Normat 60:4, 145–169 (2012) 145
Euclid’s straight lines
Christer O. Kiselman
Uppsala University,
Department of Information Technology
Division of Visual Information and Interaction
Computerized Image Analysis and Human-Computer Interaction
Box 337
SE-75106 Uppsala
kiselman@it.uu.se, christer@kiselman.eu
1 Two questions
Stoikheia (Stoiqeÿa)byEuclid(EŒkle–dhc) is the most successful work on geometry
ever written. Its translation into Latin, Elementa ‘Eleme nts’, became better known
in Western Europe. It can still be read, analyzed—and understood. Nevertheless,
I experienced a difficulty when trying to understand some results.
The First Question. Euclid’s Proposition 27 in the first book of his Stoi-
qeÿa does not follow, strictly speaking, from his postulates (axioms)—or is
possibly meaningless. Its proof relies on Proposition 16, which suffers from
the same difficulty. There must to be a hidden assumption. What can this
hidden assumption be?
Proposition 27 says:
If a straight line falling on two straight lines make the alternate angles equal to one
another, the straight lines will be parallel to one another. (Heath 1926a:307)
Proposition 16 says:
In any triangle, if one of the sides be produced, the exterior angle is greater than
either of the interior and opposite angles. (Heath 1926a:279)
Some subsequent res ults will also be affected.
In this note I shall try to save Euclid by reexamining the notions of straight line
and triangle, and expose a possible hidden assumption.
I shall also prove that if we limit the size of the triangles suitably, Proposition
16 does become valid even in the projective plane (see Proposition 7.1).
The Second Question. What does the word eŒjeÿa (eutheia) mean? It is
often translated as ‘straight line’, which in English is usually understood as
an infinite straight line, but in fact it must often mean instead ‘rectilinear
segment, straight line segment’. Which are the mathematical consequences of
these meanings, which we nowadays often prefer to perceive as different?
Michel Federspiel observes:
146 Christer O. Kiselman Normat 4/2012
La d´efinition de la droite est l’un des ´enonc´es math´ematiques grecs qui ont suscit´e le
plus de recherches et de commentaires chez les math´ematiciens et chez les historiens.
(Federspiel 1991:116)
For a thorough linguistic and philosophical discussion of this term, I refer to his
article. He does not discuss there—maybe because the answer is all too evident
for him—whether eutheia means an infinite straight line, a ray, or a rectilinear
segment, meanings that C harles Mugler records in his dictionary:
1
Ligne droite ind´efinie ; aussi demi-droite. [. . . ] 2
Segment de droite. (Mugler
1958–1959:201–202)
This is what I will discuss in Section 4. Before that, however, I shall fix the ter-
minology concerning two models for Euclid’s axioms, the Euclidean plane and the
projective plane. I will discuss the determination of triangular domains in the two
models in Section 6, the proof of Proposition 16 in Section 7, and the notion of
orientability in Section 9.
2 Approaches to this paper
The following convictions have been driving forces behind this paper.
(1) Geometry is fascinating, especially its logical content—I owe this to Bertil
Brostr
¨
om, my first mathematics teacher.
(2) Languages are fascinating—I owe this to Karl Axn
¨
as, my teacher of German and
my most inspiring teacher all categories. Much later I wanted to understand
Euclid and studied Clas sical Greek for Ove Strid.
(3) History is fascinating—I owe this to my history teacher Nils Forssell.
This means that the present text might be difficult to classify: I combine
(A) verbatim quotes from Euclid’s text to show exac tly how the terms were used;
with
(B) a critical look at the logic, where I fee l free to use the knowledge I have now,
without implying anything about what Euclid could have known.
To prove that a stateme nt, like that of Proposition 16, does not follow from ce rtain
axioms, a standard method is to exhibit a model where the axioms are true while
the statement is not. The nature of the model is not important: it can come from
any time and any place, and does not allow any conclusions relevant for history.
This argument should be compared with the proof by Lobaˇcevski˘ı, Bolyai and
Gauss that the Pos tulate of Parallels is independent of the other axioms.
As Ulf Persson remarked, history shares with mathematics the fact that its sub-
ject does not exist (any longer), while the subject of mathematics has never existed,
except perhaps in some world where Plato lives. For other thoughts comparing hi-
story and mathematics, see his essay (2007) on Robin George Collingwood’s book
The idea of history (1966). The present study combines history and mathematics,
hopefully so that both perspectives are discernable.
Normat 4/2012 Christer O. Kiselman 147
3 The Euclidean plane and the projective plane
3.1 Straight lines and rectilinear segments in the Euclidean plane
In this paper I shall use E
2
to denote what is now known as the Euclidean plane.
This is an affine space which can be equipped with coordinates which are pairs of
real numbers, in other words elements on R
2
. More precisely, given three points
a, b, c œ E
2
which do not lie on a straight line, we can give a point p œ E
2
the
coordinates (x, y) œ R
2
if p = a + x(b ≠a)+y(c ≠a). (Note that in an affine space,
where there is no origin, a linear combination ⁄a + µb + flc has a good meaning if
⁄ + µ + fl = 1, which is the case here.) In order to be able to speak about angles
and areas, we nee d to equip the associated vector space with an inner product.
In the sequel I shall use the following terms.
A straight line is given by {(1 ≠t)a + tb œ R
2
; t œ R},werea ”= b;itisinfinite
in both directions.
1
A rectilinear segment is given by {(1 ≠t)a + tb œ R
2
; t œ R, 0 Æ t Æ 1}.SinceI
want to avoid a point being declared as a rectilinear segment, I require that a ”= b.
A ray is given by {(1 ≠ t)a + tb œ R
2
; t œ R, 0 Æ t},wherea ”= b;itisinfinite
in one direction.
We note in passing that the sam e distinctions are made in Contemporary Greek:
euTe–a gramm† (f) ‘straight line’; euT‘grammo tm†ma (n) ‘rectilinear segment’; akt–na
(f) ‘ray’; ‘radius’ (Petros Maragos, p ers onal communication 2007-10-12; Takis
Konstantopoulos, personal communication 2012-01-20).
Given two points a, b on a straight line L in E
2
, the complement L r {a, b} has
three components, one of which is bounded. So the rectilinear segment with a and b
as endpoints can be recognized as the union of {a, b} with the bounded component
of L r {a, b}.
3.2 Straight lines and rectilinear segments in the projective plane
The projective plane, which I shall denote by P
2
, is a two-dimensional manifold
which can be obtained from the Euclidean plane by adding a line, called the line at
infinity, thus adding to each line a point at infinity. For a brief history of projective
geometry se e Torretti (1984:110–116). Johannes Kepler was, according to Torretti,
the first in modern times to add, in 1604, an ideal point to a line.
There are no distinct parallel lines in P
2
. Still I shall c onsider that it satisfies
Postulate 5:
e'.
2
That, if a straight line falling on two straight lines make the interior angles on the
same side less than two right angles, the two straight lines, if produced indefinitely,
meet on that side on which are the angle s less than the two right angles. (Book I,
Postulate 5; Heath 1926a:202)
This postulate, of course, must b e subject to interpretation in the new structure,
and therefore the statement that P
2
is a model is not an absolute truth.
3
1
Heath (1926a) uses straight line and Fitzpatrick (2011) straight-line as hypernyms for three
currently used terms: straight line in the sense just defined, which is the sense I shall use, rectilinear
segment,andray.
2
Statements are numbered by letters marked by a keraia (kera–a): a' =1,b' =2,..., ' (stigma)
=6,...,ia' =11,ib' =12,...,ke' =25,... .
3
AbetterknownmanifoldistheM
¨
obius strip, which can be obtained from P
2
by removing
apoint,asBoG
¨
oran Johansson points out (personal communication 2012-02-14). Now there are
148 Christer O. Kiselman Normat 4/2012
The projective plane can be given coordinates from points in R
3
as follows. A
point p œ P
2
is represented by a triple (x, y, z) ”=(0, 0, 0), where two triples (x, y, z)
and (x
Õ
,y
Õ
,z
Õ
) denote the same point if and only if (x
Õ
,y
Õ
,z
Õ
)=t(x, y, z) for some
real number t ”= 0. In other words, we may identify P
2
with (R
3
r {(0, 0, 0)})/≥,
where ≥ is the equivalence relation just defined.
We can also say, equivalently, that a point in P
2
is a straight line through the
origin in R
3
and that a straight line in P
2
is a plane through the origin in R
3
.
Alternatively, we can think of P
2
as the sphere
S
2
= {(x, y, z) œ R
3
; x
2
+ y
2
+ z
2
=1},
with point meaning ‘a pair of antipodal points’ and straight line meaning ‘a great
circle with opposite p oints identified’. Thus with this representation, P
2
= S
2
/ ≥.
As pointed out by Ulf Persson, we can construct the projective plane also as the
union of a disk and a M
¨
obius strip, identifying their boundaries.
