CLAA – Eucild's Elements - Book I, Postulates

QRV-321 Lesson 002 – Euclid Elements, Book I, Postulates

I. INTRODUCTION

This lesson studies the five postulates set down at the beginning of the first book of Euclid’s Elements.

Geometry is the science of continuous magnitude — of lines, angles, surfaces and solids considered according to their measure, shape and position. Like every science, geometry proves its conclusions. But nothing can be proved out of nothing. Every proof rests on something already granted. The postulates are among the things granted at the beginning of geometry, before any proposition is proved, so that everything afterward may be proved from them. They stand, therefore, at the very foundation of the subject. A student who does not know the postulates does not know where any geometrical demonstration comes from.

Euclid taught at Alexandria in Egypt about 300 years before Christ, in the reign of Ptolemy I. Our chief ancient witness for his life and for the arrangement of his book is Proclus, who wrote a commentary on this first book of the Elements in the fifth century after Christ. Proclus reports the well-known answer Euclid gave to King Ptolemy, who asked whether there were not some shorter way to geometry than the Elements: there is no royal road to geometry. Euclid did not invent all the matter of his book. He gathered what Thales, Pythagoras and their followers, Eudoxus, Theaetetus and others had discovered, and he ordered it into a single chain of demonstration, in which nothing is used before it has been proved, and the whole rests upon a small number of stated principles. That ordering is Euclid’s own work, and it is why the Elements displaced every earlier treatise and has been used as the first book of geometry, without interruption, for more than two thousand years.

The text came down to us in Greek manuscripts, most of them descending from the edition prepared by Theon of Alexandria in the fourth century. It passed into Syriac and Arabic, and returned to the Latin West chiefly through the twelfth-century translation of Adelard of Bath and the edition of Campanus of Novara. It was among the first mathematical books printed, at Venice in 1482. Christopher Clavius, the Jesuit mathematician, produced the great Latin edition used in the schools after the Council of Trent; Robert Simson’s English edition of 1756 governed English teaching for a century and a half, and Thomas Heath’s edition of 1908 remains the standard English scholarly text. The book has been kept for two reasons. First, its matter is true. Second, its form is the clearest example we possess of a demonstrative science, such as Aristotle described in the Posterior Analytics, actually carried out from beginning to end.

The lesson supplied to me is the heading only; the five postulates of Book I are therefore set out below in the traditional English wording and expounded in order, and nothing else is treated as part of the lesson text.

Objectives of this lesson. When you have finished it, you will be able to:

State the five postulates of Euclid’s first book, in order, in their proper wording.
Define the word postulate, and distinguish a postulate from a definition and from a common notion.
Explain what each of the five postulates grants, and name the definitions each one depends upon.
Explain why the postulates are not proved, and why this does not make geometry uncertain.
Identify, in a simple construction such as the first proposition of Book I, which postulates are used and at which step.
State the special difficulty of the fifth postulate, the history of the attempts to prove it, and the outcome of those attempts.

II. EXPOSITION

THE THREE KINDS OF PRINCIPLE AT THE HEAD OF BOOK I

Euclid opens Book I with three sets of statements, and you must know the difference between them, because the postulates are only one of the three.

First come the definitions — twenty-three of them. A definition says what a word means. It does not assert that anything exists. When Euclid says, “A point is that which has no part,” he is telling you how he will use the word point. Definitions give the terms of the science.

Second come the postulates — five of them. The Greek word is aitema, from the verb aiteo, to ask or to demand. The Latin postulatum, from postulare, to demand, translates it exactly. A postulate is something the teacher demands that the learner grant at the outset, which is not proved and cannot be proved within the science, and which belongs to that science alone. Euclid’s postulates are demands proper to geometry. They mostly grant that certain things can be done: that a line can be drawn, that it can be extended, that a circle can be described.

Third come the common notions, which the older writers also call axioms. A common notion is a self-evident truth which is not proper to geometry but is used in many sciences: for example, “Things which are equal to the same thing are equal to one another,” and “The whole is greater than the part.” These are common; the postulates are proper.

