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MODULE 6
Non-Euclidean Geometry
Revolutions are not made; they come. – Wendell Phillips
I have created a new universe out of nothing. – J´onos Bolyai
Recall our deﬁnition of parallel lines: two lines ℓ and m are parallel if they do not intersect,
that is, no point P lies on both ℓ and m.
Threedistinct branchesofgeometryarisefromthewayweanswerthefollowingquestion:
Given a line ℓ and a point P not on ℓ, how many lines through P are
parallel to ℓ?
There are three possible answers:
Choice 1. There is one and only one parallel through P. This statement is the parallel
postulate (Theorem ??) and is easily shown to be equivalent to Euclid’s 5th Axiom. The
resulting geometry is the standard Euclidean geometry, studied by school children and
mathematicians for the past two thousand years, and the main focus of this text.
Choice 2. No parallel lines exist. This branch leads to spherical geometry. Our model
of spherical geometry will be the surface of the earth, discussed in the next two sections.
Choice 3. There is more than one parallel through P. This branch leads to hyperbolic
geometry. Our model of hyperbolic geometry was ﬁrst proposed by the mathematician
Poincar´e and was later made famous by the artist Maurice Escher.
For thousands of years, mathematicians tried to prove the parallel postulate from the
other axioms of Euclid. Their minds were already made up that the only possible kind of
geometry is the Euclidean variety—the intellectual equivalent of believing that the earth is
ﬂat. In truth, the two types of non-Euclidean geometries, spherical and hyperbolic, are just
as consistent as their Euclidean counterpart. The theorems in these branches look strange
147
148 6. NON-EUCLIDEAN GEOMETRY
to us because we are so used to the theorems of the geometry we were taught since grade
school. When you’re traveling in a foreign country, their currency never looks as real as your
own.
1. Planet Earth and the Longitude Problem
We start our study of spherical geometry by looking at the surface of the earth and
considering the practical problem of navigation, especially from the point of view of a
seafaring nation like Great Britain in the 1700’s. In the next section we examine the basic
geometric concepts: distance, lines, segments, rays, angles, triangles, angle sum, rectangles,
similarity, circles, and area in this new and strange setting of spherical geometry. The last
section of this module takes a close look at the equally strange hyperbolic world.
We consider the earth to be a sphere. It’s actually slightly ﬂatter at the North and
South Poles. The equatorial radius of the earth is 6378.137 km or 3963.19 miles, while the
polar radius is 6357 km or 3950 miles. Many books use 4000 miles as a rough approximation
for the radius of the earth.
Lines on a sphere are great circles, that is, circles whose center is the center of the
sphere. Longitudes, extending from the North to the South Pole, are great circles. East–
west latitudes are not—except for the equator. Great circles can “tilt” at any angle. The
great circle connecting Los Angeles and Chicago, for example, is certainly not a longitude.
Nowweseewhytherearenoparallel lines on a sphere. Suppose one line is the equator.
Then no matter what great circle you use for the second line, it must cross the equator at
two points. No great circle can be drawn entirely in the northern or southern hemisphere.
Connection to Navigation
Aship’s position at sea is speciﬁed by giving its longitude and latitude. Latitude tells
how far north (or south) you are and longitude gives your east–west position. Latitude can
be determined by the length of day or by the height above the horizon of the sun in daytime
or known guide stars at night. Unfortunately, there is no such natural way to determine
longitude!
The Longitude Prize of 1714
In an attempt to spur scientists, artisans, and great thinkers to ﬁnd a solution to the
longitude problem, the Parliament of England established the Longitude Prize of 1714.
1. PLANET EARTH AND THE LONGITUDE PROBLEM 149
Parliament promised a prize of 20,000 pounds—an enormous sum for the time—to anyone
who could accurately determine longitude at sea. They required a test voyage to the West
Indies—a 40 day sailing, and the solution needed to be accurate to within 1 a degree.
2
Using time to determine longitude.
Dividing the 360 degrees of the earth’s rotation into 24 time zones means that the earth
rotates 15 degrees every hour. Thus, the change in longitude of a seagoing vessel is directly
related to the diﬀerence of the time at its home port and the local time at its current location
at sea. To convert this time diﬀerence into geographical signiﬁcance, use the formula:
change in longitude = time diﬀerence×15◦.
If the ship is traveling along a ﬁxed latitude, then the distance traveled is
distance = time diﬀerence ×circumference of latitude circle.
24
Hands-on Problem. You are on the equator (r = 3963 miles).
(a) How many miles comprise 90◦ of longitude?
(b) How many miles comprise a single time zone?
(c) How many miles have you traveled and how many degrees of longitude are you from
the longitude of your home port if the times are
home time local time change in longitude miles traveled
8:00 am 9:00 am
2:45 am 4:30 am
5:40 am 1:20 pm
9:30 pm 1:55 am
From this discussion we see that to determine longitude, all you need to do is compare
the local time at your current location with the time at your home port. Computing local
time was well known when the Longitude Prize was oﬀered. Local noon occurs when the sun
is directly overhead and could be ascertained by the navigational tools of 1700. Computing
port time was the problem!
150 6. NON-EUCLIDEAN GEOMETRY
You’re probably wondering, “Why couldn’t the ship’s navigator just take a clock on
board at the start of the voyage, set to the port time?” In 1700 most accurate clocks were
pendulumclocks. Thoughdependableonland, theyrequiredfavorableweatheratseadueto
the swaying of ship with the waves—especially in storms. Early clocks using a spiral balance
spring were not suﬃciently accurate, for many reasons. Spring powered mechanisms varied
according to temperature and humidity. Metal parts expanded with heat or contracted with
cold while the viscosity of the lubrication changed with temperature. In 1700 the best spring
mechanized time pieces lost up to a minute a day.
Hands-on ProblemJust how accurate did a timepiece need to be? Suppose a clock loses
3 seconds per day. On a 40 day vogage along the equator, how much error in miles does
this amount to?
The Astronomy Solution.
The list of leading astronomers and scientists who contributed to the longitude problem
reads like a Who’s Who in Science of the era. Galileo advocated using the eclipses of the
moons of Jupiter, which occur thousands of times in a year. This idea requires accurate
calculations in order to predict when the moons will disappear and reappear relative to a
ﬁxed viewing point on earth. He developed a special headgear with a telescope attached to
oneeyepieceandanemptyeyeholeintheother,tolocatethesteadylightfromJupiter. Later
the astronomer Ole Roemer discovered that the eclipses of the moons of Jupiter occurred
ahead of schedule when Jupiter was closest to the earth and behind schedule when it was
farthest away. This discrepancy was resolved when it was discovered that it takes light a
diﬀerent time to to travel from Jupiter to Earth, depending on the position of the planets
in their respective orbits. After a while astronomers abandoned the use of the moons of
Jupiter as a navigation tool and began to concentrate on our own moon. Sir Issac Newton
solved problems about the moon’s orbit with his theory of gravitation, paving the way for
Nevil Maskelyne, 4th Astronomer Royal of England, who advanced the lunar method for
navigation and his own extensive almanac of the exact times of the rise and fall of the moon.
Maskelyne was the nemesis of the hero of our story, the self taught genius destined to settle
the Longitude problem: John Harrison.
John Harrison and his chronometers.
When he was twenty years old, John Harrison built a wooden pendulum clock, using
woodworking techniques he learned from his father. Nine years later he built a tower clock
in Brocklesbe Park which has been running continuously for 280 years, except for brief

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