The projective plane can be covered by coordinate patches which are diffeomor-
phic to R
2
. For any open hemisphere, we can project the points on that hemisphere
to the tangent plane at its center. Then all points exce pt those on the boundary of
the hemisphere are represented.
On the sphere, angles are well-defined, but not in the projective plane. To illu-
strate this, take for example an equilateral triangle with vertices at latitude Ï>0
and longitudes 0, 2fi/3 and ≠2fi/3, respectively. Then its angles ◊ on the sphere
can be obtained from Napier’s rule, and are given by
sin Ï = c os
1
fi
2
≠ Ï
2
= cot
fi
3
cot
◊
2
=
1
Ô
3
cot
◊
2
, 0 <Ï<
fi
2
.
Thus ◊ tends to fi as Ï æ 0 (a large triangle close to the equator). The same is true
of the angle at a vertex if we use the coordinate patch centered at that very vertex.
But ◊ tends to fi/3 as Ï æ fi/2 (a small triangle close to the north pole). The
projection of the triangle onto the tangent plane at (0, 0, 1) is a usual equilateral
triangle, thus with angles equal to fi/3 for all values of Ï,0<Ï<fi/2. Thus we
cannot measure angles in arbitrary coordinate patches, only in coordinate patches
with center at the vertex of the angle; equivalently on the sphere.
It is convenient to use this way of measuring angles in the projective plane as
a means of controlling the size of triangles. So, although it is meaningless to talk
about angles in the projective plane itself, the sphere can serve as a kind of premodel
for the projective plane, and the angles on the sphere can serve a purpose.
Given two p oints a, b on a straight line L in P
2
, the complement L r { a, b} has
two components, and we cannot distinguish them. So to define a segment in P
2
we need two points a, b and one more bit of information, viz. which component of
L r {a, b} we shall consider. Since it seems that Euclid lets two points determine
a segment without any additional information, shall we conclude already at this
point that he excludes the projective plane? Anyway, in the projective plane, two
distinct points determine uniquely a straight line, but not a rectilinear segment.
some parallel lines. However, this interesting structure does not satisfy Postulate 5 if we measure
angles as described later in this subsection.
Normat 4/2012 Christer O. Kiselman 149
Explicitly, in the projective plane a point is given by the union of two rays R
+
a
and R
≠
a in R
3
,wherea is a point in R
3
different from the origin, and where
R
+
denotes the set of positive real numbers, R
≠
the set of negative real numbers.
Given two points, we can define two rectilinear segments, corresponding to two
double sectors in R
3
. These are given as
cvxh(R
+
a fiR
+
b) ficvxh(R
≠
a fiR
≠
b)
and
cvxh(R
+
a fiR
≠
b) ficvxh(R
≠
a fiR
+
b),
respective ly, where cvxh(A) denotes the convex hull of a set A.Thereisnoway
to distinguish them; to get a unique definition we must add some information as
to which one we are referring to.
So the cognitive content of a segment is different in E
2
and P
2
: a segment in P
2
needs one more bit of information to be defined.
4 What does eutheia mean?
Charles Mugler writes:
[. . . ] l’instrument linguistique de la g´eom´etrie grecque donne au lecteur la mˆeme
impression que la g´eom´etrie elle-mˆeme, celle d’une perfection sans histoire. Cette
langue sobre et ´el´egante, avec son vocabulaire pr´ecis et diff´erenci´e, invariable, `a
quelques changement s´emantiques pr`es, `a travers mille ans de l’histoire de la pens´ee
grecque, [. . . ]
and continues
la diction des
´
El´ements, qui fixe l’expression de la pens´ee math´ematique pour des
si`ecles, se rel`eve `a l’analyse comme un r´esultat auquel ont contribu´e de nombreuses
g´en´erations de g´eom`etres. (Mugler 1958–1959:7)
May this suffice to show that we are not trying to analyze here s ome ephemeral
choice of terms.
4.1 Lines
Euclid defines a line second in his first book:
b'. Grammò d‡ m®koc ÇplatËc. (Book I, Definition 2) — Une ligne est une longueur
sans largeur (Ho
¨
uel 1883:11) — A line is a breadthl ess length. (Heath 1926a:158) —
Une ligne est une longueur sans largeur (Vitrac 1990:152). — And a line is a length
without breadth. (Fitzpatrick 2011:6)
There is no mentioning of lines of infinite length here; also Heath does not take
up the subject. The lines in this definition are not necessarily straight, but in the
rest of the first book, most lines, if not all, are straight, so to get sufficiently many
examples we turn to these now.
4.2 Straight lines: eutheia
Euclid defines the concept of eutheia in the fourth definition in his first book thus:
d'. EŒjeÿa gramm† ‚stin, °tic ‚x “sou toÿc ‚f’ ·aut®c shme–oic keÿtai. (Book I, Definition
4) — La ligne droite est celle qui est situ´ee semblablement par rapport `a tous ses
150 Christer O. Kiselman Normat 4/2012
points (Ho
¨
uel 1883:11) — A straight line is a line which lies evenly with the points
on itself. (Heath 1926a:165) — Une ligne droite est celle qui est plac´ee de mani`ere
´e g a l e p a r r a p p o r t a u x p o i n t s q u i s o n t s u r e l l e (Vitrac 1990:154) — A straight-line
is (any) one which lies evenly with points on itself. (Fitzpatrick 2011:6)
Ho
¨
uel adds that the definition is “con¸cue en termes assez obscurs”.
Euclid’s first postulate states:
a'. >Hit†sjw
4
Çp‰ pant‰c shme–ou ‚p» pên shmeÿon eŒjeÿan grammòn Çgageÿn. (Book
I, Postulate 1) — Mener une ligne droite d’un point quelconque `a un autre p oint
quelconque ; (Ho
¨
uel 1883:14) — Let the fol lowing be postulated : to draw a straight
line from any point to any point. (Heath 1926a:195) — Qu’il soit demand´e de mener
une ligne droite de tout point `a tout point. Vitrac (1990:167) — Let it have been
postulated [. . . ] to draw a straight-line from any point to any point. (Fitzpatrick
2011:7)
The term he uses for straight line in the fourth definition and the first postulate is
eŒjeÿa gramm† (eutheia gramm¯e ) ‘a straight line’,
5
later, for instance in the se cond
and fifth postulates, shortened to eŒjeÿa ‘a straight one’,
6
the feminine form of
an adjective which means ’straight, direct’; ’soon, immediate’; in masculine eŒj‘c;
in neuter eŒj‘. This brevity is not unique; see Mugler (1958–1959:18) for other
condensed expressions.
Curiously, ac cording to Frisk (1960), the adjective eŒj‘c has no etymological
counterpart in other languages: “Ohne außergriechische Entsprechung.”
4.3 Straight lines: ex isou keitai
A key element in Definition 4 is the expression ‚x “sou [. . . ] keÿtai (ex isou [. . . ]
keitai). It is translated as ‘situ´ee semblablement’, ‘lies evenly’, ‘plac´ee de mani`ere
´egale’. The adverbial evenly is a translation of the prepositional expression ‚x “sou,
which functions like an adverbial—or actually is an adverbial (Federspiel 1991:120).
Michel Federspiel would like to create (“j’aimerais cr´eer”) an adjective isoth´etique
in analogy with homoth´etique—he argues that homoth´etique corresponds to the
Greek Âmo–wc keÿsjai
7
“ˆetre plac´e semblablement”, and that isoth´etique would cor-
respond to the Greek ‚x “sou keÿtai,
8
which occurs in Definition 4, and gives the
translation (which he calls a
ÈÈ
translation
ÍÍ
within quotation marks)
La droite est la ligne qui est isoth´etique de ses points. (Federspiel 1991:120)
4
This verb form, written ö
ÿ
t†sjw in lower case letters, is in middle voice, perfect imperative,
singular third person of the verb ateÿn ’to demand’, atËw ’I demand’. Since it is in the perfect
tense, Fitzpatrick’s translation, ”Let it have been postulated,” with the alternative ”let it stand
as p ostulated,” is more faithful than Heath’s.
5
Liddell & Scott (1978) gives gramm† as ‘stroke or line of a pen, line,asinmathematical
figures’, and eŒj‘c as ‘straight, direct,whetherverticallyorhorizontally’.Bailly(1950)gives
gramm† as ‘trait, ligne’, [. . . ] ‘trait dans une figure de math´ematiques’, and eŒj‘c as ‘droit, direct’.
Menge (1967) defines gramm† as ‘Strich, Linie (auch mathem.)’, eŒj‘c as ‘gerade (gerichtet)’, and
eŒjeÿa (gramm†)as‘geradeLinie’.InMill´en(1853)Idonotfindgramm†,onlygràmma ‘bokstaf’;
‘det som
¨
ar skrifvet, skrift, bok, bref’; eŒj‘c ‘rak, r
¨
at’; ‘strax’; ‘snart’. Linder & Walberg (1862)
translates Linie as ‘gramm†’; r
¨
at l. as ‘eŒjeia’; Rak as ‘eŒj‘c’.
