I
THE LIBRARY
OF
THE UNIVERSITY
OF CALIFORNIA
LOS ANGELES
1
THEORIES
OF
PARALLELISM
CAMBRIDGE UNIVERSITY PRESS
FETTER LANE, E.G.
C. F. CLAY, MANAGER
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AH rights reserved
THEORIES
OF
PARALLELISM
AN HISTORICAL CRITIQUE
BY
WILLIAM BARRETT FRANKLAND, M.A.
SOMETIME FELLOW OF CLARE COLLEGE, CAMBRIDGE:
VICAR OF WRAWBY
Cambridge :
at the University Press
1910
.. .s.
- 1 o ¥ r
PRINTED BY jofȣ CLAY, M.A.
AT THE UNIVERSITY PRESS
IDs ic J
%
Mathemitfoat
Sciences
Libraiy
IN PIAM MEMOEIAM
ELIZABETHAE DE BURGH
801780
ilATIlEUATJCS
CONTENTS
PAGE
APOLOGY . . vii
INTRODUCTION xi
THEORIES OF PARALLELISM :
B.C.
300 EUCLID 1
80 POSIDONIUS 4
70 GEMINUS (? AGANIS) 5
A.D.
150 PTOLEMY 6
450 PROCLUS 7
900 ANARITIUS (AN-NAIRIZI) . .- . . . . 10
1000 GERBERT (POPE SYLVESTER II) 10
1250 NASREDDIN (NASIRADDIN AT-Tusi) .... 11
1570 BlLLINGSLEY . 12
1574 CLAVIUS 13
1604 OLIVER 14
1621 SAVILE 15
1654 TACQUET ......... 15
1655 HOBBES 1 16
1663 WALLIS 17
1679 LEIBNIZ 19
1680 DA BITONTO . . . . . . . . . 19
1733 SACCHERI ..... ^ ... 20
1756 SIMSON ........... 23
1766 LAMBERT 24
1778 BERTRAND 26
1795 PLAYFAIR 30
1796 LAPLACE 31
x Contents
A.D. PAGE
1799 GAUSS 32
1803 CARNOT 35
1804 W. BOLTAI . . 36
1809 THIBAUT 37
1825 TAURINUS 37
1832 J. BOLYAI 39
1833 LEGENDRE 40
1840 LOBACHEWSKI 42
1844 MEIKLE 44
1850 BOUNIAKOWSKI 45
1854 KIEMANN 46
1859 CAYLEY 47
1866 VON HELMHOLTZ .... : .... 47
1868 BELTRAMI . 48
1870 CLIFFORD 48
1871 KLEIN 49
1877 NEWCOMB 50
1895 DODGSON 52
FIRST ADDITIONAL NOTE :
ANALYTICAL GEOMETRY FOR THE METABOLIC HYPOTHESES . 53
SECOND ADDITIONAL NOTE :
PLANETARY MOTION FOR THE METABOLIC HYPOTHESES . 62
POSTSCRIPT 70
INTRODUCTION
A BRIEF preliminary investigation of the bases of Geometry
is necessary in order to furnish the means of testing the various
Theories of Parallelism with which we shall deal. Knowing,
for instance, whether the enunciation of a proposition is correct
or incorrect, we know where most especially to search for
mistakes of reasoning; and a careful study of the following
Introduction should therefore make it a pleasurable task to
criticise the work of the earlier Geometers. It will be observed
that this short Introduction is not Euclidean in style nor
elementary in method ; nor again is it in itself thoroughly
complete and satisfactory. On the one hand, the student in-
terested in the future rather than the past of Geometry will
turn to the masterly summaries in Dr Whitehead's two tracts
(The Axioms of Projective Geometry, 1906 ; The Axioms of
Descriptive Geometry, 1907 ; Cambridge). On the other hand,
a strictly Euclidean exposition of the Fundamental Theorem
of Geometry has been given in the Appendix to Euclid Book I
with a Commentary (Cambridge, 1902); and the line of thought
projected by Meikle, Kelland and Chrystal has been success-
fully followed out by Dr Manning (Non-Euclidean Geometry,
Boston, 1901). However, the ensuing discussion will supply
what is indispensable for critical purposes in this book. Some
things in it, and in the Additional Notes at the end, may
perhaps be called original work. Biographical information will
be found in the pages of Mr Rouse Ball's Histoi'y of Mathe-
matics or Dr Cantor's Geschichte der Mathematik.
Without further preface let us assume that straight lines
are freely applicable to themselves and to one another; and
xii Introduction
that there is a plane in which they are freely raoveable ; and
let us investigate the parallelism of such straight lines in such
an even plane. The principal theorem thus obtained may be
allowed to accredit itself after the fashion in which the diamond
approves itself the hardest stone. Whatever defects our proof
of it may possess, its credentials have been shown by Beltrami
to be irreproachable (below, page 48).
This Principal Theorem of Plane Geometry is as follows :
If two polygons are called equal in area when one can be
dissected into triangles which can be pieced together to cover
the other exactly ; and if, in the uncertainty (apart from pre-
judice) about the angle-sum of any triangle being precisely TT,
we define the divergence of a polygon having n sides to be the
difference between its angle-sum and (n — 2) TT ; then the area
of any polygon is proportional to its divergence (Hilbert).
For a proof of this, consider first this Lemma :
In any compact agglomeration of rectilinear cells in the
plane, if v is the number of the cells, a- the number of vertices,
and «r the number of walls, then a- + v = «r + 1.
For if a broken line of £ rectilinear pieces be introduced
to connect two vertices of any cell, a- is increased by £— 1,
v by 1, and or by £ This being always true, and the equation
a + v = -cr + 1 holding good for the external boundary, this
same equation is always true.
Hence it follows that the divergence of the boundary of the
agglomeration is equal to the sum of the divergences of the
individual cells composing it. For if S be the angle-sum oi
the external boundary, supposed a polygon of n sides; and ii
Si, s% ... sv are the angle-sums of the v cells, of which the sides
number^, pz ... p, respectively, then
sum of divergences of cells
= S + (<r - n) 2-7T - ir£pv +
= S + (o- - n + i/) 2?r - TT (2*r - w)
= 8-(n-2)7r
= divergence of external boundary of agglomeration,
• Introduction xiii
Thus polygons of equal area have equal divergence.
Hence with any area is associated a divergence. Let A be
the area associated with the divergence 8. We may then
write A=/(8); and as there is no reason to doubt the
applicability of Euclidean Geometry to infinitesimal areas,
A is a continuous function of 8. But, considering an agglo-
meration of two cells only, with divergences 8 and e,
Differentiating for 8,
/'(S + e)=/'<S).
Hence /' (8) is a constant, the same for every polygon, say
A. Therefore, by integration for 8,
where A and B are constant over the whole plane.
I.e. & = A8 + B.
But when A -»- 0, § -»• 0 also ; for the geometry of an
infinitesimal area can be accepted as Euclidean.
Thus A = 4 8,
i.e. Area is proportional to Divergence.
According to the value of A for the plane, there are thus
suggested three Hypotheses of equal rank, for A may be
positive, or negative, or even (in the extreme case) infinite.
In these cases 8 > 0, 8 < 0, or 8 = 0, throughout the plane.
Accordingly the Geometry of the Plane may be either Elliptic,
or Hyperbolic, or Parabolic ; and for these three mutually
exclusive Hypotheses, the angle-sum of a triangle is greater
than, or is less than, or is equal to TT, respectively. It may
be said that the irrefragable demonstration of this Theorem is
the greatest triumph ever won by Geometrical Science.
We can now prove that for the elliptic, parabolic, and
hyperbolic hypotheses, there are no, one and two parallels, re-
spectively, to any given straight line through any given point.
For let M be the given point, and BB' the given straight line.
xiv Introduction
Draw MN perpendicular to BB', and AM A' perpendicular to
MN. Within the angle AMN let a straight line through M,
meeting NB in P, begin to rotate from MN towards MA.
Then,
(i) for the elliptic hypothesis, there are no parallels.
For the area of the triangle MPN is proportional to the
divergence of that triangle ; say,
- TT)
so that /.MPN> /.AMP; and as MP rotates further, and
A MPN increases in area, the difference Z MPN — Z AMP
increases. Hence when /.AMP reaches zero, /.MPN is finite
still; and therefore MP continues to intersect NB, even in
the position MA. Thus for the elliptic hypothesis there are
no parallels.
Next,
(ii) for the parabolic hypothesis, there is one duplex
parallel.
For in this special case the divergence of the triangle MPN
is zero. Thus Z MPN = Z A MP ; and as MP rotates into the
position MA, and Z AMP decreases to zero, so also does Z MPN
pari passu decrease to zero. Therefore MA is parallel to NB
(see Corollary below). And likewise MA' is parallel to NB'.
But AM A' is one straight line; and so there is one duplex
parallel for the parabolic hypothesis.
Lastly,
(iii) for the hyperbolic hypothesis, there are two parallels.
For the area of the triangle MPN is proportional to the
divergence of that triangle ; say,
AMPN=K*(7r-
so that Z MPN <z AMP; and as MP rotates, and A MPN
increases in area, the difference Z AMP - Z MPN increases.
Hence before Z AMP reaches zero, Z MPN has vanished ;
and therefore (see Corollary below) MP is parallel to NB in
Introduction xv
some position MH within the angle NMA. Similarly there is
another distinct parallel MH' within the angle NMA'. Thus
for the hyperbolic hypothesis there are two parallels.
To strengthen this argument, we choose a suitable definition
of parallelism, and deduce a Corollary in the manner of
Lobachewski,
DEFINITION. A parallel to a given straight line through a
given point is that straight line which never intersects the
given straight line, however far both are produced in the
direction of parallelism, whereas any other straight line drawn
through the given point, and inside the parallel, does intersect
the given straight line.
COROLLARY. The angle MPN decreases to zero as MP
becomes parallel to NB.
We will show that if e be any assigned angle, however
small, then Z MPN < e, before MP reaches the position of
parallelism. Let MN be perpendicular to JBNB', as usual, and
let MH be as before the parallel to NB through M ; and let
MP make with MH on the inside an angle e. Then, by the
definition of parallelism, MP meets NB, say at P. Along NB
measure off PQ = PM. Then MQ is a straight line meeting
NB, and lying within the angle PMH. But by an indubitable
proposition, Z MQP = z QMP < e. Thus a straight line MQ
has been found such that Z MQN < e, however small e may be.
This Corollary is of course intended for the parabolic and
hyperbolic hypotheses only.
With whatever defects, this discussion of Parallelism is
sufficient in this place. It is necessary also to observe care-
fully the nature of equidistants to a given straight line. So
far from equidistants being identical with parallels, equidistants
are not rectilinear, save in the infinitely special case of the
parabolic hypothesis.
Consider two straight lines MA, NB which possess a
common perpendicular MN. Let a perpendicular to NB at
any point Q meet M A in P. Then the angle MPQ is not
xvi Introduction
right, except for the parabolic hypothesis ; for the ellipti
or hyperbolic hypotheses, it is obtuse or acute respectively
Furthermore, as Q moves along NB, and area PQNM increases
the angle MPQ continually increases, decreases, or remains th
same, according as the elliptic, hyperbolic, or parabolic hype
thesis holds good, respectively. It will also be useful to prov
(as in the Appendix to Euclid Book I with a Commentary) tha
the length of the perpendicular PQ decreases, increases, c
remains the same, respectively.
PROPOSITION. If AM, BN are straight lines, to which MN i
a common perpendicular ; and as Q moves in NB from N to 1
PQ is always drawn perpendicular to NB to meet M A in jP
then according as the elliptic, parabolic, or hyperbolic hype
thesis is maintained, the length of PQ continually decrease!
remains the same, or increases, respectively.
For suppose PiQi, P2Q2, PSQ3 to be three positions of P(
as Q moves from N towards B. If P2Q2 is greater than P^
and P3Q3, or again if P2Q2 is less than both the others, contrar
to the enunciation, then it will be possible to find two position
of PQ, namely, pq between PjQt and P2Q2, and p'q' bet wee
P2Q2 and P8Q3, such that pq and p'q' are of equal lengt
(Principle of Continuity).
Consider now nothing else but the quadrilateral pqq'p' i
which angles q, q are right, and the sides pq, p'q' equal. Bisec
qq' in k, and draw kh perpendicular to qq' to meet pp' in )
Fold the quadrilateral pqq'p' about hk, so that kq covers kq
Then qp covers q'p'. And since h and p have now the position
h and p', hp covers hp'. Hence the angles khp and khp' ar
equal; and therefore the angle khp is right.
Hence the divergence of the quadrilateral MhkN is zero
and therefore the area/divergence constant is infinite; and s
the geometry of the plane is parabolic, and all the perpen
diculars PQ equal, contrary to supposition. Thus in th
elliptic and hyperbolic hypotheses the length of PQ continuall
decreases and increases, respectively.
It is in fact evident that PQ decreases for the ellipti
hypothesis, for at finite distance along NB it vanishes. An
Introduction xvii
it should be noted, that for the hyperbolic hypothesis PQ
becomes infinite at finite distance from MN \ indeed, con-
siderations of area and divergence suggest that Z MPQ must
presently vanish. If b, x, y are the lengths of MN, NQ, QP,
then for the elliptic hypothesis,
y x L b
tan j- = cos r tan T ;
and for the hyperbolic hypothesis,
tanl
so that y is infinite for
tanh -C = cosh j=. tanh -^
K. Ji. J\.
cosh -jf. = coth -~ .
A /\
To these results may be added the following, of which
satisfactory proof will be found in Dr Manning's Non-Euclidean
Geometry :
If two straight lines OP, OQ enclose an angle /3, aod if PQ
is perpendicular to OQ at Q, then writing x, y for the lengths
OQ, QP respectively,
for the elliptic hypothesis :
7/ 3C
tan £ = sin T tan /8 ;
and for the hyperbolic hypothesis :
'II IP
tanh -^ = sinh -~ tan /3.
A A
The parabolic metric is obtained by "making k (or K) infinite."
Comparing the equation of an equidistant
y = b
with the former pair of equations above, it at once appears
that for the elliptic and hyperbolic hypotheses, equidistants are
not rectilinear. In these two general cases, the equidistant is
a curve, convex and concave, respectively, to its base.
It will also be convenient to add the formulae connecting
XV111
Introduction
the sides a, b, c and the angles A, B, C of a triangle right-
angled at A :
ELLIPTIC HYPOTHESIS
a be
cos T = cos T cos T
. o . a . „
sin r = sin T sin B
k k
tan T = tan T cos B
k k
b . c
tan T = sin T tan .B
k k
cos B = cos - sin
ft
HYPERBOLIC HYPOTHESIS
, a . b . c
cosh — = cosh ^.cosh —
A K. K
sinh — = sinh — sin B
K. K.
tanh -^ = tanh — cos B
A K.
b c
tanh — = sinh — tan B
K. K.
cos B = cosh — sin C
A
These formulae contain the whole of the metric of the
non-Euclidean geometries. Thus, if the angle B diminish to
zero, the angle of parallelism C is determined by the last
formula on the right:
1 = cosh sin C.