So: definitions give meanings; common notions give truths used everywhere; postulates give the grants proper to geometry.

WHY PRINCIPLES ARE NOT PROVED

A beginner naturally asks: why should I grant these five things without proof? Aristotle answers this in the Posterior Analytics. To demonstrate a conclusion is to draw it from premises that are true, primary, immediate, better known than the conclusion, and causes of it. Now if every premise had itself to be demonstrated, then either the demonstrations would go back without end, or they would run in a circle. If they went back without end, nothing would ever be proved, because you would never reach a starting point. If they ran in a circle, a thing would be used to prove what was used to prove it, and again nothing would be proved. Therefore every demonstrative science must begin from principles that are not themselves demonstrated in that science.

This is not a weakness. The principles are not doubtful; they are more evident than anything proved from them. We do not accept “a straight line may be drawn from any point to any other point” because someone argued for it. We accept it because it is plain to anyone who understands what a point and a straight line are. Our knowledge of them begins in the senses: we see edges, corners, taut cords, the rims of round things, and from these sensible things the mind abstracts the point, the line and the circle, and sees at once what can be done with them. Euclid does not ask you to believe anything obscure. He asks you to notice what you already grant whenever you use a ruler and a compass, and to say so plainly at the start, so that afterwards nothing will be smuggled in unnoticed.

THE DEFINITIONS PRESUPPOSED BY THE POSTULATES

The postulates use words that Euclid has already defined. Since this lesson must stand on its own, here are the definitions you need, in brief.

A point is that which has no part — that is, it has no size at all, but only position.

A line is length without breadth. Its ends are points.

A straight line is a line which lies evenly with the points on itself. Understand well: for Euclid a straight line is finite, what we should call a straight segment, having two endpoints. When he wants a longer one, he must produce it, and that is exactly what the second postulate grants.

A plane surface is a surface which lies evenly with the straight lines on itself. All of Book I takes place in one plane.

A plane angle is the inclination to one another of two lines in a plane which meet and are not in a straight line. When the lines are straight, it is called a rectilineal angle.

When a straight line set up on another straight line makes the two adjacent angles equal to one another, each of the equal angles is a right angle, and the line standing on the other is called a perpendicular to it. Notice that this definition makes the right angle by comparison of two angles at one and the same place. It says nothing yet about right angles in different places.

A circle is a plane figure contained by one line, such that all the straight lines drawn from a certain point within the figure to the containing line are equal to one another. That point is called the centre. Any of those equal straight lines is what we call a radius.

Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

Now the postulates themselves.

POSTULATE 1

“Let it be granted that a straight line may be drawn from any one point to any other point.”

What this grants is the possibility of joining. Take any two points whatever, however placed, however far apart. Euclid demands that you grant that the straight line joining them may be drawn. In practice this is the use of the ruler or straight-edge: given two marked points, you lay the edge to them and draw.

Three things are to be noticed.

First, this is a grant of possibility, not a statement about some figure already drawn. The postulate does not say that a certain line exists in a certain place; it says that, whenever two points are given, the joining straight line may be had.

Second, the grant is unrestricted. Any point to any other point. There is no distance too great and none too small.

Third, the tradition understands the postulate as granting also that there is only one such straight line between the two points. Euclid never states this separately, but he uses it constantly: whenever he says “the straight line AB,” he treats it as a single determinate line, and in several proofs he takes it as absurd that two distinct straight lines should enclose a space between the same two points. This uniqueness follows from the definition of the straight line as the line which lies evenly with its own points, for if two different lines both lay evenly between the same two points they would not differ at all.

POSTULATE 2

“That a terminated straight line may be produced to any length in a straight line.”

Terminated means having ends; a terminated straight line is a segment, with two endpoints. To produce a line means to extend it beyond its endpoint, continuing in the same straightness.

This postulate grants that no straight line is trapped at its ends. Whatever segment you have, you may lengthen it, in the same direction, as far as the proof requires.