6
Similarly, une droite is very often used for une ligne droite in French, and prma (pryam´aya)
for prma lini (pryam´aya l´ınya)inRussian.
7
The verb form keÿsjai means ‘to be placed’; middle or passive voice (here most likely passive),
present infinitive.
8
The verb form keÿtai means ‘it lies, it is lying’ or perhaps ‘it is laid, placed’; middle or passive
voice, present indicative, singular, third person.
Normat 4/2012 Christer O. Kiselman 151
He does not offer a mathematical definition of the new term, and it probably
does not mean the same thing as in the e xpress ion isothetic polygon. Perhaps it is
intended to preserve the vagueness of the original.
4.4 Straight lines: s¯emeion
Vitrac (1990:189–190) points out that Euclid treats points as marks which one can
place on straight lines or in relation to straight lines. That points are actually marks
is further developed in two papers by Federspiel, who discusses in detail the meaning
of the word shme–oic in Definition 4, plural dative of shmeÿon. He had expected the
word pËrasi ‘extr´emit´es’ at the place of shme–oic here (1992:387), and argues that,
although in general shmeÿon certainly means ‘point’, in this particular definition it
has a pre-Euclidean meaning, viz. ‘rep`ere,
9
extr´emit´e’ (1992:388), ‘signe distinctif’
(1992:389), or ‘marque, rep`ere’ (1998:67) (perhaps to be rendered as reference mark,
guide mark, landmark, benchmark, extremity, mark, distinctive sign in English).
The word shmeÿa has the meaning (sens) ‘rep`eres’ and the referent ‘les extr´emit´es’
(1998:56). The re ferent is almost always the vertex of an angle in a polygon or a
polyhedron, and there is, curiously, no explicit occurrence of the word shmeÿa with
the endpoints of a rectilinear segment (1998:67). It seems that the only occurrence
is in Definition 4 (1992:388), but it is not explicit there, since it is in a definition
without explanation.
In fact, we are dealing with “un v´eritable archa¨ısme” (1998:61), whose meaning
‘extremity’ later disappeared (1998:62). Howe ver, in spite of this, the word shmeÿon
was still understood in Euclid’s time—if Euclid had found shme–oic to be incom-
prehensible in that sense, he would have replaced it by the contemporary pËrasi
‘extr´emit´es’ (1998:62).
The argument is supported by the use of shmeÿon in the sister scie nce astronomy
(1998:391–395), where it designates stars which delineate a constellation, in other
words are in extreme positions relative to the constellation, essentially like the ver-
tices of a polygon (1992:395), in particular a pentagon (1998:58), a cube (1998:58),
or an icosahedron (1998:59). On the other hand, it is not necessary to consider
astronomy as an intermediary; the meaning can appear directly in mathematics
(1992:396); there is no reason to c onsider astronomy as a mother science.
The word shmeÿon was, according to Federspiel (1992:400) adopted very early in
mathematics in the concrete sense of ‘marque’, and at any rate before the creation
of the concept of point.
At this point comes to mind the statement by Reviel Netz that the lettered
diagram is a combination of the continuous (the diagram itself) and the discrete
(the letters) as well as a combination of visual resources (the diagram) and finite,
manageable models (the letters) (Netz 1999:67).
Federspiel therefore modifies his translation from 1991 quoted above in Subsec-
tion 4.3 to the following.
La ligne droite est la ligne qui est isoth´etique de ses extr´emit´es. (Federspiel 1992:404)
And then to:
9
“Toute marque servant `a signaler un point, un enplacement `a des fins pr´ecises” (Grand Larous-
se 1977).
152 £rister O. Kiselman Normat 4/2012
La ligne droite est la ligne qui est isoth´etique de ses rep`eres. (Federspiel 1998:56)
10
In his argument, a straight line thus lies evenly between its extremities. This presup-
poses that a straight line does have two endpoints, which is a possible interpretation
of Definition 3 (which is actually a proposition rather than a definition):
g'. Gramm®c d‡ pËrata shmeÿa. (Book I, Definition 3) — Les ex tr´emit´es d’une ligne
sont des points. (Ho
¨
uel 1883:11) — The extremities of a line are points. (Heath
1926a:165) — Les limites d’une ligne sont des points (Vitrac 1990:153) — And the
extremities of a line are points. (Fitzpatrick 2011:6)
However, there are lines which do not have endpoints (circles, ellipses, and infi-
nite straight lines). Heath therefore argues that Definition 3 “is really no more
than an explanation that, if a line has extremities, those extremities are points.”
(1926a:165). Vitrac agrees (1990:153): “Il faut certainement comprendre que la
pr´esente d´efinition signifie simplement : lorsqu’une ligne a des limites, ce sont des
points.”
It seems plausible that the definition was primarily thought of as defining a recti-
linear segment, but that later, a wider use of the term eŒjeÿa forced mathematicians
to accept a broader interpretation.
4.5 Discretization
Zeno of Elea (Z†nwn  >Eleàthc) formulated four paradoxes about motion, discussed
in detail by Segelberg (1945) and Ferber (1981). The first of these is called the
Dichotomy paradox since it uses division into halves. It says, according to Aristotle
(>A r is to t Ë l h c ):
...pr¿toc m‡n  (scil.
11
lÏgoc) per» to‹ mò kineÿsjai diÄ t‰ prÏteron ec t‰ °misu
deÿn ÅfikËsjai t‰ ferÏmenon ´ pr‰c t‰ tËloc,. . . — The first says that motion is
impossible, because an object in motion must reach the half-way point before it gets
to the end. (Quoted after Segelberg 1945:16)
By re peating the argument, we conclude that the object, if we agree that it is
supposed to move from 0 to 1, must reach
1
4
before reaching
1
2
, and
1
8
before
1
4
, and
so on. We see that the object must in fact reach all points with a binary coordinate
k/2
m
, k =1,...,2
m
≠1, m =1, 2,..., thus infinitely many. Euclid does construct
the midpoint of a segment (Book I, Proposition 10, quoted in Subsubsection 4.9.4),
so also for him there are infinitely many points on any given segment. We can think
of these points as forming a potential infinity, because we can find the finitely many
points k/2
m
for a certain m and then proceed to m + 1, but the object cannot
move in this order; for the object, the points represent an actual infinity—hence
the alleged impossibility of motion (see, e.g., White (1992:147)).
In his third paradox, on the arrow which cannot move, Zeno can be seen as a
precursor of a discretization of time, and therefore also of the line.
10
Note the indefinite article in the two English translations and the definite article in four of
the five French translations of Definition 4; in the Greek original there is no article. Federspiel
(1995:252; 2005:105, note 29) explains that at the first occurrence of a mathematical term, it is
given without article; at the second occurrence and later, it appears with the article. He calls
this the Loi fondamentale for the use of the article in Classical Greek mathe mati cal texts. When
it comes to translations into French, Vitrac (1990:194, footnote 1) says with reference to his
translation of Proposition 1 quoted in Subsubsection 4.9.4 below: “L’habitude fran¸caise moderne
est d’utiliser l’article ind´efini pour souligner la validit´e universelle de la proposition.”
11
Abbreviation for scilicet ‘it is permitted to know’.
Normat 4/2012 Christer O. Kiselman 153
It would be interesting to know what Euclid thought about this paradox. As I
understand it, his lines are neutral with respect to the consequences that Zeno’s
discretized time or line lead to. The points are without parts and thus are atoms:
a'. ShmeÿÏn ‚stin, o
˜
ÕmËrocoŒjËn. (Book I, Definition 1) — Un point est ce qui
n’a pas de parties. (Ho
¨
uel 1883:11) — A point is that which has no part. (Heath
1926a:155) — Un point[. . . ] est ce dont il n’y a aucune partie (Vitrac 1990:151)
— A point is that of which there is no part. (Fitzpatrick 2011:6)
A line does not consist of points; the points are, as we have seen in Subsection 4.4,
special marks, rep`eres, on the line. And in a construction we can hardly have an
infinity of rep`eres, like all those with coordinates k/2
m
.
The two ideas—that the line is infinitely divisible while time consists of moments
which cannot be further divided—are not easy to reconcile: we cannot arrive at
the atoms by subdividing a segment. White (1992) discusses this difficulty; se e in
particular the section “The Quantum Model: Spatial Magnitude.” Islamic thinkers
in the middle ages resolved the conflict by making time divisible to a high degree
while giving up infinite divisibility. A prominent advocate of these ideas, Mosheh
ben Maimon, a Sephardic Jewish philosopher who was born in C´ordoba in 1135
or 1138 and died in Egypt in 1204, and who is now better known under his Greek
name Maimonides, wrote that an hour is divisible by 60 ten times or more: “at last
after ten or more succes sive divisions by sixty, time-elements are obtained which
are not subject to division, and in fact are indivisible” (Whitrow 1990:79). So we
can arrive at the time atoms! Now 60
≠10
hours is about 6 femtoseconds, 60
≠11
hours is about 100 attoseconds, and we are then down at the time scale of some
chemical reactions studied nowadays in femtochemistry.