It will be observed that the first formulae correspond to the
Pythagorean Theorem (Euc. I. 47).
With these preliminaries, we may now commence the task
of criticising Theories of Parallelism in the order of their
appearance. Further developments of this brief Introduction
will be found in the First Additional Note at the end of the
volume.
EUCLID
Elements of Geometry, B.C. 300.
Venerated in Geometry more than Aristotle in Philosophy,
the Elementist has enjoyed a fame excelled by none in the
domain of pure knowledge. This well-merited fame has un-
fortunately hampered and impeded research into the Science of
Space, and the more so on account of the strange transmutations
of the text of the Elements, which have cloked the weaknesses
of Euclid's method. This strange confusion as to first principles
is displayed by the table below :
Editor
Date
Place
Postulates
Axioms
EUCLID
B.C. 300
Alexandria
5
5
Proclus
A.D. 450
Athens
5
5
Grynaeus
1533
Basle
3
11
Billingsley
1570
London
6
9
Gregory
1703
Oxford
3
12
Playfair
1795
Edinburgh
3
11
Peyrard
1814
Paris
6
9
Todhunter
1862
London
3
12
Referring to Dr Heath's magnificent edition of the Elements
(Cambridge, 1908) for detailed information, we will reproduce
here Euclid's first principles as he himself laid them down :
(A) Necessary Concessions:
(1) Let it be conceded that from every point to every
point a straight line can be drawn ;
F. E.
1
2 Theories of Parallelism
(2) And a limited straight line can be produced con-
tinually in a straight line ;
(3) And for every centre and distance a circle can be
described ;
(4-) And all right angles are equal to each other ;
(5) And if a straight line falling on two straight lines
make the angles within and towards the same parts less than
two right angles, then the two straight lines being indefinitely
produced meet towards the parts where are the angles less than
two right angles.
(B) Universal Ideas :
(1) Equals to the same are also equal to each other ;
(2) And if to equals equals are added, the wholes are
equal ;
(3) And if from equals equals are subtracted, the
remainders are equal ;
(4) And things coinciding with each other are equal to
each other;
(5) And the whole is greater than the part.
How then were these employed by Euclid to uphold his
view that in space as we know it there is through any given
point a single duplex parallel to any given straight line ? — On
the one hand, he unconsciously assumed the infinitude of space ;
or rather, he was altogether unconscious of the validity of the
hypothesis of the finite extent of space. It is practically im-
possible that he should ever have seriously entertained the
idea that a straight line is a re-entrant line. Scarcely anyone
had a rational glimpse of such a possibility before Riemann's
day, that is, before the middle of the nineteenth century.
Although the Third Postulate might be so understood as to bar
out the elliptic hypothesis, it was not so devised (we must
think) by Euclid himself. So the Elementist overlooked the
possibility of there being no such thing at all as parallelism.
Then on the other hand the possibility of double parallelism
(hyperbolic hypothesis) was excluded by the Fifth Postulate,
Theories of Parallelism 3
which is remarkably distinct in character from any other of the
Postulates and Axioms. It ought not to be there, the student
feels ; still less ought it to be among the Axioms, for surely its
occurrence there is one of the most ludicrous follies of which
the human mind has ever had to plead guilty. Attempts at
remedy have continually occupied the attention of the best
Geometers of later times; and their curious failure quite to
satisfy even themselves has hastened on the radical revision of
the bases of Geometry.
The Parallel-Postulate (or Fifth Postulate, as we shall call it,
avoiding the description of it as " Parallel-Axiom ") was used
by Euclid in his I. 29 in order to prove the converse of I. 27, 28.
Let us restate these Theorems in their simplest forms, for
purposes of criticism :
(i. 27, 28.) If two straight lines A A', BB' have a common
perpendicular MN, they never intersect.
It is noteworthy that the Fifth Postulate supplies what is
lacking here, in order to make MA, NB satisfy the strict
definition of parallelism given above (page xv), which requires
any straight line near MA within NMA to intersect NB.
Euclid did not mention this ; but gave a proof by means of his
I. 16. Contrast Ptolemy's method below (page 6). But Euc.
1. 16 is only universally valid in space of infinite extent. If
space is of finite extent, and the straight line re-entrant (elliptic
hypothesis), let ABO be) a triangle so great that if D is the
middle point of BC, AD is half the entire length of a straight
line. Producing AD to A' such that DA' = DA, we shall find
A' to be at A. Thus the congruence of the triangles BDA,
CD A' shows that, if Ca is the prolongation of AC, the angle
aCB or A CD is equal to the angle DBA or CBA. In this case,
therefore, the exterior angle of the triangle would be equal to
an interior opposite angle.
We conclude that the enunciation above would constitute a
suitable Postulate for excluding the elliptic hypothesis ; but it
cannot be proved without assuming or postulating the infinitude
of space. The companion enunciation now is :
1—2
4 Theories of Parallelism
(i. 29.) // two straight lines AA, BB' do not intersect, they
have any number of common perpendiculars like MN.
For if N is any point in BB', and NM perpendicular thereto,
then also NM is perpendicular to AA '; for, otherwise, the
Parallel-Postulate would make either MA intersect NB or else
MA' intersect NB' , contrary to the enunciation.
Evidently, from the form in which they can be enunciated,
Euc. I. 27, 28 and Euc. I. 29 are not completely converse. It is
significant also that if one single pair of straight lines were
allowed to possess one single pair of common perpendiculars, both
the elliptic and hyperbolic hypotheses would be cancelled. In
such a case the area/ divergence constant would be infinite.
The plane would then be of infinite size ; and through any
point there would be one duplex parallel to any straight line,
as by (ii) page xiv.
In conclusion, the Fifth Postulate debars the hyperbolic
hypothesis in an effective way, but it reads too much like
a theorem and positively invites attempts at its proof.
POSIDONIUS
A lost work on Geometry, B.C. 80.
This Geometer was perhaps one of the earliest writers who
defined parallels as equidistants. The assumption of the
rectilinearity of equidistants disposes at once of the elliptic and
hyperbolic hypotheses (see above, page xvii). " Posidonius says
that parallel lines are such as neither converge nor diverge in
one plane, but have all the perpendiculars drawn from points of
one to the other equal. On the other hand such straight lines
as make their perpendiculars continually greater or less will
meet somewhere or other, because they converge towards each
other" (Proclus, Friedlein's edition, page 176). The latter
notion, only correct for the parabolic hypothesis, was re-affirmed
as self-evident by Nasreddin (below, page 11).
Theories of Parallelism 5
GEMINUS
The Doctrine of Mathematics, B.C. 70.
Geminus was certainly a critic of first-rate ability. His
scientific attitude is indicated in the often quoted words of his,
recorded by Proclus : " We learned from the very pioneers of
the Science never to allow our minds to resort to weak plausi-
bilities for the advancement of geometrical reasoning"(Friedlein's
edition, page 191). A theory of parallelism attributed to Aganis
is found not in Proclus but in the Commentary of Anaritius
(Curtze's edition, pages 66-73). So disappointing is this piece
of work that it is a relief to find Dr Heath maintaining that
Aganis was almost certainly not Geminus but some writer
contemporary with Simplicius (A.D. 500). For instance, the
following definition of equidistant lines is singularly lacking in
precision: "These are such as lie in one surface, and when
produced indefinitely have one space between them, and it is the
least line between them." Assuming Euc. 1. 1-26, Aganis would
prove that : " If two straight lines are equidistant, the space
between them is perpendicular to each of these lines." Here
the rectilinearity of equidistants is definitely but tacitly assumed.
Enunciated as a Postulate, this assumption is satisfactory,
except that it assumes more than is necessary; more (for
instance) than the existence of a rectangle. Aganis' proof
was :
" Let A A', BB' be two equidistant lines, and let MN be the
space between them ; then the line MN is perpendicular to each
of the lines A A', BB'. (Could a Greek mind think thus ?) For
if the line MN were not perpendicular to each of the two lines
A A', BB', the angles at the point M would not be right. Let
therefore one of them AMN be an acute angle. Let me draw
then from the point N a perpendicular Nm to the line A A', and
let it fall on the side A. Then NM will be longer than Nm,
from the proof of the tenth proposition. But this is to show that
there is a line less than MN drawn between the lines AA , BB' \
which is contradictory and impossible. Therefore the line MN
is perpendicular to each of the two lines A A ', BB'."
6 Theories of Parallelism
The reference to Euc. I. 10 is very ingenious ; the construc-
tion suggests the greater length of the oblique.
PTOLEMY
Tractate on Fifth Postulate, A.D. 150.
Ptolemy's interesting method of approaching the subject of
Parallelism has been preserved in summary by Proclus (Fried-
lein's edition, pages 362-367). He assumed Euc. I. 1-26,
including 1. 16 which was unassailed until the latter half of last
century. This application of Euc. I. 16 to unlimited areas
banishes the elliptic hypothesis altogether. Although not
perhaps quite so cogent in its original form, the first of
Ptolemy's theorems to establish the Fifth Postulate was
virtually as follows:
(1) If two straight lines AA' and BB' are crossed by
a transversal MNso that the interior angles AMN, BNM on the
same side of it are together equal to two right angles, then they
cannot ever intersect.
For angles AMN, MNB' are equal. And if MA and NB
were to meet at 0, then by superposition of AMNB upon
B'NMA', MA' and NB' would be found meeting at 0' under-
neath 0. But " two straight lines do not enclose a space " ; and
therefore it is impossible that A A' and BB' intersect at all.
Of course, 0 and 0' may in their actual positions be the
same point, and indeed are so in the (single) elliptic hypothesis.
The postulate, not Euclidean but unobjectionable nevertheless,
that two straight lines cannot intersect in two distinct points,
does not prevent the elliptic hypothesis from being upheld and
maintained. Methods being, if anything, more important than
results, it may be observed that the above process by super-
position is a fine artifice. The letter H provides a suitable
figure. If the side-strokes meet above, they meet also below.
If these points are somehow identical, space is singly elliptic ;
otherwise, doubly elliptic, an alternative only requiring mention.
Theories of Parallelism 7
Reference may be pardoned to the writer's Story of Euclid,
despite its many faults, where a correct view-point (the greatest
of all difficulties in these matters) is aimed at.
(2) Conversely, if A A' and BB' are parallel, and MN is
a transversal, then the angles AMN, BNM are together equal
to two right angles.
Ptolemy proved this thus : If the angles AMN, BNM are
not together equal to two right angles, they are either greater,
or else less. If they are together greater than two right angles,
so also are the angles A'MN, B'NM, since "in no wise are
MA and NB parallel more than M A and NB'." But this
would be impossible, because the four angles altogether make
up four right angles ; etc.
This proof presupposes one and only one parallel to BB'
through M. It assumes what has been called the duplex
character of parallelism, an unique feature of the parabolic
hypothesis. This would have been better expressed explicitly
as an alternative to the Fifth Postulate, after the manner of
Proclus or Playfair. For in the hyperbolic hypothesis, when
A A' and BB' are parallel in the strict sense, A' A and B'B are
not parallel but only asecant (see below, page 43).
Two further theorems by Ptolemy in this connexion are
given by Dr Heath in his Elements (pages 205-6, Vol. I.) in
a note upon the Fifth Postulate, of which this volume must
appear as an expansion.
PROCLUS
Commentary on the First Book of the Elements, A.D. 450.
The following passage from Proclus' Commentary might
almost have been written by a nineteenth century Geometer
like Lobachewski or J. Bolyai.
"It cannot be asserted unconditionally that straight lines
produced from less than two right angles do not meet. It is of
course obvious that some straight lines produced from less than
8 Theories of Parallelism
two right angles do meet, but the (Euclidean) theory would
require all such to intersect. But it might be urged that as
the defect from two right angles increases, the straight lines
continue asecant up to a certain magnitude of the defect, and
for a greater magnitude than this they intersect."
Anyone who has been accustomed with M. Tannery to
estimate Proclus' work at the level of industry rather than
ingenuity should certainly compare the following very able
treatment of a difficult problem with any other found in these
pages. In the first place, Proclus adduced (Friedlein's edition,
pages 371-373): Aristotle's Axiom'. If from any point are drawn
two straight lines enclosing an angle, then as they are indefinitely
produced the distance between them exceeds every finite magni-
tude (De Caelo, Bk. I.). Although incapable of proof, this is
well expressed, and excellently fitted to form a simple and
useful Postulate. Sharply and clearly it excludes the elliptic
hypothesis. If the elliptic hypothesis must needs.be rejected
at the outset, scarcely anything could be better.
In the second place, how did Proclus evade the hyperbolic
hypothesis ? He laid down a proposition, again clear, sharp,
and sufficient. The only objection, and it is certainly a serious
one, is that he attempted a proof, and did not (like his disciple
Playfair) content himself with leaving it in the region of
postulate. It was as follows:
Proclus' Proposition : If a straight line intersect one of two
parallel straight lines, it will also intersect the other.
" For let A A' and BB' be parallels, and let aMaf cut A A' in
M. I say that aMa' cuts BB'. For if the two straight lines
MA, Ma are produced indefinitely from the point M, they
have a distance greater than every magnitude, so that it
is greater than the space between the parallels." Therefore
Ma cuts BB'.
If instead of enunciating this proposition in the form of
a postulate, the proof is to be made valid, some postulate must
be laid down that the distance between A A' and BB' remains
finite. The fact is doubtless that Proclus shared that con-
ception of parallelism, as a single duplex relationship, which
Theories of Parallelism 9
goes beyond the Euclidean definition. A suitable Postulate
would be :
If two straight lines have a common perpendicular of
finite length, the perpendiculars from points of one upon the
other are all less than some assignable magnitude however
great.
This is not the case in hyperbolic geometry. We know
that the length of the perpendicular, at finite distance even,
exceeds every finite magnitude, for by page xvii, the length
K tanh"1 (cosh -^ tanh ^)
V -K. A/
6
\
becomes actually infinite for
x = K cosh"1 ( coth
This is the very curious property of the hyperbolic plane, that
a quadrilateral with three right angles may have two sides
parallel, and the fourth vertex "at infinity." Thus the con-
ception of the hyperbolic plane is fraught with difficulty, as
is more readily confessed for the elliptic plane. Compare the
criticism of Dodgson's figure below (page 52).
Proclus proved the Parallel-Postulate in the following
manner :
"Let A A' and BB' be two straight lines, and let MN fall
across them and make the angles AMN, BNM together less
than two right angles.... Let the angle aMA be made, equal to
this defect from two right angles, and let aM be produced
to a'. Then since MN falls across aa' and BB', and makes the
interior angles aMN, BNM together equal to two right angles,
the straight lines aa', BB' are parallel. But AA' cuts aa.
Therefore by the proposition above, AA' cuts BB' also. There-
fore AA' and BB' intersect on the side where are the angles
together less than two right angles."
An obiter dictum by Proclus (Friedlein's edition, page 374)
that " Parallelism is similarity of position, if one may so say "
illustrates admirably the Euclidean conception of parallelism ;
but the idea is vague, and useless for scientific purposes.
10 Theories of Parallelism
Modern geometrical research has discarded all such language.
None the less, non-Euclidean Geometry has been stigmatised
as a mere facon de parler (Dr Karagiannides, Die Nicht-
euklidische Geometrie von Alterthum zwr Gegenwart, 1893),
an unjust censure.
ANAEITIUS
Commentary on the Elements, A.D. 900.
The Commentary of this Arab Geometer was translated into
Latin by Gherard of Cremona about A.D. 1150, and this
translation has been recently edited by Curtze (Leipzig, 1899).
It is in this work that Aganis is so often mentioned (above,
page 5). Apparently citing from the writings of Simplicius
(A.D. 500), Anaritius reports the famous definition of a straight
line, made more familiar in later days by Leibniz and Saccheri:
Linea recta est quaecumque super duos ipsius extremitates rotata
non movetur de loco suo ad alium locum. Anarifcius did not
treat independently of the Parallel-Postulate. On page 34
we read : " On this Simplicius said : This postulate is certainly
not self-evident, and so it was necessary to prove it by lines."
GERBERT
Mathematical Works, A.D. 1000.
As would be expected of one who was not an Arab at this
period, the works of Gerbert (afterwards Pope Sylvester II)
contain no penetrative theory of parallelism. In his Geometria
(iv. 10) he wrote : " Two straight lines distant from each other
by the same space continually, and never meeting each other
when indefinitely produced, are called parallel, that is, equi-
distant." Thus Gerbert assumed the rectilinearity of equi-
distants. At this time the results of Euclid, rather than his
methods, were studied ; and so were laid the foundations for
Theories of Parallelism 11
Savile's severe rebuke : Homines stulti et perridiculi, quasi ullus
unqiiam artifex suas edi voluerit conclusiones, nullis adjectis
probationibus.
NASREDDIN
Principles of Geometry, A.D. 1250.
Nasreddin's attempt to prove the Parallel- Postulate was
explained by Wallis in a public lecture at Oxford on
February 7th, 1651 (Opera Mathematica, Oxford, 1693; Vol. n.
page 669). The Persian Geometer gave two Lemmas, the
former of which he considered self-evident. These were sub-
stantially as follows:
(1) If AA' and BB' are two straight lines, and PQ is a
perpendicular to BB terminated by AA', and if these per-
pendiculars meet AA' at acute angles on the side of A, B',
then the straight lines AA', BB' will approach each other
towards A, B (so long as they do not intersect) and recede
towards A', B' ; and the perpendiculars will grow less on the
side A, B, as far as the intersection of AA' and BB' ; and
greater on the side A', B'. And conversely.
" These two propositions are self-evident; and are so familiar
to Geometers both ancient and modern that they are to be
regarded as obvious."
This is scarcely axiomatic to-day. It is known that the
perpendiculars may increase to a maximum (elliptic hypothesis),
or decrease to a minimum (hyperbolic hypothesis), according
to formulae containing circular or hyperbolic functions (see
above, page xvii). It is true that the angles change their
character in accordance with the Lemma. But the possibility
of a maximum or minimum perpendicular intervening between
AB and A'B' vitiates the proof of the second Lemma. If
this possibility were expressly negated, then the first Lemma
would make an efficient substitute for the Parallel-Postulate.
(2) If AB, A'B' are equal perpendiculars to BB', then the
angles at A and A' are right.
12 Theories of Parallelism
For if the angle A'AB is not right, suppose it (for instance)
acute. Then by (1) AA' approaches BB' towards A', B' ; and
therefore A'B is of lesser length than AS, contrary to the
supposition.
Having thus secured the existence of a rectangle, Nasreddin
proved Euc. I. 32 and the Fifth Postulate. See Dr Heath's
Elements (Vol. I. page 209).
BILLINGSLEY
The Elements of Geometrie, A.D. 1570.
Afterwards Lord Mayor of London, Sir Henry Billingsley
edited the first printed edition of Euclid's Elements in English.
He was familiar with Campanus' translation from Arabic into
Latin, but his notes on I. 16, 28 were derived from Proclus.
Under the heading : " Peticions or requestes," we read :
" 5. When a right line falling upon two right lines, doth
make on one and the selfe same syde, the two inwarde angles
lesse then two right angles, then shal these two right lines
beyng produced at length concurre on that part, in which are
the two angles lesse then two right angles.
...For the partes of the lines towardes (the one side) are
more inclined the one to the other, then the partes of the lines
towardes (the other side) are...."
Billingsley did not separate out what-is-defined from what-
is-proved sufficiently. His 35th Definition ran :
" Parallel or equidistant right lines are such ; which being
in one and the selfe same superficies, and produced infinitely on
both sydes, do never in any part concurre."
But as Dr Henrici well says : "A good definition must
state as many properties as are sufficient to decide whether
a thing belongs to a class or not, but not more than are
necessary for this purpose" (Congruent Figures, London, 1891,
page 33).
Theories of Parallelism 13
CLAVIUS
Euclidis Elementorum Libri xv., Rome, 1574.
Clavius followed Proclus in revising the theory of Parallelism,
and endeavoured to prove the Aristotelian Axiom that the
distance between two intersecting straight lines increases
beyond limit. Thus:
Let OMM' and ONN' be two intersecting straight lines, and
let MN, M'N' be perpendiculars upon ONN'. If possible, let
MN and M'N' be of equal length. Take along N'NO the
length N'Qf equal to NO. Then the triangles MNO, M'N'O'
are congruent. Therefore the angles MON, M'O'N' are equal,
contrary to Euc. I. 16.
Or again, if possible, let MN exceed M'N' in length, although
0, N, N' are in this order. Take in that case a length N'm'
along N'M' equal to NM. Then as before the angles m'O'N'
and MON are equal. That is, if m'O' meets OMM' in m, the
angles mON, mO'N' are equal, contrary to Euc. I. 16.
This is excellently arranged ; but it suggests only what is
already known, that Euc. I. 16 is universally valid only in an
infinite space. Euclid's 16th proposition and Aristotle's axiom
contain the same assumption, the same principle, namely, that
the plane is of infinite size. Such three statements are equiva-
lent ; and any one of them is sufficient to free Geometry from
the elliptic hypothesis.
Clavius also laboured to make evident from general con-
siderations (which is often an unprofitable task in Geometry)
that equidistants are rectilinear. " For if all the points of the
line AA' are equally distant from the straight line BB', its
points will be disposed evenly (ex aequo)," — not crookedly,
therefore rectilinearly. But the straight line is not the only
uniform line. For instance, as Clavius himself remarked, the
equidistant to a circle is a circle.
14 Theories of Parallelism
OLIVER
De Rectarum Linearum Parallelismo, Cambridge, 1604.
Thomas Oliver, Physician, of Bury, gave two somewhat
weak attempts at the proof of the Fifth Postulate. These were
on the following lines :
First Method : To prove that if MN is a perpendicular to
BB', and of unvarying length, then NM is always perpendicular
to " the right line described by the other extremity," M ; and if
NM is always perpendicular to A A', then N describes BB' in
like manner.
For let MN, AB be positions of the perpendicular, then
MAB is to be proved a right angle. Take Nff equal to NB
along BN, and let A'B' be the position of the perpendicular at
B'. Then the triangle ABN can be superposed upon the
triangle A'B'N, and is congruent thereto. Hence AN, A 'N are
equal; and angles ANM, A'NM are equal. Hence the triangles
ANM, A'NM are congruent. And so MN is perpendicular to
AA' — if (we must add) AMA' is one straight line.
Second Method: If AM and BN are both perpendicular to
MN, then any perpendicular from a point A of A M upon BN
is equal in length to MN, and is perpendicular to MA.
For let NM be moved so as to lie along BA, then shall M
fall upon A. Otherwise, let NM assume the position Ba.
Produce BN to B', making NB' equal in length to NB ; and let
MN move along NB' into the position a'B' perpendicular to
NB' at B'. Join Na, Na'. Then the triangles NBa, NB'a' are
superposable, and so congruent. Hence the lengths Na, Na'
are equal ; and the angles MNa, MNa' are seen to be equal.
Thus the triangles MNa, MNa' are superposable, and congruent.
Therefore the angles NMa, NMa' are equal, and hence are right
angles ; so that the angles AMN, aMN are equal, which is
impossible.
It is assumed that aMa' is one straight line, which is
another working of the assumption that equidistants are
rectilinear.
Theories of Parallelism 15
SAVILE
Praelectiones XIII. in Principia Euclidis Elementorum,
Oxford, 1621.
Sir Henry Savile founded Chairs of Geometry and Astronomy
at Oxford. He commended two problems for consideration by
the professor of Geometry. One of these was the Euclidean
theory of parallelism ; the other, the Euclidean theory of pro-
portion. He expressed his dissatisfaction in epigrammatic
form: In pulcherrimo Geometriae corpore duo sunt naevi,
duae lobes. These " blots " were the Fifth Postulate, and the
Fifth Definition of Book vi. (the latter not really Euclid's ; see
Dr Heath's Elements, ad locum). Savile died the year after his
lectures were published ; and a generation elapsed before John
Wallis brought his talents to bear upon the two difficulties.
TACQUET
Elementa Geometriae Planae et Solidae, Amsterdam, 1654.
The illustrious Whiston stated in his Memoirs that " it was
the accidental purchase of Tacquet's own Euclid at an auction
which occasioned his first application to the Mathematics,
wherein Tacquet was a very clear writer."
Tacquet defined parallels to be equidistants, giving as his
reason for so doing : Euclidaea deftnitio parallelismi naturam
non satis explicat. But he went further, and inserted as
Axioms :
(11) Parallel lines have a common perpendicular,
(12) Two perpendiculars cut off equal segments from each
of two parallels.
Concerning these he remarked that " their truth is immedi-
ately apparent from the definition of parallelism." Nevertheless
he condemned the Parallel-Postulate, at that time often reckoned
Eleventh Axiom : non axioma sed theorema. His proof of it
16 Theories of Parallelism
was neither very elegant nor very cogent. And still further, he
posited yet another " axiom," that two parallels to the same
straight line do not intersect. However clear a writer, there-
fore, Tacquet would not be considered convincing by a modern
critic.
HOBBES
De Corpore, 1655 : et caetera.
"There is in Euclid a definition of strait-lined parallels ; but
I do not find that parallels in general are anywhere defined ;
and therefore for an universal definition of them, I say that any
two lines whatsoever, strait or crooked, as also any two super-
ficies, are parallel, when two equal strait lines, wheresoever they
fall upon them make equal angles with each of them. From
which definition it follows ; first, that any two strait lines,
not inclined opposite ways, falling upon two other strait
lines, which are parallel, and intercepting equal parts in both of
them, are themselves also equal and parallel" (De Corpore,
c. 14, § 12).
Hobbes' definition of parallelism contains too much, when
applied to straight lines. A rectilinear parallel to a straight
line given should be determinate by a single characteristic (e.g.
a single act of construction, as would be the case if Euc. I. 31
were made to furnish a definition). When thus defined, its
other properties should be then proved. Otherwise, something
has been defined of which the existence is not immediately
clear. It is wrong to assume two straight lines A A' and BB'
such that if a transversal MN meet them in M and N, then if
M'N' is any other transversal equal to MN in length, the angle
M'N'B' can always be equal to the angle MNB'. The common-
sense philosopher should not veil such assumptions under
definitions.
Hobbes failed to disentangle the knots in the prevalent
scheme of Geometry. He justly urged against Euclid's defini-
Theories of Parallelism 17
tion : " How shall a man know that there be strait lines
which shall never meet though both ways infinitely produced?"
(Collected Works, Vol. 7, page 206). But the challenge applies
still more forcibly to his own definition given above, which
Hobbes had the hardihood to reproduce in this place : "Parallels
are those lines and superficies between which every line drawn
in any angle is equal to any other line drawn in the same angle
the same way." How shall a man know that there be such
straight lines or plane surfaces?
WALLIS
Demonstratio Postulati Quinti Euclidis, 1663.
Of different character from the preceding is the cautious,
penetrative work of this Savilian Professor, delivered as a public
lecture at Oxford on the evening of July llth, 1663, and
published in his Collected Works (Oxford, 1693, Vol. 2, pages
674-678).
"It is known," he began, " that some of the ancient geometers,
as well as the modern, have censured Euclid for having postu-
lated, as a concession required without demonstration, the Fifth
Postulate, or (as others say) the Eleventh Axiom, or, with the
enumeration of Clavius, the Thirteenth Asciom... But those who
discover this fault in Euclid do themselves very often (at least,
as far as I have examined them) make other assumptions in
place of it, and these appear to me no easier to allow than what
Euclid postulates... Since nevertheless I observe that so
many have attempted a proof, as if they esteemed it necessary,
I have thought good to add my own effort, and to endeavour to
bring forward a proof that may be less open to objection than
theirs."
Wallis first laid down seven Lemmas of a simple and un-
exceptionable character ; but the eighth contained something of
radical significance for the Theory of Parallelism.
p. E. 2
18 Theories of Parallelism
"VIII. At this stage, presupposing a knowledge of the
nature of ratio and the definition of similar figures, I assume as
an universal idea: To any given figure whatever, another figure,
similar and of any size, is possible. Because continuous
quantities are capable both of illimitable division and illimit-
able increase, this seems to result from the very nature of
quantity ; namely, that a figure can be continuously diminished
or increased inimitably, the form of the figure being retained."
Although Wallis' argument is insufficient, this Principle of
Similitude might make a suitable substitute for Euclid's Fifth,
postulate for postulate ; and moreover it has the advantage of
negating the elliptic hypothesis. Wallis however stated more
than is really necessary for a proper postulate. If a single pair
of triangles can be found in a plane, of different areas but of
equal angle-sums, then the geometry of that plane is Euclidean.
For let A, A' be the areas, and 8 the common divergence, then
as on page xiii above, A = A . 8, A' = A . S, where A is a con-
stant over the plane. But A =}= A'; and therefore S must be zero,
and A infinite ; and the geometry of the plane is parabolic.
Wallis' proof of the Fifth Postulate was effected thus : Let
MA, NB be two straight lines making with MN the angles
AMN, BNM together less than two right angles. Suppose the
angle AMN to slide along MN until M reaches N. Then
because the angle AMN is less than the supplement of the
angle BNM, AM in its final position will be outside BNM.
