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Spherical Geometry Essay

Euclid is probably the most famous figure in geometric history. His book, The Elements, is the most famous book about geometry, and probably one of the most famous books in all history. Ask almost anyone on the street, and he or she will know about The Elements. The most famous part of The Elements is the five postulates that Euclid introduces in his book to prove the rest of his theorems. These postulates are:
  1. To Draw a straight line from any point to any other.
  2. To produce a finite straight line continuously in a straight line.
  3. To describe a circle with any centre and distance.
  4. That all right angles are equal to each other.
  5. That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, if produced indefinitely, meet on that side on which are the angles less than the two right angles

As you can see, the fifth postulate is radically different from the first four, and Euclid was not satisfied with it, so he tried to avoid using it in The Elements. In fact, the first 28 theorems in The Elements are proved without the use of the fifth postulate.

Proclus (410-485) wrote a commentary on The Elements where he comments on attempted proofs to deduce the fifth postulate from the other four, in particular he notes that Ptolemy had produced a false 'proof'. Proclus then goes on to give a false proof of his own. However he did give the following postulate which is equivalent to the fifth postulate.

Playfair's Axiom:- Given a line and a point not on the line, it is possible to draw exactly one line through the given point parallel to the line.
Although known from the time of Proclus, this became known as Playfair's Axiom after John Playfair wrote a famous commentary on Euclid in 1795 in which he proposed replacing Euclid's fifth postulate by this axiom.

Many attempts were made to prove the fifth postulate from the other four, many of them being accepted as proofs for long periods of time until the mistake was found. Invariably the mistake was assuming some 'obvious' property which turned out to be equivalent to the fifth postulate. One such 'proof' was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:-

To each triangle, there exists a similar triangle of arbitrary magnitude.
One of the attempted proofs turned out to be more important than most others. It was produced in 1697 by Girolamo Saccheri. The importance of Saccheri's work was that he assumed the fifth postulate false and attempted to derive a contradiction.

You can see a picture of the Saccheri quadrilateral

In this figure Saccheri proved that the summit angles at D and C were equal.The proof uses properties of congruent triangles which Euclid proved in Propositions 4 and 8 which are proved before the fifth postulate is used. Saccheri has shown:

a) The summit angles are > 90 (hypothesis of the obtuse angle).
b) The summit angles are < 90 (hypothesis of the acute angle).
c) The summit angles are = 90 (hypothesis of the right angle).

Euclid's fifth postulate is c). Saccheri proved that the hypothesis of the obtuse angle implied the fifth postulate, so obtaining a contradiction. Saccheri then studied the hypothesis of the acute angle and derived many theorems of non-Euclidean geometry without realising what he was doing. However he eventually 'proved' that the hypothesis of the acute angle led to a contradiction by assuming that there is a 'point at infinity' which lies on a plane.

In 1766 Lambert followed a similar line to Saccheri. However he did not fall into the trap that Saccheri fell into and investigated the hypothesis of the acute angle without obtaining a contradiction. Lambert noticed that, in this new geometry, the angle sum of a triangle increased as the area of the triangle decreased.

Legendre spent 40 years of his life working on the parallel postulate and the work appears in appendices to various editions of his highly successful geometry book Eléments de Géométrie . Legendre proved that Euclid's fifth postulate is equivalent to:-

The sum of the angles of a triangle is equal to two right angles.
Legendre showed, as Saccheri had over 100 years earlier, that the sum of the angles of a triangle cannot be greater than two right angles. This, again like Saccheri, rested on the fact that straight lines were infinite. In trying to show that the angle sum cannot be less than 180Legendre assumed that through any point in the interior of an angle it is always possible to draw a line which meets both sides of the angle. This turns out to be another equivalent form of the fifth postulate, but Legendre never realised his error himself.

Elementary geometry was by this time engulfed in the problems of the parallel postulate. D'Alembert, in 1767, called it the scandal of elementary geometry.

The first person to really come to understand the problem of the parallels was Gauss. He began work on the fifth postulate in 1792 while only 15 years old, at first attempting to prove the parallels postulate from the other four. By 1813 he had made little progress and wrote:

In the theory of parallels we are even now not further than Euclid. This is a shameful part of mathematics
However by 1817 Gauss had become convinced that the fifth postulate was independent of the other four postulates. He began to work out the consequences of a geometry in which more than one line can be drawn through a given point parallel to a given line. Perhaps most surprisingly of all Gauss never published this work but kept it a secret. At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.