4.6 The chord property in the sense of Euclid
A property which is relevant for this discussion is what I called the chord property
in the sense of Euclid (2011:359): for any two points a, b in the set A considered, the
rectilinear segment (chord) [a, b] is contained in A. This agrees with the translations
of Definition 4 given in Subsections 4.2 and 4.3. To reconcile it with Federspiel’s
later translations quoted in Subsection 4.4, one has to note that, for every two
points p, q belonging to a chord [a, b], the segment [p, q] is contained in [a, b].
In fact, the strongest chord property is obtained when we start with the two
endpoints of a rectilinear segment. However, on a straight line one can start quite
naturally with any pair of points as rep`eres and consider for these two points the
segment determined by them using the chord property.
The chord property in the sense of Euclid has a counterpart in digital geometry,
viz. the chord property in the sense of Rosenfeld introduced by Azriel Rosenfeld in
1974 and mentioned in my paper (2011:359). Moses Maimonides would have liked
it.
4.7 The mathematical meaning of eutheia
What does eutheia mean mathematically? Proclus (PrÏkloc  Diàdoqoc), in his
commentary to Euclid’s first book (Proclus 1948:92, 1992:83) notes that eutheia
has what we now usually perceive as three different meanings: a straight line;
a rectilinear segment; and a ray. “La ligne est donc prise de trois mani`eres par
154 Christer O. Kiselman Normat 4/2012
Euclide” (Proclus 1948:92); “our geometer makes a threefold use of it” (Proclus
1992:83). Thus already Proclus writes about three different meanings.
Euclid often refers to extension of straight lines, for instance in the famous Po-
stulate 5, the Axiom of Parallels, quoted in Subsection 3.2, which was to keep
mathematicians busy for more than two millennia. The postulate implies that the
two straight lines do not necessarily meet initially, so he must be talking about rec-
tilinear segments. We may conclude that, here at least, eutheia means a rectilinear
segment, not an infinite straight line.
The Greek original has ‚kballomËnac
12
[. . . ] ‚p’ äpeiron, which Heath translates
as ‘produced indefinitely’. Similarly, Definition 23 has ‚kallÏmenai
13
ec
äpeiron, translated in the same way. Fitzpatrick (2011:7) translates both as ‘being
produced to infinity’. However, Heath (1926a:190) explicitly warns against that
interpretation. Similarly, Vitrac (1990:166) makes the distinction between being
extended “ind´efiniment” and being extended “`a l’infini” and maintains that the
expressions ec äpeiron and ‚p’ äpeiron refer to the former.
4.8 Infinitely long lines vs. equivalence classes of segments
On the other hand, when two points are given, they determine uniquely a straight
line. Actually, Postulate 1 does not explicitly say so, but the discussion in Heath
(1926a:195), which leads to the conclusion that this is what is meant, is quite
convincing. He re it would be natural for us in the twenty-first century to think
about an infinite straight line, but it is also possible to limit the consideration to
rectilinear segments by forming the family of all segments which contain the two
given points—or at least a family of rectilinear segments which go out arbitrarily
far in both directions. If so, we can avoid here actual infinity, and work only with
potential infinity by looking at one segment at a time rather than at an infinitely
long line. Vitrac (1990:169) mentions this possibility: “la droite peut ˆetre envisag´ee
comme ind´efinie ou potentiellement infinie.”
Michel Federspiel states quite categorically: “Il n’y a pas d’infini actuel dans
la g´eom´etrie grecque.” (1991:118, Note 10). This should be contrasted with an
assertion by Reviel Ne tz: “[. . . ] Archimedes [>A r q i m † d h c ] calculated with actual in-
finities in direct opposition to everything historians of mathematics have always
believed about their discipline.” The quotation refers to the calculation of a volume
in the palimpsest now at the Walters Art Museum in Baltimore, MD, USA (Netz
& Noel 2007:199). It seems the basis for this assertion is not very firm. More to the
point is Euclid’s own statement in his Book X: g'. [. . . ] Õpàrqousin eŒjeÿai pl†jei
äpeiroi [. . . ] (Book X, Definition 3) — [. . . ] there exist an infinite multitude of
straight-lines [. . . ] (Fitzpatrick 2011:282).
We may note that Proclus makes the distinction between “partie infinies en acte”
(actual infinity) and “en puissance seulement” (potential infinity) (1948:140); “The
latter statement [an infinite number of parts] makes an infinite number actual,
the former [a magnitude is infinitely divisible] only potential; the latter assigns
existence to the infinite, the other only genesis” (1992:125).
12
Middle or passive voice, present participle, plural, feminine, accusative. Of the many meanings
of the verb ‚kbàllein (ekballein;activevoice,present,infinitive),thebasiconeis‘tothrowout’.
Liddell & Scott (1978) and Menge (1967) explicitly mention the mathematical sense of extending
aline.
13
Middle or passive voice, present participle, plural, feminine, nominative.
Normat 4/2012 Christer O. Kiselman 155
However, if we act like this—whether under the pressure of Aristotle or not—
there will be a lot of rectilinear segments that contain the two given points: perhaps
one with a length of one hemiplethron, then one with a length of one plethron, one
stadion, one hippikon, then one with a length of a parasang, and one with a length
of one stathmos, and so on—it do es not stop. But all of these segments represe nt
the same line: there has to be only one line. That the segme nts all represent the
same line is today conveniently expressed in the parlance of equivalence classes.
The formation of an equivalence class is a means of obtaining uniqueness—to unite
the many s egm ents into one single entity.
Let me emphasize again that two points determine a straight line segment if
we are in E
2
, and that, conversely, a straight line segment uniquely determines
two points, viz. its endpoints. If this were all there is to it, we would have perfect
uniqueness in both directions. But if we extend a se gment to a longer segment, we
have two different segments, which, however, represent the same straight line. What
does then represent mean? And what does the same mean? If we nowadays can
speak about equivalence classes, this is a convenient way to understand the verb
represent, but it is only there as a help to the modern reader. I do not know how
Euclid thought, but he must have been aware of this problem of nonuniqueness.
As for actual vs. potential infinity, we may compare with prime numbers: it is
sometimes said that Euclid proved that there are infinitely many prime numbers,
but actually he proved in his ninth book, Proposition 20, that, given three prime
numbers, he can find a fourth. Clearly the proof works for any finite set of primes:
with the idea of the proof we can go from n primes to n + 1 primes for any n. All
prime numbers need not exist at once. So this is an instructive example of potential
infinity; we need not believe in the existence of an actual infinity.
Aristotle expressed a very clear opinion on the need to consider infinite straight
lines:
I have argued that there is no such ting as an actual infinite which is untraversable,
but this position does not rob mathematicians from their study. Even as things are,
they do not need the infinite, because they make no use of it. All they need is a finite
line of any desired length. (Physics, Book III, Part 7, quoted here from Aristotle
1996:75–76)
The uniqueness requirement then leads to the need of forming an equivalence class
of all these segments.
Not only is an actual infinity unnecessary for geometry; it is even impossible in
the physical world:
[. . . ] there can be no magnitude which exceeds every specified magnitude: that
would mean that there was s om ething larger than the universe. (Physics, Book II,
Part 7, quoted from Aristotle 1996:75)
However, as Rosenfeld (1988:183) points out, Aristotle’s doctrine “that mathemati-
cal concepts are obtained by abstracting from objects of the real world enables one
to disengage oneself from the finiteness of physical magnitudes.” Ibn Rushd (Aver-
roes) wrote that a geom ete r can admit “an arbitrarily large magnitude—something
a physicist cannot do [. . . ]”.
We should also add that on the sphere, a straight line in the plane corresponds to
a great circle, mËgistoc k‘kloc (megistos kuklos; Mugler 1958–1959:19). Ce rtainly
Aristotle would not object to considering a circle on a sphere as a complete, existing
156 Christer O. Kiselman Normat 4/2012
entity.
14
But I guess he did not see a great circle as a compactification of a straight
line as we now do quite easily—after so many years.
Since every rectilinear segment determines a unique straight line, it might appear
that there is no big differenc e whether we say that two distinct points determine a
straight line or that two distinct points determine a rectilinear segment. However,
the latter assertion is untenable (if we keep ourselves strictly to the axioms) in
view of the fact that, as noted in Subsection 3.2, two points in the projective plane
determine not one segment but two.