Notice carefully what is not granted. Euclid does not grant an actually infinite line. He does not ask you to imagine a line already stretched out without end. He grants only that any given finite line may be produced further, as far as you please. This is precisely Aristotle’s doctrine of the infinite: the infinite in magnitude exists in potency, not in act. There is no line of infinite length in nature or in geometry; there is a line which can always be made longer. Euclid’s geometry is built entirely on that potential extension, and this is one of the marks of its soundness. When later, in Definition 23 and in the fifth postulate, he speaks of lines “produced indefinitely,” he means produced as far as one pleases, not laid out infinitely at once.

POSTULATE 3

“That a circle may be described from any centre, at any distance from that centre.”

This grants the use of the compass. Given any point to serve as centre, and given any distance, you may describe the circle. In practice: set the compass point at the centre, open it to the given distance, and turn it round.

Here again the grant is unrestricted as to place and as to size: any centre, any distance.

But there is a limitation of another sort, and it is important. The “distance” Euclid intends is a distance already given at that centre — that is, the circle is described about a given point and passing through another given point, or with a given line, one of whose ends is at the centre, as radius. The postulate does not grant that you may take a length lying somewhere else in the figure, carry it across the plane, and plant it at your new centre. In the older discussions this is put by saying that Euclid’s compass “collapses” when lifted. Because the postulate is thus limited, Euclid must prove, as the second proposition of Book I, that from a given point a straight line may be drawn equal to a given straight line elsewhere; and the third proposition then shows how to cut off from the greater of two lines a part equal to the less. Only after those two propositions are proved may lengths be freely transferred. This is a good example of Euclid’s exactness: he takes as little as he can at the start, and proves the rest.

POSTULATE 4

“That all right angles are equal to one another.”

The first three postulates grant that certain things may be done. The fourth does not; it asserts a truth. Why is it needed?

Recall the definition. A right angle is made when one straight line stands on another so as to make the two adjacent angles equal. That definition is entirely local: it tells you when the two angles at one place are right, by comparing them with each other. By itself, it gives no reason at all why a right angle made in one corner of the figure should be equal to a right angle made in another corner, or in another figure, or at another place in the plane.

Yet geometry cannot proceed without that. Euclid must be able to compare angles that are far apart, and the right angle is his standard of comparison: he speaks constantly of angles less than a right angle, greater than a right angle, equal to two right angles. If right angles in different places might differ in magnitude, all such comparison would be worthless, and the fifth postulate itself, which speaks of interior angles “less than two right angles,” would have no fixed meaning.

So the fourth postulate lays down that the right angle is one determinate magnitude everywhere. Wherever it is made, and however it is turned, a right angle is equal to a right angle. Understand what this really asserts: that figures do not change their magnitudes by being in one place rather than another, and that the plane is everywhere of the same character. That is a real assumption about geometrical magnitude, and Euclid was right to state it and not to hide it.

POSTULATE 5

“That, if a straight line falling upon two straight lines make the interior angles on the same side less than two right angles, these straight lines, being produced indefinitely, shall meet on that side on which are the angles less than two right angles.”

This is the longest of the postulates, and the most famous. Read it slowly, word by word.

A straight line falling upon two straight lines: a third line that crosses both of them. It is called a transversal.

The interior angles on the same side: the transversal crosses the first line at one point and the second line at another point. At each crossing, four angles are made. The interior angles are the ones lying between the two lines that are crossed. Of these four interior angles, two lie on one side of the transversal and two on the other side. Take the two that lie on the same side.

Less than two right angles: add those two interior angles together. Two right angles is the standard against which the sum is measured — and this is why the fourth postulate had to come first.

These straight lines, being produced indefinitely, shall meet: extend both lines as far as is needed, by the second postulate, and they will come together at a point.

On that side on which are the angles less than two right angles: and they will meet on that side, not on the other.

Here is the statement in plain words. If a line crosses two other lines, and on one side the two inside angles together fall short of two right angles, then the two lines lean toward each other on that side, and if you keep extending them they will eventually cross on that side.

Let me set out a literal instance. Let AB be one straight line and CD another. Let the straight line EF cross AB at the point G and cross CD at the point H. The interior angles at G and H on the side toward B and D are the angle BGH and the angle DHG. Suppose the angle BGH is a right angle and the angle DHG is less than a right angle — say the two together fall short of two right angles by a small amount. Then the postulate declares that AB produced beyond B, and CD produced beyond D, will meet in some point.

The figure may be represented so:

A ——G—— B

C ——H—— D

with the line CD tilted slightly so that the angle at H on the side of D is a little less than a right angle. Produced far enough to the right, the lines AB and CD meet.

Now three things must be said about this postulate.

First, it is true, and it is plainly seen to be true whenever the shortfall is large. If the two interior angles together are much less than two right angles, the lines meet quickly and near at hand. The difficulty is only that when the shortfall is very small the meeting point may lie very far away — beyond anything that can be drawn or seen. So the postulate cannot be verified by looking, and yet it is stated for all cases. That is why it does not have the immediate obviousness of the first three.

Second, Euclid himself clearly saw its special standing, and treated it with reserve. He does not use it at all in the first twenty-eight propositions of Book I. He proves without it that in any triangle the exterior angle is greater than either opposite interior angle, that any two angles of a triangle are together less than two right angles, and that if a transversal makes the alternate angles equal, the two lines are parallel. Only at Proposition 29, where he must prove the converse — that a transversal cutting parallels makes the alternate angles equal — does he invoke the fifth postulate. From Proposition 29 flow the theorem that the three angles of a triangle are equal to two right angles, the whole doctrine of parallelograms, and the theorem of Pythagoras. So the fifth postulate is not a decoration; the greater part of what is useful in Book I depends on it.

Third, from antiquity onward, many mathematicians thought the fifth postulate too complicated to be a principle, and tried to prove it from the other four. Ptolemy attempted it; Proclus reports and criticizes that attempt and offers one of his own; the Arabic geometers, among them Thabit ibn Qurra and Nasir al-Din al-Tusi, tried again; in modern times Saccheri, Lambert and Legendre made careful and elaborate attempts. Every attempt failed. Each was found either to contain a mistake, or to assume, somewhere in the course of the argument, something that is simply the fifth postulate under another name. In the nineteenth century the matter was settled: it was demonstrated that the fifth postulate cannot be deduced from the other four, and that consistent bodies of geometrical propositions can be constructed in which the first four hold and the fifth does not. That result vindicated Euclid’s judgment. He had put the statement among the postulates and not among the propositions, and it belongs there, because it can be neither proved nor dispensed with.

One equivalent form of the fifth postulate should be known, because most modern books use it in place of Euclid’s wording. It is called Playfair’s axiom, from John Playfair, who published it in 1795: through a given point, not on a given straight line, only one straight line can be drawn parallel to the given line. Granted the other postulates, this statement and Euclid’s fifth postulate each follow from the other, so that either may be taken as the principle. Euclid’s own form is the more useful in the actual proofs, because it tells you not merely that parallels are unique but on which side two leaning lines will meet.

HOW THE POSTULATES ARE USED: A WORKED INSTANCE

The best way to see what the postulates are for is to watch them at work. Take the very first proposition of Book I: On a given finite straight line to construct an equilateral triangle — that is, a triangle whose three sides are all equal.

Let AB be the given finite straight line.

Step 1. With centre A and distance AB, describe the circle BCD. This step is taken by Postulate 3, and by nothing else. The postulate grants exactly this: any centre, any distance.

Step 2. With centre B and distance BA, describe the circle ACE. Again Postulate 3, used a second time. The two circles cut one another; let C be a point where they cut.

Step 3. From the point C draw the straight lines CA and CB. This is taken twice by Postulate 1, which grants that a straight line may be drawn from any point to any other.

