself so that the line
BD
falls
upon
AC and AC upon
BD.
Halve
AB
and erect at the mid-point
E
the line
EF
perpendicular to
AB.
This line must also be perpendicular to CE, since the quadrilat-
erals
CAEF
and
hat the line
EF
FDBE
fit
one another
if
not be parallel to AB, but the parallel to
CG, must incline toward
ular
BD
Since
nears
a part
C
is
we
so place one on the other
remains in the same position.
AB
AB
Hence the
line
for the point C,
CD
can
namely
(Theorem 16) and cut from the perpendic-
BG < CA.
a random point in the line CG,
AB the more the farther it is prolonged.
it
follows that
CG itself
THEORY OF PARALLELS.
22
Two
25.
which are parallel
straight lines
to
a third are
also parallel to
each other.
Fig. 12.
We will first assume that the three lines AB,
If two of them in order, AB and
one plane.
AB
outmost one, EF, so are
to prove this, let fall
and
CD
CD
side
toward EF, the
in
some point
C (Theorem
parallel to
A perpendicular AG- let
fall
fall
CD
the outer line
AE, which
3),
at
In order
CD
DCE <
|-7r
middle
on the
22).
from the same
within the opening of the acute angle
AB upon the
will cut the
an angle
(Theorem
upon
(Fig. 12) lie in
parallel to the
parallel to each other.
A of
from any point
other outer line FE, the perpendicular
line
EF
CD, are
CD,
ACG
point,
(Theorem
A, must
9);
every
AH from A drawn within the angle BAC must cut EF, the
AB, somewhere in H, how small soever the angle BAH may
be; consequently will CD in the triangle AEH cut the line AH someIf AH from
should meet EF.
where in K, since
is impossible that
cut the prothe point A went out within the angle CAG, then must
other line
parallel to
it
it
it
longation of
CD
between the points
C and G
in the triangle
CAG.
AB and CD are parallel (Theorems 16 and 18).
"Were both the outer lines AB and EF assumed parallel to the middle
Hence follows
line
that
CD, so would every
angle
B AE,
the angle
Upon
cut the line
line
AK
from the point A, drawn within the
CD somewhere in the point K, how small soever
BAK might be.
the prolongation of
AK take at
random a point
L
and join
it
23
THEORY OF PARALLELS.
C by
with
EF somewhere
the line CL, which must cut
ing a triangle
in
M, thus mak-
MCE.
of the line AL within the triangle MCE can cut
CM a second time, consequently it must meet EF some*
therefore AB and EF are mutually parallel.
The prolongation
AC
neither
where
in
nor
H;
c ^1
Fig. 13.
Now let
the parallels
intersection line
fall
is
a perpendicular
AB
and
From
EF.
CD
a
(Fig. 13) lie in
two planes whose
E
of this latter let
random point
EA upon one of
the two parallels,
then from A, the foot of the perpendicular EA,
AC
dicular
upon the other
parallel
of the two perpendiculars by the
CD and
line
e. g.,
let fall
a
upon AB,
new
EC.
The angle
BAC
acute (Theorem 22), consequently a perpendicular CG- from
perpen-
E
join the end-points
and
C
must be
C
let fall
AB meets it in the point G- upon that side of CA on which the
AB and CD are considered as parallel.
Every line EH [in the plane FEAB], however little it diverges from
upon
lines
EF, pertains with the
AB
the two parallels
AB
EC to a plane which must cut the plane of
CD along some line CH. This latter line cuts
line
and
somewhere, and in fact in the very point
H which is common to all
three planes, through which necessarily also the line
quently
EF
is parallel
to
In the same way we
EH goes;
conse-
AB.
may show
EF and CD.
EF is parallel to one of two other
considering EF as the intersection
the parallelism of
Therefore the hypothesis that a line
parallels,
of
AB
two planes
and CD,
in
is
the same as
which two
parallels,
AB, CD,
lie.
Consequently two lines are parallel to one another if they are parallel
to a third line,
The
last
though the three be not
co-planar.
theorem can be thus expressed:
Three planes
intersect in lines
parallelism of two
is
presupposed.
which are
all parallel to
each other if the
THEORY OF PARALLELS.
24
Triangles standing opposite to one another on the sphere are equiva-
26.
lent in surface.
By
opposite triangles
we here understand such
as are
made on both
by the intersections of the sphere with planes; in such
therefore, the sides and angles are in contrary order.
sides of the center
triangles,
In the opposite triangles
ABC
and
A'B'C
(Fig. 14,
where one of
them must be looked upon as represented turned about), we have the
A'B', BC
C'A', and the corresponding angles
sides AB
B'C, C A
=
=
=
b'
A
Fig. 14.
C are
C.
at the points A, B,
points A', B',
th
Through the three
it
from the center
likewise equal to those in the other triangle at
points A, B, C, suppose a plane passed,
of the sphere a perpendicular
longations both ways cut both opposite triangles in the points
of the sphere.
The
distances of the
arcs of great circles on the sphere,
first
D
and upon
dropped whose pro-
D
and D'
from the points ABC, in
must be equal (Theorem 12) as well
D'C, on the other
to each other as also to the distances D'A', D'B',
triangle
(Theorem
6),
consequently the isosceles triangles about the points
D and D' in the two spherical triangles ABC and A'B'C' are congruent.
In order to judge of the equivalence of any two surfaces in general,
I take the following theorem as fundamental:
Two
surfaces are equivalent
when
they arise from the mating or separating
of equal parts.
27.
(ess
A
three-sided solid aqgle equals the half
sum of
the surface angles
a right-angle.
In the spherical triangle
nate the angles
by A,
ABC
(Fig. 15),
where each
B, C; prolong the side
ABA'B'A is produced;
side
<
jr,
desig-
AB so that a whole circle
this divides the sphere into
two equal
parts.
25
THEORY OF PARALLELS.
In that half in which
sides
circle in
A' and
the triangle
is
common
through their
ABC, prolong now the other two
C until they meet the
intersection point
B'.
c
Fig. 15.