4.9 Examples
4.9.1 Eutheia bounded
That the English term straight line or straight-line can denote a rectilinear seg-
ment is explicitly mentioned by Heath “if t wo straight lines (‘rectilinear segments’
as Veronese would call them) have the same extremities [. . . ]” (1926a:195); “what
modern Italian geometers aptly call rectilinear segment, that is, a straight line ha-
ving two extremities.” (1926a:196). For both the Greek term and the English term,
this is clear as well from several examples, e.g., the first few propos itions in Book
I:
b'. Pr‰c t¿
ÿ
dojËnti shme–w
ÿ
t®
ÿ
doje–sh
ÿ
eŒjeÿa
ÿ
“shn eŒjeÿan jËsjai. (Book I, Propo-
sition 2) — Apartird’unpointdonn´eA[...],placer une droite ´egale `a une droite
donn´ee BC (Ho
¨
uel 1883:16) — To place at a given point (as an extremity) astraight
line equal to a given straight line. (Heath 1926a:244) — Placer, en un point donn´e,
une droite ´egale `a une droite donn´ee. (Vitrac 1990:197) — To place a straight-line
equal to a given straight-line at a given point (as an e xtremity). (Fitzpatrick 2011:8)
Equality of lines here means equality of their lengths.
g'. D‘o dojeis¿n eŒjei¿n Çn–swn Çp‰ t®c me–zonoc t®
ÿ
‚làssoni “shn eŒjeÿan
Çfeleÿn. (Book I, Proposition 3) —
´
Etant donn´ees deux droites in´egales, AB, C
[. . . ], retrancher de la plus grande AB une droite ´egale `a la plus petite C(Ho
¨
uel
1883:17) — Given two unequal straight lines, to cut off from the greater a straight
line equal to the less. (Heath 1926a:246) — De deux droites in´egales donn´ees, re-
trancher de la plus grande, une droite ´egale `a la plus petite. (Vitrac 1990:199) —
For two given unequal straight-lines, to cut off from the greater a straight-line equal
to the lesser. (Fitzpatrick 2011:9)
d'. >EÄn d‘o tr–gwna tÄc d‘o pleurÄc [taÿc] dus» pleuraÿc “sac Íqh
ÿ
·katËran ·katËra
ÿ
ka» tòn gwn–an t®
ÿ
gwn–a
ÿ
“shn Íqh
ÿ
tòn Õp‰ t¿n “swn eŒjei¿n perieqomËnhn, [. . . ] (Book
I, Proposition 4) — Si deux triangles ABC, DEF [. . . ] ont les deux cˆot´es AB, AC
respectivement ´egaux aux deux cˆot´es DE, DF,etsilesanglesBAC, EDF,compris
entre les cˆot´es ´egaux, sont ´egaux, [. . . ] (Ho
¨
uel 1883:18) — If two triangles have
the two sides equal to two sides respectively, and have the angles contained by the
equal straight lines equal, [. . . ] (Heath 1926a:247) — Si deux triangles ont deux cˆot´es
´e g a u x `a d e u x c ˆo t ´e s , c h a c u n `a c h a c h u n [. . . ], et s’ils ont un angle ´egal `a un angle,
celui contenu par les droites ´egales, [. . . ] (Vitrac 1990:200) — If two triangles have
two sides equal to two sides, respectively, and have the angle(s) enclosed by the
equal straight-lines equal, [. . . ] (Fitzpatrick 2011:10)
We note that here the sides of a triangle are sometimes called sides, cot´es; someti-
mes straight lines, straight-lines, droites.
e'. T¿n soskel¿n trig∏nwn a… pr‰c t®
ÿ
bàsei gwn–ai “sai Çll†laic es–n, ka» prosek-
blhjeisw~n t¿n “swn eŒjei¿n a… ÕpÏ tòn bàsin gwn–ai “sai Çll†laic Ísonvtai. (Book I,
14
For the history of spherical geometry, see Rosenfeld (1988: Chapter 1).
Normat 4/2012 Christer O. Kiselman 157
Proposition 5) — Dans tout triangle isosc`ele ABC [. . . ], 1
les angles `a la base ABC,
ACB sont ´egaux entre eux ; 2
si l’on prolonge les cˆot´es ´egaux AB, AC,lesangles
form´es au-dessous de la base, DBC, ECB,serontaussi´egauxentreeux.(Ho
¨
uel
1883:18–19) — In isosceles triangles the angles at the base are equal to one another,
and, if the equal straight lines be produced further, the angles under the base will be
equal to one another. (Heath 1926a:251) — Les angles `a [. . . ] la base des triangles
isosc`eles sont ´egaux entre eux, et si les droites ´egales sont prolong´ees au-del`a, les
angles sous la base seront ´egaux entre eux. (Vitrac 1990:204) — For isosceles tri-
angles, the angles at the base are equal to one another, and if the equal sides are
produced then the angles under the base will be equal to one another. (Fitzpatrick
2011:11)
In Book I, Proposition 12, eŒjeÿa receives the attribute äpeiroc (apeiros) ‘unboun-
ded, infinite’:
ib'. >Ep» tòn dojeÿsan eŒjeÿan äpeiron Çp‰ to‹ doj‡ntoc shme–ou, Á m† ‚stin ‚p’ aŒt®c,
kàjeton eŒj e ÿan grammòn Çgageÿn. (Book I, Proposition 12) — D’un point donn´e
C[...],abaisser une perpendiculaire sur une droite ind´efinie donn´ee AB. (Ho
¨
uel
1883:24) — To a given infinite straight line, from a given point which is not on it,
to draw a perpendicular straight line. (Heath 1926a:270) — Mener une ligne droite
perpendiculaire `a une droite ind´efinie [. . . ] donn´ee `a partir d’un poin t donn´e qui
n’e st pas s u r cel l e- ci . (Vitrac 1990:219) — To draw a straight-line perpendicular to
a given infinite s traight-line from a point which is not on it. (Fitzpatrick 2011:17)
Here the qualification äpeiroc would not be necessary if an eŒjeÿa were always
something unb ounded in both directions.
Apollonius (>A p o l l ∏ n io c ) mentions an eŒjeÿa in a context that clearly indicates
that it refers to a segment; he needs to extend it in both directions:
>EÄn ÇpÏ tinoc shme–ou pr‰c k‘klou perifËreian, Ác oŒk Ístin ‚n t¿
ÿ
aŒt¿
ÿ
‚pipËdw
ÿ
t¿
ÿ
shme–w
ÿ
,eŒjeÿa‚pizeuqjeÿsa‚f>·kàteraprosekblhj®
ÿ
,[...] (>A p ol l ∏ n io c , Kwnik¿n a'.
VOroi pr¿toi. Apollonius, Conics, Book 1, First definitions) — If a point is joined by
a straight line with a point in the circumference of a circle which is not in the same
plane with the point, and the line is continued in both directions, [. . . ] (Rosenfeld
2012:3)
4.9.2 Segment
The C lassic al Greek word tm®ma (n) (tm¯ema) is translated by Liddell & Scott
(1978) as ‘part cut off, section, piece’; ‘segment of a line, of a circle (i.e. portion
cut off by a chord), also of the portion cut off by radii, sector’ [. . . ] ‘of segments
of other figures cut off by straight lines or planes; and of segm ents bounded by a
circle and circumscribed polygon’. Bailly (1950) translates it as ‘morceau coup´e,
section, part, segment de ce rcle’, and Menge (1967) as ‘Schnitt’; ‘Abschnitt’.
In all case s it is about some part cut out from a given object. This object could
be a disk or a rectilinear segment, viz. when a rectilinear segment is given, and one
then cuts out a part of it (Book II, Propositions 3 and 4). As I understand it, the
term is not used for a rec tilinear segment per se, only for a certain part cut out
from something els e in the course of a construction (in Section 5 we shall take a
look at how the Greek viewed geometric c onstructions). So in general an eŒjeÿa is
not thought of as being cut out from a straight line.
The term tm®ma is used for a segme nt of a circle
15
in Book III:
15
Here it does not really matter whether k‘kloc means ‘circle’ or ‘circular disk’.
158 Christer O. Kiselman Normat 4/2012
ke'. K‘klou tm†matoc dojËntoc prosanagràyai t‰n k‘klon, o
˜
ÕpËr ‚sti tm®ma. (Book
III, Proposition 25) — Given a segment of a circle, to describe the complete circle
of which it is a segment. (Heath 1926b:54) — Etant donn´e un segment de cercle,
d´ecrir e compl`etement [. . . ] le cercle duquel il est un segment. (Vitrac 1990:440) —
For a given segment of a circle, to complete the circle, the very one of which it is a
segment. (Fitzpatrick 2011:94)
The meaning ‘segment of a disk’ occurs, e.g., in Definition 6 in Book III:
'. Tm®ma k‘klou ‚st» t‰ perieqÏmenon sq®ma ÕpÏ te eŒje–ac ka» k‘klou perifere–ac
(Book III, Definition 6) — A segment of a circle is that contained by a straight
line and a circumference of a circle. (Heath 1926b:1) — Un segment de cercle est la
figure contenue par une droite et une circonf´erence de cercle (Vitrac 1990:388) —
A segment of a circle is the figure contained by a straight-line and a circumference
of a circle. (Fitzpatrick 2011:70)
A definition of segment has also been “interpolated” after Definition 18 in Book I;
see Definition 19 in Euclid (1573:39), Ho
¨
uel (1883:12), and the remark on Definition
18 in Heath (1926a:187). It s ee ms that the term is not used for a chord.
In conclusion, tm®ma is related to the verb tËmnein ‘to cut’, tËmnw ‘I cut’, and
is firmly attached to the act of cutting. Therefore it is not used for rectilinear
segments in general, which are just there, not being the result of any cutting.
