Download Geometrical researches on the theory of parallels.

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GEOMETRICAL RESEARCHES
OJf
THE THEORY
OF PARALLELS.
BY
NICHOLAUS LOBATSCHEWSKY,
IMPERIAL RUSSIAN REAL COUNCILLOR OP STATE AND REGULAR PROFESSOR
OF
MATHEMATICS IN
THE
UNIVERSITY
Translated
from
OP KASAN.
1840.
BERLIN,
the
Original
by
GEORGE BRUCE
A.
M.,
Ph.
D.,
Ex-Fellow of Princeton
HALSTED,
College
Professor of Mathematics in the
AUSTIN:
and Johns
University
Hopkins University,
of Texas.
FOURTH EDITION".
ISSUED FOR THE COUNCIL
OF THE
ASSOCIATION FOR THE IMPROVEMENT
GEOMETRICAL TEACHIXG-.
AND
DEDICATED
To the President of the
J. J.
SYLVESTER,
A. M. Cam.; F. R. S., L., and E. ;
of France; Member
Association,
Academy
Corresponding
Member Institute
of Sciences in
Berlin, Goettingen, Na
ples, Milan, St. Petersburg, etc.; LL. D., University of Dublin, and
U. of E.; D C. L Oxford; Hon. Fellow of St. John's
College, Cam.;
Fellow of New College, Oxford; Savilian Professor of
Geometry in
,
the
University
of Oxford,
By
his former
pupil,
THE
Jan. 1, 1892.
TRANSLATOR.
TRANSLATOR'S PREFACE.
Lobatschewsky was the fi>st man ever to publish a non-Euclidian 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 geom
spirit. I
perusal can not
ought to call your attention to this book, whose
give you the most vivid pleasure."
Clifford says, "It is quite simple, merely Euclid without the vicious
assumption, but the way things come out of one another is quite lovely."
*
*
*
''What Vesalius was to Galen, what Copernicus was to Ptolemy,
that was Lobatschewsky to Euclid."
Says Sylvester, "In Quaternions the example has been given of Al
gebra released from the yoke of the commutative principle of multipli
cation an emancipation somewhat akin to Lobatschewsky's of Geometry
from Euclid's noted empirical axiom."
Cay ley says, "It is well known that Euclid's twelfth axiom, even in
Playfair's form of it, has been considered as needing demonstration;
and that Lobatchewsky constructed a perfectly consistent theory, where
in this axiom was assumed not to hold good, or say a system of nonEuclidian plane geometry. There is a like system of non-Euclidian solid
eter's
think I
fail to
—
geometry."
GEORGE BRUCE HALSTED.
2407 San Marcos
Street,
Austin, Texas.
May 1, 1891.
[81
NOTE TO THE FOURTH EDITION.
The Translator has been
for this essay in its
for
permission
University
He has
gratified by
the
and
by
English dress,
to issue
an
application
from
Japan
of the
Imperial
of John
Bolyai's
use
of Tokio.
already
Saggio
be followed
the
edition there for the
in press
a
translation into
celebrated "Science Absolute of
trami's
unexpectedly great demand
by
di
Space,"
Interpretazione
English
and is at work upon Bel
della Geometria
his Teoria fondamentale
Non-Euclidea;
degli Spazii
to
di Curvatura
costante.
He may mention
as a
very
practical
result of his
Euclidean spaces, that since 1877 he has
Pure
as
Spherics,
or
Two-dimensional
new
Elementary Synthetic Geometry.
studies in
regularly taught
Spherics,
outlined in Book IX of his Elements of
own
or
say Plane
Geometry
and
non-
his classes
Spherics,
given
in his
TRANSLATOR'S INTRODUCTION.
things, hold fast that which is good," does not mean dem
From nothing assumed, nothing can be proved.
onstrate everything.
"Geometry without axioms," was a book which went through several
editions, 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
paradoxer.
The set of axioms far the most influential in the intellectual history
of the world was put together in Egypt: but really it owed nothing to
the Egyptian race, drew nothing from the boasted lore of Egypt's
priests.
The Papyrus of the Rhind, belonging to the British Museum, but
given to the world by the erudition of a German Egyptologist, Eisenlohr, and a German historian of mathematics, Cantor, gives us more
knowledge of the state of mathematics in ancient Egypt than all else
previously accessible to the modern world. Its whole testimony con
firms with overwhelming force the position that Geometry as a science,
strict and self-conscious deductive reasoning, was created by the subtle
"Prove all
intellect of the
Venus of
In
same race
the
Milo,
whose bloom in art still
Apollo Belvidere,
overawes us
in the
the Laocoon.
geometry occur the most noted set of axioms, the geometry of
Euclid, a pure Greek, professor at the University of Alexandria.
a
Not
its very birth did this typical product of the Greek genius
sway as ruler in the pure sciences, not only does its first efflor
only at
assume
escence
ages,
us
through
the
splendid days
latter,
fanatics
can
not murder
carry
unlike the
can
with the
not drown it.
new
Like the
birth of culture.
of Theon and
Hypatia,
it; that dismal flood, the
phcenix of its native Egypt, it rises
Anglo-Saxon, Adelard of Bath,
An
finds it clothed in Arabic vestments in the land of the Alhambra.
clothed
but
dark
it and the new-born
Then
press confer honor
on
printing
each other.
Finally back again in its original Greek, it is published
first in queenly Venice, then in stately Oxford, since then everywhere.
The latest edition in Greek is just issuing from Leipsic's learned presses.
m
Latin,
[5]
6
THEORY
How the first translation into
PARALLELS.
OF
our
cut-and-thrust, survival-of-the-fittest
English was made from the Greek and Latin by Henricus Billingsly,
Lord Mayor of London, and published with a preface by John Dee the
Magician, may be studied in the Library of our own Princeton College,
where they have, by some strange chance, Billingsly's own copy of the
Latin version of Commandine bound with the Editio
and enriched with his
system
emendations.
Princeps in Greek
to-day in the vast
Even
autograph
by Cambridge, Oxford, and the British gov
proof will be accepted which infringes Euclid's order, a
of examinations set
ernment,
no
sequence founded upon his set of axioms.
The American ideal is
expects
fection.
to be
In
twenty years the American maker
improved upon, superseded.
Epic and Lyric poets,
The Greek
unmatched.
for
success.
The axioms of
The Greek ideal
was
per-
the Greek
sculptors, remain
the Greek geometer remained unquestioned
twenty centuries.
How and where doubt
interest,
for this doubt
came
in the
epoch-making
axioms was one differing
Among Euclid's
lixity, whose place fluctuates
in Euclid's first
brought
to look toward them is of
was
in the
ordinary
from the others in pro
manuscripts, and which is not used
twenty-seven propositions.
in to prove the inverse of
history
no
of mind.
Moreover it is
of these
only
then
demonstrated.
already
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 in
verse, 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 make the two in
terior angles on the same side of it 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 that succeeding geometers wished by demonstration
one
All this
"
to
push
this
unwieldly thing
from the set of fundamental axioms.
TRANSLATOR'S
Numerous and
desperate
were
7
INTRODUCTION.
the attempts to deduce it from
about the nature of the
reason
line and
ings
straight
plane angle. In the
und
Kunste; Von Ersch und GruEncyclopcedie der Wissenschaften
ber;" Leipzig, 1838; under "Parallel," Sohncke says that in mathe
matics there is nothing over which so much has been spoken, written,
and striven, as over the theory of parallels, and all, so far (up to his
"
time),
without
reaching
a
definite result and decision.
defeat
Some
acknowledged
example the stupid one,
tant," still given on page 33
by taking
definition of
parallels, as
everywhere equally dis
of Schuyler's Geometry, which that author,
like many of his unfortunate prototypes, then attempts to identify with
Euclid's definition by pseudo-reasoning which tacitly assumes Euclid's
postulate, e. g. he says p. 35: "For, if not parallel, they are not every
where equally distant; and since they lie 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 if produced
farther, and if sufficiently produced, must meet." This is nothing but
Euclid's assumption, diseased and contaminated by the introduction of
for
a new
"Parallel lines
are
the indefinite term "direction."
"
How much better to have followed the third class of his
who
honestly
not in
assume a
essence.
new
axiom
differing
Of these the best is that
called Playfair's; "Two lines
the
same
The German article mentioned is followed
a
which intersect
can
not both be
parallel
to
predecessors
from Euclid's in form if
line."
by carefully prepared
subject. In English an account of
like attempts 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,
list of
ninety-two
authors
on
the
Oxford.
All this shows how
forth of
genius
ready the world was for the extraordinary flaming-
from different
parts of the world which
was
overturn, explain, and remake not only all this subject but
quence all
philosophy,
all ken-lore.
As
was
the
case
at once to
as conse
with the dis
covery of the Conservation of
of
genius,
whether in
Energy, the independent irruptions
Russia, Hungary, Germany, or even in Canada
gave everywhere the same results.
At first these results were not fully understood
even
by
the
brightest
8
THEORY
PARALLELS.
OF
publication of the book he mentions,
we see the brilliant Clifford
writing from Trinity College, Cambridge,
April 2, 1870, "Several new ideas have come to me lately: First I
have procured Lobatschewsky, 'Etudes Geometriques sur la Theorie
des Parallels'
a small tract of which Gauss, therein quoted,
intellects.
Thirty
-
says:
L'auteur
years after the
-
a
-
traite la matiere
esprit geometrique.
dont la lecture
ne
en
Je crois devoir
main de maitre et
appeler
peut manquer de
avec
votre attention
vous causer
le
plus
le veritable
sur ce
vif
livre,
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 first axiom doubted is called
"vicious
assumption," soon no
assumptions and none
clearly
vicious. He had been reading the French translation by Houel, pub
lished in 1866, of a little book of 61 pages published in 1S40 in Berlin
under the title Geometrische Untersuchungen zur Theorie der Parallellinien by a Russian, Nicolaus Ivanovitch Lobatschewsky (1793-1856),
the first public expression of whose discoveries, however, dates back to
a discourse at Kasan on
February 12, 1S26.
