here you go phya, pages 191-194 of Heath's Euclid's Elements, vol 1
"DEFINITION 23.
Parallel straight lines are straight lines 'which, being in the same plane and
being produced indefinitely in both directions, do not meet one another in either
direction.
IIapa.AATJAo~ (alongside one another) written in one word does not appear
in Plato; but with Aristotle it was already a familiar term.
Ei~ U1rEtpOV cannot be translated "to infinity" because these words might
seem to suggest a region or place infinitely distant, whereas El~ a1rEtpov, which
seems to be used indifferently with E1r' <'f1rEtpOV, is adverbial, meaning "without
limit," i.e. "indefinitely." Thus the expression is used of a magnitude being
"infinitely divisible," or of a series of terms extending without limit.
In both directions, Ecf>' £KaTEpa TO. fJ-EpTJ, literally "towards both the parts"
where "parts" must be used in the sense of "regions" (cf Thuc. II. 96).
It is clear that with Aristotle the general notion of parallels was that of
straight lines which do not meet, as in Euclid: thus Aristotle discusses the
question whether to think that parallels do meet should be called a
geometrical or an ungeometrical error (Anal. post. 1. 12, 77 b 22), and (more
interesting still in relation to Euclid) he observes that there is nothing
surprising .in different hypotheses leading to the same error, as one might
conclude that parallels meet by starting from the assumption, either (a) that
the interior (angle) is greater than the exterior, or (b) that the angles of a
triangle make up more than two right angles (Anal. prior. n. 17, 66 a II).
Another definition is attributed by Proclus to Posidonius, who said that
"parallel lines are those which, (being) in one plane, neither converge nor diverge,
but have all the perpendiculars equal which are drawn from the points 0/ one
line to the other, while such (straight lines) as make the perpendiculars less and
less continually do converge to one another; for the perpendicular is enough
to define (opi'Etv OVVaTaL) the heights of areas and the distances between lines.
For this reason, when the perpendiculars are equal, the distances between the
straight lines are equal, but when they become greater and less, the interval is
lessened, and the straight lines converge to one another in the direction in
which the less perpendiculars are" (Proclus, p. 176, 6- 17).
Posidonius' definition, with the explanation as to distances between straight
lines, their convergence and divergence, amounts to the definition quoted by
Simplicius (an-Nairizi, p. 25, ed. Curtze) which described straight lines as
parallel if,'when they are produced indefinitely both ways, the distance between
them, or the perpendicular drawn from either of them to the other, is always
equal and not different. To the objection that it should be proved that the
distance between two parallel lines is the perpendicular to them Simplicius
I. DEF. 23] NOTES ON DEFINITIONS 22, ~3
replies that the definition will do equally well if all mention of the perpendicular
be omitted and it be merely stated that the distance remains equal,
although" for proving the matter in 'question it is necessary to say that one
straight line is perpendicular to both" (an-Nairizi, ed. Besthorn-Heiberg, p. 9)'
He then quotes the definition of "the philosopher Aganis": ," Parallel
straight lines are straight lines, situated in the same plane, the distance between
which, if they are produced indefinitely in both directions at the same time, is
everywhere the same." (This definition forms the basis of the attempt of
"Aganis" to prove the Postulate of Parallels.) On the definition Simplicius
remarks that the words "situated in the same plane" are perhaps unnecessary,
since, if the distance between the lines is everywhere the same, and one does
not incline at all towards the other, they must for that reason be in the same
plane. He adds that the "distance" referred to in the definition is the
shortest line which joins things disjoined. Thus, between .point and point,
the distance is the straight line joining them; between a point and a straight
line or between a point and a plane it is the perpendicular drawn from the point
to the line or plane; "as regards the distance between two lines, that distance
is, if the lines are parallel, one and the same, equal to itself at all places on
the lines, it is the shortest distance and, at all places on the lines, perpendicular
to both" (ibid. p. 10).
Tb-e same idea occurs in a quotation by Proclus (p. 177, II) from
Geminus. As part of a classification of lines which do not meet he observes:
" Of lines which do not meet, some are in one plane with one another, others
not. Of those which meet and are in one plane, some are always the same
distance from one another, others lessen the distance continually, as the hyperbola
(approaches) the straight line, and the conchoid the straight line (i.e. the
asymptote in each case). For these, while the distance is being continually
lessened, are continually (in the position of) not meeting, though they converge
to one another; they never converge entirely, and this is the most paradoxical
theorem in geometry, since it shows that the convergence of some lines is nonconvergent.
But of lines which are always an equal distance apart, those
which are straight and never make the (distance) between them smaller, and
which are in one plane, are parallel."