The English word segment, from the Latin segmentum ‘a piece cut out’, formed
from secare ‘to cut’, also carries this connotation, like the Russian prmoline⇢ny⇢
otrezok (pryamolin´e˘ıny˘ı otr´ezok) ‘rectilinear segment’, from rezat~ (r´ezat
Õ
)‘to
cut’. This connotation is completely absent in the German Strecke, the Esperanto
streko, and the Swedish str
¨
acka.
4.9.3 Radius and chord
In a circle there are rectilinear segments which have received special names in
many languages: radii and chords.
The Greeks had no distinct word for radius, which is with them [. . . ] the (straight
line drawn) from the centre ô‚kto‹kËntrou(eŒjeÿa)[h¯e ek tou kentr ou (eutheia)]
(Book III, Definition 1; Heath 1926b:2)
Mugler (1958–1959:17) gives the full expression for radius as ô‚kto‹kËntrou(sc.
16
pr‰c tòn perifËreian ögmËnh eŒjeÿa gramm†).
There is also a word diàsthma (n) (diast¯ema) used for ‘radius’, or often for ‘the
length of a radius’ (Mugler 1958–1959:17).
Federspiel (2005:98, note 5) opposes the statement by Heath quoted above: he
says that the Greek had two words for ‘radius’, viz. the two just mentioned.
He explains that the first expression needs the article ô, and in a situation where
one needs the indefinite form, it cannot be used; here the word diàsthma comes in,
a fact which also explains why they are in complementary distribution (2005:105).
In Contemporary Greek, the word used for radius is akt–na (f) (Petros Maragos,
personal communication 2007-10-12; Takis Konstantopoulos, personal c ommunica-
tion 2012-01-20). However, this word also means ‘ray’.
Similarly, they did not have a simple word for chord (in a circle): it is ô‚nt¿
ÿ
k‘klw
ÿ
eŒjeÿa (h¯e en t¯o kukl¯o eutheia) as used not by Euclid but later by Heron
(Erik Bohlin, personal communication 2012-01-18; cf. Mugler 1958–1959:202) and
by Ptolemy (1898:48), who in the he ading of Table ia' (11) writes: KanÏnion t¿n
‚n k‘klow
ÿ
eŒjei¿n. With Euclid, not the expression itself but the words used in
16
This abbreviation stands for scilicet ‘it is permitted to know’.
Normat 4/2012 Christer O. Kiselman 159
referring to a chord appear in Definition 4 in Book III, see Heath (1926b:3); and
in Proposition 14 in Book III, see Heath (1926b:34).
The word qord† (f) (khord¯e ) is given by Liddell & Scott (1978) as ‘guts, tripe’
[. . . ] ‘string of gut, ‘string of musical instrument’. Bailly (1950) translates it as
‘boyau’, [. . . ] ‘corde `a boyau, corde d’un instrument de musique’. Frisk (1960) as
‘Darm, Darmsaite, Saite, Wurst’ and Menge (1967) as ‘Darm, Darmsaite’. Frisk
(1960) states that it is “Ohne genaue Außergreich. Enstprechung”. Linder & Wal-
berg (1862) translate Str
¨
ang p˚a ett instrument as ‘qord†’, and Tarm as ‘Ínteron,
qord†’. B u t qord† is missing in Mill´en (1853).
In Contemporary Greek the word used for chord and string is qord† (f ) (Takis
Konstantopoulos, personal communication 2012-01-20).
4.9.4 Eutheia unbounded
However, sometimes eŒjeÿa carries another qualification:
b'. Ka» peperasmËnhn eŒjeÿan katÄ t‰ suneq‡c ‡p’ eŒje–ac ‚kbaleÿn.
17
(Book I, Po-
stulate 2) — Prolonger ind´efiniment, suivant sa direction, une ligne droite finie ;
(Ho
¨
uel 1883:14) — To produce a finite straight line continuously in a straight line.
(Heath 1926a:196) — Et de prolonger continˆument en ligne droite une ligne droite
limit´ee. (Vitrac 1990:168) — And to pro duce a finite straight-line continuously in
a straight-line. (Fitzpatrick 2011:7)
From this it is obvious that an eŒjeÿa can be explicitly qualified as bounded, which
indicates that the term could refer also to an unbounded line. Or, with a potential
infinity, a family of rectilinear segments! In other words, we can interpret Postulate
2 to mean that we can extend a given segment to another segment, as long as we
wish, but still of finite length.
a'. >Ep» t®c doje–shc eŒje–ac peperasmËnhc tr–gwnon sÏpleuron sust†sasjai. (Book
I, Proposition 1) — Sur une droite finie donn´ee AB [. . . ], construire un triangle
´e q u i l a t ´e r a l . (Ho
¨
uel 1883:15) — On a given finite straight line to construct an equila-
teral triangle. (Heath 1926a:241) — Sur une[. . . ] droite limit´ee donn´ee, construire
un triangle ´equilat´eral. (Vitrac 1990:194) — To construct an equilateral triangle on
a given finite straight-line. (Fitzpatrick 2011:8)
i'. Tòn dojeÿsan eŒje ÿ an peperasmËnhn d–qa temeÿn. (Book I, Proposition 10) — Par-
tager une droite finie donn´ee AB [. . . ] en deux parties ´egales. (Ho
¨
uel 1883:22) —
To bisect a given finite straight line. (Heath 1926a:267) — Couper en deux parties
´e g a l e s [. . . ] une droite limit´ee donn´ee. (Vitrac 1990:216) — To cut a given finite
straight-line in half. (Fitzpatrick 2011:15)
The attribute peperasmËnh ‘finite, bounded’ (passive voice, perfect participle, s in-
gular, feminine, nominative) would not be necessary here if eŒjeÿa always meant
‘rectilinear segment’.
In the pro of of Proposition 12, Euclid uses the fact that an eutheia divides the
plane into two half planes. This of course must imply that the line is infinite in
both directions.
4.9.5 Eutheia as ray
Finally, we note that sometimes eŒjeÿa can mean ‘ray’:
>Ekke–sjw tic eŒjeÿa ô DE peperasmËnh m‡n katÄ t‰ D äperoic d‡ katÄ t‰ E, [. . . ]
(Book I, Proof of Proposition 22) — Tirons une droite DE, termin´ee en D, ind´efinie
vers E. (Ho
¨
uel 1883:31) — Let there be set out a straight line DE , terminated at
D but of infinite length in the direction of E, [. . . ] (Heath 1926a:292) — Que soit
17
The verb form ‚kbaleÿn is in active voice, strong aorist, infinitive.
160 Christer O. Kiselman Normat 4/2012
d’abord propos´ee une certaine droite DE, limit´ee d’un cˆot´e au point D, illimit´ee
de l’autre en E, [. . . ] (Vitrac 1990:237) — Let som e straight-line DE be set out,
terminated at D, and infinite in the direction of E. (Fitzpatrick 2011:25)
In the statement of this proposition the lines are of finite length, but in its proof
there suddenly appears a ray.
5 Constructions
The discussion on segments in Subsubsection 4.9.2 opens up the question what the
Greek mathematicians could have meant when they talked about constructions.
Hellenistic mathematics was certainly constructive (every new figure introduced by
Euclid come s with a description of its construction), but in a sense much stronger
than that of modern constructivism, because the construction was not just a meta-
phor used for providing a demonstration of existence, but the actual goal of the
theory, just as the machine described by Heron was constructed to lift weights and
not just to prove a “theorem of existence” about the machine. (Russo 2004:186)
Who is constructing?
Le g´eom`etre grec ne reconnait qu’exceptionnellement des constructions dans le sens
que nous attachons commun´ement `a ce terme, c’est-`a-dire dans le sens de la r´ealisa-
tion progressive d’une figure au moyen de lignes et de points ajout´es success ivement
aux lignes et aux points qui constituent les donn´ees primitives du probl`eme. Pour le
g´eom`etre grec la figure, mˆeme si ses propri´et´es sont encore `a d´emontrer, pr´eexiste
`a toute intervention humaine [. . . ] (Mugler 1958–1959:19)
Proclus (1992:64), Mugler (just quoted), Vitrac (1990:134) and Federspiel (2005:
106) all state that the Ancient Greek never constructed anything. The figures are
already there for all eternity:
Proclus nous avertit en effet que certains soutenaient que toutes les propositions
´etaient des th´eor`emes, en tant que propositions d’une science th´eor´etique portant
sur des objets ´eternels, lesquels n’admettent, en tant que tels, ni changement, ni
devenir, ni production : ce qu’on appelle
ÈÈ
construction
ÍÍ
n’est tel, de ce point de
vue, qu’au regard de la connaissance que nous prenons des choses ´eternelles (Vitrac
1990:134)
[. . . ] une th`ese fondamentale de Platon et de ses successeurs [. . . ] : en math´ema-
tiques, on ne construit pas : les figures sont en r´ealit´e d´ej`a construites de toute
´eternit´e ; il n’y a donc pas d’avant ni d’apr`es. (Federspiel 2005:105–106)
So any movement in time refers only to the way we learn about these things.
