Back to main course page

Non-Euclidean Geometry

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

Euclid and his Elements

Here's an introductory puzzle. In the totality of our intellectual heritage, which book is most studied and most edited? The answer is obvious: the Bible. But which is the most studied and edited work after it? That is a little harder to say. The answer comes from a branch of science that we now take for granted, geometry. The work in Euclid's Elements. This is the work that codified geometry in antiquity. It was written by Euclid, who lived in the Greek city of Alexandria in Egypt around 300BC, where he founded a school of mathematics. Since 1482, there have been more than a thousand editions of Euclid's Elements printed. It has been the standard source for geometry for millennia. It is only in recent decades that we have started to separate geometry from Euclid. In living memory--my memory of high school--geometry was still taught using the development of Euclid: his definitions, axioms and postulates and his numbering of them.

We can identify two reasons for the importance of Euclid's Elements in our understanding of the foundations of science: its structure and the certitude of its results.

How Do We Organize Our Knowledge?

First, Euclid's Elements solved an important problem. When we have a large body of knowledge, such as we have in geometry, how are we to organize it? We know many simple things in geometry: the sum of the angles of a triangle are always 180 degrees. And we know more complicated things. A 3-4-5 sided triangle is a right angled triangle. And even more complicated things. As Pythagoras found, in a right angled triangle, the sum of the areas of the squares erected on the two shorter sides is equal in area to of a square erected on the hypotenuse.

So, as our knowledge grows, how are we to organize it so that we capture in it all the truths that we want and do not let in things that don't property belong there? Euclid employed a quite profound method, deductive systematization. His elements were structured according to a series of propositions:

Definitions.

This is the response to the simple injunction: "define your terms"--else you cannot know precisely what you are talking about. There are 35 definitions. They include such familiar ideas as:
1. A point is that which has no part.
2. A line is a breadthless length.
3. The extremities of lines are points.
...
22. Quadrilateral figures are bounded by four straight lines.
...
and so on.

Axioms or Common Notions

These are general statements, not specific to geometry, whose truth is obvious or self-evident. There are 12. For example:
1. Things which are equal to the same thing are equal to one another.
2. If equals be added to equals, the wholes are equal.
and so on.

Postulates

These are the basic suppositions of geometry. They reflect its constructive character; that is, they are assertions about what exists in geometry. The first of the five simply asserts that you can always draw a straight line between any two points.

Theorems or Propositions

These are the consequences deduced logically from the definitions, axioms and postulates. They form the bulk of geometrical knowledge and include Pythagoras' famous result above concerning the areas of squares on the sides of right angled triangles.

All the definitions, axioms, postulates and propositions of Book I of Euclids Elements are here.

Once this structure is adopted, the problem of knowing just what really belongs in geometry is reduced to matters of deductive inference. Is this or that a truth of geometry? The question is answered by determining whether it can be deduced from Euclid's postulates and axioms. Do you doubt that this is a truth of geometry? Then you must show where Euclid's proof broke down. Eventually, as you trace the proof's back to their sources, you end up seeing that the truth of the result derives ultimately from the truth of postulates and axioms. And their truth is so obvious as to admit no doubt. Who wants to say that you cannot always draw a straight line between any two given points?

In the seventeenth century, with new-found confidence, natural philosophers rebuilt all learning from scratch, discarding the wisdom of antiquity as flawed. In that effusion of new investigation, one achievement stood unchallenged. That was Euclid's Elements. Indeed its premier position was reinforced when the structure it gave to geometrical knowledge was adopted by Newton to codify his new mechanics. Like Euclid, Newton listed definitions and, where Euclid gave axioms and postulates, Newton gave his celebrated three laws of motion. Euclid's Elements became the template for organizing knowledge, be it a new science such as Newton's or even knowledge outside science.

Knowing with Certainty

Second, the enduring success of Euclid's Elements assured us that some things could be known with certainty. While the knowledge of antiquity collapsed, geometry thrived as the method central to Newton's discovery and also the template for his organization of his new mechanics. The idea of that sort of certainty is familiar today. We are used to the idea that some branches of study do not need fragile experiments to verify them. There is no point in counting two apples and then two apples into a basket to verify that 2+2=4; and then doing it again with pears or pineapples, just to be sure. Arithmetic does not need it. 2+2 does not just happen to be 4; it has to be 4. There is no other possibility.

