Elizabeth Barrat
April 12, 2002
Bryan Smith
Geometry 300
Saccheri Quadrilaterals: The Discovery of
Non-Euclidean Geometry
The value of non-Euclidean geometry lies
in its ability to liberate us from preconceived ideas in preparation for the
time when exploration of physical laws might demand some geometry other than
the Euclidean.
-Bernhard
Reimann
Throughout the centuries many
mathematicians have tried to prove Euclid’s parallel postulate. One such Logician was the Jesuit priest
Girolamo Saccheri (1667-1733). Saccheri
was a Professor of Grammar in Milan, lectured on Philosophy in Turin, and in
Mathematics and Theology in Pavia and most of his books concerned
theology. Before he died, he published
a book entitle Euclides ab omni naevo
vindicatus (Euclid Freed of Every Flaw).[1] His book went unnoticed for over a century
and half until the Italian mathematician Eugenio Beltrami discovered it. Saccheri wished to prove Euclid’s fifth
postulate using a reductio ad absurdum
argument, that is, by using other axioms to prove the postulate. He assumed the negation of the Parallel
Postulate and tried to arrive at a contradiction.[2] Euclid's fifth postulate runs:
If
two lines are intersected by a transversal in such a way that the sum of the
degree measures of the two interior angles on one side of the transversal is
less than 180°, then the two lines meet on that side of
the transversal.[3]
In
order to prove this, Saccheri studied a family of quadrilaterals that was
studied many centuries earlier by the poet Omar Khayyam (1048-1131), who
started with two parallel lines A B and D C, formed the sides by
drawing lines A D and B C perpendicular to A B, and then Khayyam
considered three hypotheses for the internal angles at C and D: to be either
right, obtuse, or acute (pictured to the right).[4] These quadrilaterals have come to be known
as Saccheri quadrilaterals.
Euclid’s
postulate number V, known as the Parallel Postulate, was flawed by the fact
that was incomplete. At present, in
order for a postulate to be considered valid, it needs to follow three major
requirements.
CONSISTENCY so that it is impossible to derive two contradictory theorems from postulates,
INDEPENDENCE
so that no postulate can be derived from the others, and
COMPLETENESS
so that everything that will be used to derive the theory is stated in the
premises, leaving no tacit assumptions.[5]
For
twenty-one centuries, geometers tried to prove the postulate and they even
expressed it in many different ways in an effort to make it less intimidating. The three cases that Khayyam and then later
Saccheri considered for their hypothesis are shown in the following figure:
(from left to right) “Through a point in a plane one unique line can be drawn
parallel to a given line”, “If a
straight line falling on two straight lines makes the interior angles on the
same side together less than two right angles, the two straight lines, if
produced indefinitely, meet on that side which the angles are together less
than two right angles” (this is Euclid’s version), “The angles of a triangle
sum to 180°.”
It has been found that Euclid was correct in not providing a proof for
postulate V because there is none: it is independent of the other four
postulates as well as his first twenty-eight theorems.[6] Eugenio Beltrami discovered this in 1868 by
exhibiting a model of non-Euclidean geometry that existed in Euclidean
Geometry.
Saccheri
was the first geometer to impose laws of logic in his attempt to eliminate the
flaw in Euclid’s proof.[7] He formulated the problem in terms of three
hypotheses, only one of which can be correct.
Case
1: The summit angles are right angles.
Case
2: The summit angles are obtuse.
Case
3: The summit angles are acute.
D Summit C
A Base B
Figure: Saccheri Quadrilateral
Saccheri
wanted to prove the first case, so in order to prove Case 1 Saccheri set out to
contradict the other two cases. He
succeeded in showing that Case 2 leads to a contradiction: if the summit angles
were obtuse, the angle sum of the quadrilateral would be more than 360°, contradicting the corollary 2 to the
Saccheri-Legendre theorem (See Appendix A).[8]
As Saccheri called it “the inimical acute angle hypothesis,” he found it
impossible to find a contradiction in order to solidify Euclid V. Saccheri penetrates far deeper into
Lobacevskian geometry than his predecessors to contradict case 3. Nikolai Ivanavic Lobacevskii was the first
to publish a paper, “On the Principles of Geometry,” about the discovery of
non-Euclidean geometry, which he called imaginary geometry. He introduced the basic concepts of geometry
that do not depend on the parallel postulate.[9]
[Saccheri]
shows that under the acute-angle hypothesis two straight lines can intersect,
or have a common perpendicular on each side of which they diverge, or diverge
in one direction and come asymptotically close to one another in the other
direction. In the latter case, Saccheri
concludes that these straight lines must have a common point and a common
perpendicular at infinity.[10]
Though
he was able to produce many results, he was not pleased. He exclaimed in frustration, “The hypothesis
of the acute angle is absolutely false, because [it is] repugnant to the nature
of the straight line!” Saccheri
declared his dissatisfaction with a concluding remark:
It is well to consider here a notable difference between the foregoing refutations of the two hypotheses. For in regard to the hypothesis of obtuse angle the thing is clearer than midday light…
But
on the contrary I do not attain to proving the falsity of the other hypothesis,
that of acute angle, without previously proving that the line, all of whose
points are equidistant from an assumed straight line lying in the same plane
with it is equal to this straight line.[11]
Although
he had not realized it, Saccheri had discovered non-Euclidean geometry. Whether the right, obtuse, or acute
hypotheses are true, the sum of the angles of a triangle respectively equals,
exceeds, or falls short of 180°. Some of the
theorems that followed from Saccheri quadrilaterals are listed below:
1. The summit angles of a Saccheri
quadrilateral are equal and acute. (Proof in Appendix B)
2. The line joining the midpoint of the base
and summit – called the altitude – of a Saccheri Quadrilateral is
perpendicular. (Proof in Appendix B)
3. Two Saccheri quadrilaterals are congruent
if they have equal summits and equal summit angles.
4. The sum of the angles of any triangle is
less than two right angles.
5. If the angles of a triangle are equal to
the angles of another then the two triangles are congruent. [12]
6. In a Saccheri quadrilateral the summit is
greater than the base and the sides are greater than the altitude. [13]
Instead of leading to a proof for
postulate V, Girolamo Saccheri uncovered the consequences of the acute angle
hypothesis. Though Saccheri carefully
plotted a course through his proofs, his 33rd theorem contains a
flaw. He breaks away from his rigorous
logic and remarks, “…but this is contrary to our intuitive knowledge of a
straight line.” Saccheri ends his book
by admitting that he has not completely proven the “acute case” and for this
reason is said to have withheld publication of the book during his lifetime.[14]
I do not attain to proving the falsity of the other [acute angle] hypothesis without previously proving that the line, all of whose points are equidistant from an assumed straight line lying in the same plane with it, is equal to this straight line, which itself finally I do not appear to demonstrate from the viscera of the very hypothesis, as must be done for a perfect refutation…. But this is now enough.
Alberto Dou, S.J., who has studied
Saccheri’s Euclides for years, demonstrates that Riemann, Lobachevsky, Bolyai
and Gauss not only had direct or indirect access to Saccheri’s Euclides but
they also used his method and theorems.
Duo also notes that among Saccheri’s flaws, he includes the assumptions
that all lines are infinite in length, that in every case the exterior angle is
greater than an interior angle and that a point at infinity possess the same
properties as an ordinary point.[15]
Today, the non-Euclidean geometries
are referred to as elliptic geometry and hyperbolic geometry. Elliptic geometry involves the new
topological notion of “nonorientability,” since all the points of the elliptic
plane not on a given line, lie on the same side of that line.[16] The difference between hyperbolic and
Euclidean geometry lies in Hilbert’s parallel postulate, which is equivalent to
Euclid’s parallel postulate. Euclidean
geometry includes all the axioms on neutral geometry and Hilbert’s parallel
postulate whereas hyperbolic geometry assumes all the axioms of neutral
geometry and the negation of Hilbert’s parallel postulate, which is sometimes
called the “hyperbolic axiom.” Saccheri quadrilaterals are used in many of the
proofs for the theorems in hyperbolic geometry.
