HPS 0410 Einstein for Everyone

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# Euclidean Geometry The First Great Science

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

Euclid's Postulates and Some Non-Euclidean Alternatives

The definitions, axioms, postulates and propositions of Book I of Euclid's Elements.

## 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.

Oxyrhynchus papyrus showing fragment of Euclid's Elements, AD 75-125 (estimated)

Title page of Sir Henry Billingsley's first English version of Euclid's Elements, 1570

Oliver Byrne's 1847 edition of the first 6 books of Euclid's Elements used as little text as possible and replaced labels by colors.

A recent edition from Dover.

This long history of one book reflects the immense importance of geometry in science. We now often think of physics as the science that leads the way. In the seventeenth century, Newton found one simple system of physics that worked for both the heavens and the earth. That set a standard of achievement that the other sciences sought to emulate. Newton, however, was learning from another science that already set an enduring standard of achievement: geometry.

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.

 One of the most influential thinkers of all time, the eighteenth century philosopher Immanuel Kant, provided an enduring answer. There are some types of knowledge that are both synthetic and a priori, he declared. They are synthetic in the sense that they say more of their subjects than are given by the subject's definition; they are a priori in the sense that they can be know prior to experience of the subject. Arithmetic and geometry were Kant's premier examples of synthetic a priori knowledge. According to Kant, it is a synthetic, a priori truth that 7+5=12; and it is a synthetic, a priori truth that the sum of the angles of all triangles is 180 degrees. "A triangle has three sides." is analytic, since the definition of triangle includes the idea that is has three sides. "A triangle's angles sum to 180 degrees." is synthetic, since this summation to 180 degrees is not part of the definition of a triangle. It is an addition.

Immanuel Kant

### To Come

These ideas provide our starting point. We shall see in later chapters 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.

All this is coming in later chapters. Now, it's back to Euclid.

## Euclid's Postulates

 The geometry of Euclid's Elements is based on five postulates. They assert what may be constructed in geometry. Let us start by reviewing the first four postulates. The first postulate is: For a compact summary of these and other postulates, see Euclid's Postulates and Some Non-Euclidean Alternatives

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.

 It just says that whenever we create a right angle by erecting perpendiculars, the angles so created are always the same. Definition 10. When a straight line standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle; and the straight line which stands on the other is called perpendicular to it.

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

 It may seem that a postulate like this is superfluous. Isn't it completely obvious that all right angles made this way are the same? Yes it is--but that is the essence of the postulates, to assert what is so unproblematic as to make them unchallengeable. Nonetheless, the equality of all right angles still does need to be asserted, since it will be assumed throughout everything that is to follow in Euclid's Elements. Could it really fail? Yes. The size of a right angle is the arc of a circle it subtends divided by the radius in a neighborhood close to the point. The postulate depends upon the ratio of the circumference of the circle enclosing the point to the radius being the same everywhere in the limit of arbitrarily small circles. While this sameness obtains in all the geometries we are about to look at, once it is stated this simply, one can see that in principle in could fail. In one part of space, the ratio might be the familiar 2π; in other parts, it may be more or less.

### 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 consider both circles together. They intersect at some point. Let us call it c. The assumption that they meet is not guaranteed by Euclid's postulates. It is an additional assumption that tacitly presupposes that the surface is an ordinary two dimensional surface. This is one of several well known points in Euclid's system where the deductions are less rigorous than we would expect.

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.

 Consider the triangle ABC. From the Definitions 15 and 16 of a circle, we know that the two radii AB and AC of the circle centered at A are equal. AB=AC Similarly, we know that the two radii AB and CB of the circle centered at B are equal. AB=CB So, by Axiom 1, we know that all three are equal AB=AC=BC and the triangle is equilateral. QED Definition 15. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another; Definition 16. And this point is called the center of the circle. Axiom 1. Things that are equal to the same thing are equal to one another. QED = quod erat demonstrandum = "which was to be proved"

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, so the efforts continued.

The eighteenth century closed with Euclid's geometry justly celebrated as one of the great achievements of human thought. The awkwardness of the fifth postulate remained a blemish in a work that, otherwise, was of immortal perfection. We knew the geometry of space with certainty and Euclid had revealed it to us.

## What you should know

• How Euclid organized geometry into a deductive structure.
• An idea of what his definitions, axioms, postulates and theorems look like.
• A sense of how Euclidean proofs work.
• The sense of certainty scholars of earlier eras assigned to Euclid's geometry.
• Why the fifth postulate is an awkwardness for Euclid's geometry.

Copyright John D. Norton. December 28, 2006, February 28, 2007; February 2, 9, 14, September 22, 2008; February 2, 2010