So Euclid's geometry and Newton's physics bequeathed to thinkers the problem of understanding just how this level of certitude was possible. Our modern minds are steeped in the idea that knowledge of the world comes from experience and new experience can always overthrown old learning. By the eighteenth century, the sense was widespread that Euclid and Newton had found the final truths of geometry and mechanics. The philosophical problem was to determine how this was possible.

To Come

These ideas provide our starting point. We shall see that matters take a very different turn in the nineteenth century.
Nineteenth century mathematicians realized that the eighteenth century certainty of geometry was mistaken. Geometry was an empirical science. It reported the way our space happened to to be, not the way it had to be. If that was so, other geometries were possible and our experience of space might well have been different. In the nineteenth century, these were regarded as possibilities that were unrealized. Nature had many choices but, they thought, she chose Euclid's system.

This realization of the mere possibility of geometries other than Euclid's was shocking. Greater shocks were in store. In the twentieth century, Einstein delivered the final insult to Euclid. He found through his general theory of relativity that a non-Euclidean geometry is not just a possibility that Nature happens not to use. In the presence of strong gravitational fields, Nature chooses these geometries.

Euclid's Postulates

1. To draw a straight line from any point to any point.

This postulates simple says that if you have any two points--A and B, say--then you can always connect them with a straight line.

It is tempting to think that there is no real content in this assertion. That is not so. This postulate is telling us a lot of important material about space. Any two points in space can be connected; so space does not divide into unconnected parts. And there are no holes in space such as might obstruct efforts to connect two points.

The second postulate is:

2. To produce a finite straight line continuously in a straight line.

It tells us that we can always make a line segment longer. That means that we never run out of space; that is, space is infinite.

The third postulate is:

3. To describe a circle with any center and distance.

It allows for the existence of circles of any size and center--say center A and radius AB.

Note that this sort of postulate is not superfluous. A definition can tell us what a circle is, so we know one if ever we find one. But the definition does not assert their existence. Analogously, we can give a definition of a unicorn; that doesn't mean they exist. This postulate says circles exist, just as the first two postulates allow for the existence of straight lines.

The fourth postulate says:

4. That all right angles are equal to one another.

Sameness here means that were we to manipulate the angles by sliding them over the page, they would coincide.

Deriving a Theorem

So far everything is going very well. The postulates we have seen are utterly innocuous and readily accepted. But once they are accepted, a lot follows. The simplest is the existence of equilateral triangles. Their construction is the burden of the first proposition of Book 1 of the thirteen books of Euclid's Elements.

The problem is to draw an equilateral triangle on a given straight line AB.

Postulate 3 assures us that we can draw a circle with center A and radius B.	Analogously, Postulate 3 also assures us that we can draw a circle with center B and radius BA.

Now connect A and C with a straight line; and B and C with a straight line. That each straight line can be drawn is asserted by Postulate 1.

This illustrates the power of Euclid's system. Every step is guaranteed by an axiom or a postulate, so that one cannot accept the axioms and postulates without also accepting the proposition.

The Fifth Postulate

So far everything has been going very well. However these first four postulates are not enough to do the geometry Euclid knew. Something extra was needed. Euclid settled upon the following as his fifth and final postulate:

5. That, if a straight line falling on two straight lines make 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.

It is very clear that there is something quite different about this fifth postulate. The first four were simple assertions that few would be inclined to doubt. Far from being instantly self-evident, the fifth postulate was even hard to read and understand.

5. That, if a straight line falling on two straight lines...

... make the interior angles on the same side less than two right angles... [in this case, side on the right]

...the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.

Or, in an animation:

Attempts to Eliminate the Odd Man Out

From antiquity, there had been discomfort with this fifth postulate, an odd man out among the postulates. The obvious remedy was to find a way to deduce the fifth postulate from the other four. If that could be done, then the fifth postulate would become a theorem and the awkwardness of needing to postulate it would evaporate.

Many tried. The famous astronomer Ptolemy of the first century AD tried. The great mathematician John Wallis tried in the 17th century. The most famous of all attempts was published by Girolamo Saccheri in 1733, Euclides ab Omni Naevo Vindicatus, ("Euclid Cleared of Every Defect”). Yet even this massive work did not achieve its goal and the efforts continued.

One of most important by-products of these efforts were simpler, alternative formulations of Euclid's fifth postulate that could be used in place of Euclid's original. Many were found, including:

There exists a pair of coplanar straight lines, everywhere equidistant from one another.
There exists a pair of similar, non-congruent triangles.
If in a quadrilateral a pair of opposite sides are equal and if the angles adjacent to the third side are right angles, then the other two angles are also right angles. (Saccheri)
There is no upper bound to the area of a triangle.

Of all the reformulations, one proves to be most useful. It was stated by an 18th century mathematician and physicist, Playfair. His postulate, equivalent to Euclid's fifth, was:

5.^ONE Through any given point can be drawn exactly one straight line parallel to a given line.

This formulation made it easy to state what the alternatives were. In place of ONE, we could have NONE or MORE than one.

5^MORE. Through any given point MORE than one straight line can be drawn parallel to a given line.

The other possibility is:

5^NONE. Through any given point NO straight lines can be drawn parallel to a given line.

Once again the import of the alternative postulate is hard to draw since the screen is a Euclidean surface. In the figure, the line through the point is a straight line. The postulate tells us that no matter which straight line we pick through the point, the outcome is the same. It is not parallel to the line not on the point. If extended it will eventually meet the other line.

As we passed from the eighteenth to the nineteenth century, the way we understood the problem changed.

In the eighteenth century, as in the centuries before, the project had been to rid Euclid of the flaw, the need to assert his fifth postulate independently. That could be done if, against the background of the other postulates, the alternatives proved contradictory. So the common project was to presume one of these alternatives and show that impossible results followed. Ideally what was sought was a flat-out logical contradiction. Then, by a simple reductio, the alternative postulate was refuted. However the logical contradiction proved elusive. Time and again, the investigations produced one odd result after another. They were odd, but they were not yet logical contradictions.

In the nineteenth century, the reason for this frustrating failure was recognized by Gauss, Riemann, Bolyai, Lobachevsky and others. The early geometers had found results that were odd but not contradictory simply because they were exploring new geometries. The geometries they were exploring were odd and so were the results recovered; but they were not contradictory.

Exploring the Geometry of 5^NONE

Let us join the explorers of the nineteenth century and take the first steps into the new space of these odd geometries. Let us explore the space of 5^NONE.

A Trip Around Space

The alternative postulate, 5NONE, assures us that these perpendiculars, if projected, will eventually meet at some point. Let us call that point O. There is a perfect symmetry in the figure; we could switch A and B and nothing would change, so we can infer that AO = BO Note that in the figure the lines AO and BO are straight lines. In a Euclidean geometry, they could not possibly meet. However this is not a Euclidean geometry, so odd things will happen. This is just the beginning...

Now find the midpoint of AB and call it Q. Erect a perpendicular to AB at Q. Project it until it eventually meets AO and BO. (It must meet them since there are no parallels in this geometry.) Where will in meet them? It cannot be to either side of the point O since then there would be an asymmetry. The midpoint Q and its perpendicular do not favor either side. So the perpendicular must pass through the point O.

Now repeat the construction. Bisect AQ and from that point erect a perpendicular that will pass through O. Bisect QB and from that point erect a perpendicular that will pass through O. By repeating this process indefinitely, we can divide the original interval AB into as many equal sized parts as we like. Perpendiculars raised from each of these points will all pass through the point O. Clearly the point O has a special significance for the entire straight line through AB. Recall that every line in the figure to the right is a straight line!

Let us now do essentially the same construction but in a way that extends past AB. As before, we have points A and B on the line we chose earlier with the two perpendiculars erected at A and B. On AB produced through B we pick a point C such that AB=BC.

We now erect a perpendicular at C. As before, it must intersect the perpendiculars at A and B at the same point O. The argument is essentially the same as before. If it the perpendicular at C did not pass through O, it would intersect the perpendicular at A at some other point O' on AO. But that would now mean that the perpendicular at B no longer respects the symmetry in the large triangle AO'C.

This construction could be continued with points D, E, F, ... each a distance AB advanced from the point before. At each we erect a perpendicular, which will intersect the others at the same point O. Since the triangles produced by this construction, OAB, OBC, OCD, ODE and OEF are congruent, the angles at the apex are all the same:

angle AOB = angle BOC = angle COD = angle DOE = angle EOF.

That is, by extending the base of the triangle, AB to AC to AD etc. we can make the angle at the apex grow as large as we like. So we can certainly make it as big as a right angle. Let us say that this happens with a base AG. And we can keep extending the base to G' until we have a second right angle at GOG'. And we can extend to G" so that we have a third right angle at G'OG''. And finally we can extend the base to G''' so we have a fourth right angle at G''OG'''.