Therefore at some position of M A before M reaches N, say ma,
MA will intersect NB, say at p. Now by the principle of
similitude, it is possible to draw on MN a triangle similar to
the triangle mNp. Therefore corresponding to the point p
there exists a point P wherein MA and NB intersect. Q.E.D.
Theories of Parallelism 19
LEIBNIZ
Characteristica Geometrica, 1679; In Euclidis HPflTA; etc.
Leibniz's geometrical tracts have been gathered together in
the first volume of his collected mathematical writings (Ger-
hardt's edition, Halle, 1858). Like his contemporary and rival
Newton, Leibniz was a thorough-going Euclidean at heart.
His criticisms are acute and his suggestions valuable. A plane
surface results when a solid is cut in two so that the surfaces of
section are exactly similar, even when reversed. A straight
line is obtained by cutting a plane so that the lines of division
are superposable in any position, including the reversed. On
the Euclidean definition of parallelism Leibniz wrote : " This
definition seems rather to describe parallels by means of a more
remote property than that which they most evidently display ;
and one might doubt whether the relationship exists, or whether
all straight lines in the plane do not ultimately intersect each
other " (page 200). This is a logical premonition of the elliptic
hypothesis. Leibniz gave an attempt at a demonstration of the
existence of rectangles ; but the times were not yet ripe.
DA BITONTO
Euclide Restitute, 1680.
Like Leibniz, this Italian geometer would prefer a positive
definition of parallelism. To justify his own definition of
parallels as equidistants, he laid down fifteen or more proposi-
tions, analysed and discussed by Camerer in the edition of
Euclid only surpassed by Dr Heath's new work. Da Bitonto's
fifth proposition was : The perpendiculars let fall from points of
any curve upon any straight line cannot all be equal. This
would imply the rectilinearity of equidistants ; but the proof
does not hold good for any straight line whatever.
2—2
20 Theories of Parallelism
SACCHERI
Euclides ab omni naevo Vindicatus, Milan, 1733.
Saccheri's work was forgotten until about twenty years ago.
The remarks of Dr Stackel in this connexion are well worth
repeating at considerable length, from the preface to that
scholarly and compact volume in which the reader will find a
full recension of the Jesuit professor's apology for Euclid
(Engel and Stackel, Die Theorie der Parallel -linien von Euklid
bis auf Gauss^ Leipzig, 1895, pages 41-136).
" Quite thirty years have elapsed since by the publication of
Riemann's Inaugural Dissertation, and by the appearance of
Helmholtz's Memoir (on the Hypotheses at the basis of Geo-
metry), the space-problem and the associated parallel-question
became matters of general and abiding interest. About the
same time it became known that Gauss had realised long ago
the possibility and validity of a geometry independent of the
Parallel-Axiom, and the rescue from oblivion of the writings of
Lobachewski and J. Bolyai was effected, wherein this (hyperbolic)
geometry was systematically developed. Gauss, Lobachewski
and J. Bolyai were now the putative originators of non-
Euclidean Geometry, of which the further development and
firmer establishment were the work of Riemann and Helmholtz.
Then a certain stir was created in 1889 when Beltrami pointed
out that as early as 1733 an Italian Jesuit, Gerolamo Saccheri,
in an attempt to prove Euclid's Fifth Postulate, had been led
to a series of propositions hitherto ascribed to Lobachewski and
J. Bolyai.... Then the idea occurred to me whether possibly
Saccheri's Euclides ab omni naevo Vindicatus might not prove
to be a link in the chain of an historical development, so that
the fundamental principle of continuity might perhaps prevail
in the evolution of non-Euclidean Geometry.... In the first
yearly issue (of the Magazin fur die reine und angewandte
Mathematik) a memoir on the theory of parallels by Johann
Heinrich Lambert excited my attention, and a fuller examina-
Theories of Parallelism 2i
tion led to the startling conclusion that Lambert must be con-
sidered a hitherto disregarded forerunner of Gauss, Lobachewski
and J. Bolyai."
Saccheri protested against the procedure of early Geometers
who "assume, not without great violence to strict logic, that
parallel lines are equally distant from each other, as though
that were given a priori, and then pass on to the proofs of the
other theorems connected therewith " (page 46).
It would be a lengthy task to make a detailed discussion
of the Euclides Vindicatus. Saccheri promises that he will
apply Euc. I. 16, 17 only to triangles limited in every direction,
yet has no suspicion about the infinitude of space. He employs
Euc. i. 1-15, 18-26 with entire freedom. The hyperbolic hypo-
thesis survives the elliptic for some pages. We may sketch
a few of his earlier propositions :
(1) If two equal straight lines AB, A'B' make equal angles
with any straight line BE on the same side, the angles of the
quadrilateral at A and A' are equal. This might be proved by
superposition ; but Saccheri employs Euc. I. 4, 8 mediately.
(2) If the sides AA', BE' of such a quadrilateral are
bisected in M, N, then MN is perpendicular to AA' and BB'.
This follows from (1) by Euc. I. 4, 8. A simple and direct proof
by superposition would suffice.
(3) If two equal straight lines AB, A'B' are perpendicular
to BB', the side A A' of the quadrilateral is equal to, or less than,
or greater than, the opposite side BB', according as the equal
angles at A, A' are right, or obtuse, or acute, respectively.
This highly important General Theorem is proved in the
following way :
CASE I. When the equal angles at A, A' are right.
For AA' is not greater than BR ; since if it were so, a
length A'a might be taken along A' A equal to B'B. Then
angles B'Ba, BaA' would be equal (Prop. 1). But angle B'Ba
is less than B'BA ; and angle BaA' is greater than BAA'
22 Theories of Parallelism
(Euc. I. 16); so that angle HBa is less than BaA'. Nor is
AA' less than BB ', in like manner. And so AA' is equal
to BB'.
CASE II. When the equal angles at A, A' are obtuse.
Bisect A A', BB' in M, N; then MN is perpendicular to both
A A' and BB'. Now AM is not equal to BN; for if so, angles
MAB, NBA would be equal ; and they are not. Moreover AM
is not greater than BN. For if so, take along MA a length Ma
equal to NB. Then angles MaB, NBa are equal (Prop. 1).
But angle NBa is less than a right angle ; and angle MaB is
greater than aAB (Euc. I. 16), which is obtuse, so that angle
MaB is decidedly greater than a right angle. [Consideration of
the figure will show that Euc. I. 16 is applicable even in this
case of the elliptic hypothesis, because the join of B to the
middle point of A a is less than BP, P being the intersection
of this join with NM\ and BP is less than BO, 0 being the
intersection of BA and NM. That is, BP is less than half
the complete length of a straight line ; because, if BP were
equal to BO, angle BON would be right, and B coincide with
B' ; and if BP were greater than BO, angle BON would be
obtuse, and BB' overlap itself.] Thus angle MaB exceeds
NBa. Therefore AM cannot be greater than BN. And A M is
not equal to BN. Hence AM is less than BN.
CASE III. When the equal angles at A, A' are acute.
The proof is similar to that of Case II.
These three Cases Saccheri termed the Hypotheses of the
Right Angle, of the Obtuse Angle, and of the Acute Angle,
respectively. They were afterwards called the Parabolic,
Elliptic, and Hyperbolic Hypotheses by Dr Klein (see below,
page 50).
(5), (6), (7) If the Hypothesis of the Right, the Obtuse, or
the Acute Angle holds good for one quadrilateral of the kind
under consideration, it holds good for every such quadrilateral
throughout the entire plane.
The somewhat lengthy proofs make repeated use of Euc. I.
16, 17 (pages 54-58).
Theories of Parallelism 23
" (14) The Hypothesis of the Obtuse Angle is completely
false, because self-contradictory."
The second of Saccheri's proofs ran as follows :
" Since we have proved, with the Hypothesis of the Obtuse
Angle, that the two acute angles of a triangle ABC, right-
angled at By are together greater than a right angle, it is
evident that an acute angle BAD can be assumed (on the outer
side of BA) to make with them two right angles. Then the
straight line AD, by the foregoing Proposition (asserting the
Parallel-Postulate) for the Hypothesis of the Obtuse Angle,
ultimately meets CB. This however clearly contradicts Euc. I.
17," because if AD meets CB in E, then the triangle EAG has
its angles at A and C together equal to two right angles.
Saccheri failed to observe the possible re-entrance of straight
lines into themselves, a main feature of the elliptic hypothesis
of space.
It may be noted that Mansion has given the name Saccheri's
Theorem to the proposition that in the hyperbolic plane two
straight lines which do not intersect have in general a common
perpendicular. The exceptional case is when they are not
merely asecant but parallel (Mathesis, Vol. 16 Supplt, 1896).
We may also observe in connexion with Saccheri's work that
Clairaut in 1741 enunciated an alternative to the Parallel-
Postulate in the simple form, that a rectangle exists.
SIMSON
Euclidis elementorum libri priores sex... Glasgow, 1756.
The famous Robert Simson offered a proof of the " Eleventh
Axiom " by a method involving a new Axiom, in the following
manner :
Definition 1. The distance of a point from a straight line is
the perpendicular from the point upon the straight line.
Definition 2. A straight line is said to approach towards, or
recede from, another straight line according as the distances of
24 Theories of Parallelism
points of the former from the latter decrease or increase. Two
straight lines are equidistant, if points of the one preserve the
same distance from the other.
Axiom : A straight line cannot approach towards, and then
recede from, a straight line without cutting it; nor can a
straight line approach towards, then be equidistant to, and then
recede from, a straight line ; for a straight line preserves always
the same direction.
Simson appealed to common sense and common experience,
a proper course for educational purposes. Scientifically, it
would have been better to postulate this property of straight
lines which is peculiar to the parabolic hypothesis ; and also to
refrain from introducing the idea of direction for support. The
suggestion of Peletarius (1557) that even Axioms themselves be
reckoned as Definitions (defining the relations spoken of in
them) has been generally accepted to-day for scientific purposes.
The tendency of the educationalist as such is in the opposite
direction, of making as much as possible present an axiomatic
appearance to the inexperienced eye of the child.
Simson's first proposition, proved by means of his axiom,
was:
If two equal straight lines AB, A'B' are perpendicular to
any straight line BR, and if from any point M of the join A A'
a perpendicular MN is let fall upon BB', then AB and MN and
A'B' are equal to one another in length.
This proposition puts an Euclidean impress upon the
geometry of the plane.
LAMBERT
Die Theorie der Parallel-linien, 1766.
If two explorers, strangers to each other, enter an unknown
land beyond the mountains by the same pass, they may quite
possibly be found to have chosen the same route for some
distance into the interior; and so considerable likeness exists
Theories of Parallelism 25
between the work of Clavius, Lambert and J. Bolyai and the
earlier work of Nasreddin, Saccheri and Lobachewski. The
similarity between Saccheri's treatise and Lambert's memoir is
easily seen in the pages of Engel and Stackel's volume. Lambert
employed for his fundamental figure a quadrilateral with three
right angles, which is the half of one of Saccheri's isosceles
quadrilaterals. He stated expressly that the opposite sides of
his quadrilateral did not meet, a fatal blow to the elliptic
hypothesis. Thus, for instance, in § 33 he wrote (with reference
to a standard quadrilateral A BB' A', wherein three of the angles
at A, B and E' are right angles), "the fact that AA' and BB'
do not intersect leaves it unsettled whether the distances AB
and A'B' are always equal or greater or less" (page 178).
In § 39 Lambert considered a figure in which BN, NB' are
equal distances along BB' ; AB, MN, A'B' perpendiculars to
BNB'; and AM A' perpendicular to NM at M. He then
wrote :
" The question now arises about the angle at A ; and there-
fore we are bound to formulate three Hypotheses. For it may
be that
( equal to 90°, (i)
the angle MAB is < or greater than 90°, (ii)
[ or less than 90°. (iii)
These three Hypotheses I will adopt in order, and educe their
consequences."
These are again the parabolic, elliptic and hyperbolic
hypotheses of Dr Klein. The elliptic hypothesis (ii) soon
dropped out of Lambert's hands, owing to his extension to the
entire (infinite) plane of results into which Euc. I. 16 enters.
But the hyperbolic hypothesis (iii) was successfully worked out
to conclusions implying an absolute standard of length. On
this Lambert remarked (§§ 80, 81) :
"This consequence possesses a charm which makes one
desire that the Third Hypothesis be indeed true !
" Yet on the whole I would not wish it true, notwithstand-
ing this advantage (of an absolute standard of length), since
innumerable difficulties would be involved therewith. Our
26 Theories of Parallelism
trigonometrical tables would become immeasurably vast (com-
pare pages xviii and 57, above and below) ; the similitude and
proportionality of geometrical figures would wholly disappear, so
that no figure can be represented except in its actual size;
astronomy would be harassed (see our Second Additional
Note) ; etc.
" Still these are argumenta ab amore et invidid ducta, which
must be banished from Geometry as from every science.
"I revert accordingly to the Third Hypothesis. In this
hypothesis, as already seen, not only is the sum of the three
angles of every triangle less than 180° ; but this difference from
180° increases directly with the area of the triangle ; that is to
say, if of two triangles one has a greater area than the other,
then in- the first the sum of the three angles is less than in the
second."
Then in § 82 Lambert surmised that the area of a triangle is
actually proportional to the difference between its angle-sum
and two right angles. This has been called Lambert's Theorem
by Mansion (Mathesis, Supplt 1896, Premiers Prindpes de la
Metageome'trie).
BERTRAND
Ddveloppement Nouveau de la Partie EUmentaire des
MatMmatiques, Geneva, 1778.
We furnish here a few all too brief extracts from the
elegant and perspicuous disquisitions of this Swiss Geometer.
" Geometry, like every other science, has its roots in ideas
common to all. From this fount of ideas the first originators
derived those principles and germs of knowledge which they
bestowed upon mankind. Hence it appears that in every
science two parts can be distinguished ; the first, consisting of
the assemblage of principles or primitive conceptions from which
the science proceeds ; and a second, comprising the develop-
ment of the consequences of the principles. In respect of these
Theories of Parallelism 27
principles as they exist in every mind, science would seem to
encounter no resistance or difficulty. Yet the choice that has
to be made of first principles, the degree of simplicity and
elegance to which they have to be reduced, and the necessity of
enunciating them in precise terms, capable of clear compre-
hension,— all this is very difficult."
Bertrand now adduced the practical illustration of a Hunter,
having shot a deer, measuring the length of his shot in bow-
lengths, and meditating upon the sense of direction exercised
when he aimed the fatal arrow. He then proceeded with much
eloquence : " The spectacle of the universe displays before our
eyes an immense space. In this immensity bodies exist and
change continually their shape, size and position ; and mean-
while space itself, invariable in all its parts, remains like a sea
always calm, in everlasting repose. So the idea we form of
space is that it is infinite and limitless ; homogeneous and like
itself at every time and in every place. Space is without
bounds, for any we might assign to it would be contained in it,
and therefore would not bound it. Space is homogeneous, in
that the portion of space occupied by a body in one plane
would not differ from that which would be occupied by it
elsewhere."