In this way the hemisphere
B'CA', A'CB, whose
dent that here
The
ABC,
may be
size
having a side
goes from
C
ABO, ACB',
designated by P, X, Y, Z.
It is evi.
P + X=B, P + Z = A.
size of the spherical triangle
third angle
it
divided into four triangles,
is
lies at
C through
AB
in
Y equals that of the opposite triangle
common
with the triangle P, and whose
the end-point of the diameter of the sphere which
the center
D
of the sphere (Theorem 26).
Hence
follows that
P -\- Y — C, and since P-|-X-j-Y-|-Z = therefore we have also
P = f(A + B + C-;r).
7r,
We may attain to the same conclusion in another way, based solely
upon the theorem about the equivalence of surfaces given above. (Theorem
26.)
ABC (Fig. 16),
mid-points D and
In the spherical triangle
and through the
E
draw a great
fall
circle;
AB and BC,
this let
from A, B, C the perpendiculars
AF, BH, and CG.
ular
upon
halve the sides
from
B
falls at
If the perpendic- j^
H
between
E, then will of the triangles so
D
and
made
BDH = AFD, and BHE = EGC (Theorems 6 and
15),
the surface of the triangle
(Theorem
26).
*
4
whence follows that
ABC equals that of the
Fig. 16.
quadrilateral
AFGO
26
THEORY OP PARALLELS.
If the point
H coincides with the middle point B of the side BC (Fig.
^B
17),
ADF
only two equal right-angled triangles,
and BDE, are made,
"by
whose interchange the
equivalence of the surfaces of the triangle
and the quadrilateral
If, finally,
ABC
Fig. 17.
H falls outside the triangle
(Fig. 18), the perpendicular
ABC
ABC
is established.
CG
goes, in
consequence, through the triangle, and so
over from the triangle
triangle
the point
AFEC
to the quadrilateral
AFGC
we go
by adding the
FAD = DBH, and then taking away the triangle CGE — EBH.
AFGC a great circle passed
Supposing in the spherical quadrilateral
A and G, as also through F and C, then will their
AG and FC equal one another (Theorem 15), consequently
also the triangles FAC and ACG be congruent (Theorem 15), and the
through the points
arcs between
angle
FAC
Hence
equal the angle
ACG.
follows, that in all the
preceding
angles of the spherical triangle equals the
in the quadrilateral
Therefore
we
the three angles
cases, the
sum
of the
sum
of
all
three
two equal angles
which are not the right angles.
can, for every spherical triangle, in
is S,
which the sum of
find a quadrilateral with equivalent surface, in
which are two right angles and two equal perpendicular
where the two other angles are each £S.
sides,
and
27
THEORY OF PARALLELS.
ABCD (Fig. 19) be the spherical quadrilateral, where the
AB = DC are perpendicular to BC, and the angles A and D
Let now
sides
each £S.
f
Prolong the sides
further beyond E,
of
BC
until they cut
DE = EF
the perpendicular FG.
mid-point
The
AD and BC
make
and
let fall
Bisect the whole arc
H by great-circle-arcs with A and
triangles
EFG and DCE
one another in E, and
upon the prolongation
BG
and join the
F.
are congruent (Theorem 15), so
FG =
DC = AB.
The
triangles
ABH and HGF are likewise congruent,
since they are
right angled and have equal perpendicular sides, consequently
AH and
AHF = n, ADEF likewise = n, the
angle HAD = HFE = |S - BAH = £S - HFG = £S - HFE-EFG
— £S— HAD-7r+|S; consequently, angle HFE = |(S— or what
AF
pertain to one circle, the arc
7r);
is
the same, this equals the size of the lune
equal to the quadrilateral
ABCD,
as
we
AHFDA,
easily see
if
we
which again
is
pass over from
by first adding the triangle EFG and then BAH
and thereupon taking away the triangles equal to them DCE and HFG.
Therefore £(S— n) is the size of the quadrilateral ABCD and at the
the one to the other
same time
also that of the spherical triangle in
three aDgles
is
equal to S.
which the sum of the
28
THEORY OF PARALLELS.
28.
If
three
planes cut each other in parallel
then the
lines,
sum of
the
three surface angles equals two right angles.
Let AA', BB'
section of planes
CC
be three
(Fig. 20)
(Theorem
made by the
at random
parallels
inter-
Take upon them
25).
three
A
Fig. 20.
points A, B, C,
and suppose through these a plane
passed,
which con-
sequently will cut the planes of the parallels along the straight lines
AB, AC, and BC.
D
on
Further, pass through the line
AA'
the parallels
CC
we
CC
and BB',
and DC, and whose
and
whose
the BB', another plane,
and any point
two planes
and BB' produces the two
lines
by
The angles between the
w.
three planes in which the parallels
A as center suppose a sphere described,
sections of the straight lines AC, AD A A' with
About
upon which the
it
with the sides
lie will
AA', BB',
at the lines
CC;
BDC = a, ADC — b, ADB = c.
finally call the linear angles
p, q,
and
r.
Call
inter-
determine a spherical
its size a.
Opposite the side
X, and consequently opposite p
the angle ^--j-2a—w—X, (Theorem 27).
q
lies
In
the angle w, opposite r
like
manner CA, CD,
determine a triangle of size
opposite
q',
Finally
the lines
is
Z
opposite
r',
lies
CC
8,
«J
—
v>
7i-\-2j3
and the
angles,
a spherical triangle,
whose
—
2«,
w
— w—Z opposite p'.
sides are
D with
1,
m,
n,
and
and Y. Consequently
= £(X-f-Y+Z—n)—a—ft+w.
Decreasing
a-\-ft
p', q', r',
and consequently
the angles opposite them w-\-7t— 2ft, w-J-X
its size
lies
cut a sphere about the center C, and
with the sides
determined by the intersection of a sphere about
DA, DB, DC,
of
AD
AA'
inclination to the third plane of the parallels
will designate
be designated by X, Y, Z, respectively
triangle,
AC
intersection with the
w
lessens also the size of the triangles
can be made smaller than any given number.
a and
ft,
so that
THEORY OF PARALLELS.
In the triangle 3 can likewise the sides
1
and
m
29
be lessened even to
vanishing (Theorem 2 1), consequently the triangle 3 can be placed with
one of
its
sides
1
or
m upon a great circle of the sphere as often as
choose without thereby
filling
up the
half of the sphere, hence
you
3 van-
whence follows that necessarily we must have
ishes together with w;
X+Y+Z^JT
29.
In a
of the sides
rectilineal triangle, the perpendiculars erected at the mid-points
do not meet, or they
either
all three cut each other in one point.
Having pre-supposed in the triangle ABC (Fig. 21), that the two perpendiculars ED and DF, which are erected upon the sides AB and BC
at their mid points E and F, intersect in the point D, then draw within
the angles of the triangle the lines DA, DB, DC.
ADE and BDE
In the congruent triangles
AD— BD, thus
triangle
ADC
BD — CD;
follows also that
is
hence
(Theorem
10),
we have
the
consequently the
isosceles,
perpendicular dropped from the vertex
D upon the
AC falls upon G the mid point of the base.
The proof remains unchanged also in the case
when the intersection point D of the two perpen-
base
diculars
falls
ED and FD
falls in
the line
AC
itself,
or
FlG 21
without the triangle.
In case we
not intersect, then also
30.
-
-
two of those perpendiculars do
the third can not meet with them.
therefore pre-suppose that
The perpendiculars which are
triangle at their mid-points,
must
as the parallelism of two of them
In the triangle
ABC
erected
upon
the sides
of a
rectilineal
all three he parallal to each other, so soon
is
pre-supposed.
(Fig. 22) let the lines
DE, FG, HK, be erected
perpendicular upon the sides at their mid-
"We
points D, F, H.
will in the first place
DE and
AB in L
assume that the two perpendiculars
FG
are parallel, cutting the line
and M, and that the perpendicular
between them.
"Within the angle
from the point L,
at
random, a straight
LG, which must cut
how
HK
FG
lies
BLE draw
line
somewhere in G,
small soever the angle of deviation
Fia. 22.
GLE may be.
(Theorem
16).
30
THEORY OF PARALLELS.
Since in the triangle LGM the perpendicular HK can not meet with
MG (Theorem 29), therefore must cut LG somewhere in P, whence
follows, that HK is parallel to DE (Theorem 1
and to MG (Theorems
it
6),
18 and 25).
BC —
Put the side
2a,
AC = 2b, AB = 2c,
by A, B,
gles opposite these sides
C, then
and designate the an-
we have
in the case just
considered
A = 77(b)—/7(c),
B= 77 (a)—
= /7(a)+/7(b),
77(c),
C
may easily show with help of the lines AA', BB', CC, which
drawn from the points A, B, C, parallel to the perpendicular HK
as one
are
and consequently
rems 23 and
to both the other perpendiculars
HK
Let now the two perpendiculars
the third
them, or
The
DE
and
FG
not cut them (Theorem 29), hence
cuts
it
last
DE
and
FG
(Theo-
25).
be
parallel,
is it
then can
either parallel to
AA'.
assumption
is
not other than that the angle
C>77(a)+/7(b.)
we lessen this angle,
while we in that way give
If
and designate the
CBQ
so that
it
becomes equal to 77(a)
AC
the
new
size of the third side
BQ
by
at the point B,
the line
which
2 c', then
must the angle
increased, in accordance with
is
-f- 77(b),
position CQ, (Fig. 23),
what
is
proved above, be equal to
77(a)— 77(c')> 77(a)— 77(c),
whence follows
c'
>c
(Theorem
23).
A
B
Fig. 23.
ACQ are, however, the angles at A and Q equal,
ABQ must the angle at Q be greater than that at
consequently is AB>BQ, (Theorem 9); that is c>c'.
In the triangle
hence in the triangle
the point A,
31.
which
We
call
boundary
line (oricycle) that curve lying in
all perpendiculars erected at the mid-points
each other.
a plane for
of chords are parallel
to
THEORY OP PARALLELS.
31
In conformity with this definition we can represent the generation of
a boundary
line, if
we draw
to a given line
AB (Fig.
24) from a given
Hi
Fig. 24.
point
A in
C
the end
we can
it,
making
different angles
of such a chord will
lie
CAB= /7(a),
on the boundary
AC = 2a;
whose points
thus gradually determine.
The perpendicular
will
be
ary
line.
DE erected
parallel to the line
upon the chord
AB, which we
AC at its
will call the
mid-point
AH, be
parallel to
AB, consequently must
peculiarity also pertain to every perpendicular
erected at the mid-point
K of any chord CH,
KL
in general
A
circle
is
between whatever points
H of the boundary line this may be drawn (Theorem 30).
be called Axes of the boundary
this
which
Such
perpendiculars must therefore likewise, without distinction from
32.
D
Axis of the bound-
In like manner will also each perpendicular FG- erected at the
mid-point of any chord
and
chords
line,
AB,
line.
with continually increasing radius merges into the boundary
line.
Given
AB
(Fig. 25) a chord of the
boundary
end-points
A and B of the chord two axes
AC
BF, which
and
make with
line;
draw from the
consequently will
the chord two equal angles
BAC = ABF == a (Theorem
Upon one
of these axes
where the point
E
31).
AC, take any-
as center of a circle,
and draw the arc AF from the initial point
A of the axis AC to its
F
intersection point
with the other axis BF.
The radius
of the circle, FE, corresponding to the point
on the one side with the chord
AF
an angle
AFE = A
F
will
make
and on the
32
THEORY OF PARALLELS.
EFD = y. It follows that the
BAF = a—ft<ft+y—a (Theorem 22);
other side with the axis BF, the angle
angle between the two chords
whence
—
a ft<$ySince now however the angle y approaches the limit zero, as well in
consequence of a moving of the center E in the direction AC, when F
follows,
remains unchanged, (Theorem
F
proach of
B on
(Theorem
position
angle
to
when
the axis BF,
AF, and hence
E
may
remains in
its
with such a lessening of the
or the mutual inclination of the two chords
also the distance of the point
F
ary line from the point
Consequently one
ft,
consequence of an ap-
the center
22), so it follows, that
the angle a
y, also
AB and
21), as also in
on the
circle,
B
on the bound-
tends to vanish.
also call the boundary-line a circle with in-
finitely great radius.
33.
Let
AA' = BB' = x
A to A',
the side from
as axes for the
(Figure
which
be two lines parallel toward
26),
*.
parallels serve
two boundary arcs
on
(arcs
two boundary lines) AB=s, A'B' =s', then is
s'
where e
is
= se —
independent of the arcs
the straight line
x,
s'
s,
and of
the distance of the arc
In order to prove
this,
Fig. 26.
from
s'
assume that the
s.
ratio of the arc s to s' is
equal to the ratio of the two whole numbers n and m.
Between the two axes AA', BB' draw yet a third axis CC, which
so cuts off from the arc
the
same
side,
a part
that of the whole
AB
AC =
a part
A'C =
numbers p and
q,
=—
s',
p
=—
t
m
Divide
parts
on
s by axes into nq equal
and np on t.
However
t
to s equal to
s.
parts,
then will there be
there correspond to these equal parts on s and
equal parts on
s'
and
t',
consequently
also
AA
wherever the two arcs
;
and
BB
;
,
the ratio of
t
mq such
likewise
we have
s
t
Hence
two axes
the ratio of
q
now
sf
and from the arc A'B' on
so that
n
s
t
Assume
t'.
t
t
and V
to
t'
may
be taken between the
remains always the same, as
THEORY OF PARALLELS.
33
long as the distance x between them remains the same.
fore for x =
i,
put
s= es',
If
we
there-
then we must have for every x
s'
= se~
x.
Since e is an unknown number only subjected to the condition 6>i,
and further the linear unit for x may be taken at will, therefore we may,
for the simplification of reckoning, so choose it that by e is to be understood the base of Napierian logarithms.
"We may here remark, that
the distance between
two
s'
=
for x
parallels decrease
=
oo
,
hence not only does
(Theorem
24),
but with the
prolongation of the parallels toward the side of the parallelism this at
last
wholly vanishes.
have therefore the character of
Parallel lines
asymptotes.
Boundary
34.
surface (orisphere)
we
from the revolution of the boundary
together with
of the
all
call that surface
line
about one of
which
arises
which,
its axes,
other axes of the boundary-line, will be also an axis
boundary -surface.
A. chord is inclined at equal angles to such axes
points, wheresoever these two end-points
Let A, B,
C, (Fig. 27),
may
be taken
drawn through
its
end'
on the boundary-surface.
be three points on the boundary-surface;
Fig. 27.
AA', the
and
AC
axis of revolution,
BB' and
CC
two other
axes,
= B'BA, A 'AC =C'CA (Theorem 31.)
AB
hence
chords to which the axes are inclined at equal angles
A AB
;
THEORY OP PARALLELS.
34
Two
axes BB',
CC, drawn through
BC, are likewise parallel and
A perpendicular
DD'
EE
dicular
will
be
the end-points of the third chord
in one plane,
(Theorem
erected at the mid-point
and in the plane of the two
three axes
lie
parallels
D
25).
AA', BB', must be
;
parallel to the three axes
BB
parallel to the
AA', BB', CC, (Theorems 23 and 25); just such a perpenupon the chord AC in the plane of the parallels AA'; CC
AA', BB', CC, and the perpendicular
Let now the angle between the plane in which the
DD'.
AB
of the chord
parallels
AA'
and the plane of the triangle ABC be designated by /7(a),
where a may be positive, negative or null. If a is positive, then erect
and
7
FD = a
lie,
ABC, and
within the triangle
the chord
in
its
plane, perpendicular
upon
AB at its mid-point D.
Were a
FD = a
a negative number, then must
be drawn outside the
on the other side of the chord AB; when a=0, the point
triangle
F
coincides with D.
In
all
cases arise
two congruent right-angled triangles AFD and DFB,
FA FB.
consequently
we have
now
angle ABC.
at
Erect
F
Since the angle
DD' and
=
FF' perpendicular
the line
D'DF —
/7(a),
to the plane of the
DF=a,
and
the line EE', with which also
it
lies in
so
FF'
is
tri-
parallel to
one plane perpendicu-
ABC.
lar to the plane of the triangle
Suppose now in the plane of the
parallels
EE', FF' upon
EF
the per*
EK erected, then will this be also at right angles to the plane
triangle ABC, (Theorem 13), and to the line AE lying in this
pendicular
of the
plane,
lar to
(Theorem
EK
(Theorem
11);
and consequently must AE, which
and EE', be
11).
The
also at the
triangles
AEF
is
perpendicu-
same time perpendicular
and
FEC
to
FE,
are congruent, since they
are right-angled and have the sides about the right angles equal, hence
is
AF = FC = FB.
A perpendicular from the vertex F
fall
upon the base BC, goes through
through
FG
this perpendicular
to the plane of the triangle
BB', CC, along the
CC, (Theorem 25);
at the
same time
line
since
also to
= B'BG, (Theorem
23).
and the
ABC, and
GG', which
now
of the isosceles triangle
its
CG
BFC
let
mid-point G; a plane passed
line
FF' must be perpendicular
cuts the plane of the parallels
is
is at
likewise parallel to
right angles to
GG', so consequently
is
BB' and
FG, and hence
the angle
C'CG
35
THEORY OF PARALLELS.
Hence
may
follows, that for the boundary-surface each of the axes
be considered as axis of revolution.
we
Principal-plant
boundary
will call
each plane passed through an axis of the
surface.
Accordingly every Principal-plane cuts the boundary-surface in the
boundary
line,
while for another position of the cutting plane this
in-
tersection is a circle.
Three principal planes whicn mutually cut each
sum
each other angles whose
These angles we
whose
is n,
(Theorem
other,
make with
28).
will consider as angles in the boundary-triangle
sides are arcs of the boundary-line,
which are made on the bound-
ary surface by the intersections with the three principal planes.
Con-
sequently the same interdependence of the angles and sides pertains to
the boundary-triangles, that
is
proved in the ordinary geometry for the
rectilineal triangle.
35.
In what follows, we
with an accent added,
e. g.
x,
which
which
is
represented
relation is given
/7(x)
letter
+
by the same
rela.
letter
by the equation
#(*')
= fr-
ABC
(Fig. 28)
be a
the hypothenuse
AB = c,
the other sides
Let now
by a
x\ in order to indicate that this has a
tion to that of another line,
without accent
will designate the size of a line
where
and the
rectilineal right-angled triangle,
AC = b, BC = a,
Fig. 28.
angles opposite them are
BAC = /7(a), ABC = Htf).
THEORY OF PARALLELS.
36
At
the point
to
A erect the line AA' at right angles to the plane of the
the points B and C draw BB' and CC parallel
ABC, and from
triangle
AA'.
The planes
the angles:
in
which these three
11(a) at
A A',
parallels lie
CC
a right angle at
make with each
other
(Theorems 11 and
BB' (Theorem 28).
of the lines BA, BC, BB' with a sphere
13),
consequently f[(a') at
The
intersections
about the point
the sides are
B
described
as center, determine a spherical triangle mnk, in
which
mn = /7(c), kn= II (ft), mh — /7(a) and the opposite angles
are /7(b), /7(«'), $„.
Therefore
sides are
we
c
must, with the existence of a rectilineal triangle whose
and the opposite angles
/7(a), II (ft) fa, also admit the
existence of a spherical triangle (Fig. 29) with the sides /7(c), Tl
a, b,
(ft),
/7(a)
and the opposite angles
/7(b), /7(a'), fa.
Fia. 29.
Of
these two triangles, however, also inversely the existence of the
spherical triangle necessitates
anew
sequence, also can have the sides
that of a rectilineal,
a, a',
ft,
which
in con-
and the oppsite angles /7(b'),
n(c), fa.
Hence we may pass over from a,
a',
ft,
b, c, a,
Suppose through the point
A
to b,
a, c,
ft,
a,
and
also to
a,
C, and whose
ary-triangle,
AA' as axis, a boundtwo other axes BB', CC, in B" and
(Fig. 28) with
ary-surface passed, which cuts the
intersections with the planes the parallels
whose
angles opposite
rem
ft,
b', c.
sides are
them
B"C"=j9, C'A
/7(a), /7(a'),
=^
form a bound-
B"A = r, and
the
fa, and where consequently (Theo-
34):
p=r
sin /7(a), q
= r cos
77(a).
Now
break the connection of the three principal-planes along the line
BB', and turn them out from each other so that they with all the lines
lying in
q,
them come
to lie in
one plane, where consequently the arcs p,
which goes through the
r will unite to a single arc of a boundary-line,
37
THEORY OF PARALLELS.
A and has AA
point
one
side will
;
for axis, in such a
manner that
the arcs q and p, the side
lie,
b
(Fig. 30)
of the triangle,
on the
which
is
Fig. 30.
perpendicular to
to
allel
AA' and
parallel to
On
at
A, the axis
CC going
from the end of b
through C* the union point of p and
CC at the point C,
pendicular to
BB'
AA'
AA'
q,
par-
the side a per-
and from the end-point of a the
axis
which goes through the end-point B" of the arc p.
the other side of
AA'
the point A, and the axis
will
BB'
lie,
the side c perpendicular to
parallel to
AA'
at
AA', and going through the
end-point B' of the arc r remote from the end point of b.
The
size of the line
In like
we
point C to
by
t,
b,
which dependence we
will
=/
describe, taking
If
CD
CC* depends upon
(b).
CC*
manner we will have
express by
CC
'
BB
ff
=/
(c).
new boundary line from the
axis BB' and designate the arc
as axis, a
D with the
BD = f (a).
BB' = BD+DB" == BD+CC,
its
then
intersection
is
consequently
/(c)=/(»)+/(b).
Moreover,
we
perceive, that
t=pe*W
(Theorem 83)
=r sin
b
/7(a) e/< )
If the perpendicular to the plane of the triangle
erected at
ABC
(Fig. 28)
were
B instead of at the point A, then would the lines c and r remain
the same, the arcs q and
t
would change to
t
and
q,
the straight lines a
38
THEORY OF PARALLELS.
and b into b and
a,
H
and the angle J] a ) into
(
(ft),
consequently
we
would have
eM,
g-=rsin II (ft)
whence follows by
substituting the value of
= sin II
cos 11(a)
and
if
we change a and
into b'
ft
sin II (b)
further,
by
multiplication with
c,
= sin U
e «a
(c)
>
e-K b >
sin IJ (b) e/( b
Hence
and
q,
e«*\
(ft)
>= sin
/7(c) e/<c>
follows also
sin
n
(a)
e'<»>=
sin /7(b)
e/W.
Since now, however, the straight lines a and b are independent of
one another, and moreover, for b=0, /(b)=0, /7(b)=^, so we have
for every straight line a
e _/(a) =sin Jj(&y
Therefore,
sin II (c)
sin II
(ft)
Hence we obtain
besides
= sin /7(a) sin /7(b),
= cos /7(a) sin /7(a).
by mutation
of the letters
= cos II sin /7(b),
cos /7(b) = cos /7(c) cos /7(a),
cos /7(a) = cos /7(c) cos
sin /7(a)
(ft)
77"
(j9).
we
designate in the right-angled spherical triangle (Fig. 29) the
sides /7(c), II
/7(a), with the opposite angles /7(b), /7(a'), by the
(ft),
If
letters a, b, c,
those which
A, B, then the obtained equations take on the form of
we know as proved
in spherical trigonometry for the right*
angled triangle, namely,
sin
a=sin c
b=sin c
cos
A=cos
a sin B,
cos
B=cos
b, sin
sin
cos c=cos
a,
sin
A,
sin B,
A,
cos bj
from which equations we can pass over to those for
all
spherical
tri*
angles in general.
Hence
spherical trigonometry is not dependent
upon whether in a
THEORY OF PARALLELS.
rectilineal triangle the
sum
of the three angles
39
is
equal to two right
angles or not.
36.
ABC
"We
will
now
consider
anew the
(Fig. 31), in which the sides are
right-angled rectilineal triangle
and the opposite angles
a, b, c,
JI{a), II (ftfcr.
Prolong the hypothenuse c through
the point B, and
point
lar
make BD=/3;
D erect upon BD
DD', which consequently
parallel to
at the
the perpendicuwill
be
BB' the prolongation of the
,
beyond the point B. Parallel to
J)jy from the point A draw AA', which
is at the same time also parallel to CB',
side a
(Theorem
25), therefore is the
angle
A'AD=/7(c+/3),
A'AC = 77 (b), consequently
/7(b)=/7(a)+//(c+#.
Fia 31
from
If
enuse
32),
B we
lay off y?
within the triangle erect upon
AB
and from
the
the perpendicular DD',
point
A
BC
will
parallel to
with
its
DAA'=/7(c
If
Fig. 32.
ular
still
c=ft
is,
ft),
/?)=/7(a)+/7(b).
then also
DD' draw AA',
prolongation
third parallel; then
(b),
on the hypoth-
then at the end point D, (Fig.
c,
valid,
angle
so
CC be the
CAA'=/7
consequently JT(c
—
The last equation is
when c=$ or c<^3.
(Fig. 33), then the perpendicu-
A A' erected upon AB at the point A
40
is parallel
THEORY OF PARALLELS.
to the side
BC=a,
with
we have 77(a)+/7(b)=^,
If c<ft, then the
end of
prolongation, CO', consequently
its
B
o
-A
whilst also
ft
77(c—$=-£;r, (Theorem
beyond the point
falls
23).
A at D (Fig.
34)
upon the prolongation of the hypothenuse AB. Here the perpendicular DD' erected upon AD, and the line A A' parallel to it from A, will
if
likewise be parallel to the side
with
prolongation
its
Here we have the angle
(ft
—
c),
BC=a,
CC\
DAA' = 77
consequently
Ji(a)+n(b) =7t-n{ft-c)=ii{c-ft),
(Theorem 23).
The combination of the two equations
found
gives,
277(b)=77(c-/3)+77(c+/2),
277(a)- 77(c-/?)-77(c+/9),
whence follows
£77(c—ft)+$ 77(c+y9)]
cos 77(b)
_cos
cos 77(a)
~cos [ £77(c—ft)—£ 77(c+/2)]
[
Substituting here the value, (Theo-
rem
35)
cos 77(b)
=cos77(c),
Fig. 34.
cos 77(a)
we have
Since here
[tan £77 (c)] 2=tan £77 (c—ft) tan £77 (c+/3).
ft
is
an arbitrary number, as the angle II (ft) at the one
THEORY OF PARALLELS.
side of c
may be
41
chosen at will between the limits
and
quently 8 between the limits
oo
so
,
and
number
consecutively ft=c, 2c, 3c, &c, that for every positive
conse-
^rr,
we may deduce by
taking
[tan£
n,
/7(c)] »=tan£/7(nc).
If
we
consider n as the ratio of two lines
cot£77(c)=e
then
we
x and
c,
,
find for every line
x
whether
in general,
be positive or nega-
it
tan£77(x)=e— *
tive,
where e may be any arbitrary number, which
77(x)=0 for x=oo
since
is
greater than unity,
.
Since the unit by which the lines are measured
may
and assume that
c
by
Of the equations found above
also understand
37.
know
so
is arbitrary,
we
e the base of the Napierian Logarithms.
in
Theorem 35
sufficient to
it is
the two following,
sin /7(c)=. sin /7 fa) sin /7(b)
sin77 (a)=sin 77(b) cos
77(/3),
applying the latter to both the sides a and b about the right angle, in
order irom the combination to deduce the remaining two of Theorem
35,
without ambiguity of the algebraic sign, since here
all
angles are
acute.
In a similar manner we attain the two equations
We will
now
(1.)
tan 77(c)=sin 11(a) tan
(2.)
cos 77(a)=cos 77(c) cos II (/?).
77(a),
consider a rectilineal triangle whose sides are
a, b, c,
(Fig. 35) and the opposite angles A, B, C.
If
A and B are acute
angles, then the
perpendicular p from the vertex of the
angle
C
falls
within the triangle and cuts
the side c into two parts,
the angle
angle B.
Fia. 35.
triangles, for
which we
obtain,
A
and c
Thus
—x on
arise
x on the
two right-angled
by application of equation
#
tan 77(a)=sin
B tan 77(p),
tan 77(b)— sin
A tan 77 (p),
side of
the side of the
(1),
42
THEORY OP PARALLELS.
which equations remain unchanged also when one of the
a right angle (Fig. 36) or and obtuse angle (Fig. 37).
angles, e.g. B,
is
c
c
Fig. 36.
Therefore
we have
(3.)
universally for every triangle
sin
A tan /7(a)=sin B tan /7(b).
For a triangle with acute angles A,
B, (Fig. 35)
we have
also (Equa-
tion 2),
cos /7(x)=cos
A cos /7(b),
cos /7(c— x)=cos
which equations
or
B
As
is
a right angle or an obtuse
example, for
the other, however,
B=|tt
we have
23),
we must take x=c, the first ©quawe have found above as Equation 2,
(Fig. 36)
is self-sufficing.
we must
cos
(Theorem
A
(Fig. 37) the first equation remains unchanged, instead
of the second, however,
but
which one of the angles
angle.
tion then goes over into that which
For B>|tt
B cos /7(a)
also relate to triangles, in
cos /7(x
and
write correspondingly
/7(x— c)=cos (tt— B) cos
/7(a);
— c) == —cos /7(c — x)
also cos (re—
B)= —cos B.
A is a right or an obtuse angle, then must c— x and x be put
x and c — x, in order to carry back this case upon the preceding.
If
for
In order to eliminate x from both equations, we notice that (Theo-
rem 36)
l-[ tan£/?(c-x)P
cos/tfc-^-
1—
go
i_j_e 2x
—
8c
1— [tan |/7(c)] 2 [cot £/7(x)]»
:
1 -f[tan £/7(c)]2[cot £/7(x)]»
cos /7(c)
1
—
—
cos/7(x)
cos /7(c)cos/7(x)
THEORY OP PARALLELS.
