TOPIC 12
Euclid’s parallel postulate
and Spherical Triangles
In any axiomatic system, postulates serve as the basis for all facts that can be derived
from original facts. Postulates are statements that are accepted without proof, but that does not
mean that they are less true than the truths which can be proved. However, it is important that
the truth of a postulate is clear and undisputable.
Euclid was a scholar who accumulated the known knowledge about geometry about 300
B.C. He arranged the notions on geometry into an axiomatic system, beginning with postulates
and working through theorems that had been proved up to his time.
One of his postulates (Postulate V) has caused a great deal of debate and disagreement
over the centuries. There were scholars who thought that the Fifth Postulate could be derived
from the others, and there were many attempts to provide a proof. That was not very successful,
but the doubt about the status of the statement as a postulate continued. A breakthrough came in
the 19th century, as a culmination in the argument occurred when some mathematicians decided
to state alternates to the postulate.
So what is the postulate? How was it phrased in 300 B.C.? Why was there so much
intellectual outcry about it?
From Wikipedia, http://en.wikipedia.org/wiki/Parallel_postulate
In geometry, the parallel postulate, also called Euclid's fifth postulate because it is the fifth
postulate in Euclid's Elements, is a distinctive axiom in Euclidean geometry. It states that:
If a line segment intersects two straight lines forming two interior angles on the same side that
sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on
which the angles sum to less than two right angles.
Euclidean geometry is the study of geometry that satisfies all of Euclid's axioms, including the
parallel postulate. A geometry where the parallel postulate cannot hold is known as a nonEuclidean geometry. Geometry that is independent of Euclid's fifth postulate (i.e., only assumes
the first four postulates) is known as absolute geometry (or, in other places known as neutral
geometry).
Topic 12: Euclid’s Parallel Postulate
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We might re-state this is our more modern terms as
If two lines are cut by a transversal so that the sum of the measures of the same-side
interior angles is less than 1800, then the lines intersect on that side of the transversal.
The contrapositive of this statement is:
If lines do not intersect on that side of the transversal, then the two lines which are cut
by a transversal produce same-side interior angles whose sum is not less than 1800.
If this condition holds on both sides of the two lines, then we would say that “if two lines
do not intersect on either side of the transversal (i.e., they are parallel), then the sum of the
measures of the same side interior angles are not less than 1800 nor is the sum more than 1800
(i.e., the side-side interior angles are supplementary).”
This is not the same postulate which we usually state that begins a study of parallel lines
and the associated angles, since we often begin with “If lines are parallel and cut by a
transversal, then the corresponding angles are congruent.” From this postulate, we can prove the
other angle relationships, including the one that Euclid stated as his Fifth Postulate. That part
does not matter: any of the statements can first be postulated, and the rest can be proved from it.
That being said, what is the controversy?
Again, from Wikipedia, http://en.wikipedia.org/wiki/Parallel_postulate
Euclid's parallel postulate is equivalent to Playfair's axiom, named after the Scottish
mathematician John Playfair, which states:
At most one line can be drawn through any point not on a given line parallel to the given line in
a plane.[1]
Many other equivalent statements to the parallel postulate or to Playfair's axiom have been
suggested, some of them appearing at first to be unrelated to parallelism, and some seeming so
self-evident that they were unconsciously assumed by people who claimed to have proven the
parallel postulate from Euclid's other postulates.
1.
2.
3.
4.
5.
6.
The sum of the angles in every triangle is 180°.
There exists a triangle whose angles add up to 180°.
The sum of the angles is the same for every triangle.
There exists a pair of similar, but not congruent, triangles.
Every triangle can be circumscribed.
If three angles of a quadrilateral are right angles, then the fourth angle is also a right
angle.
7. There exists a quadrilateral of which all angles are right angles.
8. There exists a pair of straight lines that are at constant distance from each other.
9. Two lines that are parallel to the same line are also parallel to each other.
Topic 12: Euclid’s Parallel Postulate
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10. Given two parallel lines, any line that intersects one of them also intersects the other.
11. In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of
the other two sides (Pythagoras' Theorem).
12. There is no upper limit to the area of a triangle. [1]
13. The summit angles of the Saccheri quadrilateral are 90°.
However, the alternatives which employ the word "parallel" cease appearing so simple when one
is obliged to explain which of the three common definitions of "parallel" is meant - constant
separation, never meeting, or same angles where crossed by a third line - since the equivalence of
these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth
postulate. If the word "parallel" is defined as constant separation, the Euclid's fifth postulate can
be proved from his first four postulates. However, if the definition is taken so that parallel lines
are lines that do not intersect, Euclid's fifth postulate is independent to his first four postulates.