Step 4. Now AC is equal to AB, because both are drawn from the centre A to the first circle, and by the definition of a circle all such lines are equal. Likewise BC is equal to BA, by the same reason in the second circle. Therefore AC and BC are each equal to AB.

Step 5. But things which are equal to the same thing are equal to one another. This is the first of the common notions. Therefore AC is equal to BC.

Step 6. Therefore the three lines AB, AC, BC are equal to one another, and the triangle ABC is equilateral, and it has been constructed on the given line AB. Which was to be done.

Count what was used: Postulate 3 twice, Postulate 1 twice, the definition of a circle, and one common notion. Nothing else whatever. That is the discipline of the Elements, and it is only possible because the postulates were stated plainly at the head of the book. Every one of the forty-eight propositions of Book I reduces, in the end, to these grants.

III. REFLECTION

Consider now what you have gained and where it belongs.

Geometry is one of the speculative sciences — that is, one of the sciences whose end is knowledge of the truth, not the making of a thing. The speculative sciences are ordered by their remove from matter. Natural science studies things as they exist in sensible matter and cannot be understood apart from it. Mathematics studies quantity, which the mind separates from sensible matter and considers by itself: the geometer reasons about the circle without asking whether it is of bronze or of wood, though he first came to know the circle from bronze and wooden things seen with the eyes. Metaphysics studies being as being, which is furthest of all from matter. Geometry is thus the middle degree of abstraction. It is not a contemplation of shapes in some separate world; it is the mind’s abstraction of quantity from the bodies our senses report, and its careful reasoning upon that quantity.

What faculty does this lesson perfect? The intellect, in its work of demonstration. A man knows in the strict sense when he knows the cause of a thing and knows that it cannot be otherwise. The postulates are the ultimate reasons from which the geometer’s conclusions get their necessity. To learn them is not to memorize a preface; it is to learn where geometrical certainty comes from.

Among the seven liberal arts, geometry is one of the four arts of the quadrivium — arithmetic, geometry, music and astronomy — which treat of quantity: arithmetic of discrete quantity, geometry of continuous quantity, music of number in proportion, astronomy of magnitude in motion. But this lesson also joins geometry to the trivium, and especially to logic. What Aristotle teaches in the Posterior Analytics about definitions, axioms and postulates, about the necessity of indemonstrable principles, and about the difference between what is proper to one science and what is common to many, is not merely illustrated by Euclid’s Book I; it is carried out in it. This is why the schoolmen set the Elements beside the Organon. He who studies both sees the same doctrine twice: once as a rule of reasoning, once as reasoning actually performed.

And geometry serves the pursuit of wisdom in a further way. Sacred Scripture says of God that He has ordered all things in measure, and number, and weight (Wisdom 11:21), and Saint Paul teaches that the invisible things of God are clearly seen, being understood by the things that are made (Romans 1:20). The ascent to God runs through creatures. The geometer, working from a handful of granted principles, finds that bodily magnitude is subject to fixed and necessary law, that its properties are not arbitrary, and that a mind can trace them out with certainty. Order of that kind is not self-explaining. Sobriety forbids us to claim more than the science gives: geometry does not prove God. But the intelligibility and measured order which geometry uncovers in creatures is one of the works from which the mind rises to their Author.

As for modern study and modern life, the debt is direct and large. What is now called the axiomatic method — laying down undefined terms, definitions and axioms, and deriving everything else by proof — is Euclid’s method, taken from Book I and applied to every branch of mathematics. The phrase “ruler-and-compass construction,” still used in geometry, means exactly a construction using only Euclid’s first three postulates. Surveying, architectural drafting, machine drawing, navigation and the geometry used in computer graphics all rest on constructions that reduce to these grants. The fifth postulate, through the nineteenth-century investigations of its independence, gave rise to the geometries used in modern physical theories of space.

