Aristotle and the Fifth Postulate
It is difficult to gage the significance of Euclid‟s Fifth Postulate, the Parallel Postulate, to
modernity‟s view of the objectivity of science and mathematics.
Kant‟s Copernican Revolution was formulated to constrain the dual threats of dogmatism
and skepticism. His grounding the sensual conditions for the possibility of perception on the a
priori forms of intuition, space and time, allowed for a non-dogmatic, critical realism. His
demonstration that the fundamental concepts of math and physics were the a priori conditions for
the possibility of the experience of objects, provided the bulwark against a radical, empirical
phenomenalism. Essential to this second pillar was the universal and objective priority of
Euclidean geometry and Newtonian physics.
But Euclidean geometry had long been known to harbor systematic flaws. In particular,
the Fifth or Parallel Postulate, had long been under close examination by mathematicians as to its
appropriate placement as a basic assumption (postulate) or rather something that could and
needed to be proved (theorem). The Fifth Postulate states:
That, if a straight line falling on two straight lines make the interior angle on the
same side less than two right angles, the two straight lines, if produced indefinitely, meet
on that side on which are the angles less than the two right angles.
The Fifth Postulate of Euclid, the Parallel Postulate has always been controversial. From
comments in Aristotle we know that there had been efforts to prove the fundamental properties
of parallel lines prior to Euclid, and that Aristotle found them unsatisfactory. There have also
been numerous attempts from the time of Ptolemy and Proclus up through that of Lambert and
Legendre, to prove the postulate as a theorem. None of these attempts has been considered
The greatest damage to the Euclidean legacy and that of the ontological underpinnings of
Classical rationalism has been wracked by those who have just ignored the postulate. Gauss,
quickly followed by Lobatchevsky and Bolyai realized that alternative geometries to Euclid
could be constructed without the parallel postulate altogether. Alternative self-consistent
geometries were developed around the diverse possibilities that through a point an infinite
number of lines parallel to a given line could be drawn (Lobatchevsky) to the axiom that no lines
parallel the given line could be drawn (Riemann).
With the objective uniqueness of Euclidean geometry compromised, Newtonian science
could little withstand a „parallel‟ assault. Einstein eventually won the geometry/physics wars by
establishing that Riemannian geometry was the „true‟ empirical structure of the given world.1
Yet even nearly a century after the unconditional surrender of the rational and absolute
world views Newton and Euclid, we seem to have not given up this apparently privileged
perspective. Astronomers and physicists alike still utilize these classical systems to do the
normal science of their research. General relativity has had only a marginal impact on the
practice of most physics.
Whether constrained by psychological limitations or still hindered by a yet imperfect
physics, there remains reason to suspect that the Classical world view speaks to our
imaginations in a language we are never completely free from. Perhaps if we could reconfigure
the problem of the Parallel Postulate in an ontologically more sufficient manner, we could better
understand why contemporary mathematics and science seems constrained to retain this simpler
I will try to use textual support to make three points. First, that it is clear that the
Academy in general and Aristotle in particular utilized a substantially different organizational
There remains a robust debate around whether the truth of any axiomatic system can be empirically determined.
approach to how the theorems and problems of geometry were ordered. This is no mean point,
for in an axiomatic system, the order of the theorems and problems wholly determines the origins
and genesis of the logical development of the discipline. A different order implies a distinct
Second I will show that Aristotle had serious and substantial problems with the ways in
which axioms about parallel lines were established. I will try to establish that Euclid‟s
alternative, making the nature of parallel lines into a foundational postulate, does little to
ameliorate the kinds of problems Aristotle would raise.
And finally, I will show from specific evidence that there was in the work of Aristotle,
and therefore in the Academy in general, an alternative way of dealing with the problem of the
infinite, something that the parallel postulate attempts to accomplish implicitly.
The first two points have been adequately established by the close textual analysis of
Heath. In the Prior Analytics (I. 24, 41b 13-22), Aristotle gives a proof for the equality of the
base angles of an isosceles triangle, Euclid Proposition I.5. Aristotle‟s proof is based on
principles that don‟t appear in Euclid until Book III in Euclid, and in turn rely on theorems based
There are two hypotheses we can draw from this proof. The first is that rather than
ordering all of geometry from the “bottom-up” (from rectilinear figures to curves) as in Euclid‟s
synthetic tour de force, The Elements, the Academy placed many theorems based on angle
measurement within the realm of circles and curves, in many ways a more appropriate setting.
This might imply that much of the alternative approach was “top-down,” working from the
divine nature of partless circularity down to the profane world of rectilinear figures.
Second, it would show that a parallel postulate, or some alternative theorem on parallels
would not have been necessary to prove the threshold proof about the angles of a triangle being
equal to two right angles. If we can establish some primitive equivalence between the angles of
a triangle and those of its circumscribed circle, much of Euclid Book I becomes otiose.
This second observation brings us to my second point on Aristotle‟s concern with any
theorem or postulate on parallel lines. From both the Prior (ii. 17. 66 a 11-15) and Posterior
Analytics (i. 5. 74 a 13-16), we know that Aristotle thought that the received theorems on parallel
lines contained a petitio principia.2 Heath and others speculate that Euclid‟s postulating the
definition of parallel lines is an attempt to circumvent such problems.
From ontological and logical perspectives, this speculation seems spurious. Aristotle and
the Academy are very sensitive to the dangers of theorizing about actual infinities. The Fifth
Postulate, as formulated by Euclid exhibits many of the difficulties that Aristotle and the
Academy would object to in its framing of the way lines behave when “produced indefinitely.”
The fact that the ordering implied by the passages in Aristotle could develop all of the
consequences of the Fifth Postulate, without making such a claim about the “unlimited,” would
greatly strengthen the case for supporting this alternative approach.
The relationship between the rectilinear and the curvilinear, or circular, is one of a
contained infinity. Although there is an inexhaustible functional distance between the profanity
of an inscribed rectilinear figure and the divinity of its circumscribed partless circle, that
potentially infinite gap is totally contained by the absolute wholeness of the continuous circle.3
The two great mathematical problems of the Ancient world, the Double Cube and the Squared
Heath, A History of Greek Mathematics (New York, 1981), p.339.
This principle of the contained infinite will become critical to the metaphysics of Spinoza, with his two
nonconcentric circles, and in turn Schelling and Hegel.
Circle, each exemplifies the reciprocal onto-mathematical significance of this representation,
both incarnation and resurrection.4
The ontological difficulties with the Fifth Postulate prompt us to recognize the logical
anomalies that surround it. Looking at the overall structure of Euclid Book I, we see that Euclid
needs the Fifth Postulate in order for him to establish the threshold proof of Proposition 32, that
the exterior angle of a triangle is equal to the sum of the two non-adjacent interior angles. This
theorem will then allow him to introduce the extremely useful, but ontologically provocative
technique of “carrying areas.” Euclid needs this tool if he is to succeed in crowning Book I with
the magnificent Proposition 47, the Pythagorean Theorem.
Euclid fabricates the Fifth Postulate in order to establish his monotheistically synthetic
ordering of geometry. The curvilinear and rectilinear are ontologically incommensurable.
Figures from each realm can be put into relationship, but not finally reduced to each other.
Euclid‟s attempt to construct a seamless progression of proofs from simple rectilinear
congruence theorems to the powerful proof of the Pythagorean relationship, forces him to
materially bridge an theo-logical gap, an effort the Academy might well have considered as
hubris, as well as bad metaphysics.
And there are serious ontological difficulties in introducing these concepts this early in
the geometry. Carrying areas implies a new sense of equality, one of matter disassociated from
form. Prior to Proposition 32, equality meant that of form and matter, or congruence. The
introduction of areas of different form being set equal demands new ontological distinctions.
Just as important, in order to get to Proposition 32, Euclid is forced, not only to introduce
the vulnerable Fifth Postulate, but he also must rely on the always questionable indirect proof.
The differential and integral calculi are based on the geometry of these two problems.
In the two critical steps leading up to Proposition 32, in the proofs of Proposition 13, that
the angles along a straight line equal two right angles, and Proposition 27, that if a line falling
across two given lines make the alternate interior angles equal to each other, the lines are
parallel, Euclid must resort to arguments per impossibility. In logic, Aristotle is critical of this
approach, and warns to use it only as a last resort.
In geometry the case is more serious. In an axiomatic system, positive proofs allow us to
capture the rational continuity of the full fabric of theorems in how they develop through the
system. The use of indirect proofs stands as opaque walls to such seamless transparency. A
proof should make immediately clear not just that a proof is valid, but why it is, and why it must
be so. Indirect arguments destroy such absolute authority. We may have to accept them as true,
but we can never grasp the full compulsion of their authority within the axiomatic system.
Indirect arguments show us that something is true, but not why it is so. The use of more than one
in a single demonstration, compounds that uncertainty and undermines the rational determinacy
of the proof. Such proofs serve the principles of consistency, not the positive authority of
We are finally in an adequate position to understand Aristotle‟s clearest indication of the
Academy‟s ordering of geometry. In Posterior Analytics II.11 94a28-34 and in Metaphysics
IX.9 1051a27-29 Aristotle presents a proof for why “the angle in a semicircle is a right angle
universally (Euclid III.31).” His proof is completely different than any that could appear in
Euclid. The proof assumes at least one proposition that comes later in Euclid than the proof at
hand – that an angle at the circumference of a circle is half the angle at the center. And there is
at least one proposition that comes in Book I of Euclid, that Aristotle proves as part of his
demonstration – that a straight angle is equal to two right angles.
This second assumption gives us some insight to the way in which Aristotle and the
Academy may have ordered their axiomatic approach. Figure A illustrates how he shows that
the angle formed by the diameter is equal to two right angles. He drops a perpendicular bisector,
EC to the diameter, AB, and then concludes that each of the produced right angles cuts off half a
semi-circle, or an arc equivalent to a right angle. It would seem that he needs Euclid‟s
Proposition III.20, that the angle at the center of a circle is double the angle on the circumference
when they cut off the same arc, to then show that the angle on the circumference of a semicircle
is half of two right angles. 5
But there are serious problems with this assumption. In Euclid‟s organization,
Proposition III.20 relies on Proposition 32 and the ensuing proof that the angles of any triangle
add up to two right angles, which in turn relies on the parallel postulate.
If Aristotle had followed a similar ordering schema to Euclid, he would have seemingly
developed two independent proofs for the sum of the angles of a triangle equaling two right
angles, one from the nature of circularity and another based on something like the parallel
postulate. In empirical induction such consilience is fortunate. In mathematics it can only mean
an overdetermination based on superfluous assumptions.
Another option could be that Aristotle just postulated the relationship between
circumferential angles and central angles within a circle. This approach seems to err in the other
extreme: We should never assume what can be demonstrated.
Aristotle‟s construction for this problem leads us to some reasonable possible derivations.
In a similar construction to Figure A, we could just continue the perpendicular bisector out to
intersect the circumference twice (Figure B). We could then finish the diagram by joining the
four points of circumference intersection, forming a quadrilateral. By corresponding angles of
Patrick Byrne, Analysis and Science in Aristotle (Albany, 1997), 110.
congruent triangles we can show that all eight of the angles on the circumference are equal to
one half a right angle and each cut off an arc that is also intersected by a central angle equal to a
right angle: ie. Central angles are double angles on the circumference intersecting the same arcs.
It is one step to then conclude, without the use of the Fifth Postulate, that the sum of the angles in
triangle is equal to two right angles.
Although there is no direct evidence that this is the way Aristotle proceeded, there is
more than a preponderance of indirect support that he must have pursued some similar line. For
one, most of the mathematical work at the Academy, like Aristotle‟s own logical works, was
analytic, not synthetic.
Geometrical analysis cultivates a dualistic view of the universe. Where the Euclidean
Elements is mainly concerned with establishing a Unitarian system of logical justification for the
proofs of geometry, the priorities of analysis are quite different.
In analysis it is the object of analysis that must come first. And as we have shown those
objects are necessarily polarized into two completely heterogeneous ontological types, the
rectilinear and the curvilinear. The goal of analytic geometry is not so much to overcome this
ontological disjunction as to understand it in its full and rich diversity. Analysis, like any art of
discovery must be pejoratively pluralistic in order to be adequate to the broadest possibility of
what it may uncover.
This “dualistic” approach to the ontology of constructed reality, may help to explain the
paucity of our evidence. On reading Pappus' commentary on the Treasury of Analysis, one is
struck by how few of these treasures have survived antiquity. Compared to the poetry or
philosophy of the same era, the void in original mathematical works has evoked such metaphors
as a "holocaust".6 This absence of remnants only highlights by the apparent plenitude of sources
for the almost lone work to have weathered the deluge: The Elements of Euclid. The era when
most of these works disappeared was that of the tumultuous transition from paganism to
Christianity, not a particularly healthy time for dualist ontologies.
There is also tangential support from Plato to show that such an alternative organizational
schema was the received position within the Academy.
The Timaeus is a complete and rich model for how such an ordered schema might be
developed. Like the works of mathematical analysis, the Timaeus myth is tripartite, having three
distinct beginnings, the Works of Reason, the Works of Necessity and the Cooperation of Reason
Van Waerden, B.L., Science Awakening (New York, 1963).
Also paralleling what we might expect from geometrical analysis, the works or reason
deal primarily with the origins of the soul and the heavenly. These are characterized by the
harmonic patterns of the heavens, are about circular motions.
Although Plato does engage the rational conditions of the possibilities for the atomistic
microcosm within the first start, it is within the Works of Necessity that he develops the fuller
implications and limits of the model of parts and wholes. And not surprisingly, this ontology is
that of the rectilinear basis of the construction of finite beings. This arrangement illustrates both
the radical disjunction between the rectilinear and curvilinear bases of construction, but also
allows that even the productions of necessity are “measured” by reason.
And finally, only after establishing the distinct and sufficient conditions of the two
ontologically opposed foundations of any possible reality, does Plato then consider how they
might finally be reconciled. In this third model of organic “cooperation”, the myth works out the
evolution of how reason “persuades” necessity to do its bidding. In geometrical analysis it is
within this field of complex interaction where the elegant problems of Squaring the Circle and
Doubling the Cube can finally be comprehended. For overcoming the substantial dualism of the
ontological conditions for the possible construction of reality itself, this pluralistic framework of
analytic geometry is more truly Trinitarian than the simple hierarchies of religion or modern
I am under no illusion that restoring the logical and intuitive primacy of Classical
geometry will resurrect a Golden Age of metaphysical philosophy. But I stand with Socrates‟
faith in the pragmatic value of optimism: “We must, therefore, not believe that the debater‟s
argument, for it would make us idle, and fainthearted men like to hear it, whereas my argument
makes them energetic and keen on the search (81d).” Perhaps regaining an archaic vision of the
order of space and time will at least move forward the enigmatic Kantian query of making
clearer that “hidden art concealed in the depths of the human soul.”