**Problem 2**
In Euclid's Elements, a line is defined as having
breadthless length while a straight line
is defined as a line which lies evenly with the points
on itself. A straight line on the
surface of a sphere must curve from an extrinsic or external viewpoint, but
intrisically, say for example if we are living in Kansas, we can
define what it means to feel straight. What is straight on a sphere?
Is the equator an intrinsically straight line? Is the non-equator latitude between
Tallahassee, Florida and Multan, Pakistan a straight line?

**Problem 3**
In Book 1 of Euclid's Elements, postulate 1 says that
a straight line can
be drawn from any point to any other point. Is this true on the sphere?
Although it doesn't explicitly say so,
since Euclid uses postulate 1 to say that there is a unique line between
any two points, he really ought to have stated the uniqueness explicitly.
Is it true that a unique intrinsically straight line can be drawn between any
two points on the surface of a sphere?

**Problem 4**
In Book 1 of Euclid's Elements, postulate 2 says that we can produce
a finite straight line continuously in a straight line.
(see also Problems 1 and 2). In modern language, we say that every
straight line can be continued indefinitely.
Is postulate 2 true on the surface of a sphere?

**Problem 5**
In Book 1 of Euclid's Elements, postulate 3 says that
we can describe a circle with any center and radius. Is postulate 3
true on the surface of a sphere?

**Problem 6**
In Book 1 of Euclid's Elements, postulate 4 says that all right angles
equal one another. Is postulate 4 true on the surface of a sphere?

**Problem 7**
In Book 1 of Euclid's Elements, postulate 5 says that
if a straight line falling on two straight lines makes the
interior angles 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. Is postulate 5 true on the
surface of a sphere?

**Problem 8**
In Book 1 of Euclid's Elements, proposition 31
says that given a straight line and a point off
of that line, we can construct a straight line that is parallel
to the given line and goes through the point.
A corollary to proposition 31, also known as
Playfair's axiom, says that only one such line can be drawn parallel to the
given line.
Is proposition 31 true on the surface of a sphere? Is Playfair's axiom
true on the surface of a sphere?

**Problem 9**
In Book 1 of Euclid's Elements, proposition 4 (SAS or side-angle-side)
says that
if two triangles have two sides equal to two sides respectively, and have
the angles contained by the equal straight lines
equal, then they also have the base equal to the base, the triangle
equals the triangle, and the remaining angles equal the
remaining angles respectively, namely those opposite the equal sides.
Is SAS always true for spherical triangles (a curved triangle formed by
intrinsically straight lines on the surface of the sphere)? Why?

**Problem 10**
In Book 1 of Euclid's Elements, proposition 8 (SSS or side-angle-side)
says that
if two triangles have two sides equal to two sides respectively, and also have
the base equal to the base, then they also have
the angles equal which are contained by the equal straight lines.
Is SSS always true for spherical triangles (a curved triangle formed by
intrinsically straight lines on the surface of the sphere)? Why?

**Problem 11**
In Book 1 of Euclid's Elements, the second part of proposition 32 says
that the sum of
the three interior angles of the triangle equals two right angles.
On the surface of a perfectly round beach ball,
can the sum of the angles of a
spherical triangle (a curved triangle formed by intrinsically straight lines
on the surface of the sphere)
ever be greater than 180 degrees? Why?

**Problem 12**
In Book 1 of Euclid's Elements,
proposition 47 says that
in right-angled triangles the square on the side
opposite the right angle equals the sum of the squares on the
sides containing the right angle.
Assume we have a right-angled
spherical triangular plot of land (see Problem 9)
on the surface of a spherical globe between approximately
Umanak, Greenland, Goiania, Brazil, and Harare, Zimbabwe, that
measures 300 and 400 on its short sides.
How long is the long side from Greenland to Zimbabwe? Why?

**Problem 13**
On the surface of a perfectly round beach ball representing the
earth, if we head 30 miles West, then 30 miles North, then 30
miles East, and then 30 miles South would we end up back where we started?
Why? What
about 300 miles in each direction? What about 3000 miles in each direction?
Why?

**Problem 14**
Is the surface of a sphere 2-dimensional or 3-dimensional? Why?

**Problem 15** If we slice one-half of a
perfectly round loaf of bread
into equal width slices, where width is defined as usual using a
straight edge or ruler, which piece has the most crust? Why?

Using web and book references along with experimentation, research the answers to your questions. Prepare to present your research to the rest of the class, as part of the speaking designator and as a project grade.