Under this commonplace title who would have suspected the disman sees
a
than Clifford that all
more
are
covery of a new space in which to hold our universe and ourselves.
A new kind of universal space; the idea is a hard one.
To name it,
all the space in which we think the world and stars live and move and
have their
being
was
ceded to Euclid
as
his
of
by right
description, and occupancy; then the new space and its
fellows could be called Non-Euclidean.
Gauss in
as
a
far back
Again
letter to
as
Schumacher,
dated Nov. 28, 1S46, mentions that
1792 he had started
he says:
pre-emption,
quick-following
this
path
to
"LaGeometrie non-Euclidienne
ne
on
a new
universe.
renferme
en
elle
rien de
contradictoire, quoique, a premiere vue, beaucoup de ses resultats aien l'air de paradoxes.
Ces contradictions apparents doivent etre
regardees comme l'effet d'une illusion, due a l'habitude que nous avons
prise de bonne heure de considerer la geometrie Euclidienne comme
rigoureuse."
But here
we see
Clifford's letter.
in the last word the
same
imperfection
of view
The perception has not yet come that though the
Euclidean geometry is rigorous, Euclid is not one whit less so.
as
in
non-
.
TRANSLATOR'S
9
INTRODUCTION.
A clearer idea here had
already come to the former
Hungarian Wolfgang Bolyai.
Gcettingen,
work, published by subscription, has the following title:
Gauss at
the
room-mate
His
of
principal
Tentamen Juventutem studiosam in elementa Matheseos purae, eleac sublimioris, methodo intuitiva, evidentique huic propria, in-
mentaris
Tomus
troducendi.
Primus, 1831; Secundus,
1833.
8vo.
Maros-Va-
sarhelyini.
In the first volume with
of his
special numbering, appeared the
Bolyai with the following title:
absolute
Axiomatis XI Euclidei
(a priori
tem.
Auctore Johanne
Caesareo
1832.
Regio
(26
celebrated
Johann
Appendix
Ap., scientiam spatii
son
veram
Bolyai
exhibens:
haud unquam
de
a
veritate aut falsitate
decidenda) independen-
eadem, Geometrarum in Exercitu
Captaneo. Maros-Vasarhely.,
Austriaco Castrensium
pages of
This marvellous
text).
Appendix
has been translated into
French, Italian,
and German.
In the title of
Wolfgang Bolyai's last work, the only one he com
posed in German (88 pages of text, 1851), occurs the following:
Und da die Frage, ob zwei von der driiten geschnittene Geraden wenn die
"
Summa der inneren
\Yinkelnicht=-2R,
worten
wird;
die davon
eine auf die Ja
Antwort,
sich schneiden oder
Euclid das
auf der Erde ohne ein Axiom
(wie
unabhasngige
nichtf, niemand
51) aufzustellen,
Geometrie
andere auf das Nein
beant-
abzusondern, und
so zu
bauen, dass
Formeln der letzen auf ein Wink auch
die
in der ersten
gultig seien."
Lobatschewsky 's Geometris^he Untersuchungen
Berlin, 1840, and compares it with the work of his son Johann Bolyai,
"an sujet duquel il dit'Quelques exemplaires de l'ouvrage publie ici
De
ont ete envoyes a cette epoque a Vienne, a Berlin, a Gcettingen.
du
sommet
le
des
hauteurs
Goettingen geant mathematique, [Gauss] qui
embrasse du meme regard les astres et la profondeur des abimes, a ecrit
qu'il etait ravi de voir execute le travail qu'il avait commence pour le
laisser apres lui dans ses papiers.'
Yet that which Bolyai and Gauss, a mathematician never surpassed
in power, see that no man can ever do, our American Schuyler, in the
density of his ignorance, thinks that he has easily done.
In fact this first of the Non-Euclidean geometries accepts all of Eu
clid's axioms but the last, which it flatly denies and replaces by its con-
The author mentions
.
"
10
THEORY
OF
PARALLELS.
tradictory, that the sum of the angles made on the same side of a trans
versal by two straight lines may be less than a straight angle without
the lines meeting.
A perfectly consistent 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.
THEORY OF PARALLELS.
In
son
geometry I find certain imperfections which I hold to be the rea
why this science, apart from transition into analytics, can as yet
make
As
no
advance from that state in which it has
belonging
to these
imperfections,
come
to
I consider the
us
from Euclid.
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
forts of mathematicians have been
so
of
parallels,
to fill which all ef
far in vain.
theory Legendre's endeavors hav9 done nothing, since he
only rigid way to turn into a side path and take
theorems
which he illogically strove to exhibit as
in
auxiliary
refuge
axioms.
first
My
essay on the foundations of geometry I pub
necessary
lished in the Kasan Messenger for the year 1829. In the hope of having
satisfied all requirements, I undertook hereupon a treatment of the whole
of this science, and published my work in separate parts in the "t?elehrten Schriften dxr XJniversiUet Kasan'" for the years 1836, 1837, 1838,
under the title New Elements of Geometry, with a complete Theory
of Parallels." The extent of this work perhaps hindered my country
men from following such a subject, which since Legendre had lost its
\nterest.
Yet I am of the opinion that the Theory of Parallels should
For this
was
forced to leave the
"
not lose its claim to the attention of
geometers, and therefore
I aim to
investigations, remarking beforehand that
give
of
to
the
Legendre, all other imperfections for ex
opinion
contrary
a
definition
of
the
straight line show themselves foreign here
ample,
and without any real influence on the theory of parallels.
In order not to fatigue my 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.
A straight line fits upon itself in all its positions. By this I mean
1.
that during the revolution of the surface containing it the straight line
does not change its place if it goes through two unmoving points m the
surface:
(i. e., if we turn the surface containing it about two points of
here the substance of my
—
—
the
line,
the line does not
move.)
Hi]
12
THEORY
PARALLELS.
straight lines can not intersect in
straight line sufficiently produced
2.
Two
3.
A
beyond
OF
bounds, and
all
in such way cuts
a
two
points.
both ways must go out
bounded plain into two parts.
perpendicular to a third never intersect, how
straight
far soever they be produced.
5.
A straight line always cuts another in going from one side of it
over to the other side:
(i. e., one straight line must cut another if it
has points on both sides of it.)
Vertical angles, where the sides of one are productions of the
6.
This holds of plane rectilineal angles
sides of the other, are equal.
as also of plane surface angles:
(i. e., dihedral angles.)
themselves,
among
Two straight lines can not intersect, if a third cuts them at the
7.
same angle.
In a rectilineal triangle equal sides lie opposite equal angles, and
8.
inversely.
In a rectilineal triangle, a greater side lies opposite a greater
9.
angle. In a right-angled triangle the hypothenuse is greater than either
of the other sides, and the two angles adjacent to it are acute.
10.
Rectilineal triangles are congruent if they 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.
A straight line which stands at right angles upon two other
11.
straight lines not in one plane with it is perpendicular to all straight
lines drawn through the common intersection point in the plane of those
4.
Two
lines
two.
a sphere with a plane is a circle.
right angles to the intersection of two per
pendicular planes, and in one, is perpendicular to the other.
14.
In a spherical triangle equal sides lie opposite equal angles, and
inversely.
15.
Spherical triangles are congruent (or symmetrical) if they have
sides
and the included angle equal, or a side and the adjacent
two
angles
equal.
From here follow the other theorems with their explanations and
proofs.
12.
The intersection of
13.
A
straight
line at
THEORY
All
16.
lines which in
straight
with reference to
a
into two classes
into
—
The
be
boundary
called parallel
and
culling
lines of the
one
a
plane
go out from
line in the
given straight
IS
PARALLELS.
OF
same
a
plane,
point
can,
be divided
nol-cidting.
and the other class of those lines will
to the given line.
point A (Fig. 1) let fall upon the
line BC the perpendicular AD, to which again
draw the perpendicular AE.
From the
In the
right angle EAD either will all straight
lines which go out from the
line DC, as for example AF,
point
A meet the
or some
of
them,
perpendicular AE, will not meet the
In the uncertainty whether the per
line DC.
pendicular AE is the only line which does not
meet DC, we will assume it may be possible that
there are still other lines, for example AG,
like the
which do not cut
Fig. 1.
prolonged. In pass
the not-cutting lines, as AG,
ing over from the cutting lines, as
we must come upon a line AH, parallel to DC, a boundary line, upon
one side of which all lines AG are such as do not meet the line DC,
while upon the other side every straight line AF cuts the line DC.
The angle HAD between the parallel HA and the perpendicular AD
is called the parallel angle (angle of parallelism), which we will here
p.
designate by fl (p) for AD
If // (p) is a right angle, so will the prolongation AE' of the perpen
dicular AE likewise be parallel to the prolongation DB of the line DC,
in addition to which we remark that in regard to the four right angles,
which are made at the point A by the perpendiculars AE and AD,
and their prolongations AE' and AD', every straight line which goes
out from the point A, either itself or at least its prolongation, lies in one
of the two right angles which are turned toward BC, so that except the
parallel EE' all others, if they are sufficiently produced both ways, must
DC,
how far
they
AF, to
soever
may be
=
intersect the line BC.
If J!
(p)
< >V
~»
make
a
AD, making the same
parallel to the prolonga
assumption we must also
then upon the other side of
angle DAK = H (p) will lie also a line AK,
tion DB of the line DC, so that under this
distinction of sides in parallelism.
14
THEORY
All
remaining
lines
or
their
OF
PARALLELS.
prolongations
within the two
right angles
lie within the
if
intersect,
they
parallels; they pertain on the other
hand to the non-intersecting AG, if they lie upon the other sides of the
\ ,-r
parallels AH and AK, in the opening of the two angles EAH
J
77 (p), E'AK
// (p), between the parallels and EE'-the per
pendicular to AD. Upon the other side of the perpendicular EE' will
in like manner the
prolongations AH' and AK' of the parallels AH and
AK likewise be parallel to BC ; the remaining lines pertain, if in the
angle K'AH', to the intersecting, but if in the angles K'AE, H'AE'
to the non-intersecting.
In accordance with this, for the assumption 77 (p)
\ ~- the lines can
be only intersecting or parallel ; but if we assume that 77 (p) < \ rr, then
turned toward BC
angle HAK
=
2
pertain
77
(p)
to those that
between the
=
=
—
r-
—
=
we
must allow two
in addition
ing
and
we
parallels, one on the one and one on the other side;
distinguish the remaining lines into non-intersect
must
intersecting.
For both
assumptions it
line becomes
serves as
where lies the
intersecting
parallel, so that
DC, how
the
cuts
the mark of
parallelism
that the
for the smallest deviation toward the side
small
soever
if AH is
angle
parallel
to
HAF may be.
DC,
every line AF
THEORY
17.
A.
straight
OF
15
PARALLELS.
line maintains the characteristic
of parallelism
a,t all its
points.
Given AB
(Fig. 2) parallel
to
CD,
to which latter AC is
perpendic
Fio. 2.
ular.
points taken
perpendicular.
We will consider two
its
the
at
random
on
the line AB and
production beyond
point E lie on that side of the perpendicular on which AB is
looked upon as parallel to CD.
Let fall from the point E a perpendicular EK on CD and so draw EF
that it falls within the angle BEK.
Connect the points A and F by a straight line, whose production then
Let the
(by
Theorem 1 6) must cut CD somewhere in G.
Thus
we
get a triangle
into which the line EF goes; now since this latter, from the con
struction, can not cut AC, and can not cut AG or EK a second time
ACG,
(Theorem 2),
therefore it must meet CD somewhere at H
Now let E' be
a
point
on
the
production
(Theorem 3).
of AB and E'K'
perpendic
production of the line CD; draw the line E'F' making so
small an angle AE'F' that it cuts AC somewhere in F'; making the
same angle with AB, draw also from A the line AF, whose production
will cut CD in G (Theorem 16.)
Thus we get a triangle AGC, into which goes the production of the
line E'F'; since now this line can not cut AE a second time, and also
can not cut AG, since the angle BAG
BE'G', (Theorem 7), therefore
ular to the
=
must it meet CD somewhere in G'
Therefore from whatever
points
E and E' the lines EF and E'F' go
out, and however little they may diverge from the line AB, yet will
they always cut CD, to which AB is parallel.
16
THEORY
18.
Two lines
Let AC be
if
we
CE
a
OF
PARALLELS.
always mutually parallel.
perpendicular on CD, to
are
which AB is
parallel
draw from C the line
making
any acute
angle
ECD with CD, and let fall
perpendicular AF
upon CE, we obtain a rightangled triangle ACF, in which
AC, being the hypothenuse,
from A the
is greater than the side AF
(Theorem 9.)
Make AG
=
AF,
and slide
Fig. 3.
AG, when AB and FE wiJ
figure
take the position AK and GH, such that the angle BAK
FAC, con
sequently AK must cut the line DC somewhere in K (Theorem 16), thus
forming a triangle AKC, on one side of which the perpendicular GH
intersects the line AK in L (Theorem 3), and thus determines the dis
tance AL of the intersection point of the lines AB and CE on the lino
AB from the point A.
Hence it follows that CE will always intersect AB, how small soever
nay be the angle ECD, consequently CD is parallel to AB (Theorem 1 6.)
19.
In a rectilineal triangle the sum of the three angles can not be greats,
han two right angles.
Suppose in the triangle ABC (Fig. 4) the sum of the three angles is
iqual to tz -\- a; then choose in case
)f the inequality of the sides the
imallest BC, halve it in D, draw
:rom A
through D the line AD
md make the prolongation of it,
DE, equal to AD, then join the
Fig. 4.
joint E to the point C by the
line
EC.
In
the congruent triangles ADB and
straight
CDE, the angle
ABD
DEC (Theorems 6 and 10); whence follows
DOE, and BAD
ihat also in the triangle ACE the sum of the three
angles must be equal
» 71
+ a; but also the smallest angle BAC (Theorem 9) of the triangle
ABC in passing over into the new triangle ACE has been cut
up into
,he two parts EAC and AEC.
Continuing this process, continually
EFAB until AF coincides with
the
=
=
=
THEORY
17
PARALLELS.
OF
halving the side opposite the smallest angle, we must finally attain to a
triangle in which the sum of the three angles is -\- 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 r., so must a be
either null or negative.
20.
If in any rectilineal triangle the sum of the three angles is equal to
-
two
right angles,
so
is this also the
If in the rectilineal
=
,
C, be
angles,
the three
angles
A
B upon the
angle
opposite side AC
perpendicular p. This will cut the tri- A
angle into two right-angled triangles, in each
of which the sum of three angles must also be
i»C
the
either bo
greater
ti,
i
G'
n,
since it
can
and in their combination not less than
not in
n-
right-angled triangle with the perpendicular sides p
a
quadrilateral whose opposite sides are equal and
whose adjacent sides p and q are at right angles (Fig. 6.)
By repetition of this quadrilateral we can make another with sides
np and q, and finally a quadrilateral ABCD with sides at right angles
So
we
obtain
than
a
and q, and from this
to each
other, such that AB
=
np, AD
any whole numbers.
=
mq, DC
Such
=
np, BC
=
mq, where
quadrilateral is divided by
the diagonal
congruent right-angled triangles, BAD and
rz.
BCD, in each of which the sum of the three angles
The numbers n and m can be taken sufficiently great for the rightangled triangle ABC (Fig. 7) whose perpendicular sides AB np, BC
another given (right-angled) triangle BDE
mq, to enclose within itself
m
and
n are
a
DB into two
=
=
=
as soon as
tho
2
right-angles
—
car.
fit each other.
/
-
-
triangle.
(Fig. 5) the sum of
Let fall from the vertex of
acute.
the third
every other
for
ABC
then must at least two of its
—
and
triangle
case
i
18
THEORY
Drawing
the line
successive two have
DC,
a
we
OF
obtain
side in
PARALLELS.
right-angled triangles of which every
common.
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
tt; consequently it must be equal to 7-., in order that the sum in the
compound triangle may be equal to -.
The
In the
triangle BDC consists of the two triangles DEC
DBE, consequently must in DBE the sum of the three angles be
equal to 77, and in general this must be true for every triangle, since
each can be cut into two right-angled triangles.
From this it follows that only two hypotheses are allowable :
Either
is the sum of the three angles in all rectilineal triangles equal to 77, or
same
way the
and
this
sum
21.
is in all less than
From
make with
Let fall
a
n.
given point we can always draw a straight line that shall
given straight line an angle as small as we choose.
from the given point A (Fig. 8) upon the given line BC the
a
D
E
C
Fig. 8.
perpendicular AB ; take upon BC at random
AD; make DE
AD, and draw AE.
=
the
point
D
■
draw the line
THEORY
OF
19
PARALLELS.
In the
right-angled triangle ABD let the angle ADB a; then must
triangle ADE the angle AED be either |a or less (Theo
and 20).
Continuing thus we finally attain to such an angle,
=
in the isosceles
8
rems
AEB,
as
22.
is less than any
If
perpendiculars to the same straight line are parallel to each
sum
of the three angles in a rectilineal triangle is equal to two
then the
other,
given angle.
two
right angles.
Let the lines AB and CD
pendicular
(Fig. 9)
be
parallel
to each other and per
to AC.
Draw from A the lines AE
and AF to the
which
are
points
taken
on
E and
F,
the line CD
at any distances FC > EC from
the
point C.
Suppose in the right-angled tri
angle ACE the sum of the three angles
angle AEF equal to tt
/9, then must it
—
—
ft,
where
Further,
a
and R
can
not be
let the angle BAF
=
is
in
a, in the
equal to
ACF
triangle
equal n
-
—
tri
—
a
negative.
a, AFC
=
b,
so
is
a
+/?
=
a
—
b;
now
by revolving the line AF away from the perpendicular AC we can make
the angle a between AF and the parallel AB as small as we choose; so
also
can
we
can
have
no
lessen the
angle b, consequently the two angles a and R
0 and R
0.
magnitude than a
It follows that in all rectilineal triangles the sum of the three angles
is either tt and at the same time also the parallel angle 77 (p)
\ n for
or for all triangles this sum is < tt and at the same time
line
p,
every
also 77 (p) < i -.
The first 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 in
tend here to expound as far as the development of the equations be
tween the sides and angles of the rectilineal and spherical triangle.
23.
For every given angle a we can find a line p such that II (p)
aother
=
=
=
=
Let AB and AC
section
point
(Fig. 10)
be two
A make the acute
angle
straight
a;
take
lines which at the inter.
at random
on
AB
a
point
20
OF
THEORY
PARALLELS.
B'; from this point drop B'A' at right angles
AA'; erect at A" the perpendicular A"B"; and
A.'
A"
to
so
AC;
make A'A"
continue until
r
K
a
-
-
per-
c
Fig. 10.
pendicular CD is attained, which no longer intersects AB. This must
of necessity happen, for if in the triangle AA'B' the sum of all three
2 a,
a, then in the triangle AB' A" it equals 77
angles is equal to 77
until
2a (Theorem 20), and so forth,
in triangle AA'B" less than 77
—
—
—
finally becomes negative
structing the triangle.
it
The
point
perpendicular
and
thereby
shows the
CD may be the very
A all others cut AB ; at least in the
cut to those not
such
impossibility
one nearer
of
con
than which to the
passing
perpendicular FG must
over
from those that
exist.
cutting
point F the line FH, which makes with FG the
acute angle HFG, on that side where lies the point A. From any point
H of the line FH let fall upon AC the perpendicular HK, whose pro
longation consequently must cut AB somewhere in B, and so makes a
triangle AKB, into which the prolongation of the line FH enters, and
therefore must meet the hypothenuse AB somewhere in M.
Since the
is
we wish, therefore
GFH
and
can
small
as
be
taken
as
angle
arbitrary
FG is parallel to AB and AF
p.
(Theorems 1 6 and 18.)
One easily sees that with the lessening of p the angle a increases, while,
for p
0, it approaches the value -J--; with the growth of p the angle
a decreases, while it continually approaches zero for p =00
Since we are wholly at liberty to choose what angle we will underDraw
now
a
from the
=
=
.
THEORY
stand
the
by
number,
symbol TI (p)
so we
will
OF
21
PARALLELS.
when the line p is
expressed by
a
negative
assume
77(p)+77(-P)=;r,
equation which
ative, and for p
an
=
24.
lelism,
The
the
shall hold for all values of p,
0.
farther parallel lines are prolonged
they approach one another.
positive as
on
the side
well
of
as
neg
their paral
more
If to the line AB
(Fig.
1
BE are erected
perpendiculars AC
by c
F
a
straight line, then will the quadrilat
eral CABE have two right angles at
A and B, but two acute angles at C
and E (Theorem 22) which are equal
°to one another, as we can easily see A>
^IG- *!■
by thinking the quadrilateral superimposed upon itself so that the line BE falls upon upon AC and AC
1)
two
=
and their end -points C and E joined
upon BE.
Halve AB and erect at the
mid-point D the line DF perpendicular to
perpendicular to CE, since the quadrilat
erals CADF and FDBE fit one another if we so place one on the other
Hence the line CE can
that the line DF remains in the same position.
not be parallel to AB, but the parallel to AB for the point C, namely
CG, must incline toward AB (Theorem 1 6) and cut from the perpendic
AB.
This line must also be
ular BE
a
part BG < CA.
Since C is
nears
AB the
a
random
more
point
in the line
the farther it is
CG,
prolonged.
it follows that CG itself
22
THEORY
25.
one
Two
straight
lines which
OF
are
PARALLELS.
parallel
to
a
third
are
also parallel
another.
Fig. 12.
We will first
one
plane.
assume
that the three lines
If two of them in
order,
AB and CD
AB, CD, EF (Fig. 12) lie in
AB and CD,
are
parallel
another.
to
the
In order
parallel to one
point A of the outer line AB upon the
other outer line FE, the perpendicular AE, which will cut the middle
on the
line CD in some point C (Theorem 3), at an angle DCE < £
side toward EF, the parallel to CD (Theorem 22).
A perpendicular AG let fall upon CD from the same point, A, must
fall within the opening of the acute angle ACG (Theorem 9); every
other line AH from A drawn within the angle BAC must cut EF, the
parallel to AB, somewhere in H, how small soever the angle BAH may
be; consequently will CD in the triangle AEH cut the line AH some
where in K, since it is impossible that it should meet EF.
If AH from
the point A went out within the angle CAG, then must it cut the pro
longation of CD between the points C and G in the triangle CAG.
Hence follows that AB and CD are parallel (Theorems 16 and 18).
Were both the outer lines AB and EF assumed parallel to the middle
line CD, so would every line AK from the point A, drawn within the
angle BAE, cut the line CD somewhere in the point K, how small soever
the angle BAK might be.
Upon the prolongation of AK take at random a point L and join it
outmost one,
to prove
this,
EF,
so are
let fall from any
-
THEORY
with C
ing
a
by the line CL,
triangle MCE.
The
prolongation
neither AC
where in
nor
CM
OF
which must cut EF somewhere in
of the line AL within the
a
second
H; therefore
time, consequently
AB and EF
are
Fig,
Now let the
parallels
23
PARALLELS.
From
EF.
fall
EA upon
MCE
mak
can
it must meet EF
cut
some
mutually parallel.
13.
AB and CD
intersection line is
triangle
M, thus
(Fig. 1 3)
random
lie in two
E of
planes
whose
this latter let
point
parallels, e. g., upon AB,
then from A, the foot of the perpendicular EA, let fall a new perpen
dicular AC upon the other parallel CD and join the end-points E and C
The angle BAC must be
of the two perpendiculars by the line EC.
acute (Theorem 22), consequently a perpendicular CG from C let fall
upon AB meets it in the point G upon that side of CA on which the
lines AB and CD are considered as parallel.
Every line EH [in the plane FEAB], however little it diverges from
EF, pertains with the line EC to a plane which must cut the plane of
the two parallels AB and CD along some line CH. This latter line cuts
AB somewhere, and in fact in the very point H which is common to all
three planes, through which necessarily also the line EH goes; conse
quently EF is parallel to AB.
In the same way we may show the parallelism of EF and CD.
Therefore the hypothesis that a line EF is parallel to one of two other
parallels, AB and CD, is the same as considering EF as the intersection
of two planes in which two parallels, AB, CD, lie.
Consequently two lines are parallel to one another if they are parallel
to a third line, though the three be not co-planar.
The last theorem can be thus expressed:
Three planes intersect in lines which are all piarallel to each other if the
parallelism of two is presupposed.
a
perpendicular
one
a
of the two
24
THEORY
OF
PARALLELS.
Triangles standing opposite to one another on the sphere are equiva
lent in surface.
on both
By opposite triangles we here understand such as are made
sides of the center by the intersections of the sphere with planes; in such
triangles, therefore, the sides and angles are in contrary order.
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
sides AB
A'B', BC
B'C, CA CA', and the corresponding angles
26.
=
=
=
Fig. 14.
points A, B, C are likewise equal to those in the other triangle at
the points A', B', C.
Through the three points A, B, C, suppose a plane passed, and upon
it from the center of the sphere a perpendicular dropped whose pro
longations both ways cut both opposite triangles in the points D and D'
The distances of the first D from the points ABC, in
of the sphere.
arcs of great circles on the sphere, must be equal
(Theorem 12) as well
to each other as also to the distances D'A', D'B', D'C, on the other
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.
at the
27.
A three-sided solid angle equals the half sum of the surface
angles
right-angle.
In the spherical triangle ABC (Fig. 1 5), where each side < -,
desig
nate the angles by A, B, C; prolong the side AB so that a whole circle
ABA'B' A is produced ; this divides the sphere into two equal parts.
less
a
THEORY
In that half in which is the
sides
through
their
common
OF
25
PARALLELS.
triangle ABC, prolong now the other two
intersection point C until they meet the
circle in A' and B'.
Fig. 15.
In this way the
hemisphere
is divided into four triangles,
B'CA', A'CB,
whose size may be designated
dent that here P + X
A.
B, P + Z
=
The size of the
ABC, ACB',
by P, X, Y, Z.
It is evi.
=
spherical triangle Y equals that
of the
opposite triangle
triangle P, and whose
lies at the end-point of the diameter of the sphere which
through the center D of the sphere (Theorem 26). Hence
ABC, having
third angle C
a
side AB in
with the
common
goes from C
it follows that
P
-j-
Y=
C,
and since
P-|-K-|-Y-f-Z
P
=
i(A +
B
=
;T,
therefore we have also
+ C-;r).
We may attain to the same conclusion in another way, based solely
upon the theorem about the equivalence of surfaces given above. (Theo
rem
26.)
In the
spherical triangle ABC (Fig. 16),
and through the mid-points D and
E draw a great circle; upon this let
fall from A, B, C the perpendiculars
AF, BH, and CG. If the perpendic- p
halve the sides AB and
BC,
B
ular from B falls at H between D and
E, then will of the triangles
BDH
=
orems 6
AFD,
and
and BHE
15),
the surface of the
(Theorem 26).
=
so
EGC
made
(The-
whence follows that
triangle
ABC
equals that
Fig. 16.
of the
quadrilateral AFGO
26
THEORY
If the
point
OF
PARALLELS.
H coincides with the middle
B
y/\
~~^~~~~-^
point E of
the side BC
(Fig.
1 7), only
equal right-angled triangles, ADF
and BDE, are made, by whose interchange the
of the triangle ABC
of the
two
surface.s
equivalence
and the quadrilateral AFEC is established.
/
I If, finally, the point H falls outside the triangle
-~^ ABC
in
Y~
(Fig. 18), the perpendicular CG goes,
so we go
and
the
Fig. 17.
triangle,
consequence, through
the
over from the triangle ABC to the quadrilateral AFGC by adding
/
Fig. 18.
EBH.
DBH, and then taking away the triangle CGE
triangle FAD
in
the
a
AFGC
circle
spherical
quadrilateral
Supposing
passed
great
through the points A and G, as also through F and C, then will their
arcs between AG and FC equal one another
(Theorem 15), consequently
also the triangles FAC and ACG be congruent (Theorem 15), and the
angle FAC equal the angle ACG.
Hence follows, that in all the preceding cases, the sum of all three
angles of the spherical triangle equals the sum of the two equal angles
in the quadrilateral which are not the right angles.
Therefore we can, for every spherical triangle, in which the sum of
the three angles is S, find a quadrilateral with equivalent surface, in
which are two right angles and two equal perpendicular sides, and
where the two other angles are each 4-S.
=
=
THEORY
Let
each
ABCD
now
sides AB
=
DC
OF
27
PARALLELS.
(Fig. 19) be the spherical quadrilateral,
perpendicular to BC, and the angles
are
where the
A and D
£S.
Fig. 19.
Prolong the sides AD
beyond E, make
further
and BC until
DE
=
they
another in
E, and
prolongation
BG and join the
cut one
EF and let fall upon the
perpendicular FG. Bisect the whole arc
mid-point
by great-circle-arcs with A and F.
The triangles EFG and DCE are congruent (Theorem 15),
of BC the
H
DC
=
The
so
FG
=
AB.
triangles
ABH and HGF
are
likewise congruent, since
they
are
right angled and have equal perpendicular sides, consequently AH and
AF pertain to one circle, the arc AHF
tt, ADEF likewise
tt, the
BAH
HFE
HFG
HFE-EFG
+S
HAD
|S
£S
angle
£S— HAD— tt-HS; consequently, angle HFE |(S— tt); or what
—-
=
=
-
= --
=
=
-
-
=
=
is the same, this
equals
the size of the lune
quadrilateral ABCD,
AHFDA,
if
which
again
is
from
pass
easily
by first adding the triangle EFG and then BAH
and thereupon taking away the triangles equal to them DCE and HFG.
Therefore £(S tt) is the size of the quadrilateral ABCD and at the
in which the sum of the
same time also that of the spherical triangle
three angles is equal to S.
equal
the
to the
one
to the other
—
as we
see
we
over
28
THEORY
OF
PARALLELS.
If three planes cut each other in parallel lines, then the sum of the
three surface angles equals two right angles.
Let A A', BB' CC (Fig. 20) be three parallels made by the inter
section of planes (Theorem 25).
Take upon them at random three
28.
Fig. 20.
B, C, and suppose through these a plane passed, which con
will cut the planes of the parallels along the straight lines
points A,
sequently
AB, AC, and BC. Further, pass through the line AC and any point
D on the BB', another plane, whose intersection with the two planes of
the parallels AA' and BB', CC and BB' produces the two lines AD
and DC, and whose inclination to the third plane of the parallels AA'
and CC we will designate by w.
The angles between the three planes in which the parallels lie will
be designated by X, Y, Z, respectively at the lines AA', BB', CC;
c.
finally call the linear angles BDC
a, ADC
b, ADB
About A as center suppose a sphere described, upon which the inter
sections of the straight lines AC, AD AA' with it determine a spherical
triangle, with the sides p, q, and r. Call its size a. Opposite the side
q lies the angle w, opposite r lies X, and consequently opposite p lies
the angle 7T-\-2a—w—X, (Theorem 27).
In like manner CA, CD, CC cut a sphere about the center C, and
determine a triangle of size
ft, with the sides p', q', r', and the angles, v:
opposite q', Z opposite r', and consequently 7T-{-'2ft— w— Z opposite p'.
Finally is determined by the intersection of a sphere about D with
the lines DA, DB, DC, a spherical triangle, whose sides are 1, m, n, and
the angles opposite them w-\-7i
"2ft, w-^-'K la, and Y. Consequently
=
—
=
=
—
its size
d=i (X+Y+Z—-)— a—ft+w.
Decreasing w lessens also the size of the triangles a
u-\-ft w can be made smaller than any given number.
—
and
ft,
so
that
THEORY
OF
29
PARALLELS.
In the
triangle o can likewise the sides 1 and m be lessened even to
vanishing (Theorem 21), consequently the triangle d can be placed with
one of its sides 1 or m
upon a great circle of the sphere as often as you
choose without thereby filling up the half of the sphere, hence 3 van
ishes together with w; whence follows that necessarily we must have
X+Y+Z=-;r
29.
In a rectilineal triangle, the perpendiculars erected at the mid-points
of the sides either do not meet, or they all three cut each other in one point.
Having pre-supposed in the triangle ABC (Fig. 21), that the two per
pendiculars 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.
In the congruent triangles ADE and BDE (Theorem 10), we have
AD=BD, thus follows also that BD^CD; the
triangle ADC is hence isosceles, consequently the
perpendicular dropped from the vertex D upon the
base 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
°
diculars ED and FD falls in the line AC itself, or7
A
falls without the
Fl°- 2L
triangle.
therefore pre-suppose that two of those perpendiculars do
intersect, then also the third can not meet with them.
The perpenelicidars which are erected upon the sides of a rectilineal
30.
In
case we
not
mid-points, must all three be parallal to each other, so soon
is presupposed.
as the parallelism of two of them
In the triangle ABC (Fig. 22) let the lines DE, FG, HK, be erected
triangle
at their
upon the sides at their mid
We will in the first place
H.
perpendicular
points D, F,
assume
that the two
perpendiculars
DE and
the line AB in L
FG
parallel, cutting
and M, and that the perpendicular HK lies
between them. Within the angle BLE draw
are
from the
point L,
LG, which
how small
at
random,
a
straight
line
must cut FG somewhere in
soever
the
angle
G,
Fig. 22.
of deviation GLE may be.
(Theorem 16).
30
OF
THEORY
PARALLELS.
with
triangle LGM the perpendicular HK can not meet
whence
in
P,
MG (Theorem 29), therefore it must cut LG somewhere
follows, that HK is parallel to DE (Theorem 16), and toMG (Theorems
18 and
25).
Since in the
the
2c, and designate
2b, AB
case just
the
in
have
then
we
these
sides
by A, B, C,
gles opposite
Put the side BC
2a, AC
=
an-
=
=
considered
A=
77(b)—77(c),
B= 77 (a)—77(c),
C
as one
are
may
easily
consequently
rems
23 and
Let
now
to both the other
perpendiculars
DE and FG
(Theo
25).
the two
perpendiculars
the third DE not cut them
them,
77(a)+77(b),
help of the lines AA', BB', CC, which
points A, B, C, parallel to the perpendicular HK
drawn from the
and
=
show with
parallel, then can
it either parallel to
HK and FG be
(Theorem 29),
hence is
it cuts AA'.
or
The last
assumption
is not other than that the
angle
C>77(a)+77(b.)
If
while
we
lessen this
we
angle,
give
in that way
so
that it becomes
the line AC the
new
equal to 77(a) -(-77(b),
position CQ, (Fig. 23),
designate the size of the third side BQ by 2c', then must the angle
CBQ at the point B, which is increased, in accordance with what is
proved above, be equal to
and
whence follows c' >c
77(a)-77(c')> 77(a)—77(c),
(Theorem 23).
Fig. 23.
In the
triangle ACQ are, however, the angles at A and Q equal,
triangle ABQ must the angle at Q be greater than that at
the point A, consequently is AB>BQ, (Theorem 9); that is c>c'.
31.
We call boundary line (oricycle) that curve lying in a plane
for
hence in the
which all
perpendiculars
each other.
erected at the
mid-points of
chords
are
parallel
to
THEORY
In
a
conformity
boundary line,
OF
with this definition
if
we
draw to
a
31
PARALLELS.
we can
represent the generation of
line AB
given
(Fig. 24)
from
a
given
Fig. 24.
point
A in
it, making different angles CAB
the end C of such
chord will lie
a
on
the
=
77(a),
chords AC
boundary line,
whose
=
2a;
points
gradually determine.
perpendicular DE erected upon the chord AC at its mid-point D
will be parallel to the line AB, which we will call the Axis of the bound
In like manner will also each perpendicular FG erected at the
ary line.
mid-point of any chord AH, be parallel to AB, consequently must this
peculiarity also pertain to every perpendicular KL in general which is
erected at the mid-point K of any chord CH, between whatever points
C and H of the boundary line this may be drawn (Theorem 30). Such
perpendiculars must therefore likewise, without distinction from AB,
we can
thus
The
be called Axis
of the boundary line.
continually increasing radius
A circle toith
32.
merges into the
boundary
line.
Given AB
end-points
(Fig. 25)
a
chord of the
A and B of the chord two
/
/
BF, which consequently will
make with the chord two equal angles
AC and
BAC
=
Upon
ABF
one
where the
--
■-
a
point
/
Ifcfe^
V\.
(Theorem 31).
of these
E
as
axes
AC,
ty/
take any-
center of
a
draw from the
boundary line;
axes
circle,
and draw the arc AF from the initial point
A of the axis AC to its intersection point
vJ
\
jf—
-^
—
c
Fig. 25.
F with the other axis BF.
The radius of the
on
the
one
circle, FE, corresponding
side with the chord AF
an
to the
angle
point
AFE
=
ft,
F will make
and
on
the
32
PARALLELS.
OF
THEORY
BF, the angle EFD
other side with the axis
between the two chords BAF
angle
whence
Since
follows,
now
=
y.
It follows that the
ft<ft+y— a (Theorem 22);
=
a—
ft<iy-
—
a
however the
angle
r
approaches
the limit zero,
as
well
m
when
r
moving of the center E in the direction AC,
remains unchanged, (Theorem 21), as also in consequence of an ap
in its
proach of F to B on the axis BF, when the center E remains
the
of
position (Theorem 22), so it follows, that with such a lessening
chords
angle y, also the angle .-;. ft, or the mutual inclination of the two
AB and AF, and hence also the distance of the point B on the bound
ary line from the point F on the circle, tends to vanish.
Consequently one may also call the boundary-line a circle with in
finitely great radius.
consequence of
a
—
Let AA'
33.
=
B3'
=
be two lines
(Figure 26).
x
the side from A to
A', which parallels serve
boundary arcs (arcs on
boundary lines) AB=s, A'B'=.s', then is
as axes
two
toward
>|L
for the two
s'
where
parallel
=
se
—
x
is
Fig. 26.
independent of the arcs s, s' and of
straight line x, the distance of the arc s' from s.
In order to prove this, assume that the ratio of the arc s to s' is
equal to the ratio of the two whole numbers n and m.
Between the two axes A A', BB' draw yet a third axis CC, which
e
the
so
cuts off from the
the
same
side,
a
arc
AB
part A'C
part AC
a
=
t'
.
that of the whole numbers p and q,
=
—
Divide
on
now s
by
s' and np
However there
equal parts
on
axes
on
into nq
arc
t to
*
A'B'
on
equal
to
that-
so
p
s',
I=
m
parts
from the
Assume the ratio of
n
s=
t and
—
s.
q
equal parts, then
will there be
too
such
t.
correspond to these equal parts
l', consequently we have
on s
and
t
likewise
s' and
f
t
Hence also wherever the two
two
axes
AA' and
BB',
s
arcs t
the ratio of
and V
may be taken between the
t to V
remains
always
the
same
as
THEORY
long
as
fore for
the distance
x=
i,
put
x
s=
OF
between them remains the
e.s',
then
s'
Since
e
is
an
33
PARALLELS.
we
=
unknown number
and further the linear unit for
x
must
If
same.
have for every
we
there
x
se~ x.
only subjected
may be taken at
to the condition
e>i,
therefore
may,
will,
we
for the
simplification of reckoning, so choose it that by e is to be un
derstood the base of Napierian logarithms.
We may here remark, that s'= 0 for a;= oo hence not only does
the distance between two parallels decrease (Theorem 24), but with the
prolongation of the parallels toward the side of the parallelism this at
last wholly vanishes.
Parallel lines have therefore the character of
,
asymptotes.
Boundary surface (orisphere) we call that surface which arises
from the revolution of the boundary line about one of its axes, which,
together with all other axes of the boundary -line, will be also an axis
of the boundary-surface.
A chord is inclined at equal angles to such axes drawn through its endpoints, wheresoever these two end-points may be taken on the boundary-surface.
Let A, B, C, (Fig. 27), be three points on the boundary-surface;
34.
Fig. 27.
AA' the axis of
and AC chords
—
B'BA, A AC
revolution,
to which the
=
CCA
BB' and CC two other axes, hence AB
axes are
inclined at
(Theorem 31.)
equal angles
A'AB
34
THEORY
OF
PARALLELS.
end-points of the third chord
BC, are likewise parallel and lie in one plane, (Theorem 25).
A perpendicular DD' erected at the mid-point D of the chord AB
and in the plane of the two parallels AA', BB', must be parallel to the
three axes AA', BB', CC, (Theorems 23 and 25); just such a perpendicular EE' upon the chord AC in the plane of the parallels AA', CC
will be parallel to the three axes AA', BB', CC, and the perpendicular
DD'.
Let now the angle between the plane in which the parallels AA'
and BB' lie, and the plane of the triangle ABC be designated by 77(a),
If a is positive, then erect
where a may be positive, negative or null.
FD
a within the triangle ABC, and in its plane, perpendicular upon
the chord AB at its mid-point D.
obe drawn outside the
Were a a negative number, then must FD
when
of
the
chord
AB;
a=0, the point F
triangle on the other side
Two
axes
BB', CC,
drawn
througn
the
=
=
coincides with D.
In all
cases
consequently
Erect
angle
now
arise two
we
congruent right-angled triangles AFD and DFB,
have FA
=
FB.
at F the line FF'
perpendicular
to the
plane
of the tri
ABC.
angle D'DF= 77(a), and DF=a, so FF' is parallel to
DD' and the line EE', with which also it lies in one plane perpendicu
lar to the plane of the triangle ABC.
Suppose now in the plane of the parallels EE', FF' upon EF the per
pendicular EK erected, then will this be also at right angles to the plane
of the triangle ABC, (Theorem 13), and to the line AE lying in this
plane, (Theorem 11); and consequently must AE, which is perpendicu
lar to .EK and EE', be also at the same time perpendicular to FE,
(Theorem 1 1). The triangles AEF and FEC are congruent, since they
are
right-angled and have the sides about the right angles equal, hence is
Since the
AF
A
perpendicular
fall upon the base
=
FC
=
FB.
from the vertex F of the isosceles
its
triangle
BFC let
mid-point G;
BC, goes through
plane passed
perpendicular FG and the line FF' must be perpendicular
to the plane of the triangle ABC, and cuts the plane of the
parallels
BB', CC, along the line GG', which is likewise parallel to BB' and
CC, (Theorem 25); since now CG is at right' angles to FG, and hence
at the same time also to GG', so
consequently is the angle CCG
B'BG, (Theorem 23).
through
=
this
a
THEORY
OF
35
PARALLELS.
Hence follows, that for the boundary-surface each of the
be considered
as
Principal-plane
boundary
axes
may
axis of revolution.
we
will call each
plane passed through
an
axis of the
surface.
Accordingly every Principal-plane cuts
boundary line, while for another position
tersection is
a
the
boundary-surface in the
cutting plane this in
of the
circle.
Three
principal planes which mutually cut each other, make with
each other angles whose sum is tt, (Theorem 28).
These angles we will consider as angles in the boundary-triangle
whose sides are arcs of the boundary-line, which are made on the bound
Con
ary surface by the intersections with the three principal planes.
of
the
and
sides
angles
pertains to
sequently the same interdependence
the boundary-triangles, that is proved in the ordinary geometry for the
rectilineal triangle.
In what follows, we will designate the size of a line by a letter
35.
with
an
accent
x', in order to indicate that this has a rela.
line, which is represented by the same letter
which relation is given by the equation
added,
e.
g.
tion to that of another
without accent x,
n(x) + n(x')
Let
the
now
ABC
hypothenuse
(Fig. 28)
AB
=
be
a
rectilineal
=
i-.
right-angled triangle,
c, the other sides AC
=
Fig. 28.
angles opposite
them
are
BAC=77(4ABC
=
77(/3).
b, BC
=
where
a, and the
36
THEORY
At the
point
triangle ABC,
PARALLELS.
OF
A erect the line AA' at
and from the
points
right angles
to
the
plane
B and C draw BB' and CC
of the
parallel
to AA'.
planes in which these three parallels lie make with each other
and 13),
the angles:
77(,y.) at AA', a right angle at CC (Theorems 11
at
BB'
consequently H(a')
(Theorem 28).
The intersections of the lines BA, BC, BB' with a sphere described
about the point B as center, determine a spherical triangle mnk, in which
the sides are run
77(c), kn 77 (ft), mk 77(a) and the opposite angles
The
=
=
=
are
77(b), /7(«'), **■
Therefore
sides
are
a,
we
b,
existence of
77(a)
c
must, with the existence of a rectilineal triangle whose
opposite angles 77 (a), 17 (ft) £-, also admit the>
and the
spherical triangle (Fig. 29) with
opposite angles 77(b), 77(«'), \~.
a
and the
the sides
77(c),
77 (ft),
Fig. 29.
Of these two
triangles, however, also inversely the existence of the
spherical triangle necessitates anew that of a rectilineal, which in con
sequence, also can have the sides a, «', ft, and the oppsite angles 77 (b'),
//(C), iTT.
Hence
a', ft, b',
we
may pass
over
from a,
b,
c, a,
ft,
to
b,
a, c,
ft,
a, and also to a,
c.
Suppose through the point A (Fig. 28) with AA' as axis, a bound
ary-surface passed, which cuts the two other axes BB', CC, in B" and
C", and whose intersections with the planes the parallels form a bound
ary-triangle, whose sides are ~B"C"=p, C'A {, B"A r, and the
angles opposite them 77(a), I7(o'), I-tt, and where consequently (Theo
rem
34):
=
—
p
r
sin
77(a),
q
=
r cos
Now break the connection of the three
=
11(a).
principal-planes along the line
BB' and turn them out from each other so that
they with all the lines
lying in them come to lie in one plane, where consequently the arcs p
q, r will unite to a single arc of a boundary-line, which goes through the
,
THEORY
point
one
A and has AA' for
side will
lie,
the
OF
37
PARALLELS.
axis, in such
a manner
that
q and p, the side b of the
arcs
(Fig. 30)
triangle,
on
the
which is
Fig. 30.
perpendicular
to A A' at
A, the
axis CC
going
from the end of b par
through C" the union point of p and q, the side a per
point C, and from the end-point of a the axis
pendicular
BB' parallel to AA' which goes through the end-point B" of the arc p.
AA' at
On the other side of AA' will lie, the side c perpendicular to
and
the
to
BB'
axis
the
AA',
through
going
parallel
the point A, and
allel to AA' and
to
end-point
CC at the
B" of the
arc r
remote from the end
The size of the line CC"
express by
In like
If
point
we
depends
upon
b,
point of b.
dependence
which
we
will
CC"=/(b).
manner we
will have BB"
describe, taking
CC
'
as
=/ (c).
axis,
axis
C to its intersection D with the
/».
CD by t, then is BD
BB"= BD+DB"
boundary line from the
BB' and designate the arc
a new
=
BD-f-CC", consequently
y(c) =/(*)+ /(b)Moreover, we perceive, that (Theorem 32)
t^pem =zr sin 77(a) e^K
ABC (Fig. 28) were
If the perpendicular to the plane of the triangle
lines c and r remain
would
the
then
erected at B instead of at the point A,
the
to t and q,
straight lines a
the same, the arcs q and t would change
=
38
THEORY
and b into b and a, and the
PARALLELS.
OF
angle 77(a)
into 77 (ft),
consequently
we
would have
q=r sin
whence follows
by substituting
cos
and if
we
change
a
and
77
ft
sin
(a)
II(ft) e'<»,
the value of q,
=
sin 7/ (ft)
e/«,
into b' and c,
77(b)
=
sin
77(c) e«a>;
further, by multiplication with e-^b)
sin 77
e-«b> = sin 7/ (c) e-«c>
(b)
Hence follows also
sin 77
e-«»>= sin
(a)
77(b)
e-«b>.
however, the straight lines a and b are independent of
another, and moreover, for b=0, /(b)=0, 77(b)—-^-tt, so we have
Since now,
one
for every
straight
line
a
e-/(»)=sin
77(a).
Therefore,
sin 77
(c)
sin II
(ft)
Hence
we
obtain besides
=
=
by
sin
sin
cos
77(a) sin 77(b),
11(a) sin 77 (a).
mutation of the letters
77(a) cos 77Q9) sin 77(b),
7/(b) cos 77(c) cos 77(a),
cos
77(a) cos 77(c) cos 77 (ft).
If we designate in the right-angled spherical triangle (Fig. 29) the
sides 77(c), 77 (ft), 77(a), with the opposite angles 77(b), 77 (a'), by the
=
cos
=
=
letters a, b, c, A, B, then the obtained equations take on the form of
those which we know as proved in spherical trigonometry for the right-
angled triangle, namely,
sin a=sin
c
sin
sin b=sin
c
sin
cos
cos
cos
from which
A,
B,
A=cos a sin B,
B=cos b, sin A,
c=cos a, cos b;
equations
pass
angles in general.
Hence spherical trigonometry is
t
we can
over
not
to those for all
dependent
spherical
tri
upon whether in
a
THEORY
rectilineal
angles
36.
ABC
or
triangle
the
sum
OF
39
PARALLELS.
of the three
angles
is
equal
to two
right
not.
We will
(Fig. 31),
now
consider
anew
in which the sides
the
are
right-angled rectilineal triangle
a, b, c, and the opposite angles
H(a),n(ft),iTT.
Prolong the hypothenuse c through
the point B, and make BD=y3; at the
point D erect upon BD the perpendicu
lar DD', which consequently will be
parallel to BB' the prolongation of the
side a beyond the point B. Parallel to
DD' from the point A draw AA', which
is at the same time also parallel to CB',
(Theorem 25), therefore is the angle
,
A'AD=77(c+/?),
A' AC
77 (b), consequently
U(h)=n(a)+ii(c+ft).
=
Fig gL
If from B
enuse
we
lay
off
ft
c, then at the end
on
the
hypoth
point D, (Fig.
triangle erect upon AB
perpendicular DD', and from the
point A parallel to DD' draw AA', so
will BC with its prolongation CC be the
third parallel; then is, angle CAA'=77
(b), T>AA' U(c—ft), consequently 77(c—
ft) II(a)-{- 77(b). The last equation is
then also still valid, when c=^9, or c<ft.
If c=ft (Fig. 33), then the perpendicu-
32),
within the
the
—
—
frIG. 32.
ular A A' erected upon AB at the
point A
40
is
THEORY
parallel
we
to the side
PARALLELS.
OF
BC=a, with its prolongation, CC, consequently
77(a)+77(b)=^, whilst also 77(c—$=|-, (Theorem 23).
c<;3, then the end of ft falls beyond the point A at D (Fig. 34)
have
If
upon the
prolongation
lar DD' erected upon
of the
AD,
Here the
AB.
perpendicu
parallel to it from A, will
likewise be parallel to the side BC=a,
with its prolongation CC.
Here we have the angle DAA'
77
(ft c), consequently
77(a)+ 77(b) =-_77(y3-c)=7/(c-/3),
(Theorem 23).
The combination of the two equations
hypothenuse
and the line AA'
B
=
—
found
gives,
277(b)=77(c-/3)+/7(c+;?),
277(«)=/7(c-/?)-77(c+,3),
whence follows
cos
77(b)
cos
[ jll(c—ft) +j 77(c+/?)]
cos
77(a) "cos
[ ^n(c—ft)—^ 77(c+/?)j
here the
Substituting
rem
value, (Theo
35)
cos
Fig. 34.
77(b)
=cos77(c),
77(a)
[tan £77 (c)]2=tan \U (c—ft) tan £77 (c+ft).
ft is an arbitrary number, as the angle 77 (;3)
■
cos
we
have
Since here
at
the
one
THEORY
side of
OF
41
PARALLELS.
may be chosen at will between the limits 0 and
between the limits 0 and co , so we may deduce
c
quently ft
consecutively /3=c, 2c, 3c, &c,
that for every
positive
\tz, conse
by taking
number n,
[tan£
77(c)] n=tan£77(«c).
If
consider
we
the ratio of two lines
n as
x
and c, and
assume
that
cot£77(c)=ec,
then
find for every line
we
tan£77(x)=e—
tive,
where
x
in
may be any arbitrary
77(x)=0 for x=co
e
since
general,
whether it be
positive
or
nega
-
number, which is greater than unity,
.
Since the unit
by
which the lines
measured is
arbitrary,
Napierian Logarithms.
are
the base of the
so we
may also understand by
37.
Of the equations found above in Theorem 35 it is sufficient to
e
following,
know the two
77(c)=. sin U (a) sin /7(b)
(a)=sin 77(b) cos IJ(ft),
sin
sin77
applying
the latter to both the sides
and b about the
a
right angle,
in
order .from the combination to deduce the remaining two of Theorem
without ambiguity of the algebraic sign, since here all angles are
35,
acute.
In
similar
a
manner we
tan
(1.)
(2.)
We will
(Fig. 35)
now
cos
consider
^^
p.
/
n.
1
d
^
for which
triangle
whose sides
are
a,
b,
c,
B, C.
the side
the
■pio. 35.
triangles,
rectilineal
If A and B are acute angles, then the
perpendicular p from the vertex of the
angle C falls within the triangle and cuts
c
■**
a
equations
77(c)=sin 77(a) tan 77(a),
77(a)=cos 77(c) cos 77 (ft).
0PP0site angles A,
and tlie
/ ^v
y
attain the two
we
c
into two
A and
parts,
x on
the side of
the side of the
angle
angle B. Thus arise two right-angled
obtain, by application of equation (1),
tan
tan
c
—
77(a)=sin B tan 77(p),
77(b)=sin A tan 77 (p),
x on
42
PARALLELS.
OF
THEORY
equations remain unchanged also when one of the angles,
right angle (Fig. 36) or ani obtuse angle (Fig. 37).
which
is
a
Fig. 36.
Therefore
For
tion
a
we
g.
B,
Fig. 37.
have
universally
(3.)
sin A tan
triangle
e.
with acute
for every
triangle
77(a)=sin B tan 77(b).
angles A, B, (Fig. 35) we have
also
(Equa
2),
77(x)=cos A cos 77(b),
77(c x)=cos B cos 77(a)
which equations also relate to triangles, in which one of the angles A
or B is a right angle or an obtuse angle.
As example, for B=£- (Fig. 36) we must take x=c, the first equa
cos
cos
—
tion then goes over into that which we have found above
the other, however, is self-sufficing.
For
of the
we
have
(Theorem 23),
If A is
x
Equation
77(x c)=cos (- B) cos 77(a);
cos
77(x— c)= cos 77(c— x)
also cos (tt— B)=
cos B.
a
—
—
—
and
right
or an
—
obtuse
angle,
then must
c— x
and
x
be put for
in order to carry back this case upon the preceding.
In order to eliminate x from both equations, we notice that
and
rem
2,
B>i,T (Pig- 37) the first equation remains unchanged, instead
second, however, we must write correspondingly
cos
but
as
c— x,
36)
[tan+/7(c—x)]s
l+[tan|77(c— x)]2
1—
cos77(c
—
x)=
"
i_|_e2i
:
—
2c
£/7(c)] 2 [cot £/7(x)] s
l-t-[tan£77(c)p[cot£77(x)]2
cos //(c)
cos7/(x)
1— [tan
—
1
—
cos
77(c)cos77(x)
(Theo
THEORY
If
we
substitute here the
OF
43
PARALLELS.
for
expression
cos
77(x), cos77(c
—
x), we ob
tain
cos
cos77(c)=
//(a) cos B-f
cos
77(b) cos A
l-|-cos77(a) cos//(b) cosA cosB
whence follows
cos
and
77(a) cosB=
cos
/7(c)-cosA cos/7(b)
1
cos A cos
—
77(b) cos 77(c)
finally
[sin 77 (c) ] 2 =[ 1 —cos B cos 77 (c) cos 77 (a) ] [ 1 —cos A cos 77 (b) cos 77 (c) ]
In the
same
must
we
way
also have
(*■)
[sin 77 (a) ] 2 =[ 1 —cos C cos 77 (a) cos 77 (b) J [ 1 —cos B cos 77 (c) cos 77 (a) ]
[sin 77(b) ]2 =[1— cos A cos 77 (b) cos 77(c) ] [1—cos C cos 77(a) cos 77(b) ]
From these three
equations
we
find
[sin //(b)!* ["sin/7 (cYl2
h 77
[sin (a)]3
,,,
Hence follows without
,*
If
ambiguity
a
s
(5.)
K
'
tion
we
v,,
cos
A cos
of
n/M
sign,
sin 77(c)
77(b)
y—t
^1— 1
sm
77(a)
sin 77(c)
corresponding to equa
tt,
x
cos 7/(c) A
77(b)
v ;-r
w
substitute here the value of
sin
(3.)
sin
then
we
77(c)=
tan
.
sin 0
77(a) cos 77(c)
obtain
cos
cos
but
r
=[1—
cosAcos77(b)cos//(c)1-.
L
\ I
\ )}
77(C):
by substituting
this
expression
for
cot A sin C sin
(a ";\
*•
77(a) sin C
sin A sin 77 (b)-|-cos A sin C cos77
cos
cos
17(a) cos C= 1
j-fr-r
77 (b)
In the meantime the
cos
—
B
sm-R-sin
-AJ
77(a)
equation (3) comes
cosA
—
cos
.
0
sm
11
(a).
w
equation (6) gives by changing
^-7-=cotB
77(b)
(a) cos 77(b) ;
equation (4),
77 (b)-l-cos C=
By elimination of sin 77(b) with help of the
cos
in
77(c)
sinC sin 77 (a) 4- cos C.
the
letters,
44
From the last two
equations follows,
cos
sin B sin C
_,
„
(7.)
'
v
PARALLELS.
OF
THEORY
cosC=
A+cosB
'
— —
:
.
„.
77 (a)
sm
and
equations for the interdependence of the sides a, b, c,
therefore
will
be,
opposite angles A, B, C, in the rectilineal triangle
All four
the
[Equations (3), (5), (6), (7).]
/ sin A tan
sin B tan 77 (b),
77 (a)
=
77 (b)
cos
sin
i cos
(8.)
v ;
A
cos
77 (c) +
77(b)
8in
sin
77(c)
-=
77(a)"
cos 77(b)
J
\cotAsinCsin77(b) + cosC— cos 17
v
(a)
COS
A
-f-
B
COS
COS
C
b,
77(a)
77(a)
cos 77
(a)
manner
equations
.
=
=
=
a,
1
£a"
—
a,
also for the other sides b and
8 pass
for such
over
b sin A
=
a
a2 =b2
a
sin
cos
Of these
(a)
are
cot
The
sin B sin C
:
sin 77
c, of the
sin
and in like
'
very small, we may content our
triangle
approximate determinations. (Theorem 36.)
If the sides a,
selves with the
1,
sin
-|- c2
(A -f- C)
A -4-
cos
triangles
c.
into the
following:
B,
—
=
2 be
cos
A,
b sin A,
(B + C)
=
0.
equations the first two are assumed in the ordinary geom
two lead, with the help of the first, to the conclusion
etry; the last
A
-f-
B
+C
= -.
imaginary geometry passes over into the ordinary, when
we suppose that the sides of a rectilineal triangle are very small.
I have, in the scientific bulletins of the University of Kasan, pub
lished certain researches in regard to the measurement of curved lines,
of plane figures, of the surfaces and the volumes of solids, as well as in
relation to the application of imaginary geometry to analysis.
The equations (8) attain for themselves already a sufficient foundation
for considering the assumption of imaginary geometry as possible.
Therefore the
Hence there is
no
means, other than astronomical
observations,
to
use
THEORY
OF
45
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 my
investigations, so that, for example, in triangles whose sides are attain
able for our measurement, the sum of the three angles 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) of
plane geometry pass over into the equations for spherical triangles, if
put a^/
l,bi/- l.cy'- l, instead of
change, however, we must also put
we
—
sin
and
similarly
In this
77(a)
=
the sides a,
cos
77(a)
tan
77(a)
—-
=
(a),
(\/— l)
=-.
sm a
tan a,
———
(y
—
I),
c.
pass over from equations
sin A sin b = sin B sin a,
manner we
=:
cos
cot A sin C
cos
A
=
b
cos c
-f-
cos a
c; with this
cos
also for the sides b and
cos a
b,
cos
-|- sin b
C
cos
b
sin B sin C
sin
=
—
(8)
to the
c cos
A,
sin b cot a,
cos
B
cos
C.
following:
TRANSLATOR'S APPENDIX.
ELLIPTIC GEOMETRY.
Gauss himself
published aught
upon this
fascinating subject,
Geometry
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
never
Non-Euclidean ; but when the most
themes submitted it
"
the choice of Gauss which fixed upon the one
Hypothesen welche der Geometrie zu Grunde liegen."
Ueber die
Gauss
was
then
was
recognized
I wonder if he
world.
saw
when in his sixth sentence
special
"From
of
case
a
powerful mathematician in the
pupil was already beyond him,
Riemann says, "therefore space is only a
as
the most
that here his
three-fold extensive
this, however,
it follows of
and
magnitude,"
necessity,
that the
continues:
propositions
of
general magnitude-ideas, but that
those peculiarities through which space distinguishes itself from other
thinkable threefold extended magnitudes can only be gotten from ex
geometry
can
perience.
not be deduced from
Hence arises the
problem,
to find the
simplest
facts from
a
which the metrical relations of space are determinable
problem
which from the nature of the thing is not fully determinate ; for there
—
may be obtained several systems of simple facts which suffice to deter
mine the metrics of space; that of Euclid as weightiest is for the pres
These facts are, as all facts, not necessary,
ent aim made fundamental.
only of empirical certainty; they are hypotheses. 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."
but
Riemann extends the idea of curvature to spaces of three and
dimensions.
on
the
The curvature of the
freely
figures
plane is constant and
it
can
move
sphere
is constant and
without deformation.
zero, and
on
it
The curvature of the two-dimentional
[47]
more
positive, and
The curvature of
figures slide without stretching.
space of Lobatschewsky and
48
Bolyai completes
ures can move
to the
sphere
being constant and negative, and in it fig
stretching or squeezing. As thus corresponding
without
it is called the
It would
formation.
or
we
pseudo-sphere.
live,
the
supposed
stretch
when
fact that
we
space.
suppose
If you think so, take
to
we can
move
Riemann, be
a
without de
special
case
of
At
We presume its curvature null.
interfere
to
not
does
squeeze us
space
our
move, is
But is it not absurd to
thing?
it into
us
we
then, according
space of constant curvature.
once
PARALLELS.
the group,
In the space in which
a
OF
THEORY
envisaged as a peculiar property of our
speak of space as interfering with any
a
knife and
a raw
potato,
and
try
to cut
solid.
seven-edged
on in this astonishing discourse comes the epoch-making idea,
that though space be unbounded, it is not therefore infinitely great.
"In the extension of space-constructions to the im
Riemann says:
measurably great, tne unbounded is to be distinguished from the in
finite; the first pertains to the relations of extension, the latter to the
a
Father
size-relations.
our space is an unbounded three-fold extensive manifoldness, is
hypothesis, which is applied in each apprehension of the outer world,
according to which, in each moment, the domain of actual perception is
filled out, and the possible places of a sought object constructed, and
The unboundedwhich in these applications is continually confirmed.
ness of space possesses therefore a greater empirical certainty than
any
From this however the Infinity in no way follows.
outer experience.
Rather would space, if one presumes bodies independent of place, that
is ascribes to it a constant curvature, necessarily be finite so soon as this
curvature had ever so small a positive value.
One would, by extend
the
of
the
in
a
ing
beginnings
geodesies lying
surface-element, obtain
an unbounded surface with constant positive
curvature, therefore a sur
"That
an
face which in
a
take the form of
Here
ception
we
homaloidal three-fold extensive manifoldness would
a
sphere,
and
so
is finite."
have for the first time in human
that universal space may
yet be
thought
finite.
the marvelous per
only
straight line is uniquely determined by two points, but
take the contradictory of the axiom that a straight line is of infinite
size; then the straight line returns into itself, but two having inter
sected get back to that intersection point without ever again
meeting.
Assume that
a
TRANSLATOR'S
49
APPENDIX.
Two
intersecting complete straight lines enclose a plane figure, a digon.
digons are congruent if their angles are equal. All complete
straight lines are of the same length, I. In a given plane all the per
pendiculars to given straight line intersect in a single point, whose
distance from the straight line is \l.
Inversely, the locus of all the
points at a distance -J7, on straight lines passing through a given point
and lying in
given plane, is a straight line perpendicular to all the
Two
»
a,
radiating
lines.
The total volume of the universe is I3/-The
angle by
The
ence
of the
sum
an excess
greater
the
angles of a plane triangle is greater than a straight
proportional to its area.
area of the
triangle, the greater the excess or differ
angle
sum
the
Royal
Astronomer for Ireland:
Says
"It is necessary to
to which
mers
are
from
of the
have
we
have found
tt-
large triangles, and the largest triangles
triangles which the astrono
The
measuring.
largest available triangles
measure
access
means
are, of course, the
of
those which have the diameter of the earth's orbit
It is
fixed star at the vertex.
of annual
base and
a
of the stars
precisely
the
investiga
mighty tri
*
*
*
angles had the sum of its three angles equal to two right angles.
"Astronomers have sometimes been puzzled by obtaining a negative
vestigations
parallax
as a
very curious circumstance that the in
a
are
tions which would be necessary to test whether
parallax
indeed
as
always
might result
tion.
*
*
generally or
inquiries oi
were really curved then a negative parallax
space
observations which possessed mathematical perfec
arisen from the
from
*
of these
No doubt this has
the result of their labors.
this nature, but if
one
errors
It must remain
large enough triangles
the
sum
which
an
open
are
inevitable in
question
angles
of the three
whether if
we
had
would still be two
right angles."
nothing within our experience
possibility that the space in which we
*
*
*
find ourselves may be curved in the manner here supposed.
"The subjective impossibility of conceiving of the relation of the
most distant points in such a space does not render its existence in
credible. In fact our difficulty is not unlike that which must have been
felt by the first man to whom the idea of the sphericity of the earth
Says Prof. Newcomb:
which will justify
»
"There is
denial of the
50
was
THEORY
suggested
OF
PARALLELS.
in conceiving how by
he could return to the
point
traveling
in
a
constant direction
from which he started without
during
his
of gravity."
any sensible change in the direction
In accordance with Professor Cayley's Sixth Memoir upon Quantics:
journey feeling
"
points is equal to C times the logarithm of the
joining the two points is divieled by the funda
The distance between two
cross
ratio in which the line
mental
quadric."
projective expression for distance, and Laguerre's for an angle
were in 1871 generalized by Felix Klein in his article Ueber die sogenannte Nicht-Euklidische Geometrie, and in 1872 (Math. Ann., Vol. 6)
This
equivalence of projective metrics with non-Euclidean
geometry, space being of constant negative or positive curvature ac
cording as the fundamental surface is real and not rectilineal or is im
aginary.
We have avoided mentioning space of four or more dimensions,
wishing to preserve throughout the synthetic standpoint.
For a bibliography of hyper-space and non-Euclidean geometry see
articles by George Bruce Plalsted in the American Journal of Mathe
he showed the
matics, Vol. I., pp. 261-276, 3S4, 385;
We notice that Clark
regular
courses
fessors, and
we
in non-Euclidian
presume, without
authority
on
Vol.
II,
and Cornell
geometry by
pp. 65-70.
University
are
giving
their most eminent Pro
looking, that the same is true of Har
Hopkins University, with Prof. Newcomb an origi
this far-reaching subject.
vard and the Johns
nal
University
■*&y
•c*.
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