Let's begin by reviewing what

*Euclidean*geometry is. The name "Euclidean" refers, of course, to the ancient Greek mathematician Euclid. In particular, Euclidean geometry is that in which Euclid's five postulates hold. These postulates are: http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html

**Postulate 1.**

**Postulate 2.**

**Postulate 3.**

**Postulate 4.**

**Postulate 5.**

And it's that fifth postulate that is the key. We usually think of non-Euclidean geometry as that in which Euclid's fifth postulate fails. Because Euclid uses archaic language -- for example, he doesn't use degrees, instead treating the right angle as a unit of measure -- most modern textbooks use an equivalent statement that gives the properties of parallel lines. In the U of Chicago text, this is given as Playfair's Parallel Postulate, Section 13-6:

Through a point not on a line, there is exactly one [line] parallel to the given line.

And so to create non-Euclidean geometry, we replace this postulate with one of two alternatives:

-- Through a point not on a line, there is

*more than one*line parallel to the given line.

-- Through a point not on a line, there is

*no*line parallel to the given line.

(or to put it simply, there are no parallel lines)

When we make the first choice to allow many parallel lines, it is called

*hyperbolic*geometry. When we make the second choice to forbid parallel lines, it is either

*elliptic*or

*spherical*geometry.

I first read about non-Euclidean geometry years ago -- indeed, it was mentioned in the Geometry text I had when I was a student. Back then, I was under the impression that hyperbolic and spherical geometry were equally valid -- all we had to do was replace the fifth postulate with a new postulate telling how many parallel lines there are. But as it turns out, this is incorrect. It's much easier to produce hyperbolic geometry than one in which there are no parallels.

If you think about it, you can see why hyperbolic is closer to the Euclidean ideal than spherical geometry would be. In hyperbolic geometry, like Euclidean, lines are infinite in length, but on a sphere, if we go all the way around we return to the point where we started. Hyperbolic geometry satisfies the first four postulates of Euclid -- just not the fifth. On the other hand, we will see that spherical geometry doesn't satisfy all of the first four Euclidean postulates.

There's a name for geometry that satisfies the first four Euclidean postulates --

*neutral*geometry. We see that this geometry is indeed "neutral" in that it could be either Euclidean or hyperbolic. It is definitely not "neutral" regarding spherical geometry -- it explicitly rejects it.

Mathematicians, in the name of elegance, prefer to use as few postulates as possible when coming up with the proof of a statement. This tradition goes back to Euclid, whose first 28 propositions (theorems) avoid the fifth postulate. Therefore, the first 28 theorems hold in neutral geometry.

For years, I never thought about which theorems hold in neutral geometry. Believe it or not, it was actually Common Core that led me to think about neutral geometry -- in particular, the writings of Drs. Franklin Mason and Hung-Hsi Wu opened my eyes to neutral geometry.

We see that Dr. M introduces a parallel postulate (Playfair) in Section 4.3 of his text. It then follows that all theorems proved up to Section 4.2 hold in neutral geometry. In particular, the Parallel Tests of Section 4.2 are theorems of neutral geometry. These correspond to Euclid's Propositions 27 and 28 -- the last two theorems the ancient sage proves before using his fifth postulate.

One thing that I'm considering doing for my geometry course next year is making it so that the entire first quarter teaches only neutral geometry. If I were following Dr. M's curriculum, this sounds like a reasonable pace. After all, his text consists of thirteen chapters, and so we expect that covering about 1/4 of the chapters -- Chapters 1 to 3 and the first two sections of Chapter 4 -- gets us through Chapter 13 by the last day of school. But that pace wouldn't get us through Chapter 13 by the date of the PARCC test -- and the final chapter is on the coordinate plane, a major topic tested on PARCC.

In the U of Chicago text, the first postulate equivalent to Euclid's fifth is the Parallel Lines Postulate (Corresponding Angles Consequence) of Section 3-4, so the text is neutral only up to Section 3-3. But if we cut off the rest of Chapter 3, then the text is neutral up to Section 5-4, since Section 5-5 gives the Trapezoid Angle Theorem (Same-Side Interior Angles Consequence). Notice that my pacing plan for the upcoming year has us ending the first quarter with Section 5-4 -- which means that my first quarter will indeed be neutral.