We have arrived at something remarkable in this figure. It is not just that all these lines are straight lines. It is more. The angle AOG''' is four right angles. How can that be? Rather than tell you right away, let me give you a clue. We don't need to draw all the lines as straight in the figure. We just need to remember which are straight--in this case all of them. So we can redraw the figure as:

Think once again what it means for the angle AOG''' to be four right angles. Consider the line OA as it sweeps around O. It passes one right angle to reach OG; two right angles to reach OG'; three right angles to reach OG''; four right angles to reach OG'''. But if a radial arms sweeps four right angles, it has returned to its starting point. That is, the line OG''' has returned to OA; that is OG''' is OA. Or the point G''' just is the same point as A. So the figure is more correctly drawn as:

Notice what has happened. We started with a straight line AB and extended it to G, G', G'' and then finally back to itself. So the straight line on which points A and B lie is actually a straight line that wraps back onto itself.

Now recall that there was nothing special about this line. We started with ANY STRAIGHT LINE at all. It follows that all straight lines in the new geometry wrap back onto themselves. Since these straight lines fill all of space, it follows that that this space wraps back onto itself in every direction.

Circles and Triangles

This last figure has more surprises. To begin, recall that all the lines in it are straight. So it follows that one of the quarter wedges--AOG'' say--is actually a triangle, since it is a figure bounded by three straight lines. Moreover, the angles at each corner are the same--a right angle. That means that we have a triangle the sum of whose angles is three right angles, one more than we are used to for all triangles in Euclidean geometry.

Also it is clear from the symmetry of the three angles, that each side is the same length. This triangle is also an equilateral triangle. So it is more accurately drawn as the triangle on the right.

There is also a circle in the figure. While the line AGG'G''G''' is a straight line, it also has the important property of being the circumference of a circle centered on O. Every point on AGG'G''G''' is the same distance from O. That is the defining property of a circle.

And what an unusual circle it is. It has radius AO. That radius AO is equal in length to each of the four segments AG, GG', G'G'', G''A that make up the circumference.

Radius = AO
Circumference = AG + GG' + G'G'' + G''A
AO = AG = GG' = G'G'' = G''A

That means that the circle AGG'G''G''' has the curious property that

Circumference = 4 x Radius

Contrast that with the properties familiar to us from circles in Euclidean geometry

Circumference = 2π x Radius

A longer analysis would tell us that the area of the circle AGG'G''G''' stands in an unexpected relationship with the radius AO. Specifically

Area = (8/π) x Radius²

In Euclidean geometry, the area of a circle relates to its radius by

Area = π x Radius²

The New Geometry of 5^NONE is Spherical Geometry

You would be forgiven for thinking that the new geometry of 5^NONE is a very peculiar and unfamiliar geometry. The surprising thing is that this is not so. The geometry just explored proves to be very familiar; it is just the geometry that is natural to the surface of a sphere, such as is our own earth, to very good approximation. To see how that works, we need only identify the line AGG'G'' with the equator. The perpendiculars we erected to it then just become lines of longitude all of which intersect at the North Pole, that is at, O.

It isn't quite that simple. We do need to adjust our notion of what a straight line is. The essential idea remains the same. A straight line between two points A and B is still the shortest distance between two points. But now we are forced to remain on the surface of the sphere in finding the shortest distance. There is no burrowing into the earth to get a shorter distance between two points. The curve that implements the shortest distance in the surface is known as a "geodesic".

There is a simple way of creating geodesics on the surface of a sphere. They are the "great circles." That is, they are the circles produced by the intersection of the sphere with a plane that passes through the center of the sphere.

In short, the new geometry of 5^NONE is just the geometry of of great circles on spheres.

In such a geometry, there are no parallel lines. All pairs of great circles intersect somewhere. That this is so is sometimes overlooked. People sometimes mistake a parallel of latitude for a great circle. In the figure below, points A and B of the same latitude are connected by a parallel of latitude. The parallel of latitude is a parallel to the equator. However it is not the analog of straight line in this geometry, a geodesic. For geodesics are produced by the intersection of the sphere with planes that pass through the center of the sphere. The great circle passing through points A and B is shown in the second figure. It connects A and B by a path that deviates to the North. Since it is the great circle, it is the curve of least distance in the surface of the sphere between A and B.

The great circles are the routes taken by ships and airlines over the surface of the earth, whenever possible, since they are the paths of least distance.

Circles and Triangles

We can now return to the triangles and circles visited earlier. Their properties were radically different from Euclidean triangles and circles. The triangle's angles summed to three right angles and the circle's circumference was only four times the radius. It is now easy to see that these deviations from Euclidean expectations arise only for very large figures on the surface of the sphere. A very small patch of the surface of a sphere is very close to being a Euclidean plane. The calm surface of a small lake on the Earth is very nearly a flat plane; the surface of an ocean is markedly curved. In those very small patches, circles and triangles are very nearly Euclidean in their properties.

The figure below shows a very small equilateral triangle A''B''C''. The sum of its angles will meet Euclidean expectation near enough and be two right angles. As the triangle grows larger, passing through triangle A'B'C' to the huge ABC, the sum of its angles will grow until they are three right angles at ABC.

The situation is the same with circles. The circle around the North Pole below with very small radius OA will meet Euclidean expectations, near enough, and have its circumference 2π times its radius. As the circle grows with radius increasing through OB to OC, the formula will mutate. When the radius is OC, so the circle now coincides with the equator, the circumference will have dropped to being only four times the radius.

Corrections to the Other Postulates

Now that we have identified our geometry of 5^NONE as the geometry of great circles on spheres, two small corrections are needed. The first postulate allows us to draw a straight line between any two points. In the new geometry, there are two ways of connecting any two nearby points by a great circle. One goes the short way; the other goes the long way all around the other side of the sphere.

The second correction is for the second postulate which allows us to produce a straight line indefinitely. That is not possible for great circles. They are already maximally extended. One part of the original notion of the second postulate was that a straight line never really comes to an end. Any point that looks like an end is only a temporary terminus and the line can be extended past it. That lack of a boundary point is all we need for the revised second postulate.

The two modified forms of the first and second postulates that accommodate these two alterations are:

1'. Two distinct points determine at least one straight line.

2'. A straight line is boundless (i.e. has no end).

Is the Geometry of 5^NONE Consistent?

How do we know that a more imaginative, more thorough analysis might not eventually produce a contradiction? That is, how do we know that the new geometry is consistent?

The question could be answered by a proof of the consistency of the geometry. Alas, advances in twentieth century mathematics have shown that proving the consistency of a rich system in mathematics is typically impossible. However the geometers of the nineteenth century had already supplied us with something that, for practical purposes, is good enough.

In showing that the geometry of 5^NONE is really the geometry of great circles on spheres, they provided a relative consistency proof. The idea is simple enough. In a three dimensional Euclidean space, we can recreate or simulate, the different geometry of 5^NONE by constructing a sphere. So, imagine that somehow we could generate a contradiction within the geometry of 5^NONE. That means that we could generate a contradiction within the geometry of great circles on spheres. And that means that there must be a contradiction recoverable within the geometry of three dimensional Euclidean spaces.

To get a more concrete sense of how this works, imagine that there is a way of deducing an inconsistency in the geometry of5^NONE. So a geometer sits down and begins the steps of the construction that leads to a contradiction. Perhaps the geometer draws a straight line AB; and then a perpendicular to it; and so on. Now imagine a second geometer who works in Euclidean space. That geometer clones exactly everything the first geometer does, but now replaces the first geometer's straight line AB by a great circle through AB on some sphere. The two constructions will proceed analogously for the original geometer working the space of 5^NONE and the clone geometer working in the Euclidean space. So if the first geometer finds the construction leads to a contradicition, then so must the clone geometer. But that clone geometer is working fully within Euclidean geometry. That is, if the first geometer finds a contradiction in the geometry of 5^NONE, then the second must find a contradiction in Euclidean geometry.

So, if the geometry of 5^NONE is inconsistent, then Euclidean geometry must be inconsistent. Or turning it around, if Euclidean geometry is consistent, then so must the geometry of 5^NONE. Of course the big catch is that we cannot prove that Euclidean geometry is consistent. However we can take some comfort that millennia of investigations have failed to find an inconsistency in it. The relative consistency proof assures us that we are no worse off in the geometry of 5^NONE.

The Geometry of 5^MORE

What of the geometry of 5^MORE? One might imagine that there are many distinct versions according to how many parallels can be drawn through a point not on the original straight line. One can quickly see, however, that there is only one possibility for this number. Imagine, for example, that the geometry allows two parallels AA' and BB' through the point but no more.

Then we can always bisect AA' and BB' with a third line CC'. Now AA' and BB' are parallel to the original line in the sense that they never intersect it, no matter how far they are projected. Since CC' is sandwiched between AA' and BB', the same must be true of it.

The basic idea generalizes. Any attempt to limit the maximum number of parallels allowed by 5^MORE fails; we can always add one more. So the geometry of 5^MORE is the geometry that arises when we may draw infinitely many parallels through the point not on the original line.