Gauss discussed the theory of parallels with his friend, the mathematician Farkas Bolyai who made several false proofs of the parallel postulate. Farkas Bolyai taught his son, János Bolyai, mathematics but, despite advising his son not to waste one hour's time on that problem of the problem of the fifth postulate, János Bolyai did work on the problem.

In 1823 Bolyai wrote to his father saying I have discovered things so wonderful that I was astounded out of nothing I have created a strange new world . However it took Bolyai a further two years before it was all written down and he published his strange new world as a 24 page appendix to his father's book, although just to confuse future generations the appendix was published before the book itself.

Gauss, after reading the 24 pages, described János Bolyai in these words while writing to a friend: I regard this young geometer Bolyai as a genius of the first order . However in some sense Bolyai only assumed that the new geometry was possible. He then followed the consequences in a not too dissimilar fashion from those who had chosen to assume the fifth postulate was false and seek a contradiction. However the real breakthrough was the belief that the new geometry was possible. Gauss, however impressed he sounded in the above quote with Bolyai, rather devastated Bolyai by telling him that he (Gauss) had discovered all this earlier but had not published. Although this must undoubtedly be true, it detracts in no way from Bolyai's incredible breakthrough.

Nor is Bolyai's work diminished because Lobachevsky published a work on non-Euclidean geometry in 1829. Neither Bolyai nor Gauss knew of Lobachevsky's work, mainly because it was only published in Russian in the Kazan Messenger a local university publication. Lobachevsky's attempt to reach a wider audience had failed when his paper was rejected by Ostrogradski.

In fact Lobachevsky fared no better than Bolyai in gaining public recognition for his momentous work. He published Geometrical investigations on the theory of parallels in 1840 which, in its 61 pages, gives the clearest account of Lobachevsky's work. The publication of an account in French in Crelle's Journal in 1837 brought his work on non-Euclidean geometry to a wide audience but the mathematical community was not ready to accept ideas so revolutionary.

In Lobachevsky's 1840 booklet he explains clearly how his non-Euclidean geometry works.

You can see Lobachevsky's_diagram.

All straight lines which in a plane go out from a point can, with reference to a given straight line in the same plane, be divided into two classes - into cutting and non-cutting. The boundary lines of the one and the other class of those lines will be called parallel to the given line.
Hence Lobachevsky has replaced the fifth postulate of Euclid by:-
Lobachevsky's Parallel Postulate. There exist two lines parallel to a given line through a given point not on the line.
Lobachevsky went on to develop many trigonometric identities for triangles which held in this geometry, showing that as the triangle became small the identities tended to the usual trigonometric identities.

Riemann, who wrote his doctoral dissertation under Gauss' supervision, gave an inaugural lecture on 10 June 1854 in which he reformulated the whole concept of geometry which he saw as a space with enough extra structure to be able to measure things like length. This lecture was not published until 1868, two years after Riemann's death but was to have a profound influence on the development of a wealth of different geometries. Riemann briefly discussed a 'spherical' geometry in which every line through a point P not on a line AB meets the line AB. In this geometry no parallels are possible.

It is important to realise that neither Bolyai's nor Lobachevsky's description of their new geometry had been proved to be consistent. In fact it was no different from Euclidean geometry in this respect although the many centuries of work with Euclidean geometry was sufficient to convince mathematicians that no contradiction would ever appear within it.

The first person to put the Bolyai - Lobachevsky non-Euclidean geometry on the same footing as Euclidean geometry was Eugenio Beltrami (1835-1900). In 1868 he wrote a paper Essay on the interpretation of non-Euclidean geometry which produced a model for 2-dimensional non-Euclidean geometry within 3-dimensional Euclidean geometry. The model was obtained on the surface of revolution of a tractrix about its asymptote. This is sometimes called a pseudo-sphere.

You can see the graph of a tractrix and what the top half of a Pseudo-sphere looks like.