But there are still many theorems of neutral geometry that appear after Section 5-4. Indeed, we see that Chapter 3 of Dr. M is all about congruence, including SSS, SAS, and ASA. All three of these hold in neutral geometry -- they are Propositions 4, 8, and 26, all before the fifth postulate appears in Proposition 29. This is another reason why I want to move the congruence theorems of U of Chicago's Chapter 7 up into the first quarter.

But even Dr. M doesn't include all of the theorems of neutral geometry before his Section 4.3. For one thing, we notice that AAS is a neutral theorem -- indeed, Euclid included AAS along with ASA in his Proposition 26. Yet Dr. M doesn't prove AAS until his Section 4.6. The problem here is that nearly every single geometry text proves AAS by combining ASA with the Triangle-Sum Theorem -- and Triangle-Sum is definitely not neutral. Yet even though Triangle-Sum is not neutral, AAS is neutral, and Euclid proves AAS without Triangle-Sum.

This happens in higher mathematics as well -- there may be two ways to prove something true, one using a certain postulate, the other without it. AAS has two proofs -- one with the parallel postulate, and one without it. The former is shorter, but only works in Euclidean geometry. The latter is longer, yet works in both Euclidean and hyperbolic geometry. This is why mathematicians often seek out longer proofs in order to avoid using certain postulates. The following link is an example from higher mathematics of a long, 35-page proof that would have taken less than one page if the authors had only used a certain postulate of set theory:

https://math.dartmouth.edu/~doyle/docs/three/three.pdf

(Notice that listed as an author is one of my favorite mathematicians, John Horton Conway. I don't want to get into the details of this proof, but let me give you the gist of it. The question is, is it possible to divide a set into three sets? It sounds obvious -- but just as I mentioned in the discussion of Cavalieri's principle, the problem is with

*infinite*, not finite, sets. The proof is trivial if we're allowed to use the Axiom of Choice -- an axiom (postulate) that has almost the same status in set theory as the parallel postulate has in geometry. Even though the proof at the above link is technical, you may want to read Section 9, which discusses an infinite hotel called Hilbert's Hotel, as well as Section 13, on why this Axiom of Choice is so controversial.)

Also among the theorems of neutral geometry are the inequalities of Dr. M's Chapter 5 (spread out among Chapters 1, 7, and 13 in the U of Chicago). Dr. M used to give a neutral proof of one of these, the Triangle Exterior Angle Inequality (TEAI). This is Euclid's Proposition 16. All of the other inequalities appear before Proposition 29 in Euclid and are also neutral -- Unequal Sides Theorem (Proposition 18), Unequal Angles Theorem (Proposition 19), Triangle Inequality (Proposition 20), and SAS Inequality/Hinge Theorem (Proposition 24). Proposition 25 is a sort of converse to the Hinge Theorem, and is included in many texts as the "SSS Inequality."

But by this point, trying to include all 28 neutral Propositions of Euclid into the first quarter of a high school class would overwhelm most students. It might be nice at least to include all of the congruence theorems, including AAS and HL, in the same chapter rather than separate them as Dr. M does (the first three in Chapter 3, AAS and HL in Chapter 4), but that would entail coming up with a suitable neutral proof of AAS. Recall that the U of Chicago uses transformations to prove ASA, so it might be possible to convert this proof into one of AAS.

And yes, I leave neutral geometry by reminding everyone how transformations work in Euclidean and hyperbolic geometry. Both types of geometry have reflections, rotations, and glide reflections. But translations work differently. In Euclidean geometry, the composite of reflections in parallel lines is a translation, but in hyperbolic geometry, there are

*two*types of parallel lines. If the two mirrors are "ultraparallel," then their composite is still a translation, but if they are "horoparallel," then their composite is a new transformation, called a "horolation." So the symmetry group of the hyperbolic plane has five types of transformations, while the symmetry group of the Euclidean plane has only four types.