Proclus' axiom, which states "if a line intersects one of two parallel lines, both of which are
coplanar with the original line, then it must intersect the other also", is also equivalent to the
parallel postulate.[2]
Let’s look at one consequence of the parallel postulate, especially using Playfair’s
statement.
1.
Draw any triangle.
2.
Choose one vertex of the triangle, and draw a line parallel to the side opposite that
vertex.
It can be shown that m < A = m < 1 (using a theorem derived from the parallel postulate), m< 2
= m < 2 , and m< B = m < 3. Then m < 1 + m < 2 + m < 3 = 1800, so
m < A + m < 2 + m < B = 180, by substitution. So the sum of the three interior angles of a
triangle is 1800. This result depends directly on the parallel postulate and theorems which it
produces and supports.
Euclid’s work supports our modern teaching of geometry so much so that what we study
is often called Euclidean plane geometry. “Geometry” means that we are “measuring the earth”.
But is the earth a plane? This is actually the essence of the controversy.
Topic 12: Euclid’s Parallel Postulate
Back to Wikipedia….
page 4
(http://en.wikipedia.org/wiki/Parallel_postulate)
For two thousand years, many attempts were made to prove the parallel postulate using Euclid's
first four postulates. The main reason that such a proof was so highly sought after was that,
unlike the first four postulates, the parallel postulate isn't self-evident. If the order the postulates
were listed in the Elements is significant, it indicates that Euclid included this postulate only
when he realised he could not prove it or proceed without it.[3]
Ibn al-Haytham (Alhazen) (965-1039), an Iraqi mathematician, made the first attempt at proving
the parallel postulate using a proof by contradiction,[4] where he introduced the concept of
motion and transformation into geometry.[5] He formulated the Lambert quadrilateral, which
Boris Abramovich Rozenfeld names the "Ibn al-Haytham–Lambert quadrilateral",[6] and his
attempted proof also shows similarities to Playfair's axiom.[7]
Omar Khayyám (1050-1123), a Persian, made the first attempt at formulating a non-Euclidean
postulate as an alternative to the parallel postulate,[8] and he was the first to consider the cases of
elliptical geometry and hyperbolic geometry, though he excluded the latter.[9] The KhayyamSaccheri quadrilateral was also first considered by Omar Khayyam in the late 11th century in
Book I of Explanations of the Difficulties in the Postulates of Euclid.[6] Unlike many
commentators on Euclid before and after him (including Giovanni Girolamo Saccheri), Khayyam
was not trying to prove the parallel postulate as such but to derive it from an equivalent
postulate: "Two convergent straight lines intersect and it is impossible for two convergent
straight lines to diverge in the direction in which they converge."[10] He recognized that three
possibilities arose from omitting Euclid's Fifth; if two perpendiculars to one line cross another
line, judicious choice of the last can make the internal angles where it meets the two
perpendiculars equal (it is then parallel to the first line). If those equal internal angles are right
angles, we get Euclid's Fifth; otherwise, they must be either acute or obtuse. He persuaded
himself that the acute and obtuse cases lead to contradiction, but had made a tacit assumption
equivalent to the fifth to get there.
Nasir al-Din al-Tusi (1201-1274), in his Al-risala al-shafiya'an al-shakk fi'l-khutut almutawaziya (Discussion Which Removes Doubt about Parallel Lines) (1250), wrote detailed
critiques of the parallel postulate and on Khayyám's attempted proof a century earlier. Nasir alDin attempted to derive a proof by contradiction of the parallel postulate.[11] He was also one of
the first to consider the cases of elliptical geometry and hyperbolic geometry, though he ruled out
both of them.[9]
Topic 12: Euclid’s Parallel Postulate
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Euclidean, elliptical and hyperbolic geometry. The Parallel Postulate is satisfied only for models
of Euclidean geometry.
Nasir al-Din's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), wrote a book on the
subject in 1298, based on his father's later thoughts, which presented one of the earliest
arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. "He essentially
revised both the Euclidean system of axioms and postulates and the proofs of many propositions
from the Elements."[11][12] His work was published in Rome in 1594 and was studied by
European geometers. This work marked the starting point for Saccheri's work on the subject.[11]
Giordano Vitale (1633-1711), in his book Euclide restituo (1680, 1686), used the KhayyamSaccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit
CD, then AB and CD are everywhere equidistant. Girolamo Saccheri (1667-1733) pursued the
same line of reasoning more thoroughly, correctly obtaining absurdity from the obtuse case
(proceeding, like Euclid, from the implicit assumption that lines can be extended indefinitely and
have infinite length), but failing to debunk the acute case (although he managed to wrongly
persuade himself that he had).