If
we substitute
43
—
here the expression for cos /7(x), cos/7(c
we ob-
x),
tain
cos H(*) cos B-f-cos/7(b) cos
cob TT(c\
A
l-fcos/7(a) cos/7(b) cosA cosB
whence follows
cos /7(a)
v '
and
cos ff(c)-cosA cos/7(b)
cosB=
]—cosAcos77(b) cos 77(c)
finally
[sin /7(c)
]*=[1—cosBcos /7(c) cos /7(a) ][1
-cos
A cos /7(b) cos /7(c)]
In the same way we must also have
[sin /7(a) ]«
[sin /7(b) ]«
From
=[1—cos C cos /7(a) cos /7(b) ] [1—cos B cos /7 (c) cos /7(a)]
=[1—cos A cos /7 (b) cos /7(c) ] [1—-cos C cos /7(a) cos /7(b)]
these three equations
we
find
[8in/7(b)]2[sin/7(c)]2
t
^/(a)]^
Hence
follows without ambiguity of sign,
cos
(5.)
If
we
_
ri
„
=[l-cosAcos
/7(b)cos/7(c)p.
A cos /7(b)
w cos /7(c) >* /7(b) sin Z7(c)_. L
v '
sm/7(a)
'
substitute here the value of sin /7(c) corresponding to equa-
tion (3.)
„.
.
.
sm /7(c)=
then
we
sin
.
A tan
sinC
„,
/7 (a)' cos // v(c)
'
obtain
cos /7(c)
=sm A
sin
_^
cos /7(a) sinC
/7(b)+cos A sin C cos/7 (a) cos /7(b);
but by substituting this expression for cos /7(c) in equation
cot
(6.)
A sin C sin /7 (b)+cos C==
(4),
cos/y( b)
cos /7(a)
By elimination
of sin /7(b) with help of the equation (3) comes
cos /7(a)
cos
n
„
C08C=1
81 *
8in/7 (a)'
c"osl7(b)
In the meantime the equation
COS 11
1
—
A
sIuTT
(6) .gives
by changing the
8i\
j^tt =cot B
cos /7(b)
sin C sin JI (a)-i-cos
C.
r
v
'
letters,
44
THEORY OF PARALLELS.
From
the last two equations follows,
,»\
a
t>
B
.
cos A-4-cos
(7.
J
v
'
B
cos
sin
~ —
—„.sin .-C
C—
sin 77(a)
:
All four equations for the interdependence of the sides
a, b, c,
and
the opposite angles A, B, C, in the rectilineal triangle will therefore be,
[Equations
(3), (5), (6), (7).]
A tan
'sin
,
(8.)
v
'
= sin B tan 77 (b),
sin 77(b) sin /7(c)
A cos 77(b) cos 77(c)H
cos
^Jjjaj
=
*'
cos 77(b)
v '
/
Scot
cos
77(a)
A sin C sin 77(b) + cos C= cos
A
-\-
cos
B cos C =
sin
77 (a)
'
B sin C
sin 77 (a)
If the sides
a, b, c,
of the triangle are very small,
we may content our-
(Theorem
selves with the approximate determinations.
36.)
=
— £a*
sin 77 (a) =
cos 77 (a) =
cot 77(a)
a,
1
a,
like manner also for the other sides b and c.
The equations 8 pass over for such triangles into the
b sin A = a sin B,
and in
a2
=b
a sin
cos
Of these equations the
2
(A
A
— 2bc cos A,
= b sin A,
cos (B
C) =
8
-f c
-j-
-f-
first
following:
C)
0.
-j-
two
are assumed in the ordinary geom-
etry j the last two lead, with the help of the
first,
to the conclusion
A4-B + C=tt.
Therefore the imaginary geometry passes over into the ordinary,
we
when
suppose that the sides of a rectilineal triangle are very small.
I have, in the scientific bulletins of the University of Kasan, published certain researches in regard to the
measurement of curved
of plane figures, of the surfaces and the volumes of
relation to the application of imaginary
The
geometry to
lines,
solids, as well as in
analysis.
equations (8) attain for themselves already a sufficient foundation
for considering the assumption of
Hence there
is
imaginary geometry as possible.
no means, other than astronomical observations, to use
45
THEORY OF PARALLELS.
for judging of the exactitude
which pertains to the calculations of the
ordinary geometry.
This exactitude
is
very far-reaching, as I have shown in one of
investigations, so that, for example, in triangles
able for our measurement, the
sum
whose
of the three angles
my
sides are attain-
is
not indeed dif:
ferent from two right-angles by the hundredth part of a second.
In addition,
it is
worthy of notice that the four equations
(8)
plane geometry pass over into the equations for spherical triangles,
b y'— 1, c *J— 1, instead of the
change, however, we must also put
we put a ^/—
1,
=
cos
cos /7(a) = /y/— 1) tan
sin
H(&)
v '
sides
a, b, c;
a,
tan II (a) =-.
sina^y — 1),
and
similarly also for the sides
b and
c.
we pass over from equations
sin A sin b = sin B sin a,
In this manner
(8) to the following:
= cos b cos c sin b sin c cos A,
A sinC -j-cosC cosb = sinb cot
cos A = cos a sin B sin C — cos B cos 0.
cosa
cot
-j-
a,
if
with this
(a),
(
of
TRANSLATOR'S APPENDIX.
ELLIPTIC GEOMETRY.
Gauss himself never published aught upon this fascinating subject,
Geometry Non-Euclidean; but when the most extraordinary pupil of
his long teaching life
came
to read his inaugural dissertation before the
Philosophical Faculty of the University of Goettingen, from the three
was the choice of Gauss which fixed upon the one
"Ueber die Hypothesen welche der Geometric zu Grunde liegen."
themes submitted
it
Gauss was then recognized as the most powerful mathematician in the
world.
when
I
wonder
if
he saw that here his pupil was already beyond him,
in his sixth sentence
special
case
of a
" From this, however,
it
says, " therefore space is only
Riemann
three-fold
extensive
magnitude,"
a
and continues:
follows of necessity, that the propositions of
geometry can not be deduced from general magnitude-ideas, but that
those peculiarities through which space distinguishes itself from other
thinkable threefold extended magnitudes can only be gotten from experience.