In spite of his abrupt conclusion,
Saccheri’s investigation of Euclid V and quadrilaterals was a crucial step
towards the evolution of non-Euclidean geometries. He broke ground for future geometers, which can now be realized
as his major achievement. But many past
historians missed Saccheri’s contribution because earlier writers viewed the
scholarly works of the Jesuit order with contempt. Unfortunately, Saccheri failed himself in not being able to
realize the true significance of his discovery and moving beyond the belief
that Euclid’s was the only true geometry.
Appendix A
Saccheri-Legendre
Theorem is as follows: The sum of the
degree measures of the three angles in any triangle is less than or equal to 180°.[17]
Corollary 2 to
the Saccheri-Legendre theorem: The sum of the degree measures of the angles in
any convex quadrilateral is at most
360°.[18]
Quadrilateral ABCD is called convex if it has a pair of opposite sides, e.g., AB and CD, such
that CD is contained in one of the half-planes bounded by line AB and segment
AB is contained in one of the half-planes bound by line CD.[19]
Appendix B
Theorem: In a
Saccheri Quadrilateral
1) The summit angles are congruent, and
2) The line joining the midpoints of the
base and the summit is perpendicular to both.
D N C
A M B
Proof: Let M be the midpoint of AB and let N be the midpoint of CD
- We are given that (ÐDAB)° = (ÐABC)° = 90°.
Now,
AD@BC and AB@AB, so that by SAS DDAB@DCBA, which
implies that BD@AC. Also, since CD@CD then we may apply the SSS criterion to see that DCDB@DDCA. Thus, ÐD@ÐC.
- We need to show that line MN is
perpendicular to both line AB and line CD.
Now
DN@CN, AD@BC, and ÐD@ÐC, thus by SAS DADN@DBCN. This means that
AN@BN.
Also, AM@BM and MN@MN.
By SSS DANM@DBNM and it follows that ÐAMN@ÐBMN. They are
supplementary angles; hence they must be right angles. Therefore line MN is perpendicular to line
AB.
Using the
analogous proof and triangles DDMN and DCMN, we can
show that line MN is perpendicular to line CD.[20]
Works Cited
Encyclopedia Britannica. “Non-Euclidean geometries.” 2002
Greenberg, Marvin Jay. Euclidean and Non-Euclidean Geometries:
Development and History. W. H.
Freeman and Company: New York, 2001
MacDonnell, Joseph, S.J. Girolamo
Saccheri, S.J. (1667-1733) and His Solution to Euclid’s Blemish. “Saccheri’s flaw while eliminating Euclid’s
flaw: the Evolution of Non-Euclidean Geometry.” www.faculty.fairfield.edu/jmac/sj/sacflaw/sacflaw.htm
---. Theorems of Girolamo Saccheri,
S.J. (1667-1733) and His Hyperbolic Geometry. “A Sample of Saccheri’s contribution to the evolution of
Non-Euclidean Geometry.” www.faculty.fairfield.edu/jmac/sj/sacflaw/sacther.htm
Rosenfeld, B.A. History of Non
Euclidean Geometry: Evolution of the Concept of a Geometric Space.
“Saccheri’s Theory of Parallel Lines.” Springer-Verlag Inc: New York,
1988. 98-99.
“Hyperbolic Geometry: Hyperbolic Parallel
Postulate.” http://s13a.math.aca.mmu.ac.uk/Geometry/M23Geom/NonEuclideanGeometry/NonEuclidean.htm
“Saccheri.” www.math.uncc.edu/~droyster/math3181/notes/hypergeom/node41.html