Two losses should also be named. First, most modern schoolbooks give the student results — formulas for area, rules about angles — with no account of the principles from which they follow, so that the student memorizes conclusions he cannot defend and calls this geometry. He is left unable to say why any of it is true. Second, the word postulate has been degraded in common speech, where “to postulate” now often means to guess, to suppose without warrant, or to assert arbitrarily. That is not what the word means here. A postulate is a demand made at the beginning of a science for something evident, proper to that science, and incapable of proof within it. Keep the true sense of the word, and you will read not only Euclid but the whole tradition more accurately.

IV. CONCLUSION

Take the objectives in order.

First, you were to be able to state the five postulates in order and in their proper wording. They have been set out one by one: the drawing of a straight line between two points; the producing of a terminated straight line; the describing of a circle from any centre at any distance; the equality of all right angles; and the meeting of two lines cut by a transversal on the side where the interior angles are together less than two right angles. They are collected below for memory.

Second, you were to define postulate and distinguish it from definition and common notion. A postulate is a demand made at the beginning of a science, proper to that science, granted without proof. A definition states the meaning of a term. A common notion is a self-evident truth common to many sciences.

Third, you were to explain what each postulate grants and what definitions it uses. The first grants joining, and rests on the definitions of point and straight line. The second grants extension, and rests on the straight line’s being finite. The third grants the describing of circles, and rests on the definitions of circle and centre. The fourth asserts that the right angle, defined locally by equal adjacent angles, is one and the same magnitude everywhere. The fifth asserts when and where two lines cut by a transversal must meet, and uses the right angle as its standard of measure.

Fourth, you were to explain why the postulates are not proved. Demonstration cannot proceed without end and cannot proceed in a circle; therefore every science begins from principles it does not demonstrate. Those principles are not less certain than the conclusions but more so.

Fifth, you were to identify the postulates used in a construction. In the first proposition of Book I, Postulate 3 is used twice to describe the two circles, Postulate 1 twice to join the point of intersection to the ends of the given line, together with the definition of a circle and the first common notion.

Sixth, you were to know the standing of the fifth postulate. It is true, it is indispensable from Proposition 29 onward, it was attacked for two thousand years by men who thought it should be proved rather than granted, and it was at last demonstrated that it cannot be derived from the other four. Euclid placed it rightly.

Now study this lesson for mastery, not for acquaintance. It is not enough to have read the postulates and found them reasonable. Learn them by heart in their proper wording, so that you can say them without the book; learn what each one grants and what it does not grant; and practice pointing out, in the constructions you meet, which postulate authorizes each step. Then develop and demonstrate that mastery through the assessments set on this lesson in the Classical Liberal Arts Academy. Return to the lesson until you can teach its content from its causes — until you can say not only what the postulates are, but why a science must have such principles, and why these five and no others are demanded at the head of Euclid’s first book. For, as Aristotle holds, a man knows a thing when he can explain why it is so.

MEMORY WORK

A definition states the meaning of a term; it does not assert that anything exists.
A postulate is a demand made at the beginning of a science, proper to that science, which is granted without proof and cannot be proved within that science.
A common notion, or axiom, is a self-evident truth which is used in many sciences and is not proper to geometry alone.
Every demonstrative science must begin from indemonstrable principles, because demonstration can neither proceed without end nor proceed in a circle.
The first postulate is: Let it be granted that a straight line may be drawn from any one point to any other point.
The second postulate is: That a terminated straight line may be produced to any length in a straight line.
The third postulate is: That a circle may be described from any centre, at any distance from that centre.
The fourth postulate is: That all right angles are equal to one another.
The fifth postulate is: That, if a straight line falling upon two straight lines make the interior angles on the same side less than two right angles, these straight lines, being produced indefinitely, shall meet on that side on which are the angles less than two right angles.
The first three postulates grant that certain things may be done; the fourth and fifth assert that certain things are true.
For Euclid a straight line is finite, and the second postulate grants that it may always be produced further; no line is taken as actually infinite.
Euclid does not use the fifth postulate until the twenty-ninth proposition of Book I, and it was afterward demonstrated that the fifth postulate cannot be deduced from the other four.

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