Thus the equidistance-theory of parallels (to which we shall return) is very
fully represented in antiquity. I seem also to see traces in Greek writers of a
conception equivalent to the vicious direction-theory which has been adopted
in so many modern text-books. Aristotle has an interesting, though obscure,
.allusion in Anal. prior. II. 16, 65 a 4 to a petitio principii committed by "those
who think that they draw parallels" (or "establish the theory of parallels,"
which is a possible translation of TO.'; 1rapaAA~AOV<; ypo.<!ml'): "for they unconsciously
assume such things as it is not possible to demonstrate if parallels
do not exist." It is clear from this that there was a vicious circle in the then
current theory of parallels; something which depended for its truth on the
properties of parallels was assumed in the actual proof of those properties,
e.g. that the three angles of a triangle make up two right angles. This is not
the case in Euclid, and the passage makes it clear that it was Euclid himself
who got rid of the petilio principii in earlier text-books by formulating and
premising before I. 29 the famous Postulate 5, which must ever be regarded
as among the most epoch-making achievements in the domain of geometry.
But one of the commentators on Aristotle, Philoponus, has a note on the
above passage purporting to give the specific character of the petitio principii
alluded to; and it is here that a direction-theory of parallels may be hinted at,
whether Philoponus is or is not right in supposing that this was what Aristotle
had in mind. Philoponus says: "The same thing is done by those who draw
parallels, namely begging the original question; for they will have it that it is
possible to draw parallel straight lines from the meridian circle, and they
assume a point, so to say, falling on the plane of that circle and thus they
draw the straight lines. And what was sought is thereby assumed; for he
who does not admit the genesis of the parallels will not admit the point
referred to either." What is meant is, I think, somewhat as follows. Given
a straight line and a point through which a parallel to it is to be drawn, we
are to suppose the given straight line placed in the plane of the meridian.
Then we are told to draw through the given point another straight line in the
plane of the meridian (strictly speaking it should be drawn in a plane parallel
to the plane of the meridian, but the idea is that, compared with the size of
the meridian circle, the distance between the point and the straight line is
negligible); and this, as I read Philoponus, is supposed to be equivalent to
assuming a very distant point in the meridian plane and joining ~he given
point to it. But obviously no ruler would stretch to such a point, and the
objector would say that we cannot really direct a straight line to the assumed
distant point except by drawing it, without more ado, parallel to the given
straight line. And herein is the petitio principii. I am confirmed in seeing
in Philoponus an allusion to a direction-theory by a,·remark of Schottel',) on a
similar reference to the meridian plane supposed to be used by advocates of
that theory. Schotten is arguing that direction is not in itself a conception
such that you can predicate one direction of two different lines. " If anyone
should reply that nevertheless many lines can be conceived which all have the
direction from north to south," he replies that this represents only a nominal,
not a real, identity of direction.
Coming now to modern times, we may classify under three groups
practically. all the different definitions that have been given of parallels
(Schotten, op. cit. II. p. 188 sqq.).
(I) Parallel straight lines have no point common, under which general
conception the following varieties of statement may be included:
(a) they do not cut another,
(b) they meet at infinity, or
(c) they have a common point at infinity.
(2) Parallel straight lines have the same, or like, direction or directions,
under which class of definitions must be included all those which introduce
transversals and say that the parallels make t'qual angles with a transversal.
(3) Parallel straight lines have the distance between them constant;
with which group we may connect the attempt to explain a parallel as the
geometrical locus of all poil/ts 10hich are equidistant from a straight line.
But the three points of view have a good deal in common; some of them
lead easily to the others. Thus the idea of the lines having no point common
led to the notion of their having a common point at infinity, through the
influence of modern geometry seeking to embrace different cases under one
conception; and then again the idea of the lines having a common point at
infinity might suggest their having the same direction. The" non-secant"
idea. would also naturally lead to that of equidistance (3), since our
observation shows that it is things which come nearer to one another that
tend to meet, and hence, if lines are not to meet, the obvious thing is to see
that they shall not come nearer, i.e. shall remain the same distance apart.
We will now take the three groups in order.
(I) The first observation of Schotten is that the varieties of this group
which regard parallels as (a) meeting at infinity or (b) having a common
point at infinity (first mentioned apparently by Kepler, 16°4, as a "fac;on de
parler" and then used by Desargues, 1639) are at least unsuitable definitions
for elementary text-books. How do we know that the lines cut or meet at
infinity? We are not entitled to assume either that they do or that they do
not, because "infinity" is outside our field of observation and we cannot verify
either. As Gauss says (letter to Schumacher), "Finite man cannot claim to
be able to regard the infinite as something to be grasped by means of ordinary
methods of observation." Steiner, in speaking of the rays passing through a
point and successive points of a straight line, observes that as the point of
intersection gets further away the ray moves continually in one and the same
direction (" nach einer und derselben Richtung hin"); only in one position,
that in which it is parallel to the straight line, "there is no real cutting"
between the ray and the straight line; what we have to say is that the ray is
"directed towards the infinitely distant point on the straight line." It is true
that higher geometry has to assume that the lines do meet at infinity: whether
such lines exist in nature or not does not matter (just as we deal with "straight
lines" although there is no such thing as a straight line). But if two lines do
not cut at any finite distance, may not the same thing be true at infinity also?