Division of space into two halves identical in all but position
gives the plane surface; and division of the plane similarly
gives the straight line (so also Leibniz, above, page 19).
So far, so good. But the elliptic hypothesis escapes from
the Hunter's grasp in his seventh effort of thought, as from
Ptolemy's many centuries before.
" Proposition 7. Two straight lines AM A', BMB' traced on
the same plane, and making with a third MN the interior
angles AMN, BNM of which the sum is equal to two right
angles, cannot intersect."
For AMNB can be superposed upon BNM A'. " Hence it
will follow that the straight Hues A A', BB' will intersect in two
points, or will not intersect at all. But the first is impossible,
therefore the second is true."
28 Theories of Parallelism
This is incisive; but as remarked above (page 6) these
presumably very distant points may be one and the same point.
The Hunter's thoughts are supposed to turn next to an
eighth Proposition, equivalent to the following : The plane con-
tains an infinite number of strips such as are formed by two
perpendiculars MA, NB to a limited straight line MN on the
same side of it. For if MN is produced to P, and NP is equal
to MN, and PC drawn perpendicular to NP, then the strip
AMNB can be superposed upon the strip BNPG. And space
being infinite by the preceding Proposition, this process,
repeated infinitely often, furnishes an infinite number of strips,
congruent to AMNB, over half of the infinite plane.
Then the Hunter elaborates one of the finest proofs of the
Parallel-Postulate of which we have knowledge, after this
manner :
Let MH be drawn within the strip AMNB ; then MH
meets NB. For the area included within the angle AMH is
a finite fraction of the area of the entire plane ; in fact, the
same fraction that the angle AMH is for a denominator of four
right angles. Whereas the area of the strip AMNB has been
seen to be only an infinitesimal fraction of the whole plane ; in
fact, the same fraction that the length MN is for an infinite
denominator (the whole length of a straight line). Hence the
angle AMH cannot be contained within the strip AMNB.
Therefore it must overlap. Therefore MH must intersect and
cross NB.
This is exceedingly forcible, indeed almost overwhelming.
It seems to pierce the heart of the hyperbolic hypothesis, like
a swift arrow from a sure bow. Let us see whether any light is
shed upon it by our results for the hyperbolic hypothesis
on page xvii above. The area of the angle AMH is a certain
definite fraction of the area of the infinite plane. Take then
a circle of any assigned radius R however large, its centre
being M. The area of this circle is
!R sinh ^ dx = ZirK* ( cosh § - 1 V
Jo A \ K /
Theories of Parallelism 29
When R becomes indefinitely great, even compared with the
space-constant K, the value of this expression is of the order of
R
magnitude trK^eK. Thus if the angle AMH is ft in circular
measure, the area of the sector AMH is not very different from
Now let us consider the area of the strip AMNB, and write
b for the length MN, then by considering a very great number
of very narrow strips all of length R, we have for the area
required
R
K'
b 2
which is not very different from ^ KeK, the further boundary of
/R x
cosh ^.dx = bK sinh
) -ft
the strip being an equidistant on MN of height R.
Thus the ratio of the areas of sector and strip, computed
thus, is practically j- Ky which is always finite ; and indeed may
be made as small as desired by choosing the angle ft small
enough.
There may be observed an inconsistency in the areas of the
hyperbolic plane computed by means of an infinite number of
sectors and an infinite number of strips. A circle of very great
R
radius R has area 7rK2eK approximately. But a double strip of
R
length 2R has an area of about bKeK. And as the number of
strips is certainly infinite, the area of the plane reckoned in
strips infinitely exceeds the area of the plane reckoned in
sectors. The reason is this. The space between a circle of
great diameter 2R and the equilateral circumscribing quad-
rilateral of medial dimensions 2R, is ultimately infinite com-
pared with the area of the circle, in the hyperbolic hypothesis.
There is a like point to be raised in criticism of Dodgson's
figure (below, page 52).
30 Theories of Parallelism
None the less, the above piece of rigorous analysis is very
different from that simple, convincing objection which one
would wish to raise. The only possible elementary criticism
appears to be that more is assumed about the infinite regions of
the plane than we can really know or conceive. Probably the
experts will consider the argument, on this or other grounds,
one of the most plausible and fallacious of sophisms.
For the hyperbolic hypothesis, the rectangular strip widens
out ultimately at the same rate as a circular sector; and our
mental picture of such infinitely distant regions is not legitimate
unless drawing to scale (Wallis' principle) is a legitimate and
indeed possible procedure. If there is no court of appeal from
the verdict of "commonsense," Bertrand's argument stands, and
Euclid's geometry prevails.
PLAYFAIR
Elements of Geometry, Edinburgh, 1795.
John Pluyfair was a typical expositor of Euclidean Geometry.
What has become generally known as Playfair's Axiom is only
a slightly varied form of Proclus' Proposition (above, page 8).
Playfair wrote :
"A new Axiom is introduced in the room of the 12th, for
the purpose of demonstrating more easily some of the properties
of parallel lines."
This new Axiom, assigned the llth place, was couched in
the familiar terms :
" Two straight lines which intersect one another, cannot be
both parallel to the same straight line."
As will be seen by a comparison with Lobachewski's position
(below, page 43), this prevents the hyperbolic hypothesis ; and
the elliptic hypothesis would probably have been thought by
Playfair to be sufficiently frustrated by the corollary to his
definition of the straight line, that two straight lines cannot
enclose a space.
Theories of Parallelism 31
Of this Axiom of Playfair's, Cayley said : " My own view is
that Euclid's Twelfth Axiom in Playfair's form of it, does not
need demonstration, but is part of our notion of space, of the
physical space of our experience, which is the representation
lying at the bottom of all external experience" (Presidential
Address to the British Association, 1883). Dr Russell cites and
criticises this interesting confession in his admirable Foundations
of Geometry (Cambridge, 1899, page 41).
LAPLACE
Exposition du Systeme du Monde, Paris, 1796.
This great Analyst expressed his views to a certain extent,
in connexion with the Law of Universal Gravitation (Harte's
translation, Dublin, 1830, page 321).
"The law of attraction, inversely as the square of the
distance, is that of emanations which proceed from a centre...
One of its remarkable properties is, that if the dimensions of all
the bodies in the universe, their mutual distances and velocities,
increase or diminish proportionally, they describe curves entirely
similar to those which they at present describe ; so that if the
universe be successively reduced to the smallest imaginable
space, it will always present the same appearance to all
observers... The simplicity of the laws of nature therefore only
permits us to observe the relative dimensions of space."
On the other hand, Gauss (below, page 34) remarked the
possibility of objects possessing absolute dimensions; and the
analysis of celestial mechanics becomes much more intricate if
the law of the inverse square has to be abandoned or assigned
a subordinate place. For the elliptic and hyperbolic hypotheses,
the area of a sphere of radius r is
4^sin3T and 4flT2sinh24?
k K
respectively. Thus the law of intensity of radiation issuing
32 Theories of Parallelism
uniformly from a point-source would for the three hypotheses be
expressed by the factors
T T
cosec2 y- , r"2, cosech2 -^ .
K K
The application of these extended laws to planetary motion
is the task undertaken in the Second Additional Note to this
volume (page 62).
The translator's note (page 536) may be reproduced here :
"The endeavours of Geometers to demonstrate Euclid's
Twelfth Axiom about parallel lines have been hitherto un-
successful. However no person questions the truth of this
Axiom, or of the Theorems which Euclid has deduced from it.
The perception of extension contains therefore a peculiar
property which is self-evident, without which we could not
rigorously establish the doctrine of parallels. The notion of
a limited extension (for example, of a circle) does not involve
anything that depends on its absolute magnitude ; but if we
conceive its radius to be diminished, we are forced to diminish
in the same proportion its circumference, and the sides of all
the inscribed figures. This proportionality was, according to
Laplace, much more obvious than that of Euclid. It is curious
to observe that agreeably to what is stated on page 322, this
Axiom is pointed out in the results of universal gravitation."
Certainly the utterances of great minds are always deserving
of careful attention, and repay it.
GAUSS
Letters and Reviews, 1799-1846.
For nearly fifty years the mind of Gauss was repeatedly
engaged upon the hyperbolic hypothesis, styled by him non-
Euclidean Geometry. His occasional writings, collected by
Engel and Stackel, are humanly interesting. His influence was
exerted upon the elder Bolyai, and so indirectly upon the
brilliant son, Johann Bolyai. In a letter to the father towards
the close of the year 1799, Gauss wrote :
Theories of Parallelism 33
" I have arrived at much which most people would regard as
proved, but it is in my eyes good for nothing in this respect.
For example, if it could be proved that a rectilinear triangle is
possible, of area exceeding any assigned area, I should be in
a position to prove rigorously the whole of (Euclidean) geometiy.
Now most people would regard this as axiomatic, but I do not.
It would be quite possible that however distant from each
other the vertices of the triangle were assumed to lie in space,
the area should still be less than an assignable limit. I have
more propositions of a similar character, but in none of them do
I find anything really satisfying."
In fact, the maximum area of a triangle formed by three
straight lines for the hyperbolic hypothesis would be
K*(ir-A-B-C)t
wherein A=B = C=Q, that is, TrK2, which though probably
large is strictly limited. The sides of the triangle would be
parallel in pairs, and the vertices " at infinity."
Thus Gauss seems to have worked out several fundamental
theorems of hyperbolic geometry but not so completely as to
feel ready to publish them. His letter to Taurinus, a facsimile
of which forms the frontispiece to Parallel-linien von Euklid bis
auf Gauss, reads as follows :
" I have read not without pleasure your kind letter of October
30th with the small sketch enclosed, the more so because I
have been accustomed to discover scarcely any trace of pure
geometrical spirit among the majority of people who make the
new attempts on the so-called Theory of Parallel Lines. With
reference to your attempt I have nothing, or not much, to
observe except that it is incomplete. Your presentation of the
proof that the sum of the three angles of a plane triangle
cannot be greater than 180° leaves much to be desired in
respect of geometrical rigour. This in itself could be remedied,
and beyond all doubt the impossibility can be proved most
rigorously. Things stand otherwise in the second part, that the
sum of the angles cannot be less than 180° ; this is the crucial
F. E. 3
34 Theories of Parallelism
point, the reef on which all the wrecks take place. I imagine
that you have not been long occupied with this subject. My
own interest in it has extended over 30 years, and I do not
think that anyone can have occupied himself more with this
second part than I, although I have never published anything
on it. The assumption that the sum of the three angles of
a triangle is less than 180° leads to a peculiar Geometry entirely
different from ours, — a geometry completely self-consistent,
which I have developed for myself perfectly satisfactorily, so
that I can solve any problem in it with the assumption that a
constant is determinate, this constant not being capable of
a priori specification. The greater this constant is assumed to
be, the more nearly is Euclidean Geometry approached ; and an
infinite value of the constant makes the two systems coincide.
The theorems of this geometry seem somewhat paradoxical, and
to the lay mind absurd ; but continued steady reflexion shows
them to contain nothing at all impossible. Thus, for instance,
the three angles of a triangle can be as small as we please, if
only the sides are taken sufficiently great ; and yet the area of
a triangle can never exceed a definite limit, however great the
sides are taken to be, and indeed can never reach it. All my
efforts to discover a contradiction, an inconsistency, in this non-
Euclidean Geometry have been unsuccessful ; and the one thing
in it contrary to our conceptions is that, were the system true,
there must exist in space a linear magnitude, determined for
itself albeit unknown to us. But methinks, despite the say-
nothing word-wisdom of the metaphysicians, we know far too
little, too nearly nothing, about the true nature of space, for
us to confuse what has an unnatural appearance with what is
absolutely impossible. If the non-Euclidean Geometry were
true, and that constant at all comparable with such magnitudes
as lie within reach of our measurements on the earth or in
the heavens, it could be determined a posteriori. Hence I have
sometimes expressed in jest the wish that the Euclidean
Geometry were not true, since then we should possess an
absolute standard of measurement a priori. I am not afraid
that a man who has shown himself to possess a thoughtful
mathematical mind will misunderstand the foregoing; but in any
Theories of Parallelism 35
case, please regard this as a private communication, of which
public use, or use leading to publicity, is not to be made in any
way. If at some future time I acquire more leisure than
in my present circumstances, I shall perhaps publish my
investigations.
Gottingen, November 8th, 1824"
These investigations continued in the condition of scattered
epistolary hints and suggestions until the decease of the fore-
most mathematician of his day. The solution of any problem,
referred to by Gauss as possible for him, is also, at least in
principle, possible by the use of the analytical formulae
obtained in the First Additional Note at the end of this
volume (page 59).
CAENOT
Geometric de Position, Paris, 1803.
Carnot advocated the Principle of Similitude as an alter-
native to the Parallel-Postulate, a view-point secured with
greater elaboration by Wallis (above, page 18). He wrote
(§ 435):
"The Theory of Parallels rests on a primitive idea which
seems to me almost of the same degree of clearness as that
of perfect equality or of superposition. This is the idea of
Similitude. It seems to me that we may regard as a principle
of the first rank that what exists on a large scale, as a ball,
a house, or a picture, can be reduced in size, and vice versa-,
and that consequently, for any figure we please to consider, it is
possible to imagine others of all sizes similar to it; that is to
say, such that all their dimensions continue to be in the same
proportions. This idea once admitted, it is easy to establish the
Theory of Parallels without resorting to the idea of infinity."
3—2
36 Theories of Parallelism
W. BOLYAI
Theoria Parallelarum, Maros-Vasarheli, 1804- ;
Kurzer Grundriss, 1851.
In the earlier of these tracts, reprinted in the 49th volume
of Mathematische Annalen (pages 168-204), the elder Bolyai
begau by supposing an inverted T-square to slide along a
straight line. What is the curve described by the upper end
of it ?
The discussion of this equidistant line in a rather awkward
manner led the Hungarian Geometer to a conviction of the
validity of Euclidean Geometry.
In the second tract was introduced an ingenious substitute
for the Parallel-Postulate :
Let it be conceded that " if three points are not in a
straight line, then they lie on a sphere " : and therefore on
a circle.
To prove from this the principle of the Parallel-Postulate,
let A, B, A' be the three points. Let M, M' be the middle
points of AB, BA' ; and MN, M'N' drawn perpendicular to
them. Then the sum of the angles M'MN, MM'N' is less than
two right angles by the sum of the angles BMM', BM'M.
Now MN and M'N' must meet, in the centre of the circle
(or sphere), unless ABA' is straight. That is, however close
the sum of the angles M'MN, MM'N' is to two right angles,
MN and M'N' intersect. Q.E.D.
The elder Bolyai did not entertain the possibility of MN,
M'N' continuing to intersect when ABA' is one straight line;
nor did his brilliant son work out the elliptic hypothesis. Like
Lobachewski, the younger Bolyai only elaborated a hyperbolic
geometry. They both however discussed the properties of
curves (Z-lines, horocycles) neither rectilinear nor circular,
whereof the normals are parallel. Only in Euclidean Geometry
is the straight line the limit of a circle of indefinitely increased
radius.
Theories of Parallelism 37
THIBAUT
Grundriss der reinen Matheniatik, Gottingen, 1809.
Any treatment of parallelism based upon the idea of direc-
tion assumes that translation and rotation are independent
operations, and this is only so in Euclidean Geometry. The
following is the easiest and most plausible way of establishing
the parabolic hypothesis. From Euc. I. 32 can be deduced very
readily the Euclidean theory of parallelism. Thibaut argued
for this theorem as follows :
"Let ABC be any triangle whose sides are traversed in
order from A along AB, BC, CA. While going from A to B
we always gaze in the direction ABb (AB being produced to 6),
but do not turn round. On arriving at B we turn from the
direction Bb by a rotation through the angle bBG, until we
gaze in the direction BCc, Then we proceed in the direction
BCc as far as C, where again we turn from Cc to CAa through
the angle cCA ; and at last arriving at A, we turn from the
direction Aa to the first direction AB through the external
angle aAB. This done, we have made a complete revolution, —
just as if, standing at some point, we had turned completely
round ; and the measure of this rotation is 2?r. Hence the
external angles of the triangle add up to 2?r, and the internal
angles A + B 4- C = TT. Q.E.D."
TAURINUS
Theorie der Parallel-linien, 1825.
Taurinus expressed eight objections to the wider range
of Geometry then being manifested (Engel and Stackel, pages
208-9).
"1. It contradicts all intuition. It is true that such a
system would in small figures present the same appearance
as the Euclidean ; but if the conception of space is to be
regarded as the pure form of what is indicated by the senses,
then the Euclidean system is incontestably the true one, and it
38 Theories of Parallelism
cannot be assumed that a limited experience could give rise to
actual illusion."
None the less, however, measurement of angles, if sufficiently
exact, might deal a fatal blow to Euclid's a priori system, by
subverting Euc. I. 32.
" 2. The Euclidean system is the limit of the first system,
wherein the angles of a triangle are more than two right angles.
With this procedure to the limit, the paradox in connexion with
the axiom of the straight line ceases."
In fact Taurinus did not overcome the difficulty universally
experienced in respect of the elliptic hypothesis. Further on
he wrote (Engel and Stackel, page 257):
"In this theorem (51) it is proved that with the assumption
that the angle-sum of a quadrilateral can be greater than four
right angles (or, what comes to the same thing, if the angle-
sum of a triangle can be greater than two right angles), then
all the lines, perpendicular to another line, intersect in two points
at equal distance on either side. Hence arises the evident
contradiction of the axiom of the straight line, and so such a
geometrical system cannot be rectilinear."
To continue :
" 3. Were the third system the true one, there would be
no Euclidean Geometry, whereas however the possibility of the
latter cannot be denied."
Certainly the possibility of the Euclidean system cannot be
denied, but its actuality may be doubted. It is so infinitely
special an hypothesis. By way of analogy, does a single comet
in the universe possess a strictly parabolic orbit ?
"4. In the assumption of such a system as rectilinear,
there is no continuous transition ; the angles of a triangle could
only make more or less than two right angles."
The Euclidean system, nevertheless, is the connecting link
desired between the elliptic and hyperbolic geometries. It is
the limit of each ; whether it separates them or unites them is
only a matter of words.
" 5. These systems would have quite paradoxical conse-
quences, contradicting all our conceptions ; we should have to
assign to space properties it cannot have."
Theories of Parallelism 39
This appeal to commonsense is of value in practical work,
where close approximation suffices.
" 6. All complete similarity of surfaces and bodies would
be wanting; and still this idea seems to have its roots in
intuition, and to be a true postulate."
The view of some of the world's greatest thinkers, — and
they have Wallis for spokesman.
" 7. The Euclidean system is in any case the most complete,
and its truth therefore possesses the greatest plausibility."
But all three hypotheses together supply a more complete
Theory of Space than any single one of them, and their common
basis is the assumption that space is homogeneous. More
complete still will be the Geometry of the future, contem-
plating a heterogeneous space (compare Clifford's speculation ;
below, page 49).
" 8. The internal consistency of the third system is no
reason for regarding it as a rectilinear system ; however, there
is in it, so far as we know, no contradiction of the axiom of the
straight line as in the first."
These objections are mainly of an a priori character. On
the other hand, our knowledge of space as an objectivity is
small. Increased precision of astronomical instruments might
display antipodal images of a few bright stars, and this would
then tell in favour of an elliptic hypothesis. The space-constant
is large, very large; but no experiment can ever prove it infinite,
as Euclideans assume it.
J. BOLYAI
Appendix scientiam spatii e#/w'&ews...Maros-Vasarheli, 1832.
English, German and French translations have been made
of this brilliant tractate by Halsted, Frischauf and Hoiiel, the
first under the title Science Absolute of Space (Austin, Texas,
1896). Bolyai gave independently therein a clear, brief, and
sound introduction to the study of the hyperbolic hypothesis.
Only a summary description is necessary, as the student will
40 Theories of Parallelism
prefer to peruse its 43 paragraphs for himself. Starting from
the strict definition of a parallel as the limiting position of a
secant, Bolyai proceeded to solid geometry, and deduced the
existence of Z-lines and ^-surfaces (called by Lobachewski
horocycles and horospheres), which are the limiting forms of
circles and spheres of infinite radius in hyperbolic space. He
found Euclidean geometry to obtain for horocycles drawn on
horospheres ; but for rectilinear triangles drawn on a plane,
he proved that the area was proportional to the supplement
of the angle-sum (§ 43). The work merits careful study, and
comparison with the corresponding work of Lobachewski.
LEGENDRE
Reflexions sur...la TMorie des ParallUes, Paris, 1833.
Although Dr Heath has given a very full account of this
contribution of Legendre's to the Memoires de I'Institut de
France (Vol. 12, pages 367-390), the following notes on the
work of this most popular Geometer afford opportunities for
independent criticisms. We find Legendre writing:
" After some researches undertaken with the aim of proving
directly that the sum of the angles of a triangle is equal to two
right angles, I have succeeded first in proving that this sum
cannot be greater than two right angles. Here is the proof as
it appeared for the first time in the 3rd edition of my Geometry
published in 1800."
The proof proceeded thus: Let P&Qz,
be a series of identically equal triangles with their bases
Q-zQs, QaQ* ••• collinear and contiguous, and their vertices Plt P2,
P3... on the same side of their bases. If the angle-sum of a
triangle exceeds a straight angle, the angle Q^PiQ^ exceeds the
angle P,Q,P2. Therefore P,Pa < Q,Q2 (Euc. I. 18).
Let this difference be x. Then if n is a number such that
nx is greater than the sum of the two sides of any one of the
congruent triangles, supposed now to be n in number,
i > &P, + P,P, + P2P3 + ... + Pn-i
Theories of Parallelism 41
i.e. the straight line is not the shortest distance between the
two points Q! and Qn+1.
The solution of this enigma is that the elliptic hypothesis
was actually assumed; and for this hypothesis, the straight
line is re-entrant, and Qn+l may fall, for instance, between Qj
and Q2.
Legendre then proceeded further :
"The first proposition being established, it remained to
prove that the sum of the angles cannot be less than two
right angles; but we must confess that this second proposition,
though the principle of its proof was well-known (see Note II,
page 298, of the 12th Edition of the Elements of Geometry), has
presented difficulties which we have not been able entirely to
clear away. This it is which caused us, in the 9th Edition, to
return to Euclid's procedure ; and later, in the 12th, to adopt
another method of proof to be spoken of hereafter It is doubt-
less due to the imperfection of popular language, and the
difficulty of giving a good definition of the straight line, that
Geometers have hitherto achieved little success, when they
endeavoured to deduce this Theorem (Euc. I. 32) from ideas
purely based on the equality of triangles contained in the First
Book of the Elements."
The notion that the definition of the straight line was in
some way the nodus of the Theory of Parallelism had evidently
occurred to Taurinus and others, and was expressed as a con-
viction by Dodgson in his New Theory of Parallels (1895).
But contrast Lobachewski's judgment (below, page 43).
Legendre then gave a proof of Euc. I. 32 "as it appeared for
the first time in the 1st Edition of my Geometry, published in
1794, and as reproduced in the Editions following." This was
based on the following reasoning :
By superposition it appears that two triangles are congruent
if their bases and both base-angles are equal, each to each.
That is, a, B, C determine the triangle ABC uniquely. Hence
A =f(B, C, a). Therefore a = F(A, B, C), say. But A, B, C
are pure numbers, if the right angle is taken for unit-angle.
Hence a is obtained as a pure number. This is in fact the
42 Theories of Parallelism
paradox remarked upon by Gauss in his letter to Taurinus
(above, page 34). Only in the exceptional case when a does
not enter at all into the equation A =f(B, C, a), e.g. when
A = '7r — B—C, is there no unit of length associated with any
unit of angle.
Legendre next gave an ingenious and elaborate proof that
if the angle-sum of one triangle in the plane is TT, the angle-sum
of every triangle is IT. He observed further that if the angle-
sum of a triangle were greater (less) than TT, then equidistants
would be convex (concave) to their bases; but failed to make
cogent use of this apparent paradox, as it then was.
His last attempt to dispose of the elliptic hypothesis by
manipulation of the figure of Euc. I. 16 was effected with much
greater elegance by Lobachewski (below, page 44). Legendre's
construction and argument are given by Dr Heath in the first
volume of his Elements (page 215).
LOBACHEWSKI
Geometrische Untersuchungen zur Theorie der Parallel-linien,
Berlin, 1840.
As Dr Whitehead has said in one of his tracts :
"Metrical Geometry of the hyperbolic type was first dis-
covered by Lobachewski in 1826, and independently by J. Bolyai
in 1832. This discovery is the origin of the modern period of
thought in respect of the foundations of Geometry " (Axioms
of Descriptive Geometry, page 71).
The Geometrische Untersuchungen has been done into English
by Halsted (Austin, Texas, 1891).
Characterising Legendre's efforts as fruitless, the Russian
Geometer went on to say :
" My first essay on the foundations of Geometry was published
in the Kasan Messenger for the year 1829.... I will here give the
substance of my investigations, remarking that, contrary to the
Theories of Parallelism 43
opinion of Legendre, all other imperfections, for instance, the
definition of a straight line, are foreign to the argument, and
without real influence upon the Theory of Parallels."
Then followed a number of simple theorems, independent
of any particular theory of parallelism, although the third
presents the appearance of contradicting the elliptic hypothesis,
thus :
"A straight line sufficiently prolonged on both sides proceeds
beyond every limit (iiber jede Grenze), and so separates a limited
plane into two parts."
The ninth of Lobachewski's preliminaries, however, did
distinctly assume a theorem not universally valid for the
elliptic hypothesis:
" In the rectilinear triangle, a greater angle lies opposite the
greater side."
The sixteenth step inaugurated the Hyperbolic Hypothesis :
" 16. All straight lines issuing from a point in a plane can
be divided, with reference to a given straight line in this plane,
into two classes, namely, secant and asecant. The limiting
straight line between one class and the other is called parallel
to the given straight line."
Thus if Mil is a parallel to NB on one side of the perpen-
dicular MN let fall from M upon NB, then the angle HMN was
called the angle of parallelism. Let the length of MN be p,
and denote by CT the corresponding angle of parallelism HMN,
then Lobachewski established rigorously the very curious
result :
The complement of «r is the Gudermannian of pjK, where
K is constant over the plane.
Thus the angle of parallelism was determined in all cases;
and Lobachewski's result was identical with that obtained above
on page xviii
sin vf = sech p/K.
Lobachewski's proof was intricate, however, and involved the
horocycles and horospheres mentioned already (page 40).
44 Theories of Parallelism
Very elegant is Lobachewski's application of the construction
of Euc. I. 16 (itself a pattern of elegance) to prove that the
angle-sum of a triangle cannot exceed two right angles. Let
ABC be the triangle, and let its angle-sum be TT + e. Take
the least side EG, and bisect it in D. Join AD, and produce
to E, so that DE is equal in length to A D. Then the triangles
ODE, EDA are superposable. Hence the angle-sum of the
triangle AGE is 7r + e. Now by bisecting the side opposite
the least angle, and continuing Euclid's construction inde-
finitely for each new triangle obtained, two of the angles can
at length be reduced to magnitudes each less than ^ ; and thus,
z
since the angle-sum of each triangle is TT + e, the third angle
must finally exceed TT.
There is nevertheless no serious difficulty, provided space is
of finite size, as Riemann suggested later.
MEIKLE
Theory of Parallel Lines, Edinburgh, 1844.
The work of Henry Meikle, published in the Edinburgh
New Philosophical Journal (Vol. 36, pages 310-318), merits
special attention for its sound character, comparative obscurity,
and subsequent fertility in the hands of Kelland (Transactions
R. S. E., Vol. 23, pages 433-450) and Chrystal (Non-Euclidean
Geometry, Edinburgh, 1880).
The memoir commenced :
" During the long succession of ages which have elapsed
since the origin of Geometry, many attempts have been made
and treatises written, though with little success, to demonstrate
the important Theorem which Euclid, having failed to prove,
has styled his 12th Axiom, and which is nearly equivalent to
assuming that the three angles of every triangle amount to two
right angles."
Except that the elliptic hypothesis is not excluded by the
Parallel-Postulate, one might say "exactly equivalent."
Theories of Parallelism 45
The excellence of Meikle's work, and its supreme originality,
are attached to an ingenious and effective Construction, of which
some use was made without acknowledgment in the Appendix
to our Euclid, Book I. with a Commentary. This Construction
affords a demonstration of the Theorem that triangles of equal
areas have equal angle-sums. Thus :
Let ABC be any triangle, and D, E the middle points of the
sides AC, AB. Draw AL, BM, CN perpendicular to DE.
Then the triangles BME, ALE are congruent; and so are
the triangles ALD, CND. Hence the quadrilateral BCMN
has both its area and its angle-sum equal to those of the
triangle ABC.
Thus, reversing the construction, a triangle of equal area
and angle-sum can be constructed having a side BA' of .any
desired length not less than twice BM ; and then on BA as
base can be constructed an isosceles triangle of equal area and
angle-sum. In this way, two triangles of equal areas can be
reduced to isosceles triangles of the same areas and angle-sums ;
and on the same base. But isosceles triangles of equal areas
on the same base must coincide entirely. Hence the two
original triangles with areas equal have also their angle-sums
equal. Q.E.D.