Christian Marinus Taisbak explains similarly:
When mathematicians are doing geom etry, describing circles, constructing triangles,
producing straight lines, they are not really creating these items, but only drawing
pictures of them. (Taisbak 2003:27)
Plato, in The Republic, asserts (as we could expect): “[. . . ] geometry is the know-
ledge of the eternally existent.” (Plato 1935:171, Book VII, 527B).
This Platonic idea is often reinforced by the language itself: the authors use the
passive voice, without indicating an agent, and the perfect tense, i.e., a tense which
indicates that something has occurred in the past and has a result re maining up to
the present time (Mugler 1958–1959:20; Michel Federspiel, personal communication
2012-04-16). This is in slight contradiction to Plato’s statement about the language
of geometricians:
Normat 4/2012 Christer O. Kiselman 161
Their language is most ludicrous,[. . . ] though they cannot help it,[. . . ] for they speak
as if they were doing something [. . . ] and as if all their words were directed towards
action. (Plato 1935:171, Book VII, 527B)
There are, however, some exceptions to the use of the passive voice: In Euclid’s
Data (DedomËna), the first two definitions use the pronoun we. “The use of ‘we’ in
the definitions is alien to Euclid’s style; in the Elements no person is involved in
constructions or proofs in any way [. . . ]” (Taisbak 2003:18).
Regardless of these philosophical and linguistic considerations it is convenient
for us nowadays to think of an ongoing construction, just as a way of thinking—not
implying any opinion on this interesting historical question.
6 Triangular domains
A triangular domain can be given in three different ways: using points, segments,
or straight lines, respectively.
6.1 Triangular domains in the Euclidean plane
E1. In E
2
, three points which do not lie on a straight line determine a triangular
domain: it is the convex hull of the three points. If the points are a, b, c, their convex
hull is the set
cvxh({a, b, c})={⁄a + µb + flc; ⁄, µ, fl Ø 0,⁄+ µ + fl =1}.
This is the closed triangular domain defined by a, b, c.
E2. A triangular domain can also be given by three segments [a, b], [b, c], [c, a]with
pairwise common endpoints but not contained in a straight line. The complement
of the union [a, b] fi [b, c] fi [c, a] has two components, and one is bounded—this is
the open triangular domain.
E3. Finally, a triangular domain in E
2
can be given by three straight lines L
1
,L
2
,L
3
which meet in exactly three different points. The complement of the union L
1
fi
L
2
fi L
3
has seven components, and exactly one of them is bounded; this defines
the open triangular domain.
To be precise, if the equations of the three lines are f
j
(x, y) = 0, j =1, 2, 3,
where the f
j
are affine functions, and if the signs are chosen so that f
j
(p) < 0 for
some point p in the bounded component of E
2
r {L
1
fiL
2
fiL
3
}, then the other six
components are defined by the conditions that f
j
(q) shall be nonzero for all j and
positive for one or two choices of j; there is no point q with f
j
(q) positive for all j.
The se t of points where the convex function f = max(f
1
,f
2
,f
3
) is negative is the
open triangular domain determined by the three lines.
To sum up, in E
2
we can define a triangular domain using indifferently points,
segments or straight lines.
6.2 Triangular domains in the projective plane
In P
2
the determination of triangular domains takes on a different quality.
162 Christer O. Kiselman Normat 4/2012
P1. We first look at three points in P
2
which do not lie in a straight line. They are
given by three rays in R
3
,
R
j
= R
+
a
(j)
= {ta
(j)
; t>0},j=1, 2, 3,
where the a
(j)
are three nonzero vectors in R
3
. We can now form
cvxh(R
1
fi ◊
2
R
2
fi ◊
3
R
3
) fi(≠cvxh(R
1
fi ◊
2
R
2
fi ◊
3
R
3
)),
where ( ◊
2
,◊
3
)=(±1, ±1) (four possibilites). These are the four triangular domains
that we can form in P
2
from the three points, and we see that two bits of information
are needed in addition to the information contained in the three points in order to
determine which domain we shall consider.
P2. The complement of the union of three segments which do not lie in a straight
line and have pairwise common endpoints has two components, and they are of
equal status. A triangular domain in this case is given by three segments and the
additional information which of the two components is meant. And remember that
the segments also require one bit of information each in addition to the information
contained in the endpoints.
P3. The complement of three lines in P
2
which meet in exactly three different
points has four com ponents, all of equal status. So a triangular domain is given by
three lines plus the additional information which of the four components is me ant.
Explicitly, if the lines are given by three planes in R
3
passing through the origin
with linear equations l
k
(x, y, z) = 0, the four triangular domains are
A
3
‹
k=1
Y
◊,k
B
fi
A
≠
3
‹
k=1
Y
◊,k
B
,◊=(◊
1
,◊
2
,◊
3
) œ {≠1, 1}
3
,
where Y
◊,k
is the half space
Y
◊,k
= {(x, y, z) œ R
3
r {(0, 0, 0)}; ◊
k
l
k
(x, y, z) Ø 0},k=1, 2, 3,◊œ {≠1, 1}
3
,
and where ◊ =(◊
1
,◊
2
,◊
3
)=(1, ±1, ±1) (four possibilities).
We may conclude that, just as for segments, the notion of triangular domain
comes with different cognitive content in P
2
compared with E
2
.
7 Proposition 16
Proposition 16 says, as we have seen in Section 1, that an exterior angle in a triangle
is greater than any of the two opposite interior angles. Let a triangle with vertices
a, b, c be given, and let us examine the proof that the exterior angle at c is strictly
larger than the interior angle \bac at a (see the figure on page 163). Euclid extends
the side [b, c] beyond c to a point d such that c lies between b and d (the exact
position of d is not important; it serves only to define the exterior angle \acd at
c). The problem is now to prove that the exterior angle \acd is larger than the
interior angle \bac.
Normat 4/2012 Christer O. Kiselman 163
a
b
c
d
e
f
a
b
c
d
e
f
b
a
a
b
c
d
e
f = b
a
a
b
c
d
e
f
b
a
164 Christer O. Kiselman Normat 4/2012
Euclid introduces a new point e as the midpoint of the side [a, c] and extends the
segment [b, e] to a point f, defined so that e is the midpoint of [b, f]. He therefore
obtains two congruent triangles —abe and —cfe,where\ecf = \eab. Hence the
angle at c in the triangle —cfe is equal to the angle at a in the triangle —abe.So
far everything is OK. Euclid then says:
me–zwn dË ‚stin ô Õp‰ EGD t®c Õp‰ EGZ; (Sj
¨
ostedt 1968:22; Fitzpatrick 2011:21) (But
the angle \ecd is greater than the angle \ecf;)
This is something we should see from a (deceptive) lettered diagram. (On the
significance of the lettered diagram in Greek mathematics, see Section 8.)
At this point, it is convenient to continue the argument on a sphere. We need
only look at a triangle on the sphere such that the distance ”(b, e)betweenb and
e is fi/2. (We measure as usual the length of a side by the angle subtended by it
as viewed from the center of the sphere.) Then the distance between f and b is fi,
that is, they are antipodes and will be identified in the projective plane. Hence the
great circle determined by the side [b, c] and the great circle through b and e meet
at f, and the exterior angle at c is equal to the interior angle at a.
This is the simplest example I have found; by perturbing it a little (taking the
distance between b and e to be a little larger than fi/2), we can arrange that the
exterior angle at c is smaller than the interior angle at a.
18
In fact, the crucial
quantity here is the length of the median [b, e]:
Proposition 7.1. Let a triangular domain on the sphere be given with vertices in
a, b, c. We assume that all sides and all angles are less than fi. Let e be the midpoint
on the side [a, c].
(1) If the distance between b and e is less than fi/2, then the conclusion in Euclid’s
Proposition 16 holds: the exterior angle at c is larger than the interior angle at a.
(2) If the distance between b and e is equal to fi/2, then the exterior angle at c is
equal to the interior angle at a.
(3) If the distance between b and e is larger than fi/2, then the exterior angle at c
is smaller than the interior angle at a.
It is reasonable to assume that no side or angle in the triangle is equal to fi or
larger—we avoid the trouble of defining the exterior angle of a concave angle.
Note that this result is a result on the geometry of the projective plane. I have
chosen to formulate it for the sphere only because in this way it will be easier to
visualize.
Beweis. Note that we cannot speak about the midpoint between two non-antipodal
points of the s phere, since there are two midpoints (they are antipodal). However,
if a triangular domain is given, we take the midpoint which belongs to it. This is
how we de fine e.
By the Spherical Sine Theorem applied to the triangle —bcf we obtain
sin(fi ≠ \ecd + \ecf)sin”(b, c)=sin(\bfc)sin”(b, f).
18
Also Heath (1926a:280) remarks that in order for the proof to be valid, it is necessary that
the line cf should fall within the angle \acd,andBernardVitrac(personalcommunication2012-
04-01) directs my attention to the fact that also he points this out (Vitrac 1990:228).
Normat 4/2012 Christer O. Kiselman 165
Now
sin(fi ≠ \ecd + \ecf)=sin(\ecd ≠ \ecf )=sin(\ecd ≠ \bac),
and since sin ”(b, c) and sin(\bfc)=sin(\abc) are positive by assumption, the
sine of the difference \ecd ≠ \bac has the same sign as sin ”(b, f)=sin2”(b, e).
The three cases (1), (2), (3) are obtained if ”(b, e) <fi/2, = fi/2, and >fi/2,
respective ly.
Thus if all three medians in the triangle we consider are less than fi/2, Euclid is
all right.
8 Relying on diagrams
Reviel Netz devotes the first chapter of his book (1999:12–67) to an instructive
account of the all-important role of the lettered diagram in Greek mathematics.
The lettered diagram is a combination of different elements on the logical plane, the
cognitive plane, the semiotic plane, and the historical plane; “the fertile intersection
of different, almost antagonistic elements which is responsible for the shaping of
deduction” (Netz 1999:67).
When I studied Euclidean geometry at Norra real in Stockholm some sixty years
ago, our teacher, Bertil Brostr
¨
om, repeatedly emphasize d that we we re not allowed
to draw any conclusions from the diagrams: all proofs should depend only on the
axioms and the chain of logical implications. Nevertheless, the diagrams served as
inspiration and mnemonic help—and perhaps a little bit more.
It is an interesting fact that we can actually draw some valid conclusions from
a diagram—provided it is not too special (whatever that means). And it is not
obvious where to draw the boundary b etween legitimate and forbidden uses of
visual information. This point was brought up in a disc ussion with the authors of
the paper by Avigad et al. (2009). They discuss there the role of diagrams in the
proofs, and the formal logical sys tem called E which they have constructed acc epts
Euclid’s proof considered in Section 7 without protest.
19
John Mumma explains
that the system E licenses the inference that the angle \ecd is larger that the angle
\ecf.
Similarly, one cannot generally infer, from inspecting two angles in a diagram, that
one is larger than the other, but one can draw this conclusion if the diagram “shows”
that the first is contained in the second. (Avigad et al. 2009:701)
So clearly the formal system E does ac ce pt some information from a diagram.
The relations of betweennes s and same-sidedness are primitives in the system E.
The possibility of a non-orientable plane is ruled out not by any explicit ass umption
but by the rules for reasoning with betweenness and same-sidedness (John Mumma,
personal communication 2012-04- 15). Conceivably, one could construct a similar
formal system which does not have the betweenness relation for triples of points,
nor the same-sidedness relation. (Cf. the Kernsatz of Pasch quoted in the next
section.)
19
The system E is proved to be equivalent to an earlier formal system for Euclidean geometry
due to Alfred Tarski.
166 Christer O. Kiselman Normat 4/2012
9 Orientability
Orientability of a manifold means, roughly speaking, that you can walk around
it with a watch and the hands of the watch still go around clockwise (as viewed
from the outside) when you return to the starting point after an excursion. The
Euclidean plane E
2
and the sphere S
2
are both orientable. However, the sphere is
not a model for Euclid’s axioms (postulates), since two lines in general position
will intersect in two points, not in one, and two antipodal points do not determine
a great circle uniquely. This is what forces us to identify antipodes; the projective
plane becomes a bona fide mo del—at least we so argued—but orientability is lost.
Nevertheless, it is often convenient to conduct an argument on the sphere, as I have
done in Proposition 7.1 above.
Postulate 5, the Postulate o f Parallels, quoted in Subsection 3.2, states that two
lines meet on a certain side. In the projective plane it is meaningless to talk about
the side of a straight line. Given a point on a straight line, you can define two sides
of the line in a neighborhood of the point, but if you go along the line and have
your watch on your left wrist, you come back after a while with the watch on your
right wrist (as viewed from the outside). So the ve ry fact that Euclid talks about
“the same side” and “that side” means that he assumes the plane to be orientable.
Hence projective geometry is excluded.
One can retain from Postulate 5 merely that the lines are not parallel, i.e., that
they do me et somewhere, not mentioning any side. In this modified form, Postulate
5 is true also in the pro je ctive case.
Here it is of interes t to note one of Pasch’s axioms, viz.
III. Kernsatz. — Liegt der Punkt C innerhalb der Strecke AB, so liegt der Punkt
A außerhalb der Strecke BC (Pasch 1926:5). — (III. Axiom. If the point C lies
within the segment AB, then the point A lies outside the segment BC .)
In the projective plane this can have a meaning only if we define both segments
carefully; see the discussion in Subsection 3.2.
10 Conclusion
10.1 The first question
Propositions 16 and 27 become true if we suppose orientability or introduce some
other hypothesis which will rule out the projective plane. And orientability is a
reasonable hypothesis: Euclid in his Postulate 5 talks about the sides of a straight
line, which is meaningless without orientability.
With the projective plane as a model, we can either conclude that Proposition
16 is meaningless, since we cannot compare angles, or false if we measure angles as
discussed in Subsection 3.2. Proposition 27 can be interpreted as saying that the
mentioned lines do not meet, and if so it is false whether we measure the angles
on the sphere or not. The reasonable way out of this confusion is, again, to accept
the tacit hyp othesis of orientability.
If our beloved teacher, Âstoiqeiwt†c, could see my paper, he might react in one
of two possible ways. Either
a'. Sure, my boy, I do assume orientability—I just forgot to jot it down. (I was
too busy thinking about Postulate Five.) In the next edition, which is now being
Normat 4/2012 Christer O. Kiselman 167
prepared here in the Mouseÿon, I shall include orientability as Postulate Six. Who
wants to live on a M
¨
obius strip anyway?
or
b'. >Ido‘! — Hey, that’s interesting! Seems to be a more general geometry. I shall
write about it in Book Fourteen. And I like Napier’s rule and the Spherical Sine
Theorem which you learnt from your navigating father Sam Svensson even before
you studied my geometry and plane trigonometry for Bertil Brostr
¨
om. We are all
navigators here in Africa, aren’t we? Navigare necesse est, as somebody will soon
quip.
Can you guess which?
10.2 The second question
We have observed that the term eŒjeÿa often means a rectilinear segment. Perhaps
this is its most basic meaning. In other contexts it could be interpreted as an infinite
straight line, but also, if we want to avoid an actual infinity, as a family of equivale nt
rectilinear segments, thus as a potential infinity. However, in projective geometry,
the infinite straight lines are just great circles with opposite points identified, thus
hardly infinitely large. This gives us one more reason to believe that Euclid did not
think about projective geometry. Finally, but rarely, it can me an ‘ray’.
For straight lines in the sense of Heath that are infinite in one or both directi-
ons there appears the problem of actual infinity; if we avoid that by considering
only segments, we have to obtain uniqueness by forming equivalence classes, which
is certainly an anachronistic viewpoint, but maybe was exactly what Euclid did
implicitly.
Let us listen to our beloved teacher once more, this time on eutheia:
g'. Lhreÿte! — Bah! What is straight is straight, and the wise understand. I do not
waste words in my geometry. You young people use too many. Maybe you left Africa
too early. I am afraid you will have to set up a Terminology Center in a futile effort
to control the flood.
And on infinity:
d'. Aristotle and his gang of physicists are harassing us mathematicians. We must
nowadays be careful when writing about infinity—potential infinity has rapidly
become PO—but at night I am free to think about actual infinity. I can even see it.
Acknowledgment
This paper has evolved slowly since 2007 (or perhaps even earlier) and passed
through many versions. Several people have contributed to its successive improve-
ment.
· Bo G
¨
oran Johansson commented on several of the concepts studied here, especially
on actual and potential infinity.
· Erik Bohlin, my teacher of mathematical Classical Greeek, brought Federspiel’s
article (1991) to my attention, made remarks on Proclus’s commentary, and
helped me with several mathematical terms in Classical Greek.
· Petros Maragos and Takis Konstantopoulos informed me about geometric terms
in Contemporary Greek.
· Seidon Alsaody made helpful comments which led to improvements of the geo-
metric arguments.
168 Christer O. Kiselman Normat 4/2012
· Jesper L
¨
utzen kindly sent me constructive criticism on an earlier version.
· Michel Federspiel m ade valuable comments on several of the problems considered
here and sent me three of his papers (1992, 1998, 2005).
· Ove Strid, my teacher of Classical Greek, patiently explained the use of interjec-
tions in that language (see b' and g' in Section 10).
· Bernard Vitrac sent me valuable comments on an earlier version.
· David Pierce sent me interesting comments and drew my attention to the pap er
by Avigad, Dean and Mumma (2009).
· John Mumma made interesting observations on the system E of his paper with
Avigad and Dean (2009).
· Jockum Aniansson helped me with references to Apollonius’s work, made c areful
comments and gave me good advice.
For all this he lp I am most grateful.
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Received 2013 September 20. Accepted for publication 2013 October 09.