Are lines conceivable which would not cut even at infinity but always remain
at the same distance from one another even there? Take the case of a line
of railway. Must the two rails meet at infinity so that a train could not stand
on them there (whether we could see it or not makes no difference)? It
. seems best therefore to leave to higher geometry the conception of infinitely
distant points on a line and of two straight lines meeting at infinity, like
imaginary points of intersection, and, for the purposes of elementary geometry,
to rely on the plain distinction between "parallel" and "cutting" which
average human intelligence can readily grasp. This is the method adopted
by Euclid in his definition, which of course belongs to the group (I) of
definitions regarding parallels as non-secant.
It is significant, I think, that such authorities as Ingrami (Elementi di
geometria, 1904) and Enriques and Amaldi (Elementi di geometria, 1905),
after all the discussion of principles that has taken place of late years, give
definitions of parallels equivalent to Euclid's: "those straight lines in a plane
which have not any point in common are called parallels." Hilbert adopts
the same point of view. Veronese, it is true, takes a different line. In his
great work Fondamenti di geometn"a, 1891, he had taken a ray to be parallel to
another when a point at infinity on the second is situated on the first; but he
appears to have come to the conclusion that this definition was unsuitable for
his Elementi. He avoids however giving the Euclidean definition of parallels
as "straight lines in a plane which, though produced indefinitely, never meet,"
because" no one has ever seen two straight lines of this sort," and because
the postulate generally used in connexion with this definition is not evident in
the way that, in the field of our experience, it is evident that only one straight
line can pass through two points. Hence he gives a different definition, for
which he claims the advantage that it is independent of the plane. It is
based on a definition of figures "opposite to one another with respect to a
point " (or reflex figures). "Two figures are opposite to one another with
respect to a point 0, e.g. the figures ABC ... and A'B' C' ..., if to every point
of the one there corresponds one sole point of the other, and if the segments
H. E. I',) OA, OB, OC, ... joining the points of one figure to 0 are respectively equal
and opposite to the segments OA', OB', OC', ... joining to 0 the corresponding
points of the second": then, a transversal of two straight lines being any
segment having as its extremities one point of one line and one point of the
other, "two straight lines are called parallel if one of them contains two points
opposite to two points of the other with respect to the middle point of a common
transversal." It is true, as Veronese says, that the parallels so defined and the
parallels of Euclid are in substance the same; but it can hardly be said that
the definition gives as good an idea of the essential nature of parallels as does
Euclid's. Veronese has to prove, of course, that his parallels have no point in
common, and his "Postulate of Parallels" can hardly be called more evident
than Euclid's: "If two straight lines are parallel, they are figures opposite to
one another with respect to the middle points of all their transversal segments."
(2) The direction-theory.
The fallacy of this theory has nowhere been more completely exposed
than by C. L. Dodgson (Euclid and his modern Rivals, 1879). According to
Killing (Einfiihrung in die Grundlagtn der Geomelrie, I. p. 5) it would appear
to have originated with no less a person than Leibniz. In the text-books
which employ this method the notion of direction appears to be regarded as a
primary, not a derivative notion, since no definition is given. But we ought
at least to know how the same direction or like directions can be recognised
when two different straight lines are in question. But no answer to this
question is forthcoming. The fact is that the whole idea as applied to noncoincident
straight lines is derived from knowledge of the properties of
parallels; it is a case of explaining a thing by itself. The idea of parallels
being in the same direction perhaps arose from the conception of an angle as
a difference of direction (the hollowness of which has already been exposed) ;
sameness of direction for parallels follows from the same "difference of
direction" which both exhibit relatively to a third line. But this is not
enough. As Gauss said (Werke, IV. p. 365), "If it [identity of direction] is
recognised by the equality of the angles formed with one third straight line,
we do not yet know without an antecedent proof whether this same equality
,will also be found in the angles formed with a fourth straight line" (and any
number of other transversals); and in order to make this theory of parallels
valid, so far from getting rid of axioms such as Euclid's, you would have to
assume as an axiom what is much less axiomatic, namely that "straight lines
which make equal corresponding angles with a certain transversal do so with
any transversal" (Dodgson, p. 101). .
(3) In modern times the conception of parallels as equidistant straight
lines was practically adopted by Clavius (the editor of Euclid, born at
Bamberg, 1537) and (according to Saccheri) by Borelli (Euclides restitu/us,
1658) although they do not seem to have defined parallels in this way.
Saccheri points out that, before such a definition can be used, it has to
be proved that "the geometrical locus of points equidistant from a straight
line is a straight line." To do him justice, Clavius saw this and tried to
prove it: he makes out that the locus is a straight line according to the
definition of Euclid, because "it lies evenly with respect to all the points
on it"; but there is a confusion here, because such "evenness" as the locus
has is with respect to the straight line from which its points are equidistant,
and there is nothing to show that it possesses this property with respect
to itself. In fact the theorem cannot be proved without a postulate."
the jumbled up spammish words are Greek...i'm not going to go find the words and write them out, sorry, though i can link a pdf to the whole book if anyone is interested