Hence
arises the problem, to find the simplest facts
which the metrical
relations of space are
which from the nature of the thing
may be
is
determinable — a
ent aim made fundamental.
problem
not fully determinate; for there
obtained several systems of simple facts which
mine the metrics of space; that
from
suffice to deter-
of Euclid as weightiest
These facts
are, as all facts,
but only of empirical certainty; they are hypotheses.
is
for the pres-
not necessary,
Therefore one
can investigate their probability, which, within the limits of observation,
of course is very great,
and
after this
judge of the allowability of their
extension beyond the bounds of observation, as well on the side of the
immeasurably great as on the side of the immeasurably small."
Riemann extends the idea of curvature to spaces of three and more
The curvature of the sphere is constant and positive, and
move without deformation. The curvature of
constant
and
zero, and on it figures slide without stretching.
plane
is
the
The curvature of the two-dimentional space of Lobachevski and
dimensions.
on
it
figures can freely
[47]
48
THEORY OF PARALLELS.
Bolyai completes the group, being constant and negative, and in
move without
ures can
to the sphere
it is
In the space in
formation.
It
As
stretching or squeezing.
it fig-
thus corresponding
called the pseudo-sphere.
which we
would
live,
we suppose we can move without
then, according to
"We presume
a space of constant curvature.
de-
Riemann, be a special case of
its
curvature null.
At
once the supposed fact that our space does not interfere to squeeze us
when we move,
or stretch us
space.
But
thing?
If
it
is
it
is
envisaged as a peculiar property of our
not absurd to speak of space as interfering with any-
you think
into a seven-edged
so,
take a knife and a
raw
potato,
and try to cut
solid.
Further on in this astonishing discourse comes the epoch-making idea,
that though space be unbounded, it is not therefore infinitely great.
"In the extension
measurably great, the unbounded is
Riemann
finite;
the
says:
of space-constructions to the im-
first
pertains to the relations of extension, the latter to the
to
be distinguished from the
in-
size-relations.
"That our space is an unbounded three-fold extensive manifoldness,
a hypothesis, which
is
according to which, in each moment, the domain of actual perception
filled out,
is
and the possible places of a sought object constructed, and
which in these applications
The unbounded-
continually confirmed.
is
ness of space possesses therefore a greater empirical certainty than
outer experience.
Rather would space,
is ascribes
is
applied in each apprehension of the outer world,
to
it
From
if
this
however the
Infinity in
no way
any
follows.
one presumes bodies independent of place, that
a constant curvature, necessarily be
curvature had ever so small a positive value.
finite so
soon as this
One would, by extend,
ing the beginnings of the geodesies lying in a surface-element, obtain
an unbounded surface with constant
positive curvature, therefore a sur-
which in a homaloidal three-fold extensive manifoldness would
take the form of a sphere, and so is finite."
face
Here we have
for the
first
ception that universal space
Assume
time in
may
that a straight line
is
human thought
yet be only
the marvelous per-
finite.
uniquely determined by two points, but
take the contradictory of the axiom
tjjaat
a straight line
size; then the straight line returns into itself,
sected get back to that intersection point.
is
of infinite
and two having
inter-
BIBLIOGRAPHY.
A bibliography of non-Euclidean literature down to the year 1878
was given by Halsted, "American Journal of Mathematics," vols, i, ii,
containing 81 authors and 174 titles, and reprinted in the collected
works of Lobachevski (Kazan, 1886) giving 124 authors and 272 titles.
This was incorporated in Bonola's Bibliography of the Foundations of
Geometry (1899) reprinted (1902) at Kolozsvar in the Bolyai Memorial
In 1911 appeared the volume Bibliography of Non-Euclidean
Volume.
:
Geometry by Duncan M. Y. Sommerville
London, Harrison and Sons.
The Introduction says: "The present work was begun about nine
years ago. It was intended as a continuation of Halsted's bibliography,
but it soon became evident that the growth of the subject rendered such
diffuse treatment practically impossible, and short abstracts of the
works would have
;
to be dispensed with.
The object
is to
produce as
far as possible a complete repository of the titles of all works from
the earliest times up to the present which deal with the extended
conception of space, and to form a guide to the literature in an easily
accessible form.
It includes the theory of parallels, non-euclidean ge-
ometry, the foundations of geometry, and space of n dimensions."
In 1913 Teubner issued in two parts Paul Stackers important book:
Wolfgang und Johann Bolyai. Geometrische Untersuchungen. John com-
The profound philosophic
pares Lobachevski's researches with his own.
import of non-euclidean geometry forms an integrant part of "The
Foundations of Science," by H. Poincarg; Vol.
and Education, The Science
Fress,
New York
I of the series Science
City, 1914.
The Transac-
tions of the Royal Society of Canada, Vol. XII, Section III, contains
a striking Presidential Address by Alfred Baker on The Foundations
of Geometry.
Of the cognate works issued by The Open Court Pub.
Co.,
we mention
only Euclid's Parallel Postulate by Withers.
Scores
of errors are pointed out in "Non-Euclidean Geometry in the Encyclopaedia Britannica," Science,
And now
May
10, 1912.
at last the theory of relativity has
made
non-euclidean
geometry a powerful machine for advance in physics.
Says Vladimir Varicak in a remarkable
[49]
lecture,
"Ueber die nicht-
50
BIBLIOGRAPHY
euklidische Interpretation der Relativtheorie,"
Ver., 21, 103-127),
(Jahresber. D. Math.
'
I postulated that the phenomena happened in a Lobachevski space,
and reached by very simple geometric deduction the formulas of the
Assuming non-euclidean terminology, the formulas
relativity theory.
of the relativity theory become not only essentially simplified, but
capable of a geometric interpretation wholly analogous to the interpretation of the classic theory in the euclidean geometry.
far, that
analogy often goes so
the classic theory
may
be
left
And
this
the very wording of the theorems of
unchanged.