Where Khayyám and Saccheri had attempted to prove Euclid's fifth by disproving the only
possible alternatives, the nineteenth century finally saw mathematicians exploring those
alternatives and discovering the logically consistent geometries which result. In 1829, Nikolai
Ivanovich Lobachevsky published an account of acute geometry in an obscure Russian journal
(later re-published in 1840 in German). In 1831, János Bolyai included, in a book by his father,
an appendix describing acute geometry, which, doubtlessly, he had developed independently of
Lobachevsky. Carl Friedrich Gauss had actually studied the problem before that, but he did not
publish any of his results. However, upon hearing of Boylai's results in a letter from Bolyai's
father, Farkas Bolyai, he stated:
"If I commenced by saying that I am unable to praise this work, you would certainly be surprised
for a moment. But I cannot say otherwise. To praise it would be to praise myself. Indeed the
whole contents of the work, the path taken by your son, the results to which he is led, coincide
almost entirely with my meditations, which have occupied my mind partly for the last thirty or
thirty-five years."[13]
The resulting geometries were later developed by Lobachevsky, Riemann and Poincaré into
hyperbolic geometry (the acute case) and spherical geometry (the obtuse case). The
independence of the parallel postulate from Euclid's other axioms was finally demonstrated by
Eugenio Beltrami in 1868.
Let’s see if we can investigate the angle sum in a triangle to see if Euclid’s postulate
holds up, or if the alternatives have credibility. The exercises that you have done to date have
assumed that the sum of the interior angles of a triangle. If you measure the angles with a
protractor, the conclusion seems to be that the sum is 1800.
Topic 12: Euclid’s Parallel Postulate
page 6
We will use a sphere model to measure the angles in triangles that sit on the surface of a
sphere. Note that the triangles in your homework and on the blackboard have been sitting on a
plane. Is there an essential difference? (Be open minded here….)
WORK WITH LENART SPHERE
1.
Look at the spherical rule on the Lénart sphere. It is marked in degrees (and actually in
tenths of a degree) along a great circle.
Definitions:
A circle is a set of planar points which are equidistant from one point, called the center.
A sphere is a noncoplanar set of points which are equidistant from one point called the
center.
A great circle of a sphere is a circle on the surface of a sphere whose center is the center
of the sphere. (example: The equator of a circle is a great circle, but it is not the only
location of a great circle.)
2.
Put a hemispherical transparency over the top of the Lénart sphere.
a) Put the small, round center locator at the point at the top of the sphere (North Pole)
b) Set the compass to 300 and put a transparency pen in the pen holder. Draw a circle.
c) Move the compass to 500 and draw another circle.
These are latitude lines. Measure from the equator up to the latitude circle to get the
latitutde measurement (Note: Latitude is not measured south from the North Pole, but
north from the equator. Latitude is also measures south from the equator toward the
South Pole as well.)
Question: Are latitude lines parallel? _______
Are latitude lines great circles? _________
d) Use the spherical ruler. Place it so that it passes through the North Pole (NP) and
intersects the equator. Draw an arc (defn: An arc is a part of a circle.) from the North
Pole to the equator. (Stay on the transparency, please.) Call the point on the equator A.
Move the rule and draw another arc from the North Pole to a second point on the equator.
Call this point on the equator B.
NP to A and NP to B are longitude lines.
Topic 12: Euclid’s Parallel Postulate
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Question: Are longitude lines parallel? _______
Are longitude lines great circles? _________
Imagine one complete longitude line, drawn from the North Pole to the equator and
continuing to the South Pole. This will be a great circle.
One of these longitude lines will be called the prime meridian, passing through
Greenwich, England. The other half of that longitude line is the International Date
Line, which passes through the central Pacific Ocean.
How many degrees are there, along the equator, from the prime meridian to the
International Date Line? _________
Cut this amount into 12 equal parts. How many degrees is each part? ____ Each
part is an hour of time, called a time zone. Draw the longitudes for one hour from
the NP to the equator. What happens to the time zones as you move to the north
(or to the south from the equator to the South Pole ( SP)?
__________________________
3.
Go back to the points NP, A, and B.
Measure the angle in the spherical triangle at the NP. (It equals the arc on the equator
between A and B which is opposite the angle at the NP)
m< NP = _______
m < A = _______ m < B = ________
m < NP + m < A = m < B = ________________
The triangle is called a spherical triangle. Describe the sum of the three interior angles
in a spherical triangle from the NP to two points on the equator.
4.
Topic 12: Euclid’s Parallel Postulate
page 8
Describe the arcs AP, BP, CP, etc., where P is the NP.
Are these arcs parallel? ____________
Are the same-side interior angles supplementary? ________
Among the longitude lines, how many are parallel? ________
So, using longitude PA (a geodesic), how many geodesics are there through B which are
parallel to geodesic PA? ___________
This is a different parallel postulate. This is an alternative to Postulate V, and it is not the
only alternative!
What happens to one of the consequences of the parallel postulate?
The sum of the interior angles of a spherical triangle is __________ .
Where is Euclid’s Fifth Postulate “violated” on the surface of the earth? Consider the
triangle formed by the North Pole and two points on the equator. The “lines” (which are called
geodesics, which are defined “the shortest path on the surface between two points”) connecting
each point on the equator with the North Pole are longitude lines. They are perpendicular to the
equator, which is the 00 latitude line. So the angles formed at the equator with the longitude lines
are each 900, and there is still the angle formed at the North Pole. It seems that the same-side
interior angles are supplementary, and yet the lines (“geodesics”) are not parallel, since they
meet at the North Pole. This is a geometry where there are no parallels to a line from a point
outside the line, to use Playfair’s language.
We will now use a calculator program which uses some formulas from spherical
trigonometry to calculate the angles in a spherical triangle. (Note: We have not yet studied
plane trigonometry with these kinds of formulas, so the formulas will be used without
explanation. After we have dealt with plane triangles and trigonometry in a few weeks, we may
return to see the spherical analogy to calculations.)
To locate each of the points on the surface of the sphere (the earth), we will use latitude
and longitude. Latitude is measured in degrees, with a maximum of 900 to the north and 900 to
the south. Latitude is a north-south measurement, but latitude lines are horizontal on the globe.
Longitude is measured in degrees, also, with a maximum of 1800 to the east and 1800 to the
west.
As an aside, longitude has a very interesting history, as there were not easy ways to
measure longitude for ocean navigation, and it was important ins seafaring days and for time-
Topic 12: Euclid’s Parallel Postulate
page 9
keeping to know where you were in an east-west direction. Another fascinating aspect of
longitude for our purposes is the fact that the North Pole is located at all longitudes
simultaneously. What direction must you move, if you are standing on the North Pole and take
one step, regardless of which “direction” you move?
So, using the calculator program SPHANGLE (for “spherical angle”), you will enter the
latitude and longitude of three points on the earth, and the program will give you the measures of
the three interior angles of the spherical triangle. We are investigating the effect that one form of
the denial of the parallel postulate takes, namely on the theorem which follows pertaining to the
sum of the angles of a triangle.
A suggested website to find latitudes and longitudes is
www.itouchmap.com/latlong.html
Record the measures of the three angles, and the sum of the measures of the three angles.
latitude
1.
longitude
North Pole
One point on the equator
A second point on the equator
(Don’t look up latitudes and longitudes for this one)
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
2.
longitude
North Pole
Baltimore
Seattle
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
longitude
Topic 12: Euclid’s Parallel Postulate
3.
page 10
North Pole
Pt. Barrow, Alaska
Town in northern Finland
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
4.
longitude
North Pole
Reykjavik, Iceland
Vik, Iceland
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
5.
longitude
Baltimore
Houston, TX
Vancouver, BC
m < 1 = ________
m <2 = _________
m< 1 + m < 2 + m < 3 = _______________
m < 3 = ________
Topic 12: Euclid’s Parallel Postulate
page 11
latitude
6.
longitude
London, UK
Rio de Janeiro
Nome, Alaska
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
7.
longitude
Tokyo
Buenos Aires
Moscow
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
8.
longitude
New York City
Chicago
Atlanta
m < 1 = ________
m <2 = _________
m< 1 + m < 2 + m < 3 = ________________
m < 3 = ________
Topic 12: Euclid’s Parallel Postulate
page 12
latitude
9.
longitude
New York City
Philadelphia
State College, PA
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
10.
longitude
Bryn Mawr
Gilman
RPCS
m < 1 = ________
m <2 = _________
m < 3 = ________
m< 1 + m < 2 + m < 3 = ________________
latitude
11.
longitude
Hardy 27
Hardy 27
Hardy 27
(Use three locations within Hardy 27. Estimate the latitudes and longitudes.)
m < 1 = ________
m <2 = _________
m< 1 + m < 2 + m < 3 = ________________
m < 3 = ________