We could continue the exercise of discovering the geometry 5^MORE through step by step inference. Since we've seen it done once for the geometry of 5^NONE, let us just skip to the final result. It turns out that the geometry of 5^MORE is the geometry of a negatively curved surface like a saddle or potato chip.

In this geometry, lines can have infinite length, just as in familiar Euclidean geometry.

However there are differences that are analogous to those of the geometry of a spherical space:

In very small parts of the space, circles and triangles behave like Euclidean circles and triangles, near enough.

As the circles and triangles get larger, deviations from Euclidean behavior emerge. The circumference of circles becomes more than 2π times the radius; and the sum of the angles of a triangle become less than two right angles.

The perpendiculars to the equator on the surface of a sphere converge to a single point, the North Pole. On this surface of negative curvature, perpendiculars to a straight line diverge.

The Geometries Generalized To Three Dimensions of Space

So far, we have explored the geometries of 5^NONE and 5^MORE for the case of two dimensional spaces. We can also consider each in three dimensional spaces. The results we would arrive at are summarized in the table (duplicated in Euclid's Postulates and Some Non_Euclidean Alternatives).

In very small regions of space, the three geometries are indistinguishable. For small triangles, the sum of the angles is very close to 2 right angles in both spherical and hyperbolic geometries.

Dropping the Embedding Space

What made visualizing these non-Euclidean geometries easy was that we embedded the non-Euclidean space in a higher dimensioned Euclidean space. That took an unfamiliar and even disquieting geometry and made it familiar. However in the end, we must dispense with these higher dimensional embedding spaces and simply take the new geometries as worthy geometries in their own right. There are three problems.

One is technical. Sometimes the embedding cannot be implemented fully. The two dimensional negatively curved saddle shape can only be embedded into a three dimensional space in pieces; the full surface cannot be embedded.

Another is practical. The real gain is to our imagination. Imagine a three dimensional curved space that is curving into the fourth dimension of a four dimensional Euclidean space. Well--that's the problem. You cannot imagine it. So the practical gain to visualization is lost in this case. It is replace by a new problem: how are we to visualize the curving of the three dimensional space into a four dimensional space?

The final problem is the most serious. If our geometry turns out to be factually one of the curved geometries, then the supposition of a higher dimensioned Euclidean space is a falsehood and a potentially very misleading one. For, if we take it seriously, we end up believing that space is really Euclidean after all, but only in some higher dimension to which we have no access. If all we know is the three dimensions of space in which we measure, then we have no license to conjure up an otherwise inaccessible higher dimensioned Euclidean space for it to curve into. What makes us think such a higher dimensioned space exists?

Einstein's Moral

Let us return to our starting point. Euclid's achievement appeared unshakeable to the mathematicians and philosophers of the eighteenth century. The great philosopher Immanuel Kant declared Euclid's geometry to be the repository of synthetic, a priori truths, that is propositions that were both about the world but could also be known true prior to any experience of the world. His ingenious means of justifying their privileged status came from his view about how we interact with what is really in the world. In our perceiving of the world, we impose an order and structure on what we perceive; one manifestation of that is geometry.

The discovery of new geometries in the nineteenth century showed that we ought not to be so certain that our geometry must be Euclidean. In the early twentieth century Einstein showed that our actual geometry was not Euclidean. So what are we to make of Kant's certainty? Einstein gave this diagnosis in his 1921 essay "Geometry and Experience."

"... an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things? In my opinion the answer to this question is, briefly, this: as far as the propositions of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality..."

To restate Einstein's point in terms closer to Kant's terminology: in so far as geometry is synthetic its propositions are not certain; they are empirical claims about the world to be investigated by science like any other claim and we can never be absolutely certain of them. In so far as a geometry's propositions are a priori, they are not factual claims about the world; they are "if-then" statement of logic within some logical system whose initial propositions are the postulates of the geometry.

What you should know

• How Euclid organized geometry into a deductive structure with definitions, axioms, postulates and theorems.
• The efforts to deduce his fifth postulate from the other four.
• How their failure led to the discovery of new geometries.
• How to use the alternative postulates of these geometries in simple geometric constructions.
• How the different geometries are associated with surfaces of various curvature.
• How these new geometries changed our view of the certitude of geometrical knowledge.

Copyright John D. Norton. December 28, 2006, February 28, 2007; February 2, 9, 14, September 22, 2008.