By the aid of this magnificent piece of reasoning, Meikle
proved that the area of a triangle is proportional to what has
been called its divergence ; but he rejected the hyperbolic
hypothesis on the ground that it involved triangles of finite
area with zero angles, — the paradox which aroused Gauss'
interest (above, page 33).
BOUNIAKOWSKI
Memoires de l'Acaddmie...de St Petersbourg, 1850.
This writer criticised the work of Legendre and Bertrarid,
and endeavoured to improve upon it. He presented a simple
proof of the Parallel-Postulate in the following form :
46 Theories of Parallelism
Let MNP be a transversal crossing MA, NB\ and making
the angles AMN, BNM together less than two right angles.
Then the angle BNP is greater than the angle AMP] and
therefore the infinite sector BNP is of greater area than the
infinite sector AMP. Now if MA did not meet NB, the
infinite sector BNP would be contained wholly within the
infinite sector AMP, and so would be of less area. Therefore
MA cannot but meet NB.
If however we try to draw a figure introducing the infinitely
distant regions in any reasonable way, the proof collapses entirely.
Far more powerful and forcible is Bertrand's demonstration given
above (page 28).
RIEMANN
Habilitationsrede, 1854.
Riemann's brief but brilliant and epoch-making Essay was
translated by Clifford (Collected Papers, page 56). The possible
combination of finite size and unbounded extent, as properties
of space, was indicated in the words :
"In the extension of space-construction to the infinitely
great, we must distinguish between unboundedness and infinite
extent', the former belongs to the descriptive category, the
latter to the metrical. That space is an unbounded threefold
manifoldness is an assumption that is developed by every
conception of the outer world.... The unboundedness of space
possesses... a greater empirical certainty than any external
experience. But its infinite extent by no means follows from
this; on the contrary, if we assume bodies independent of
position, and therefore ascribe to space constant curvature, it
must necessarily be finite, provided this curvature has ever so
small a value."
The expression curvature of space is an unfortunate metaphor,
derived from the analogy between the elliptic geometry of the
plane and the parabolic geometry of surfaces of uniform curva-
Theories of Parallelism 47
ture. The plane is not curved for the elliptic, nor for the
hyperbolic hypothesis. Considerable misconception has arisen
in this way. The study of Beltrami's analogies rectifies such
an error.
Biemann further noted the conceivable heterogeneity of
space. The space-constant may be different in different places.
It may also vary with the time.
CAYLEY
Sixth Memoir upon Qualities, London, 1859.
Cayley developed an analytical theory of Metric which could
be coordinated to the three hypotheses which constitute the
geometry of a homogeneous space. The hypotheses presented
themselves as the three cases when the straight line has no,
one, or two real points at infinite distance from all other points
on itself. These three cases are allied to the hypotheses of no,
one, or two parallels from a given point to the straight line.
VON HELMHOLTZ
The Essential Principles of Geometry, 1866, 1868.
Von Helmholtz endeavoured to give new and precise expres-
sion to the Axioms or Postulates upon which a science of spatial
relationships could be constructed, e.g. perfectly free moveability
of rigid bodies. The whole of the ground has been gone over
with great thoroughness by succeeding Geometers, notably Lie.
The final results of these labours have been summarised by
Dr Whitehead in his Tracts.
48 Theories of Parallelism
BELTRAMI
Attempt to interpret non-Euclidean Geometry, 1868.
Beltrami's Saggio, of which a French translation was made
by Hoiiel (Annales de l'£cole Normale Superieure, Vol. 6, pages
251-288), is of supreme importance in the development of the
Science of Space. The Italian Geometer proved conclusively
the right of the elliptic and hyperbolic hypotheses to rank
equally with the Euclidean system as theories of a homogeneous
space ; and, empirically, they are clearly superior. Beltrami
showed this by pointing out that all the elliptic (and hyper-
bolic) geometry of the plane was characterised by the same
metrical relationships as hold good in the parabolic geometry
of surfaces of uniform positive (and negative) curvature. Any
flaw in the former would necessarily be accompanied by a flaw
in the latter. If there is no flaw in the Euclidean geometry
of geodesies on a surface of uniform curvature, then there can
be no flaw in the metabolic geometry of straight lines on a
plane surface, for the metrical relationships are identical. This
conclusive argument can scarcely be refuted ; Poincare' does
not meet it in his La Science et I'Hypothese. It was there-
fore Beltrami's labours which first established Non-Euclidean
Geometry on the firm foundation whereon it rests to-day,
despite every kind of prejudice and misconception which it has
encountered hitherto.
CLIFFORD
The Space-Theory of Matter, 1870.
Somewhat beyond theories of parallelism, but suggestive
like everything else of his, the fragment of Clifford's here re-
produced shows the freedom of the Geometer released from the
fetters of traditionalism. It is chosen from a paper embodied
in his Collected Works (page 22) :
Theories of Parallelism 49
" I wish here to indicate a manner in which these specula-
tions (of Riemann's) may be applied to the investigation of
physical phenomena. I hold in fact :
"(1) That small portions of space are of a nature analogous
to little hills on a surface which is on the average flat ; namely,
that the ordinary laws of geometry are not valid in them.
"(2) That this property of being curved or distorted is
continually passed on from one portion of space to another after
the manner of a wave.
" (3) That this variation of the curvature of space is what
really happens in that phenomenon which we call the motion
of matter whether ponderable or ethereal.
" (4) That in the physical world nothing else takes place
but this variation, subject, possibly, to the law of continuity.
" I am endeavouring in a general way to explain the laws of
double refraction on this hypothesis, but have not yet arrived at
any results sufficiently decisive to be communicated."
The boldness of this speculation is surely unexcelled in the
history of thought. Up to the present, however, it presents the
appearance of an Icarian flight.
KLEIN
Ueber die sogenannte nicht-Euklidische Geometric, 1871.
To Dr Felix Klein, Professor at Gb'ttingen, are owed two
monographs (Mathematische Annalen, Vol. 4, pages 573-625 ;
Vol. 6, pages 112-145), which have been followed up by two
volumes of lectures on non-Euclidean Geometry. The names of
Cayley, Clifford and Klein will always be associated with a
certain view-point, which may be styled the analytical theory
of metrical relations.
In the first of his classical Memoirs Dr Klein introduced
that terminology which has won its way to general acceptance :
F. E. 4
50 Theories of Parallelism
" The three Geometries have been called hyperbolic, elliptic,
and parabolic, respectively, according as the two infinitely
distant points of the straight line are real, imaginary, and
coincident" (see page 47 above).
The second paragraph opens thus:
" All spatial metric rests upon two fundamental problems,
as we know : the determination of the distance of two points
and of the inclination of two straight lines."
The fundamental laws of linear and angular measurement
are, in fact : xy + yz = xz, with the proviso that xx = 0 ; so that
xy + yx = xx = 0, and therefore xy = — yx. These first principles
of metric have been discussed in a simple but ingenious way by
Sir Robert Stawell Ball in the last chapter of his Theory of
Screws (Cambridge, 1900).
Klein followed Riemann and Cayley in the use of coordinates
to define position, and deduced formulae for lengths and angles
from the first principles suggested above. The results, that
length and angle are proportional to the logarithms of certain
anharmonic ratios estimated with reference to an Absolute
formed of infinitely distant elements, cannot be assessed by
Euclidean standards, but belong to a higher sphere of research
than is here explored. This remark applies also to Klein's
second Memoir, and indeed to most of the best modern work in
Non-Euclidean Geometry.
NEWCOMB
Elementary Theorems relating to the Geometry of a Space
of three Dimensions and of uniform positive Curvature
in the fourth Dimension, 1877.
The American Astronomer first assumed the homogeneity
of space (Crelles Journal, Vol. 83, pages 293-299). Then :
"2. I assume that this space is affected with such curva-
ture that a right line shall always return into itself at the end
Theories of Parallelism 51
of a finite and real distance 2Z), without losing, in any part of
its course, that symmetry with respect to space on all sides of it
which constitutes the fundamental property of our conception
of it."
This definition of rectilinearity and the assumption of
firiitude are faultless ; but no more needs to be assumed.
Newcomb might now have proved that
area/divergence = 4Z)2/7r2 ;
and the theorem below might have been furnished with a
demonstration. Instead of this, however:
"3. I assume that if two right lines emanate from the
same point, making the indefinitely small angle a with each
other, their distance apart at the distance r from the point
of intersection will be given by the equation
_ rir
: n
But, as was shown by Dr Chrystal, since area is by (2)
necessarily proportional to divergence,
, 4D* I(M
sdr = d -- — tan"1 -j- ,
7T2 drj
ITT TTT
Hence s = A sin =-=: + B cos =-= ;
— / ' 2iL)
and when r — »-0, s — »-0. and ^ -- >-«: so that
dr
B.O.
mu C •
Therefore s = -- sin
TT
4—2
52 TJieories of Parallelism
DODGSON
A New Theory of Parallels, London, 1895.
The amiable author of Alice in Wonderland contributed to
the Theory of Parallelism a pretty substitute for the Fifth
Postulate, as follows:
" In every circle, the inscribed equilateral tetragon is greater
than any one of the segments which lie outside it."
Dodgson's Axiom was aimed at the exclusion of the hyper-
bolic hypothesis, in which the assertion is not universally
correct.
For consider, on the hyperbolic plane, a circle of very great
radius nK, where n is a large number, and two perpendicular
diameters A OA ',BOB'.
The area of the circle is
rnK x
2-7T I K sinh ^dx= 2?rif2 (cosh n — 1),
Jo -"-
which approaches the limit irK2en, as n increases indefinitely.
On the other hand, the area of the quadrilateral formed by
the points A, A', B, B' is
4K2 1| - 2 tan-1 (sech
AO
because cos ABO = cosh -~- sin OAB (above, page xviii). And
as n increases, this approaches the limit
(20-"), i.e.
Whence it appears that the area of a segment is ultimately
infinitely greater than the area of the tetragon.
FIRST ADDITIONAL NOTE
ANALYTICAL GEOMETRY FOR THE METABOLIC HYPOTHESES
FROM very simple synthetic results, the most comprehensive
analytical formulae for Metabolic Geometry can be secured.
These formulae of general application will be proved for the
elliptic hypothesis, and the corresponding forms for the
hyperbolic hypothesis will be interpolated as occasion arises.
The principal assumptions are (i) that the area of a triangle
is k? (or K2) times its divergence, and (ii) that the straight line
is in general a line of minimum length between any two of its
points. The groundwork of our Introduction, along with an
argument similar to that of Euc. I. 16-20, justifies these
assumptions, to which may be added the observation that
Euclidean geometry holds good within any infinitesimal area
of a homogeneous plane.
Adhering then, for convenience, to the Elliptic Hypothesis,
let r be the radius vector, & the vectorial angle, and <f> the
inclination of the tangent to the radius vector for any point of
a curve. Let 0 be the pole, and P, P1 the points of contact
of two consecutive tangents to the curve, inclined at an
infinitesimal angle d\}r. Let RdO be the area of the triangle
OPP', where R is a function of r at present unknown. Then
by (i), to the first order of infinitesimals,
RdO = k*(d8 + TT - (f> + <j> + d(f> + TT - d^r - 2-Tr)
54 Theories of Parallelism
Let t be a parameter for the point P of the curve, then this
result becomes
In particular, for the points of a straight line ^ = 0, and so
Again by (ii) we have for the straight line
t = 0,
where s is the length of the arc of the curve measured from
one of its points. And the infinitesimal length of the per-
pendicular from P upon OP' is R'dO, where the dash denotes
differentiation for r, — because the product of dr and R'dO gives
the correct value dRdO for the element of area in polar co-
ordinates. Thus, for a straight line,
)
Hence, by the Calculus,
d r R'R"62
dil~ § '
and — = 0,
at s
where dots denote differentiation for the parameter t.
r R'd
But cos 6 = - , sin d> = ,
s s
so that we have
-T- (cos </>) = R"6 sin <£, -r.(R' sin <£) = 0.
Thus j> = -R"0, R'sm(j) = P, where P is the value
assumed by the function R' when r=p, p being the per-
Theories of Parallelism
55
pendicular from 0 on the straight line. These two results are
not independent, of course.
But, for a straight line, k* (j> + (k* - R) 6 = 0.
Therefore k2R"+ R — k2 = 0 ; and by differentiation for r,
I I
Integrating, R' = L cos r + M sin 7, where L and M are
K K
independent of r.
Remembering that for r=0, R' = 0; and that for r small,
T
R' = r, we see that R — k sin T .
K
Hence for the straight line, as considered,
j • r . i • P
k sin j sin 9 = k sin j- ;
rC /C
that is, in a right-angled triangle of hypothenuse r, wherein p
is the length of the side opposite the angle <f>,
. r . , . p
sin r sin d> = sin ,- .
/ • / •
A/ iv
T . T)
[For the Hyperbolic Hypothesis, sinh ~sin <£ = smh ^..
Also we find now for any curve
ds* = dr* + A? sin2 T . d&\
K
, . T dB
tan <p = k sin T . ^— ,
k dr
T
= COS y .
K
[For the Hyperbolic Hypothesis,
.
zt
„ . , r dO
tan <b = K smh -TT • 'f~ >
K dr
T ' *
ir = cosh j. 8 + <>.
56 Theories of Parallelism
/j\ fp
We prove next that tan ^ = tan j- cos 0.
K K
d0 1 T . . TO T
In fact, -J- — -J- cosec r tan <f> ; and sin <£ = sin ^ cosec T 5
CtT* K fC 1C K
so that
T dv r
k -j- = cosec -r
dr k
n T
= cosec2 T / A / cosec2 y — cosec2 -r
7 / A / V^WU^V-' T V-*-/O^S-- ,
«/ V k k
T I I n T
= cosec2 T / A / cot2 y- - cot2T ;
«/ V A; A;'
/ T I 1)\
whence 6 — a. = cos"1 cot T / cot \ •
V */ k)
f) T
and so tan^- = tan j cos (0 — a).
K K
n T
[For the Hyperbolic Hypothesis, tanh ^ = tanh ^ cos (6 — a).
A xt
Hence if the perpendicular p from 0 upon any straight
line is inclined at angle a to the axis Ox from which the
vectorial angle 0 is measured,
D T
tan ^ = tan T cos (6 — a).
We can next find the length P'P" between the points
(r'6r) and (r"6"). Using the result just obtained, along with
T '
$ = k sin T cosec <£ . 0, we get
K
P'P" = k tan? sec?
•6"
d0
uuo ^i/ — w/ i uodi y
J &
Adopting tan ^ tan (0 — a) for variable, and integrating,
P'P" = k I tan-1 {sin £ tan (0 - a)l .
Jv \ k ']
Theories of Parallelism 57
sin ^ (tan 0" - a. - tan 6' - a)
I.e., tan — y-
1 + sin2 1 tan (ff - a) tan (0" - a)
a result presently transformed into
P'P" r' r" r> r" t
cos — j — = cos y cos y— + sin — sin y- cos (0 — 6 ),
A? K K K K
which manifests the analogy between the elliptic geometry of
the plane and the parabolic geometry of the sphere.
[For the Hyperbolic Hypothesis,
P'P" r' r" r' r"
cosh „ = cosh -^ cosh -^ — sinh ^ sinh -~ cos (ff— 0").
J\. Ji. Ji XL XL
Let us now adopt Escherich's Coordinates. Let Ox, Oy be
two perpendicular axes ; PM and P^ the perpendiculars upon
, . , p , .. OJf 0^
them trom any point r ; and write x = tan —7—, y = tan — y—.
A? K
, OM . ON
[For the Hyperbolic Hypothesis, x= tanh -j=-, y = tanh -^-.
We then have at once
PF \+xx' + yy
COS —j— = -p=== . (I).
l.t ./-i.o.n./-i.frt«/rt ^ '
[For the Hyperbolic Hypothesis,
cosh
PP' \—xx— yy'
V 1 - of - y* V 1 - x* - y'2 '
Choosing consecutive points, x' = x + dx, y' = y + dy, we
have
[For the Hyperbolic Hypothesis,
, _ rr2 da? + dy* - (xdy - ydxf
58 Theories of Parallelism
1} T
Also, from tan j- = tan y cos (6 — a), obtained above, we now
1C K
have
T T fir \ n
tan T cos 0 cos a + tan r cos ( - — 0 } sin a = tan y ,
k k \2 J k
showing that the equation of the straight line in these
coordinates is x cos a + y sin a = constant ; or, in general,
ax + by+c = Q (II)
where a, b, c are any constants.
We may next determine the angle of intersection of the two
straight lines ax + by + c = 0, a'x + b'y + c = 0.
Let P be their point of intersection, with coordinates x, y ;
and Q, ty consecutive points of the two lines, with coordinates
(x + e&, y — ea) and (x + e'b', y — ea), where e and e' are y^ery
small.
Then the angle ^ required is determinable from the
application of the Euclidean formula to the infinitesimal
triangle QPQ',
QQ>* = PQ* + PQ'* - 2PQ . PQ'. cos QPQ.
Hence, omitting a common denominator,
- e'6')2 + (ea - eV)2 + (x . ea - eV + y.eb- e'bj
= (e&)2 + (ea)2 + (a? . ea + y . eft)2 + (e'6')2 + (e'aj
+ (x . ea' + y . e'&')2 - 2 cos x
x Ve2^ + e2a2 + (ea# + efa/)2 . V e/26'2 + e/2a'2 + (e'a'x + e'6't/)2.
But ax + by= — c) and a'x + b'y = — c' ; and so
aa + bb' + cc'
cosv= .. = — . =
Va2 + 62 + c2. Va'2 + 6/2 + c'2
[For the Hyperbolic Hypothesis,
aa' + bb' — cc'
cos v =
In particular, then, the straight lines are perpendicular, if
aa' + bb' + cc' = 0.
Theories of Parallelism 59
[For the Hyperbolic Hypothesis,
aa' + bb' - cc = 0.
Calling u, v the coordinates of a straight line ux + vy — 1 = 0,
the angle between the consecutive lines (u, v) (u + du, v + dv)
is given by
2 _ du* + dv1 + (udv — vduf
[For the Hyperbolic Hypothesis,
, 2_ du2 + dv2 — (udv — vduf
X = (1 - w2 - v2)2
We now only need further (and only for convenience' sake)
to determine the length of the perpendicular p' from (x'y)
upon ax + by + c = 0. This is derivable from results I, II
and III ; suggesting that these contain the entire metric of the
elliptic plane in analytical form. The coordinates of the foot
of the perpendicular are found by II and III to be
(Dx^P^a Dy' - P'fr
VlJ^TFc" ' D-P'c ) '
where D = a2 + b2 + c-, P' = ax + by' + c.
Thus oos^-- AD(l +#'2 + 2/'2) — -f2
1 11U> LfUa , — / — 2 — — .
k V (1 + a;'2 + y'*)D
, . p ax' + by' + c ,T VN
whence sin y- =- — — (IV).
* v a2 + 62 + c2 V 1 + x'* + y'2
[For the Hyperbolic Hypothesis,
. , p ax' + by' + c
smh £= = - f
Any problem in Metabolic Geometry may be attacked by
means of results I — IV thus secured.
For instance, the. equation of a Circle of centre (£>;) and
radius p is by I
(1 + far + rjyy- = cos2 . (1 + f + rf) (1 + & + y2).
60 Themes of Parallelism
[For the Hyperbolic Hypothesis,
[Letting £ = 0, tj = tanh ~, and p become indefinitely great,
XL
the equation of a Horocycle touching Ox at 0 is obtained:
(l-2/)2=l-^-2/2.
The equation to an Equidistant of axis ax + by + c = 0 at
distance OT is by IV
(ax + by + c)2 = sin2 ? . (a2 + 62 + c2) (1 + a? + f).
[For the Hyperbolic Hypothesis,
(ox + % + c)2 = sinh2 J. . (a2 + 62 - c2) (1 - a? - f).
A result of considerable beauty and interest is an expression
for the measure of curvature. If de is the angle between the
normals at the extremities of an infinitesimal arc ds of any
curve, and the radius of curvature p is defined by -7- = Arsin £
then if t is the parameter of a point on the curve, formulae
I and III give
.o . ......
cot I = <^ - *y
[For the Hyperbolic Hypothesis,
p ... ., f l-tf2-v2
coth ^. = (xy — xy) \ — — -. — , . .
K 'x* + *-x-x
Applied to the equation of the equidistant to Ox at distance
8, namely :
s
2/2 = (l+#2)tan2 -,
rC
or parametrically,
x = tan t, y = tan j- . sec i,
A>
Theories of Parallelism 61
the curvature-formula gives
suggesting that the curve is a circle of radius -= — 8.
i!
Evidently the results given in this brief Note are sufficient
to furnish occupation for innumerable leisure hours, in the
extension of the usual results of Analytical Geometry to the
metabolic hypotheses. The following Note will also furnish
the materials for extending Analytical Dynamics so that it
may apply to a space whose geometry is metabolic.
SECOND ADDITIONAL NOTE
PLANETARY MOTION FOR THE METABOLIC HYPOTHESES
THE difficulties introduced into the Newtonian theory of
planetary orbits by the adoption of either of the metabolic
hypotheses of space are far from insuperable. Let us assume
LL T*
j- cosec2 T as the law of gravitational force, for the elliptic
Kr K
hypothesis. We can then prove that the polar equation of a
planetary orbit, with the sun for pole, and an apsidal distance
d for axis, is simply
k _ 1 4- e cos 6
~d~ ~l+e '
COt y
K
and that if
major axis of orbit
-
then the periodic time is
^ 1 + a- '
[For the Hyperbolic Hypothesis,
T
coth j, ., a
_ K 1 + e cos 6
7d= 1+6 ;
coth -T*.
Ji
periodic time,
Vjil-o"
where
/major axis of orbitN
Theories of Parallelism 63
In what follows we adhere to the Elliptic Hypothesis.
The results obtained in the First Note (page 55) give for the
radial velocity : r ; and for the transverse velocity : k sin -r 6 ;
K
in the case of a particle whose polar coordinates are (r, 6).
Applying the usual method, we find for the increase of
radial velocity in time dt from P to P'
rdt — k sin y Q cos G>,
where to is the angle made with OP by the perpendicular to
OP' at P'.
OP' r r •
But cos w = cos —j— sin POP' = cos -r.d& = cos y 6 . dt, to
the first order, by the last formula on page xviii above. Thus
the radial acceleration is
V T '
r — k sin y cos y . 6* (I).
Similarly again the increase of transverse velocity in time
dt is
d I T '\
-r- 1 k sin T 0 } . dt + r cos «',
dt\ K J
where a/ is the angle made with OP' by the perpendicular to
OP at P.
OP r •
But cos o>'= cos -j- . d6 = cos y 6 dt (page xviii).
Hence the transverse acceleration is
/ rdf'«rA\ TT
k cosec 7 -7; sin2 T 0} (11).
k dt \ k J
It is clear also that if <£ is the inclination of the velocity v
to the radius r,
r 0
tan <j> = k sin j - (III).
K T
The areal velocity is
(IV).
64 Theories of Parallelism
And the kinetic energy of a particle of mass m is
(V).
j
Thus for a particle of unit mass acted upon only by a radial
force R towards the origin,
T T '
r — k sin -r cos T^ = — R,
T d f T "\
k cosec 7- -7- ( sin2 T 6 } = 0.
k at \ k J
T '
Hence &* sin2 y 0 = h, where h is a constant for the orbit ;
and the energy-equation is found to be
r" + fc2 sin2 £ 02 = C- 2 f ' Rdr.
It appears that if a metabolic hypothesis of space holds
good, equal areas are not described in equal times in central
orbits. What transverse force must be introduced for the
retention of the Newtonian Law ? If we are to secure that
the transverse force is determined by the value of the trans-
verse acceleration to be
r d ( . „ r i\
k cosec T -T- sin2 T 6
k dt\ k J
r d f. r
= kH cosec r -j- 1 -f cos T
k dt\ k
= -Hr;
so that there must be a transverse retardation proportional to
the radial velocity. This would not be produced by frictional
forces of the usual kind.
Theories of Parallelism 65
Considering now the earth's orbit about the sun, let us
ill V
assume R = ~ cosec2 r , corresponding to the law of potential
K K
(extended from Poisson's):
where p is the density of a material medium.
r
j
k
• h i*
Then when we substitute Q = j^ cosec2 7, the equation of
energy becomes
r 2 + J cosec2 _ cot = 2E, say.
Thus
h r ,
TT cosec2 T dr
n n
whence dO =
j
T h2
1 r
Write conveniently -ar for T cot -r, then
2E
0-a =
Write also o- for OT — 2, and /2 for -- — + -, then
and so <r = f cos ($ ~~ a)-
If we measure ^ from the apsidal line, and write d, u for
an apsidal distance and corresponding velocity, then
h = kusm-r, 2E = u*— -77 cot T.
F. E. 5
66 Theories of Parallelism
Thus
, I d fj, -d
/= 7cot T -- %-j- cosec2 j ;
J k k i(2k2 k
and we get
, r /* -d / .d At d\
cot T — c? cosec2 T = cot T -- -r cosec2 r cos 0.
A; u?k k \ k u*k k}
Now write
u2k . d d
e — — sin T cos T — 1,
/* « «
so that e is positive or negative according as the centripetal
force at the apse is less or greater than for the description of a
circle ; then
(1 + e) — -5 = 1 + e cos 0,
, d>
COt 7
k
the equation of the earth's orbit about the sun, if space is of
finite extent.
The equation of this orbit in Escherich's coordinates, with a
suitable change of origin along the apsidal line, is reducible to
the form
so that the orbit has a geometrical centre.
The curve then exhibits several properties, precisely
analogous to those of the ellipse in parabolic geometry, obtain-
able by formulae I-IV of the First Note.
Three of these properties may be indicated thus :
(1) SP+ST = AA',
SP PM
(2) sin -y- cosec — : — = constant,
k k
SY . S'Y'
(3) sin -r- sm — j-— = constant.
Theories of Parallelism 67
/major axis of orbitA
If now we write a = tan ( - — =T — - ) , we have the
formula analogous to a well-known result in Newtonian
dynamics :
Let us proceed further to find the periodic time in the
orbit, T ; that is, the length of the year if space is of finite
extent.
h { T\
Since 6 = r^ 1 1 + cot2 -r ) , we have by use of the polar
K \ K /
equation of the orbit found above
d 2ir
Acot2r
A ,/•
(1 + ef tan2 ^ + (1 + e cos &?
i ™
In addition to a as defined already, let us introduce
OS
c = tan -j- , where G is the centre of symmetry and S the centre
K
of force for the orbit.
Then the four following equivalences are true :
u, a + c 1 4- a2 , „ ak a2 — c2
t •— • "T -- ~ j- , — »
K a — c a al + c2
^dl + ac cl+a2
cot 7 = , e = - = .
fc a — c al+c2
The value of T can now be transformed thus :
It is therefore necessary to evaluate the definite integral
f "
where
P=-^
68 Theories of Parallelism
Express then this integral as the difference of two
conjugate integrals:
2" de *" d6
_L_ I f2" de [*
2iP\Jo RcosB + Q-iP J0
each of which may be evaluated by the application of Cauchy's
Theorem to a contour in the 2-plane (z = ei6), consisting of a
circle of unit radius about the origin.
Thus
M = 2_ f dz
cose + Q-iP iR - 9Q-iP '
I Z + £t - jj - •2 + 1
/ Jt
where the second integral is along the unit-circle in the
2-plane.
Q — iP
Now z* + 2 „ — z + 1 is identically equal to
Q-iP H .\/ Q-iP H .
~
where H< = (Q2- E2 - P2)2 + (2P£)2, and tan 2^, =
The product of the moduli of the poles (a, /3) of the integral
is unity, since a/3 = 1 ; and therefore only one of these poles
lies within the unit-circle.
Moreover, the integrand
Q-iP H . Q-iP H . y
Hence, by Cauchy's Theorem, inasmuch as there is but one
simple pole within U| = l, the contour-integral above has the
value
2 .Re*" 2-7T ,
Theories of Parallelism
Hence also
_ a — *<»
'"~ H '
Therefore
de 2?r
f
- o
.
toa _ />— i<a\
27T
Thus
'2* de
] a*~^
V(i +
-C2 )2 . L , Cl + a\030\
(1 + a2)^ ' a2 - c2 ' (a2 - c2)^ (1 + a2
(l+a2)(a2-c2)
Finally,
V/A 1 + a2
If the straight line had a complete length as great as
25,000 light-years, the quantitative difference between the
length of the year deduced from this formula and the
Newtonian result for Euclidean space would be an incon-
ceivably small fraction of a second.
But the formal difference between the old and new results
is considerable.
70
Theories of Parallelism
POSTSCRIPT
It will be observed that by use of the formula
0 27T
r
Jo
COS i
the preceding analysis may be more rapidly worked out as
follows :
a* (1 + a2)2 ac2 „,
~~ i • J- ==
f
Jo
dd
c 1 + aV c2 (1+ a2)2
. c (1 + a2) f2jr
i» a2 - c2 I
7 o
cos
a(l+c2)-i(a2-c2)
c(l+a2)
cos ^ +
+ a2)
1 c(l+a2)r 27
"2t a2-c2 L(0i_
27rac2(l +a2)
27rc(l+t'a)
(a2 -
which furnishes the desired result.
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