In fact Beltrami's model was incomplete but it certainly gave a final decision on the fifth postulate of Euclid since the model provided a setting in which Euclid's first four postulates held but the fifth did not hold. It reduced the problem of consistency of the axioms of non-Euclidean geometry to that of the consistency of the axioms of Euclidean geometry.

Beltrami's work on a model of Bolyai - Lobachevsky's non-Euclidean geometry was completed by Klein in 1871. Klein went further than this and gave models of other non-Euclidean geometries such as Riemann's spherical geometry. Klein's work was based on a notion of distance defined by Cayley in 1859 when he proposed a generalised definition for distance.

Klein showed that there are three basically different types of geometry. In the Bolyai - Lobachevsky type of geometry, straight lines have two infinitely distant points. In the Riemann type of spherical geometry, lines have no (or more precisely two imaginary) infinitely distant points. Euclidean geometry is a limiting case between the two where for each line there are two coincident infinitely distant points.

References (23 books/articles)

Credit Due to The Mac Tutor History of Mathematics Non-Euclidean geometry page.



Spherical Geometry

Danielle Frailey


Spherical Geometry in History

At the time when Earth was discovered to be round rather than flat, spherical geometry began to emerge to aid navigators in mapping the land and water. However, even before Columbus, ancient Greek and Phoenician mariners used the ideas of spherical geometry in naval explorations of the world they knew.  Contrary to popular belief, it was indeed Ptolemy, not Christopher Columbus, who discovered that the Earth was spherical.  Ptolemy, the man shown to the left, said, “If the earth were flat from east to west, the stars would rise as soon for westerners as for orientals, which is false.  Also, if the earth were flat from north to south and vice versa, the stars, which were always visible to anyone, would continue to be so wherever he went, which is false.  But it seems flat to human sight because it is so extensive.”  As much as 2000 years ago, the discovery of the Earth’s curved surface was making huge impacts on the way the Earth was viewed and mapped. 

What is Spherical Geometry?

The basic element of spherical geometry is the sphere, a three-dimensional surface made up of the set of all points in space at a given distance from a fixed point called the center. If we take an arbitrary plane and sphere lying in the plane, there are three possibilities for their intersections. First, the plane and the sphere never intersect.  Secondly, the plane may lay tangent to the sphere forming one distinct point of intersection.   In the final case where the plane passes through the sphere, the intersection forms a circle.  Within this final case there lies a unique case.  If the plane passes through the center of the sphere, the circle formed is known as a great circle.  In the diagram to the right, Point O represents the sphere’s center, and the great circle is represented by circle E. The points N and S represent the two antipodal points. Antipodal points are the points formed by the intersection of a sphere and a line that passes through the center of the sphere. One recognizable example of antipodal points would be the North and South poles of the Earth.

Great circles are defined as those circles of intersection that share the same radius and same center as the sphere it intersects, and divide a sphere into two equal halves.  The great circle is the largest possible circle able to be formed along a sphere, such as the Earth’s equator.  Interestingly, all of Earth’s lines of longitude are great circles, yet the Equator is the only line of latitude that is a great circle. 

Spherical geometry is defined as the study of figures on the surface of a sphere, and it can be viewed as the three-dimensional version of Euclidean or planar geometry.  Although closely related, the two ideas of spherical and planar geometry are completely different. Most students who have taken a high school level geometry course can recite some of the basic facts of planar geometry: two parallel lines never intersect, the sum of the interior angles of a triangle is equal to one-hundred and eighty degrees, and the shortest distance from one point to the next is a straight line.  Yet, these concepts do not hold in spherical geometry. 

Properties of Spherical Geometry

In spherical geometry, there are no parallel lines.  In fact, there are no straight lines.  Therefore, in spherical geometry, a great circle is comparable to a line.  There are no straight lines in spherical geometry. Instead, the shortest distance from one point to the next lying on a sphere is along the arc of a great circle.  On a sphere, the angle between two curved arcs is measured by the angle formed from the intersection of the lines lying tangent to the two arcs. 

When three curved arcs intersect one another, a spherical triangle is formed. A spherical triangle is any three-sided region enclosed by sides that are the arcs of great circles. Spherical triangles can have angle sums that range between one hundred eighty degrees and five hundred forty degrees. Spherical triangles can have angle sums larger than the usual one hundred eighty degrees found in a triangle because lines connecting points have a slight curve to them. Even so, spherical triangles can have ninety-degree angles just like triangles in the plane.