If I wanted to motivate students to learn non-Euclidean geometry, I wouldn't want to start with hyperbolic geometry, with a hyperbolic plane that's difficult to describe. I'd want to begin with spherical geometry, because we live on a planet whose shape approximates a sphere -- the earth.

The problem is that the geometry of the sphere is

*not*neutral geometry -- that is, it doesn't satisfy the first four postulates of Euclid. Indeed, Euclid's first postulate is commonly stated as "two points determine a line segment" -- that is, through any two points, there is exactly one line. In the U of Chicago, this is given as part (a) of the Point-Line-Plane Postulate. We see that on the sphere, this postulate is

*false*. If we stood at the North Pole, then every direction is south -- so every direction gives a line that eventually leads to the South Pole. Therefore, there are many lines between the North and South Pole, thereby contradicting Euclid's first postulate.

So spherical geometry can't be a neutral geometry. There's actually another type of geometry, called

*elliptic*geometry, where the North and South Pole count as the same point. Depending on how we interpret the postulates, elliptic geometry now satisfies the first postulate, but now it contradicts the second postulate. I won't go into the details here, but this is related to the Plane Separation Postulate that Dr. Wu mentions on his website. At any rate, I don't want to confuse students by saying that the North Pole equals the South Pole. So I stick only to spherical geometry.

Spherical geometry isn't neutral geometry, since the first postulate fails. On the blog, I once argued that spherical geometry arguably satisfies the

*fifth*postulate. The version of Playfair mentioned by Dr. M is stated:

Through a point not on a given line, there's

*at most*one line parallel to the given line.

[emphasis mine]

The phrase "at most" is included since "at least" is already implied by the first four postulates -- so "at least" holds in neutral but not spherical geometry. The Corresponding Angle Consequence, which follows from Playfair, holds

*vacuously*in spherical geometry. (All unicorns are white.) Finally, even Euclid's original fifth postulate holds in spherical geometry -- if the same-side interior angles add up to less than 180, then

*of course*the lines will surely meet on that side because

*all*lines intersect in spherical geometry. The postulate says

*nothing*about what happens on the

*other*side of the transversal, nor does it say anything about what happens if the angles

*do*add up to 180. So nothing in Euclid's fifth postulate is refuted in spherical geometry. Spherical geometry isn't neutral -- much less Euclidean -- because of Euclid's

*first*postulate, not his

*fifth*.

And so we wish to begin our study of spherical geometry. As I mentioned before, I'm getting the information from Legendre's 19th century geometry text

*Elements of Geometry*-- the first such widely used text since Euclid himself. Here is a link to Legendre on Google books:

https://books.google.com/books?id=gs5JAAAAMAAJ&pg=PR1&source=gbs_selected_pages&cad=3#v=onepage&q&f=false

We begin on page 157 of this text. "Section Third: Of the Sphere." I've decided to cover two pages each day that I post about spherical geometry on the blog, so today is pages 157-158. Most of these propositions given on these two pages are definitions -- only the last is actually a theorem. So let us get started. The first proposition dealing with spherical geometry is numbered 437.

437. A

*sphere*is a solid terminated by a curved surface, all the points of which are equally distant from a point within called the

*centre*.

The sphere may be conceived to be generated by the revolution of a semicircle

*DAE*about its diameter

*DE*; for the surface thus described by the curve

*DAE*will have all of its points equally distant from the centre

*C*.

So Legendre begins spherical geometry by defining the word "sphere." The word "sphere" is also defined in Section 9-4 of the U of Chicago text:

Definition:

A

**sphere**is the set of points in space at a fixed distance (its radius) from a point (its center).

We notice a slight difference between Legendre's definition and the modern U of Chicago definition -- nowadays the word "sphere" refers to the surface, but back then "sphere" referred to the entire solid. I will now proceed with Legendre's definitions, except that I freely replace Legendre's Anglo-French spelling "centre" with the modern American spelling "center" without annotation.

438. The

*radius of a sphere*is a straight line drawn from the center to a point on the surface; the

*diameter*or

*axis*is a line passing through the center, and terminated each way by the surface.

All radii of the same sphere are equal; the diameters are also equal, and each double of the radius.

The words "radius" and "diameter" are defined similarly in the U of Chicago text. The text points out that "the terminology of a circle extends to spheres," and points out that these words "radius" and "diameter" refer to both the segments themselves as well as their lengths (which are numbers).

439. It will be demonstrated, art. 452, that every section of a plane made by a sphere is a circle. This being supposed, we call a

*great circle*the section made by a plane which passes through the center, and a

*small circle*the section made by a plane which does not pass through the center.

The phrases "great circle" and "small circle" are defined similarly in the U of Chicago text. Legendre writes that his Proposition 452 will be the proof that a plane section of a sphere is a circle -- thereby justifying the names "great circle" and "small circle." Notice that Proposition 452 is the only one whose proof we will get to today. In the U of Chicago text, the proof is given as Exercise 20 in this current Section 9-4.

440. A

*plane*is a

*tangent*to a sphere, when it has one point only in common with the surface of the sphere.

The U of Chicago text waits to define "tangent to a sphere" until Section 13-5. In the text, either a line or a plane may be tangent to a sphere.

441. The

*pole of a circle*of the sphere is a point in the surface of a sphere equally distant from every point in the circumference of the circle. It will be shown, art. 464, that every circle, great or small, has two poles.

This is the first term in Legendre that isn't defined in the U of Chicago. For such terms, a good way to visualize it is to consider the shape of the earth. We consider the lines of latitude -- that is, the curves along which the latitude is constant. These are circles wrapping around the earth -- the Equator is a great circle, while all the others are small circles. All of these circles share the same two poles -- the actual North and South Poles. It's this fact that justifies the name "pole" for these two points.

Of course, any circle other than the latitudes will have two other points as its poles, rather than the actual North and South Poles. Because of this, I will write the word "pole" in lowercase to denote the poles of a circle, and capitalize "Pole" when it refers to either the North or South Poles.

442. A

*spherical triangle*is a part of the surface of a sphere comprehended by three arcs of great circles.

These arcs, which are called the

*sides*of the triangle, are always supposed to be less than a semicircumference. The angles, which their planes make with each other, are the angles of the triangle.

An example of a spherical triangle appears in Section 5-7 of the U of Chicago text:

"The surface of the earth can be approximated as a sphere. A triangle is formed by two longitudes (north-south lines) and the equator is isosceles with two right base angles!"

This sentence also demonstrates some of the terms defined in the next Legendre proposition:

443. A spherical triangle takes the name of

*right-angled*,

*isosceles*and

*equilateral*, like a plane triangle, and under the same circumstances.

So the triangle that appears in Section 5-7 is described as a right isosceles triangle. Notice that the sides of this triangle are the arcs of great circles -- unlike lines of latitude, meridians of longitude are always great circles. The pole of any meridian lies on the Equator -- the longitudes of the poles are 90 degrees to the east and west of the original meridian.

If the angle at the North Pole happens to be 90 degrees, we obtain a very special triangle. The right triangle now becomes equilateral -- and furthermore, each side has the opposite vertex as one of its two poles.

444. A

*spherical polygon*is a part of the surface of a sphere terminated by several arcs of great circles.

It's because of the requirement that the sides of a polygon be the arcs of great circles that the states of Colorado and Wyoming can't be polygons, much less rectangles. The sides that follow longitudes are great circles, but the sides that follow latitudes are small circles.

By this point, one may wonder why we insist that the sides of spherical polygons be great circles. As it turns out, using only great circles as sides makes Propositions 459-460 work out -- but we won't get to either of those today.

445. A

*lunary surface*is the part of a surface of a sphere comprehended between two semicircumferences of great circles, which terminate in a common diameter.

Many modern texts abbreviate "lunary surface" by simply calling it a

*lune*. The region between two longitudes is a lune. The time zones into which the earth's surface is divided are nominally lunes -- for example, my Pacific time zone is the lune between longitudes 112.5 and 127.5 degrees West. In practice, though, none of the time zones are actually lunes because they have all been adjusted to follow state or national borders, for convenience. For example, the Pacific time zone ends at the border between Nevada and Utah, even though the longitude is 114, not the idea 112.5 degrees West.

The names "lunary surface" and "lune," of course, refer to the moon. The portion of the moon visible at any given time is a lune. The smallest lunes are crescents approaching the new moon, while the largest lunes are gibbous moons approaching the full moon.

446. We call a

*spherical wedge*the part of a sphere comprehended between the halves of two great circles, and to which the lunary surface answers as a base.

When I hear the word "wedge," I can't help but think about an orange. Each section of the fruit is a spherical wedge, and the peel that covers the wedge represents a lune. On the earth, a wedge is a lune dug all the way to the center of the earth.

447. A

*spherical pyramid*is the part of a sphere comprehended between the planes of a solid angle whose vertex is at the center. The

*base*of the pyramid is the spherical polygon intercepted by these planes.

Spherical pyramids actually appear in Section 10-9 of the U of Chicago text, where the text is trying to derive the relationship between the volume and surface area of a sphere. The text shows a sphere being divided into "almost pyramids," and writes:

The solid is not exactly a pyramid because its base is not exactly a polygon.

Well, if we follow Legendre, then the bases can be made into exact

*spherical*polygons, and so the resulting solids are exact

*spherical*pyramids. On the earth, a pyramid is a polygon dug all the way to the center of the earth.

448. A

*zone*is the part of a surface of a sphere comprehended between two parallel planes, which are its

*bases*. One of these planes may be a tangent to the sphere, in which case the zone has only one base.

Despite their name, the time zones of the earth are not zones by Legendre's definition -- recall that these are actually lunes. On the other hand, the temperate zones of the earth are actually zones. The north temperate zone lies between the latitudes of the Tropic of Cancer and the Arctic Circle, since these circles are bounded by parallel planes. (Notice that the word "parallel" in the phrase "parallels of latitude" actually refers to parallel

*planes*-- these are not parallel lines because they are small circles, not great circles. Recall that there are no parallel lines on the globe.)

The entire continental U.S. lies in this north temperate zone -- although sometimes one defines yet another zone, the

*subtropical*zone, as the zone bounded by Tropic of Cancer and the 38th parallel. We see that most of California (as far north as San Francisco, but not as far north as Sacramento) lies in the subtropical zone by this definition.

Legendre states that zones may also be defined with one of the planes tangent to the sphere. An example of such a zone on earth is the north frigid zone, bounded by Arctic Circle and the plane tangent to the earth at the North Pole. Notice that the so-called rectangular states such as Colorado and Wyoming are each actually the intersection of a lune and a zone.

449. A

*spherical segment*is the portion of a sphere comprehended between two parallel planes which are its bases. One of these planes may be a tangent to the sphere, in which case the spherical segment has only one base.

For this one we think about our orange again -- but this time we divide it into slices, not wedges. We then observe that each slice is a spherical segment. The peel surrounding the fruit slice now represents a zone.

450. The

*altitude of a zone*or

*of a*

*segment*is the distance between the parallel planes which are the bases of the zone or segment.

Notice that this altitude is measured in

*space*, not along the sphere. When a recipe asks us to divide an orange into half- or quarter-inch slices, we are most likely measuring the fractions in space, so these would be the altitudes of the segments.

451. While the semicircle

*DAE*turning about the diameter

*DE*, describes a sphere, every circular sector, as

*DCF*or

*FCH*, describes a sector, which is called a spherical sector.

A spherical sector is a solid of revolution, On the earth, a spherical sector is a zone dug all the way to the center of the earth.

We now reach our first theorem:

452.

*Every section of a sphere made by a plane is a circle.*

*Demonstration*. Let

*AMB*be a section, made by a plane, of the sphere of which

*C*is the center. From that point

*C*draw

*CO*perpendicular to the plane

*AMB*, and different oblique lines

*CM*,

*CM'*, to different points of the curve

*AMB*which terminates the section.

The oblique lines

*CM*,

*CM'*,

*CB*, are equal, since they are radii of the sphere; consequently they are at equal distances from the perpendicular

*CO*(329); whence all the lines

*OM*,

*OM'*,

*OB*are equal; therefore the section

*AMB*is a circle of which the point

*O*is the center. QED

Here is the same proof given in the U of Chicago text, Question 20 of Section 9-4. Notice that the labels are different -- here

*O*is the center of the

*sphere*(not the section as in Legendre) and

*M*is the name of the

*plane*(not a point on that plane as in Legendre). I'll keep it as a paragraph proof, just as it appears in the text.

Proof:

Let

*P*be the foot of the perpendicular from point

*O*to plane

*M*. Let

*A*be a fixed point and

*X*be any other point on the intersection. Then

*OPX*and

*OPA*are congruent by HL. Thus, due to CPCTC,

*PX*=

*PA*. Thus any point

*X*lies at the same distance from

*P*as

*A*does. So by the definition of circle (sufficient condition), the intersection of sphere

*O*and plane

*M*is the circle with center

*P*and radius

*PA*. QED

Notice that both proofs only deal with the case where the plane doesn't pass through the center of the sphere -- we proved that the intersection is a small circle. The case where the plane

*does*pass through the center is trivial -- it simply follows from the definitions of sphere and circle that the intersection is a great circle.

And so we've completed the first two pages of Legendre's section on spheres. But there is one more term that doesn't appear in Legendre yet is important enough for me to define here. For every point on the sphere, there exists exactly one point directly opposite it, called

*antipodes*, or

*antipodal point*. The endpoints of any diameter of the sphere are antipodal points.

The significance of antipodal points in spherical geometry is as follows -- we know, and will at some point prove, that every circle has two poles. Well, the poles of any circle are antipodal points. In particular, the North and South Poles are antipodal to each other.

An interesting exercise is to locate antipodal points on the earth. As it turns out, the transformation mapping each point to its antipodes is an isometry -- point inversion about the center of the sphere. I also point out that the antipodal map is the composite of a reflection about the Equator and a rotation of 180 degrees about the axis passing through the Poles. This means that given the latitude and longitude of a point, we can find its antipodes by reflecting its latitude about the Equator (which switches North and South) and rotating its longitude 180 degrees. The rotation is itself the composite of two reflections -- one about the longitude 0 degrees, or Prime Meridian (which switches East and West) and the other about the longitude 90 degrees (which subtracts the longitude from 180).

I live at approximately 34 degrees North, 118 degrees West. (In fact, the exact confluence is not that far from one of the districts where I work.) We calculate the antipodes as follows -- we reflect 34 degrees North about the equator to obtain 34 degrees South, we reflect 118 degrees West about the Prime Meridian to obtain 118 degrees East, and we reflect 118 degrees East about the 90th meridian to obtain 62 degrees East.

This confluence is in the middle of the Indian Ocean. In fact, almost the entire continental U.S. is antipodal to the Indian Ocean. Only three uninhabited French-owned islands are actually antipodal to the continental U.S. (two antipodal to Colorado, the other antipodal to the Montana-Canada border). I point out that if we dig a hole, it would not end up, contrary to popular belief, in China.

Most land is antipodal to water. This is mainly because of the existence of the Land and Water Hemispheres -- most land lies in the Land Hemisphere, so a point there would have its antipodes in the Water Hemisphere and so likely in the ocean. California lies very close to the great circle which separates the Land and Water Hemispheres (but is indeed on the Land side) -- one of the poles of this great circle happens to be in the town of Nantes, France.

There do exist a few land antipodes. The town of Christchurch, New Zealand is almost exactly antipodal to A Coruna, Spain, and Hong Kong is antipodal to La Quiaca, Argentina. So we see that China is actually antipodal to South America, not the USA.

My next post will be tomorrow.

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