Differing from Euclidean geometry, two spherical triangles are not only similar, but congruent if they share the same angles.  If one of the angles of a spherical triangle is a right angle, the triangle is known as a spherical right triangle, and a Spherical Pythagorean Theorem exists. In a spherical right triangle, let C denote the length of the side opposite the right angle. Let A and B denote the lengths of the other two sides.  Let R denote the radius of the sphere.  Then, we can use the following formula: cos(C/R) = cos(A/R)*cos(B/R).

            To calculate the area of a spherical triangle on a unit sphere, one must sum the angles of the spherical triangle (in radians) and subtract pi. For example, say a spherical triangle had two right angles and one forty-five degree angle. To find the area of the spherical triangle, restate the angles given in degrees to angles in radians. Thus, we are working with a spherical triangle with two pi/2 angles and one pi/4 angle. Add the three angles together (pi/2 + pi/2 + pi/4). Thus, the angles total 5pi/4. Subtract pi from 5pi/4 to find the area of the spherical triangle. Therefore, we have a total area of pi/4 units squared.

It has been established that a great circle is formed from a plane that intersects a sphere through its center.  In analyzing two great circles that lie in a sphere, the two planes that form the great circles must intersect in a line, which in turn intersects the sphere at two distinct points.  Therefore, two great circles intersect each other at two antipodal points.  The planes also divide the sphere into four parts.  These regions are known as lunes, which comes from the Latin word “luna,” meaning moon.   A lune is a two-sided polygon lying on a sphere.



To find the area of a lune, one must know the lunar angle and the formula 2R2(lunar angle), where R represents the radius of the sphere. If the lunar angle is not given, find p(pi)/q, where p represents the union of the lunes and q is the number of equal lunes drawn in a hemisphere. Knowledge about a lune’s area can be helpful in solving the area of the sphere. For example, say the area of a lune is thirty-two centimeters squared, and we want to find the area of the sphere where the lune is present. The area of a sphere is 4piR2, where R represents the radius. Since we know the formula for the area of a lune, we can solve for the radius, the unknown variable. Then we can use the radius in the formula for the area of the sphere to solve for the area of the sphere. After a few calculations, we can see that the radius is four centimeters. We then can square this and multiply the new result by 4pi. Thus, we have the sphere’s area equal to 64pi centimeters squared. 

In two dimensions, a lune is a plane figure bounded by two circular arcs. Greek mathematician Hippocrates proves that lunes are quadrable, or it is possible to construct a square having equal area to that of the lune. While Hippocrates did not show that all such figures are quadrable, he did show that the particular lune he created was.

In fact, there are five quadrable lunes. In an attempt to square the circle, Hippocrates squared the lune showed above. He is also credited with finding two other squarable lunes. In 1934, Russian mathematician Tschebatorev came close to finding a solution, which his student Dorodnov completed in 1947, ending the search for other quadrable lunes.

As discussed earlier, the shortest distance between two points on a sphere is along the arc of a great circle.  In spherical geometry, there exists a formula to find this distance.  Given two points A and B, first locate the angle (measured in radians) that is created between the two pointsand the center of the sphere.  Then, multiply by the radius (R) of the sphere, giving the formula d(A,B)= Ra, where a represents the angle formed between points A and B. For example, to find the distance between two points A and B with an angle of pi/4 between them on a sphere having a radius of 4, we multiply pi/4 by the radius length of 4 units to get a spherical distance of pi.

Where can I find Spherical Geometry today?

Today, the concepts of spherical geometry are implemented in air and space travel, naval cruises, and much more.  For example, an airplane looking to travel from Florida to the Philippines would pass over Alaska.  Since the Philippines lie south of Florida, it does not seem reasonable to take this flight route.  Yet, this happens to be the shortest distance between the two points, since Florida, Alaska, and the Philippines lie relatively “collinearly” along the path of a great circle. Thus, the best path to travel from Florida to the Philippines would include a flight route over Alaska.

Picture sources:


Intersection of Sphere and Plane:


Spherical Distance: